CONSUMPTION AND PORTFOLIO DECISIONS WHEN ...pages.stern.nyu.edu/~dbackus/GE_asset_pricing...ity on portfolio choice. Kim and Omberg [1996] work with a similar framework but, by assuming

CONSUMPTION AND PORTFOLIO DECISIONS WHENEXPECTED RETURNS ARE TIME VARYING

JOHN Y CAMPBELL AND LUIS M VICEIRA

This paper presents an approximate analytical solution to the optimalconsumption and portfolio choice problem of an innitely lived investor withEpstein-Zin-Weil utility who faces a constant riskless interest rate and a time-varying equity premium When the model is calibrated to U S stock market datait implies that intertemporal hedging motives greatly increase and may evendouble the average demand for stocks by investors whose risk-aversion coeffi-cients exceed one The optimal portfolio policy also involves timing the stockmarket Failure to time or to hedge can cause large welfare losses relative to theoptimal policy

I INTRODUCTION

The choice of an optimal portfolio of assets is a classic problemin nancial economics In a single-period setting the problem iswell understood and analytical solutions for optimal portfolioweights are available in important special cases When mean-variance analysis is appropriate for example optimal portfolioweights are known functions of the rst and second moments ofasset returns

In a multiperiod setting the problem is far less tractableExplicit solutions for portfolio weights are available in the specialcases where investment opportunities are constant or the investorhas log utility and hence acts myopically but these cases aretractable precisely because they reduce to the familiar single-period problem Merton [1969 1971] and Samuelson [1969]followed more recently by Cox and Huang [1989] have shown thatin general shifting investment opportunities can have importanteffects on optimal portfolios for investors with long horizonsThese papers characterize some properties of optimal portfoliosbut do not deliver analytical solutions for portfolio weights asfunctions of state variables

Interest in long-horizon portfolio choice has recently beenstimulated by empirical evidence that the conditions under whichthe multiperiod problem reduces to the single-period problem do

Campbell acknowledges the nancial support of the National ScienceFoundation and Viceira the nancial support of the Bank of Spain We are gratefulto John Cochrane Ming Huang Andrei Shleifer Robert Stambaugh Stanley Zinseveral anonymous referees and seminar participants at the Fall 1996 NBERAsset Pricing program meeting for helpful comments on earlier drafts

r 1999 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics May 1999

433

not hold Expected asset returns seem to vary through time sothat investment opportunities are not constant the evidence forpredictable variation in the equity premium the excess return onstock over Treasury bills is particularly strong (see Campbell[1987] Campbell and Shiller [1988a 1988b] Fama and French[1988 1989] Hodrick [1992] or the textbook treatment in Camp-bell Lo and MacKinlay [1997 Chapter 7]) Economists research-ing the equity premium puzzle nd that average excess stockreturns are too high to be consistent with a representative-investor model in which the investor has log utility (see Campbell[1996] Cecchetti Lam and Mark [1994] Cochrane and Hansen[1992] Hansen and Jagannathan [1991] Kocherlakota [1996]Mehra and Prescott [1985] or the textbook treatment in Camp-bell Lo and MacKinlay [Chapter 8])

In response to these empirical ndings several recent papershave used numerical methods to solve for optimal portfolios inmodels with realistic predictability of returns Investors aregenerally assumed to have power utility dened over wealth at asingle terminal date Different papers choose different investmenthorizons and make different assumptions about investorsrsquo abilityto rebalance their portfolios Kandel and Stambaugh [1996]consider the effects of predictability on the optimal portfolio of asingle-period Bayesian investor who takes account of parameteruncertainty while Barberis [1999] extends this work to study theoptimal portfolio of a long-horizon Bayesian investor who rebal-ances annually or not at all Brennan Schwartz and Lagnado[1997] consider a long-horizon investor who rebalances fre-quently while Balduzzi and Lynch [1997a 1997b] consider along-horizon investor who faces xed and proportional transac-tions costs which reduce the frequency of optimal rebalancing1

The results in these papers though dependent on the particularparameter values they assume illuminate the effects of predictabil-ity on portfolio choice Kim and Omberg [1996] work with asimilar framework but by assuming continuous time and zerotransactions costs are able to solve the portfolio choice problemanalytically2

A limitation of these models is that they abstract from the

1 Most of these papers like our paper work with a single state variabledriving the equity premium Only Brennan Schwartz and Lagnado [1997]consider multiple state variables

2 Kim and Omberg study the choice between a riskless asset with a constantreturn and a risky asset whose expected return follows a continuous-time AR(1)(Ornstein-Uhlenbeck) process They assume that the investor is nitely lived and

QUARTERLY JOURNAL OF ECONOMICS434

choice of consumption over time Since the investor is assumed tovalue only wealth at a single terminal date no consumption takesplace before the terminal date and all portfolio returns arereinvested until that date This simplies the analysis but makesit hard to apply the results to the realistic problem facing aninvestor saving for retirement In addition these models cannoteasily be related to the macroeconomic asset pricing literature inwhich consumption is used as an indicator of marginal utility

This paper extends the previous literature in three majorrespects First and most important we consider a model in whicha long-lived investor chooses consumption as well as an optimalportfolio to maximize a utility function dened over consumptionrather than wealth3 Second we assume that the investor hasEpstein-Zin-Weil preferences [Epstein and Zin 1989 Weil 1989]This allows us to distinguish the coefficient of relative riskaversion from the elasticity of intertemporal substitution inconsumption power utility restricts risk aversion to be thereciprocal of the elasticity of intertemporal substitution but infact these parameters have very different effects on optimalconsumption and portfolio choice Third like Kim and Omberg[1996] but unlike other previous research we solve the problemanalytically This provides economic insights that are hard to getfrom numerical solutions and it enables us to distinguish generalproperties of the solution from results that depend on particularparameter values

In order to keep our problem analytically tractable we makeseveral simplifying assumptions We assume that there are twoassets a riskless asset with a constant return and a risky assetwhose expected return the single state variable for the problemfollows a mean-reverting AR(1) process The assumption that theriskless return is constant simplies our analysis and enables usto isolate the effects of time variation in the equity premium

We work in discrete time and assume that the investor is ableto rebalance the portfolio every period Our approximate solutionmethod becomes more accurate as the period length shrinks thus

has HARA utility dened over terminal wealth They nd that the optimalportfolio weight is linear and the value function is quadratic in the state variable

3 Since the rst version of this paper was circulated some numerical resultshave been obtained for the long-horizon portfolio choice problem with utilitydened over consumption Balduzzi and Lynch [1997a 1997b] consider some caseswith endogenous consumption and Brandt [1999] uses the Generalized Method ofMoments to estimate consumption and portfolio rules that best satisfy theintertemporal Euler equation given the stochastic properties of historical data

CONSUMPTION AND PORTFOLIO DECISIONS 435

our model applies to an investor who is able to rebalancefrequently We also abstract from transactions costs and restric-tions on borrowing or short sales We make these assumptions notonly for tractability but also because we want to focus on the pureintertemporal effects of return predictability on optimal consump-tion and portfolio choice for long-horizon investors Transactionscosts and portfolio restrictions while interesting in their ownright may obscure these effects

Finally we assume that the investor is innitely lived In amodel with endogenous consumption every period Fischer [1983]notes that lsquolsquothe notion of the horizon loses its crispness Date T isstill the horizon in the sense that the individual looks no furtherahead than T But now events that occur at t T matter not onlybecause they affect the situation at T but also because consump-tion at t and later depends on the state of the world at time trsquorsquo[p 155] An innite horizon is particularly convenient analyticallybecause the problem becomes one of nding a xed point ratherthan solving backward from a distant terminal date It may be anappropriate assumption for investors with bequest motives asdiscussed in the macroeconomic literature on Ricardian equiva-lence and it approximates well the situation of investors withnite but long horizons4

The endogeneity of consumption in our model makes itimpossible for us to follow Kim and Omberg [1996] and derive ananalytical solution that is exact for all parameter values5 Insteadwe nd an approximation to the portfolio choice problem that canbe solved using the method of undetermined coefficients Weapproximate the Euler equations of the problem using second-order Taylor expansions and we replace the investorrsquos intertempo-ral budget constraint with an approximate constraint that islinear in log consumption and quadratic in the portfolio weight onthe risky asset This enables us to nd approximate analyticalsolutions for consumption and the portfolio weight Like Kim and

4 Brandt [1998] compares his nite-horizon results to ours For the parame-ter values he uses the nite-horizon solution converges quickly to the innite-horizon solution and is very similar by the time the nite horizon reaches twentyyears

5 The lack of an exact analytical solution is not due to the fact that we workin discrete time Schroder and Skiadas [1998] use stochastic differential utility acontinuous-time version of Epstein-Zin-Weil preferences due to Duffie and Epstein[1992] and show that it is possible to characterize the solution in terms ofquasi-linear parabolic partial differential equationsmdashwhich are relatively easy tosolve numericallymdashbut an exact closed-form solution exists only in the samespecial cases as in discrete time


Omberg [1996] we nd that the optimal portfolio weight is linearin the state variable while the log consumption-wealth ratio andthe log value function are quadratic in the state variable

The approximate solution holds exactly in some special butimportant cases noted by Giovannini and Weil [1989] In all othercases its accuracy is an empirical issue Campbell Cocco GomesMaenhout and Viceira [1998] compare the approximate analyticalsolution to a discrete-state numerical solution and nd that thetwo are very similar except at the upper extreme of the statespace We briey summarize these ndings in subsection IV6 ofthis paper

Our solution method uses the intertemporal Euler equationas its starting point In this sense it belongs to the class ofstochastic dynamic programming methods Cox and Huang [1989]have proposed an alternative solution method which transformsan intertemporal optimization problem with complete marketsinto an equivalent static optimization problem that can be solvedusing standard Lagrangian theory He and Pearson [1991] haveextended the Cox-Huang approach to settings with incompletemarkets In a related paper [Campbell and Viceira 1998] weformulate a problem of optimal consumption and portfolio choicewith time-varying interest rates constant risk premiums andcomplete markets and we explore the relation between thelog-linear approximate solution method and the Cox-Huangapproach

Our paper builds on the work of Campbell [1993] Campbellconsiders the simpler problem where only one asset is availablefor investment and so the agent need only choose consumption Heshows that this problem becomes tractable if one replaces theintertemporal budget constraint by a log-linear approximateconstraint He uses the solution in a representative-agent modelto characterize the equilibrium prices of other assets that are inzero net supply in the spirit of Mertonrsquos [1973] intertemporalCAPM Campbell [1996] estimates the parameters of the modelfrom U S asset market data while Campbell and Koo [1997]evaluate the accuracy of the approximate analytical solution bycomparing it with a discrete-state numerical solution

The organization of the paper is as follows Section II statesthe problem we would like to solve while Section III explains ourapproximate solution method Section IV calibrates the model topostwar quarterly U S stock market data and briey discusses


the accuracy of the approximate solution Section V calculates theutility cost of suboptimal portfolio choice and Section VI concludes

II THE INTERTEMPORAL CONSUMPTION AND

PORTFOLIO CHOICE PROBLEM

II1 Assumptions

We consider a partial-equilibrium problem in which(A1) Wealth consists of two tradable assets Asset 1 is risky

with one-period log (continuously compounded) returngiven by r1t 1 1 asset f is riskless with constant logreturn given by rf Therefore the one-period return onwealth from time t to time t 1 1 is

(1) Rpt1 1 5 a t(R1t 1 1 2 Rf) 1 Rf

where R1t 1 1 5 exp r1t 1 1 Rf 5 exp rf and the portfolioweight a t is the proportion of total wealth invested inthe risky asset at time t

(A2) The expected excess log return on the risky asset isstate-dependent There is one state variable xt suchthat

(2) Etr1t1 1 2 rf 5 xt

The state variable follows an AR(1)

(3) xt1 1 5 micro 1 f (xt 2 micro) 1 h t 1 1

where h t1 1 is a conditionally homoskedastic normallydistributed white noise error that is h t 1 1 N (0 s h

2)(A3) The unexpected log return on the risky asset denoted

by ut 1 1 is also conditionally homoskedastic and nor-mally distributed It is correlated with innovations inthe state variable

(4) vart (ut 1 1) 5 s u2

(5) covt (ut1 1h t1 1) 5 s u h

(A4) The investorrsquos preferences are described by the recur-sive utility proposed by Epstein and Zin [1989] andWeil [1989]

(6) U(CtEtUt1 1) 5 (1 2 d )Ct(12 g ) u 1 d (EtU t1 1

12 g )1u u (1 2 g )

where d 1 is the discount factor g 0 is the coefficientof relative risk aversion c 0 is the elasticity of


intertemporal substitution and the parameter u isdened as u 5 (1 2 g )(1 2 c 2 1) It is easy to see that (6)reduces to the standard time-separable power utilityfunction with relative risk aversion g when c 5 g 2 1 Inthis case u 5 1 and the nonlinear recursion (6) becomeslinear

(A5) The investor is innitely livedAssumptions (A1) and (A2) on the number of risky assets and

state variables are simplifying assumptions which we adopt forexpositional purposes The approach of this paper can be appliedto a more general setting with multiple risky assets and statevariables at the cost of greater complexity in the analyticalsolutions to the problem Assumption (A3) is also a simplicationthat can be relaxed in order to study the effects of conditionalheteroskedasticity on portfolio choice Assumption (A4) on prefer-ences allows us to separate the effects on optimal consumptionand portfolio decisions of the investorrsquos attitude toward risk fromthe investorrsquos attitude toward consumption smoothing over timeFinally assumption (A5) allows us to ignore the effects of a nitehorizon on portfolio choice but this assumption too can be relaxedin future work

II2 Euler Equations and the Value Function

The individual chooses consumption and portfolio policiesthat maximize (6) subject to the budget constraint

(7) Wt1 1 5 Rpt 1 1(Wt 2 Ct)

where Wt is total wealth at the beginning of time t and Rpt 1 1 is thereturn on wealth (1)

Epstein and Zin [1989 1991] have shown that with this formfor the budget constraint the optimal portfolio and consumptionpolicies must satisfy the following Euler equation for any asset i

(8) 1 5 Et dCt1 1

Ct

2 1c u

Rpt1 12 (12 u )Rit1 1

Equation (8) holds regardless of how many tradable assets areavailable In our simple model i denotes the riskless asset thesingle risky asset or the investorrsquos portfolio p When i 5 p (8)reduces to

(9) 1 5 Et dCt1 1

Ct

2 1 c

Rpt 1 1

u


Dividing (6) by Wt and using the budget constraint we obtainthe following expression for utility per unit of wealth

(10) Vt 5 (1 2 d )Ct

Wt

12 1c

1 d 1 2Ct

Wt

12 1c

(Et[V t 1 112 g Rpt1 1

12 g ])1u1(1 2 c )

where Vt Ut Wt Epstein and Zin [1989 1991] show that thevalue function per unit of wealth can be written as a powerfunction of (1 2 d ) and the consumption-wealth ratio

(11) Vt 5 (1 2 d ) 2 c (1 2 c )Ct

Wt

1(1 2 c )

Two special cases are worth noting First as c approaches one theexponents in (11) increase without limit The value function has anite limit however because the ratio Ct Wt approaches (1 2 d )[Giovannini and Weil 1989] We consider this case in more detailin subsection III5 Second as c approaches zero Vt approachesCt Wt A consumer who is extremely reluctant to substituteintertemporally consumes the annuity value of wealth eachperiod and this consumerrsquos utility per dollar is the annuity valueof the dollar

III APPROXIMATE SOLUTION METHOD

Our proposed solution method builds on the log-linear approxi-mations to the Euler equation and the intertemporal budgetconstraint proposed by Campbell [1993] By combining the approxi-mations to these equations we can characterize the properties ofa t the optimal allocation to the risky asset We then guess a formfor the optimal consumption and portfolio policies we show thatpolicies of this form satisfy the approximate Euler equation andbudget constraint and nally we show that the parameters of thepolicies can be identied from the primitive parameters of themodel

III1 Log Euler Equations

The rst step in our proposed solution method is to log-linearize the Euler equation (9) to obtain

0 5 u log d 2u

cEt D ct1 1 1 u Etrpt 1 1 1

1

2vart

u

cD ct1 1 2 u rpt1 1

where lowercase letters denote variables in logs and D is therst-difference operator This expression holds exactly if consump-


tion growth and the return on wealth have a joint conditionallognormal distribution In our model the return on wealth isconditionally lognormal because the portfolio weight is known inadvance and so the return on wealth inherits the assumedlognormality of the return on the risky asset Consumptiongrowth however is endogenous in our model and so we cannotassume at the outset that it is conditionally lognormally distrib-uted In fact our approximate solution implies that consumptiongrowth is not conditionally lognormal unless the elasticity ofintertemporal substitution c is one or the expected return on therisky asset is constant When these conditions do not hold wemust derive the log Euler equation using both a second-orderTaylor approximation around the conditional mean of rpt1 1D c t 1 1

and the approximation log (1 1 x) lt x for small xReordering terms we obtain the well-known equilibrium

linear relationship between expected log consumption growth andthe expected log return on wealth

(12) Et D ct 1 1 5 c log d 1 vpt 1 c Et rpt 1 1

where the term vpt is a time-varying intercept proportional to theconditional variance of log consumption growth in relation to logportfolio returns

(13) vpt 5 12( u c ) vart ( D ct 1 1 2 c rpt1 1)

In a similar fashion we can log-linearize the Euler equationfor a general asset (8) If we subtract the resulting log-linearEuler equation for the riskless asset from the log-linear Eulerequation for the risky asset we nd that

(14) Etr1t1 1 2 rf 1 12 s 11t 5 ( u c ) s 1ct 1 (1 2 u ) s 1pt

where s xzt 5 covt (xt 1 1 2 Etxt 1 1 zt 1 1 2 Etzt1 1) Under assumption(A3) the conditional variance of the risky asset return s 11t 5 s u

2but we avoid making this substitution until we use (A3) to solvethe model in subsection III4 Equation (14) is the starting pointfor our analysis of optimal portfolio choice

III2 Log-Linear Budget Constraint

Following Campbell [19931996] we also log-linearize thebudget constraint (7) around the mean consumption-wealth ratioand we obtain

(15) D wt1 1 lt rpt 1 1 1 (1 2 (1 r ))(ct 2 wt) 1 k


where k 5 log ( r ) 1 (1 2 r ) log (1 2 r )r and r 5 1 2 exp E(ct 2 wt)is a log-linearization parameter Note that r is endogenous in thatit depends on the average log consumption-wealth ratio which isunknown until the model has been solved

Campbell [1993] and Campbell and Koo [1997] have shownthat the approximation (15) is exact when the consumption-wealth ratio is constant over time and becomes less accurate asthe variability of the ratio increases In our model the consumption-wealth ratio is constant when the elasticity of intertemporalsubstitution is one or the expected risky asset return is constantWhen these conditions do not hold the consumption-wealth ratiovaries and we can only solve for it by using the approximation(15) Hence to check the accuracy of the approximation we mustcompare our solution with a discrete-state numerical solution Weundertake this exercise in Campbell Cocco Gomes Maenhoutand Viceira [1998] and discuss the results briey in subsectionIV6 below

The log-linearization (15) takes the return on the wealthportfolio as given and does not relate it to the returns onindividual assets We can push the approach further by using anapproximation to the log return on wealth

(16) rpt 1 1 5 a t(r1t 1 1 2 rf) 1 rf 1 12 a t(1 2 a t) s 11t

This approximation holds exactly in a continuous-time model withan innitesimally small trading interval where Itorsquos Lemma canbe applied to equation (1) It has the effect of ruling out bank-ruptcy even when the investor holds a short position with a t 0 ora leveraged portfolio with a t 1 Thus our portfolio solutions likethose in the continuous-time models of Merton [1969 1971] allowa t to vary outside the range from zero to one6

Combining (15) and (16) we get

(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t

6 In the recent literature on long-horizon portfolio choice Balduzzi andLynch [1997a 1997b] Barberis [1999] and Brennan Schwartz and Lagnado[1997] restrict short sales and borrowing but Brandt [1999] and Kim and Omberg[1996] do not


which is linear in log returns and log consumption and quadraticin the portfolio weight a t

III3 Characterizing the Optimal Portfolio Rule

The next step in our solution method is to characterize theoptimal portfolio rule by relating the current optimal portfoliochoice to future optimal portfolio choices This will then allow usto guess a functional form for the optimal portfolio policy and toidentify its parameters

Our strategy is to characterize the covariance terms s 1ct ands 1pt that appear in the log-linear portfolio-choice Euler equation(14) Since the risky asset return is exogenous and the consump-tion-wealth ratio is stationary in our model it is convenient torewrite these moments in terms of the variance of the risky assetreturn and its covariance with the consumption-wealth ratio Werst note that using the trivial equality

(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1

and the budget constraint (17) we can write s 1ct as

s 1ct 5 covt (r1t 1 1 D ct 1 1)

5 covt (r1t 1 1ct 1 1 2 wt 1 1) 1 a t vart (r1t1 1)

5 s 1c 2 wt 1 a t s 11t

where to obtain the second equality we use the fact thatcovt (xt1 1 zt) 5 0

Similarly equation (16) implies that

s 1pt 5 covt (r1t 1 1 a tr1t 1 1 1 (1 2 a t)rf 1 12 a t(1 2 a t) s 11t)

5 a ts 11t

These expressions can be substituted into (14) to get

Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t

which can be rearranged using the fact that u 5 (1 2 g )(1 2 c 2 1)to get

(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t

This equation was rst derived by Restoy [1992] It has twoterms each one capturing a different aspect of asset demand The


rst term captures that part of asset demand induced exclusivelyby the current risk premiummdashhence the adjective lsquolsquomyopicrsquorsquo oftenused to describe it in the nance literature The myopic compo-nent of asset demand is directly proportional to the asset riskpremium and inversely proportional to the individualrsquos relativerisk aversion The second term is the lsquolsquointertemporal hedgingdemandrsquorsquo of Merton [1969 1971 1973] It reects the strategicbehavior of the investor who wishes to hedge against futureadverse changes in investment opportunities as summarized bythe consumption-wealth ratio Intertemporal hedging demand iszero when returns are unpredictable so s 1c 2 wt 5 0 or riskaversion g 5 17 It is well-known that asset demand is myopic inthese special cases but much less is known about asset demand inthe general case

Although equation (19) gives us meaningful informationabout the nature of the investorrsquos demand for the risky asset it isnot a complete solution of the model because the current optimalportfolio allocation in (19) is a function of future portfolio andconsumption decisions which are endogenous to the problem

The dependence of todayrsquos portfolio allocation on futureportfolio and consumption choices operates through the condi-tional covariance s 1c 2 wt To see this note that the approximationto the intertemporal budget constraint can be used to write the logconsumption-wealth ratio as a constant plus the discountedpresent value of the difference between expected future logreturns on wealth and consumption growth rates [Campbell1993]

ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r

This equation follows from combining the log-linear budget con-straint (15) with (18) solving forward the resulting differenceequation and taking expectations If we substitute the expression forexpected consumption growth (12) into this equation we obtain

(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )

7 Log utility is the special case in which c 5 1 as well as g 5 1 But g 5 1delivers zero intertemporal hedging demand regardless of the value of c [Giovan-nini and Weil 1989]


Hence s 1c 2 wt depends on the individualrsquos future portfolio andconsumption decisions and (19) falls short of a complete solutionto the model

III4 Solving for the Optimal Policies

The nal step in solving the dynamic optimization problem isto guess a functional form for the optimal consumption andportfolio policies and to identify the parameters of these policiesusing the method of undetermined coefficients We guess that theoptimal portfolio weight on the risky asset is linear in the statevariable and that the optimal log consumption-wealth ratio isquadratic in the state variable Hence we guess that

(i) a t 5 a0 1 a1xt

(ii) ct 2 wt 5 b0 1 b1xt 1 b2xt2

where a0a1b0b1b2 are xed parameters to be determinedUnder assumptions (A1)ndash(A5) we can show that guesses

(i)ndash(ii) are indeed a solution to the intertemporal optimizationproblem of the recursive-utility-maximizing investor and we cansolve for the unknown parameters a0a1b0b1b2 Details areprovided in Appendices 1 and 2 here we give a brief intuitiveexplanation of the solution

The linear portfolio rule (i) has the simplest form consistentwith time variation in the investorrsquos portfolio decisions Thisportfolio rule implies that the expected return on the portfolio isquadratic in the state variable xt because an increase in xt affectsthe expected portfolio return both directly by increasing theexpected return on existing risky-asset holdings and indirectly bychanging the investorrsquos optimal allocation to the risky assetEquation (20) shows that the log consumption-wealth ratio islinearly related to the expected portfolio return so it is natural toguess that the log consumption-wealth ratio is quadratic in thestate variable xt

Of course variances and covariances of consumption growthand asset returns also affect the optimal consumption and portfo-lio decisions But the homoskedastic linear AR(1) process for xt

implies that all relevant variances and covariances are eitherlinear or quadratic in the current state variable and thussecond-moment effects do not change the linear-quadratic form ofthe solution Appendix 1 states nine lemmas that express impor-tant expectations variances and covariances as linear or qua-dratic functions of the state variable


We now state two propositions that enable us to solve for theunknown coefficients of the model The propositions are proved inAppendix 2 using the lemmas from Appendix 1

PROPOSITION 1 The parameters dening the linear portfolio policy(i) satisfy the following two-equation system

a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f

Proof of Proposition 1 See Appendix 2

The rst term in each of these equations is the myopiccomponent of asset demand in equation (19) Therefore theremaining terms represent intertemporal hedging demand Theydepend on the consumption coefficients b1 and b2 divided by oneminus the intertemporal elasticity of substitution (1 2 c ) as wellas on the scaled deviation of risk aversion from one ( g 2 1) g andthe scaled covariance of the risky asset return with revisions inthe expected future return s h us u

2 There is no hedging demand ifthis covariance is zero for then the risky asset cannot be used tohedge changes in investment opportunities We discuss the effectsof these parameters on portfolio selection in more detail in ourcalibration exercise in Section IV

Proposition 1 expresses the coefficients of the optimal portfo-lio policy as linear functions of the parameters of the optimalconsumption rule Proposition 2 shows that these parameterssolve a recursive nonlinear equation system whose coefficientsare known constants

PROPOSITION 2 The parameters dening the consumption policy(ii) b0b1b2 are given by the solution to the followingrecursive nonlinear equation system

(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22


where L i j i 5 123 j 5 1 6 are constants given inAppendix 2 These constants depend on the exogenous param-eters of the model and on the log-linearization parameter r


The equation system given in Proposition 2 can be solvedrecursively starting with the quadratic equation (23) whose onlyunknown is b2 This equation has two possible roots whichare always real when g $ 1 and are real when g 1 ifg ( f 2 2 (1 r ))2s u

2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0

The existence of real roots is necessary (but not sufficient) for thevalue function of the problem given in Property 1 below to benite We argue in the next section that one of the two roots thepositive root of the equation discriminant delivers the correctsolution to the model

Once we have solved for b2 the second equation in the systembecomes a linear equation in b1 Finally given b1b2 the rstequation of the system is also linear in b0 Using the known valuesof b0b1b2 in Proposition 1 we can nd a0a1

All of these calculations are conditional on a value for r sincer helps to determine the constants L i j in (21) (22) and (23) Onecan write the parameters as functions of r for example b0( r ) b1( r )and b2( r ) to express this dependence But r itself depends on theoptimal expected log consumption-wealth ratio and hence onthe parameters r 5 1 2 exp E(ct 2 wt) 5 1 2 exp b0( r ) 1b1( r )micro 1 b2( r )(micro2 1 s x

2) The solution of the model is complete onlywhen a value of r has been found to satisfy this nonlinearequation Unfortunately an analytical solution is available onlyin the case c 5 1 where the optimal consumption policy is myopicand r 5 d In all other cases we resort to a numerical method Werst set r 5 d and then nd the optimal values of a0a1b0b1b2

given this value of r For these optimal values we then computeE(ct 2 wt) and a new value of r for which a new set of optimalpolicies is computed We proceed with this recursion until theabsolute value of the difference between two consecutive values ofr is less than 10 2 4

This procedure converges extremely rapidly whenever thereexists a solution for r between zero and one For some parametervalues however r converges to one and the implied valuefunction of our model is innite It is well-known that this canoccur in innite-horizon optimization problems Merton [1971]


and Svensson [1989] for example derive parameter restrictionsthat are required for nite value functions in continuous-timemodels with constant expected returns Unfortunately the nonlin-earity of the equation for r prevents us from deriving equivalentanalytical restrictions in our model with time-varying expectedreturns but the problem tends to arise whenever the utilitydiscount rate is too low or the expected excess equity return is toohigh on average or too variable relative to the risk of equityinvestment

III5 Properties of the Solution

Propositions 1 and 2 identify the parameters of the optimalpolicies and the value function per unit of wealth If we pick thesolution for b2 given by the positive root of the discriminant in(23) Propositions 1 and 2 also imply the following properties ofthe solution Here we merely state these properties proofs aregiven in Appendix 3

PROPERTY 1 The approximate value function per unit of wealth isgiven by

(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2

and b2(1 2 c ) 0 Therefore the value function per unit ofwealth is a convex function of xt the expected log excessreturn on the risky asset

PROPERTY 2 The slope of the optimal portfolio rulemdashthe coeffi-cient a1mdashis positive Also lim g ` a1 5 0 and lim g 0 a1 5 1 `

Property 1 characterizes the approximate value function perunit of wealth Equation (24) shows that the log value function perunit of wealth is a quadratic function of the state variable whosecoefficients are the coefficients of the log consumption-wealthfunction divided by one minus the elasticity of intertemporalsubstitution8

Property 1 tells us that the value function per unit of wealthis convex in xt so it increases with xt when xt is large enough and

8 This expression has a well-dened limit as c 1 The solutions toequations (22) and (23) imply that b1(1 2 c ) and b2(1 2 c ) are functions only of rand g and do not depend directly on c Property 3 shows that r 5 d when c 1Finally equation (21) implies that (b0 2 c log (1 2 d ))(1 2 c ) does not dependdirectly on c when r 5 d Thus (24) is well dened as c 1


decreases with xt when xt is small enough The intuition for thisresult is as follows The investor can prot from predictable excessreturns on the risky asset whether these are positive or negativeProperty 2 implies that the investor increases the allocation to therisky asset as its expected excess return increases If excessreturns are expected to be sufficiently positive the investor willprot by going long whereas if they are expected to be sufficientlynegative the investor will prot by going short Thus movementsin xt to extreme positive or negative values increase the investorrsquosutility

Property 2 generalizes a known comparative-statics result foran investor with power utility facing constant expected returns ina continuous-time model In that setting the allocation to the riskyasset is constant over time and it increases with the expectedexcess return on the risky asset In static models with moregeneral utility functions however it is possible for the allocationto the risky asset to decline with the expected excess return on therisky asset because the income effect of an increase in the riskpremium can overcome the substitution effect [Ingersoll 1987Chapter 3] Property 2 shows that this does not happen in ourdynamic model with Epstein-Zin-Weil utility The coefficient a1 isalways positive and increases from zero when g is innitely largeto innitely large values as g approaches zero

PROPERTY 3 The solution given by Propositions 1 and 2 ap-proaches known exact solutions as the parameters of utilityapproach the following special cases

a) When c THORN 1 and g 1 equation (23) becomes linear andb2(1 2 c ) 12 s u

2(1 r 2 f 2) 0 In this case the optimalportfolio rule is myopic a0 12 g and a1 1 g s u

2 Thisportfolio rule is the known exact solution of Giovanniniand Weil [1989] in which portfolio choice is myopic eventhough consumption choice is not The portfolio rulemaximizes the conditional expectation of the log return onwealth

b) When c 1 and g THORN 1 b1 0 b2 0 r d and b0

log (1 2 d ) This consumption rule is the known exactsolution of Giovannini and Weil [1989] in which consump-tion choice is myopicmdashin the sense that the consumption-wealth ratio is constantmdasheven though the optimal portfo-lio rule is not


c) When c 1 and g 1 so that utility is logarithmic b1

0 b2 0 r d b0 log (1 2 d ) a012 and a1 1s u

2This is the known exact solution for log utility in whichboth the optimal consumption rule and the optimal portfo-lio rule are myopic

d) When s n2 0 so that expected returns are constant both

the optimal consumption rule and the optimal portfoliorule converge to the known exact myopic solution Theportfolio parameters a0 12 g and a1 1 g s u

2

It is important to note that the previously known resultsmentioned in parts a) and b) of Property 3 are only partial That isthe exact portfolio rule is known for the case g 5 1 but ourapproximate solution method is still needed to determine theoptimal consumption rule The exact consumption rule is knownfor the case c 5 1 but our solution method is still needed todetermine the optimal portfolio rule In this case our solution isexact (in continuous time) since the optimal consumption-wealthratio is constant so our log-linear version of the intertemporalbudget constraint holds exactly

Property 3 holds only if we choose the positive root of thediscriminant in the quadratic equation for b2 (23) If instead wechoose the negative root of the discriminant the approximatesolutions diverge as the preference parameters approach theknown special cases This is our main reason for choosing thepositive root of the discriminant9

PROPERTY 4 The optimal portfolio rule does not depend on c forgiven r

This property holds because only the ratios b1(1 2 c ) andb2(1 2 c ) appear in the portfolio rule and these ratios do notdepend on c for given r The property shows that the mainpreference parameter determining portfolio choice is the coeffi-cient of relative risk aversion g and not the elasticity of intertem-poral substitution c Conditioning on r c has no effect on portfolio

9 We would like to be able to show analytically that the unconditional meanof the value function a measure of welfare we study in Section V below is alwayshigher when we choose the positive root of the discriminant in (23) Unfortunatelywe have been unable to do this but in our calibration exercise we have veried thatthe positive root always gives the higher unconditional mean for every set ofparameter values we consider We do have a stronger analytical result when g 1In this case the negative root of the discriminant violates a necessary andsufficient condition (derived by straightforward extension of the results of Constan-tinides [1992]) for the existence of the unconditional mean of the value function


choice However r itself is a function of c mdashrecall that r 5 1 2exp E[ct 2 wt] mdashso the optimal portfolio rule depends on c indi-rectly through r Our calibration results in Section IV show thatthis indirect effect is small

PROPERTY 5 The parameters a1 and b2 the slope of the portfoliopolicy and the curvature of the consumption policy do notdepend on micro for given r

Property 5 shows that some aspects of the optimal policymdashthesensitivity of the risky asset allocation to the state variable andthe quadratic sensitivity of consumption to the state variablemdashare independent of the average level of the excess return on therisky asset Of course other aspectsmdashthe average allocation to therisky asset the average consumption-wealth ratio and the linearsensitivity of consumption to the state variablemdashdo depend on theaverage risk premium We discuss this dependence in greaterdetail in subsection IV3

IV CALIBRATION EXERCISE

IV1 Data and Estimation

An important advantage of our approach is that we cancalibrate our model using real data on asset returns To illustratethis we use quarterly U S nancial data for the sample period19471ndash1995410 In our calibration exercise the risky asset is theU S stock market and the risk-free asset is a short-term debtinstrument To measure stock returns and dividends we usequarterly returns dividends and prices on the CRSP value-weighted market portfolio inclusive of the NYSE AMEX andNASDAQ markets The short-term nominal interest rate is thethree-month Treasury bill yield from the Risk Free File on theCRSP Bond tape To compute the real log risk-free rate thebeginning-of-quarter nominal log yield is deated by the end-of-quarter log rate of change in the Consumer Price Index from theIbbotson les on the CRSP tape Log excess returns are computedas the end-of-quarter nominal log stock return minus the begin-ning-of-quarter log yield on the risk-free asset

10 A similar exercise using annual U S data for the period 1872ndash1993 isreported in the NBER Working Paper version of this article [Campbell and Viceira1996]


The state variable is taken to be the log dividend-price ratiomeasured as the log of the total dividend on the market portfolioover the last four quarters divided by the end-of-period stockprice Campbell and Shiller [1988a] Fama and French [1988]Hodrick [1992] and others have found this variable to be a goodpredictor of stock returns We estimate the following restrictedVAR(1) model

(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1

where ( e 1t 1 1 e 2t1 1) N(0 V ) and report the maximum likelihoodestimation results in Table I Since (25) is equivalent to amultivariate regression model with the same explanatory vari-ables in all equations ML estimation is identical to OLS regres-sion equation by equation The standard errors for the slopesintercepts and the residual variance-covariance matrix are basedon Proposition 112 in Hamilton [1994] using these standarderrors which assume that the variables in the model are station-ary the slopes and the elements of the variance-covariance matrixall appear to be statistically different from zero11

The parameters in (3) (4) and (5) that dene the stochasticstructure of our model can be recovered from the VAR system (25)as follows micro 5 u 0 1 u 1b 0 (1 2 b 1) f 5 b 1 s h

2 5 u 12V 22 s u

2 5 V 11 ands h u 5 u 1V 12 Table I reports these implied parameters along withstandard errors computed using the delta method All of the derivedparameters except s h

2 are signicantly different from zero at the 5percent condence level The unconditional expected log excess returnmicro is estimated at 5 percent per year (125 percent per quarter) whilethe log real risk-free rate rf is a meager 28 percent per year12

IV2 Solution of the Model

Using the parameter estimates in Table I we compute theindividualrsquos optimal portfolio allocation and consumption-wealthratio for a range of values of relative risk aversion and elasticity of

11 Note however that b ˆ 1 is close to one Elliott and Stock [1994] have shownthat the t-ratio for u 1 under the null H0 u 1 5 0 does not have a standard asymptoticnormal distribution when the log dividend-price ratio follows a unit-root ornear-unit-root process and V 12 THORN 0 We do not pursue this issue further here andproceed to calculate standard errors under the assumption that the estimatedsystem is stationary but we note that standard errors computed under thisassumption should be treated with some caution

12 These parameter estimates differ slightly from those reported in theNBER Working Paper version of this article The reason is that there because of acomputational error we used the dividend-price ratio instead of its log whenestimating (25)


intertemporal substitution We set d the time discount parameterunder time-additive utility to 94 in annual terms This isequivalent to a 62 percent annual log time discount rate

We consider relative risk aversion coefficients g 575115024102040 and elasticity of intertemporal substitu-tion coefficients c 5 175111501214110120140 Theliterature on the equity premium puzzle has shown that highlevels of risk aversion are needed to reconcile aggregate consump-tion data with asset market data in the standard power-utilityframework here we are able to compare the portfolio allocations

TABLE IESTIMATES OF THE STOCHASTIC PROCESS FOR RETURNS (19471ndash19954)

(A) Restricted VAR(1)

r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737

Table I reports ML estimates of the stochastic process driving expected and unexpected returns in themodel These estimates are based on quarterly returns dividends and prices from CRSP for the period19471ndash19954 Stock market data are for the CRSP value-weighted market portfolio inclusive of the NYSEAMEX and NASDAQ markets and the short-term nominal interest rate is the three-month Treasury billyield from the Risk Free File on the CRSP Bond tape Panel A reports ML point estimates and standard errors(in parentheses) of a restricted VAR(1) model (see equation (24) in text) for excess log returns and the logdividend-price ratio Panel B reports estimates for the parameters dening the stochastic structure of themodel These estimates and their standard errors (in parentheses) are derived from the estimates in Panel AStandard errors are obtained using the delta method


and consumption rules implied by low and high risk aversioncoefficients We consider low elasticities of intertemporal substitu-tion both because we want to include the power-utility cases inwhich the elasticity of intertemporal substitution is the reciprocalof risk aversion and because a low elasticity of intertemporalsubstitution seems to be required to explain the insensitivity ofconsumption growth to real interest rates in postwar U S data[Hall 1988 Campbell and Mankiw 1989]

Tables II III and IV and Figures I and II report the results ofthis exercise To make it easier to interpret our results wenormalize the parameters dening the optimal portfolio andconsumption policies (i) and (ii) so that the intercepts of theoptimal policy functions are the optimal allocation to stocks andthe optimal consumption-wealth ratio when the expected simpleexcess return Et[R1t 1 1] 2 Rf is zero At this point in the statespace the risky asset is a lsquolsquofair gamblersquorsquo offering no risk premiumThus a myopic risk-averse investor would allocate no wealth to itand all the demand for the risky asset is intertemporal hedgingdemand The expected simple excess return is zero when theexpected log excess return xt is equal to 2 s u

22 Therefore theparameters reported in the tables are a0 a1 b0 b1 and b2 in

(26) a t 5 a0 1 a1(xt 1 s u

22)

and

(27) ct 2 wt 5 b0 1 b1 (xt 1 ( s u22)) 1 b2(xt 1 ( s u

22))2

where a0 5 a0 2 a1( s u22) b0 5 b0 2 b1( s u

22) 1 b2( s u44) b1 5 b1 2

b2s u2 and a1 and b2 do not have asterisks because they coincide

with the original parametersThe main diagonal of each panel in the tables corresponds to

standard power-utility preferences since the elasticity of intertem-poral substitution is the reciprocal of risk aversion along the maindiagonal The numbers reported in the tables summarize theoptimal decisions of a recursive-utility individual who observesthe true process for returns Since we do not observe the trueprocess but must estimate it we have also computedmdashbut we donot report here to save spacemdashthe standard errors for theseparameters using the delta method13 These standard errors

13 The delta method requires the computation of derivatives of the parame-ters of interest (for example a1) with respect to u 0u 1b 0b 1V Since no analyticalformulas are available we use two-sided numerical derivatives based on aproportional perturbation parameter equal to 10 2 4 The standard errors arereported in the NBER Working Paper version of this article


show that the intercepts of the optimal policies are estimated withless precision than the parameters determining the slope andcurvature of the optimal policy

IV3 The Optimal Portfolio Rule

Tables II and III and Figure I summarize the optimalportfolio decision Panel A in Table II reports a0 the optimalallocation to stocks when the expected gross excess return is zerowhile Panel B in the same table reports a1 the slope of the optimalportfolio policy Panel A in Table III reports the average totaldemand for stocks as a fraction of wealth while Panel B reportsthe share of this average total demand that is attributable to theaverage hedging demand for stocks

Figure I which is divided into four panels illustrates theportfolio rule a t Figure Ia xes c at 1075 and plots a t for a wide

TABLE IIOPTIMAL PORTFOLIO POLICY

RRA EIS

(A) Exponentiated intercept a0 3 100

175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87

Panel A reports the optimal percentage allocation to stocks when the expected gross excess return is zerofor different levels of relative risk aversion and elasticities of intertemporal substitution Panel B reports thechangemdashin percentage pointsmdashin the optimal allocation to stocks when the expected log excess returnincreases by 1 percent per quarter These numbers are all based on the parameter estimates for the returnprocess reported in Table II (sample period 19471ndash19954) The values on the main diagonal correspond to thepower utility case


range of g values Figure Ib repeats this exercise xing c at 14Figures Ic and Id on the other hand x g at 075 and 4respectively and plot a t for a wide range of c values In all thesegures we consider values of xt in the interval (micro 2 2 s xmicro 1 s x)and the horizontal axis is the log of the expected gross excessreturn ie log Et[R1t 1 1Rf] 5 xt 1 s u

22 The right vertical lineintersects the horizontal axis at the log of the unconditional meangross excess return log E[R1t1 1Rf] 5 micro 1 s x

22 1 s u22

The most striking lesson from the tables and from Figure I isthat relative risk aversion is far more important than theelasticity of intertemporal substitution in determining the opti-mal portfolio allocation to stocks The variation in parameters

TABLE IIIMEAN OPTIMAL PERCENTAGE ALLOCATION TO STOCKS AND PERCENTAGE MEAN

HEDGING DEMAND OVER MEAN TOTAL DEMAND

RRA EIS

(A) Mean optimal percentage allocation to stocksa t 5 [a0 1 a1(micro 1 s u

2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207

(B) Fraction due to hedging demand (percentage)[ a thedging(microg c )a t(microg c )] 3 100

175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655

Panel A reports the mean optimal percentage allocation to stocks for different levels of relative riskaversion and elasticities of intertemporal substitution Panel B reports the percentage mean hedging demandover mean total demand ie the fraction of the mean allocation due to hedging demand Mean hedgingdemand is calculated as a thedging (micro g c ) 5 a t(microg c ) 2 a t(micro1 c ) g These numbers are all based on theparameter estimates for the return process reported in Table II (sample period 19471ndash19954) The values onthe main diagonal correspond to the power utility case


FIG

UR

EI


across rows of the tables as g changes is far greater than thevariation across columns as c changes Similarly the a t lines inFigures Ic and Id are all close together whereas those in FiguresIa and Ib vary widely in both slope and intercept This result canbe understood by recalling Property 4 of our solution ie that a t

depends on c only through the dependence of r on c Our calibrationresults show that this indirect effect through r is very small

Panel A in Table II shows that the intercept of a t a0 ispositive when g 1 It is zero when g 5 1 as we already knowfrom the analysis of the special case with unit relative riskaversion and negative when g 1 These results hold regardlessof the value of c To interpret this behavior recall that a0 is theoptimal allocation to stocks when the expected excess gross returnis zero Since the myopic demand for stocks is zero at this level ofthe expected excess gross return a0 is completely determined byhedging demand Thus at the point in the state space where therisky asset has a zero expected excess return the sign of hedgingdemand is positive for investors with g 1

Panel B in Table II shows that the coefficient a1 the slope ofthe a t function is positive for all levels of g and c as implied byProperty 2 of our solution Like the intercept a0 the slope a1 variessubstantially across g for a given level of c but changes very littleacross c for a given level of g As g increases a1 rapidly approacheszero indicating that the optimal portfolio rule is very responsive tochanges in expected excess returns when the individual is close torisk-neutral but is almost at when the individual is highly risk-averse This nding is also implied by Property 2 of our solution14

Panel B in Table II also shows that whenever g 1intertemporal hedging demand increases the slope of the portfoliorule equivalently hedging demand itself has a positive slope Tosee this note that when g 5 1 hedging demand is zero and theslope coefficient of 1888 is entirely attributable to the myopiccomponent of asset demand For higher values of g the myopicslope shrinks in proportion to g Thus it is 944 for g 5 2 472 forg 5 4 and so forth The slope coefficients reported in Panel B inTable II shrink more slowly than this implying a positive slopecontribution from hedging demand The analytical foundation ofthis result is that from Proposition 1 the slope of hedging demand

14 The standard errors not reported here show that the slope coefficient a1 ismuch more precisely estimated than the intercept of the optimal portfolio ruleOne reason for this greater precision is that as we showed in Property 5 the slopeof the portfolio rule is not sensitive to the mean excess stock return micro whereas theintercept does depend on micro


is 2 (b2(1 2 c ))(( g 2 1) g )( s h u s u2)2 f which is positive when g 1

under our assumption of negative s h u This implies that conserva-tive long-horizon investors who are free to rebalance every periodare actually more aggressive market timers than conservativeshort-horizon investors15

The results in Table II can be explained in intuitive terms asfollows We have estimated a return-generating process which hasa negative sign for s h u the covariance between unexpected stockreturns and revisions in expected future stock returns Thisimplies that stocks tend to have high returns when their expectedfuture returns fall Since the investor is normally long in stocks adecline in expected future stock returns is normally a deteriora-tion in the investment opportunity set There are offsettingconsiderations that determine an investorrsquos attitudes towardassets that pay off when the investment opportunity set deterio-rates On the one hand an investor with low risk aversion ( g 1)wants to hold assets that deliver wealth when wealth is mostproductive that is when investment opportunities are good Thisinvestor has a negative hedging demand On the other hand aninvestor with high risk aversion ( g 1) wants to hold assets thatdeliver wealth in unfavorable states of the world that is wheninvestment opportunities are poor This investor has a positivehedging demand Interestingly the hedging demand is not mono-tonic in risk aversion because an extremely risk-averse investorlimits her exposure to the risky asset in all states of the worldThus the magnitude of hedging demand rst rises and then fallswith the coefficient of risk aversion

Although the investor is normally long in stocks if theexpected excess return becomes sufficiently negative a decline inexpected future stock returns can represent an improvement inthe investment opportunity set because it creates a protableopportunity to short stocks At this point in the state space thesign of hedging demand for stocks reverses This explains whyhedging demand has both a positive intercept and a positive slopeallowing a sign reversal of hedging demand for sufficientlynegative xt16

The average level of excess simple stock returns micro 1 s u2 2

plays an important role in this argument We have estimated micro 1s u

2 2 to be positive and quite large this leads the investornormally to maintain a long position in stocks for which a

15 Barberis [1999] nds that conservative buy-and-hold investors who knowthe parameters of the stock return process are about equally aggressive markettimers whether they have a short or a long horizon


decrease in the expected stock return represents a deteriorationin investment opportunities If micro 1 s u

22 were negative howeverthe investor would normally have a short position in stocks forwhich a decrease in the expected stock return represents animprovement in investment opportunities In this case the normalsign of hedging demand would be negative for an investor with g 1 The slope of hedging demand is unaffected by the average levelof excess returns however as shown in Property 5 so in this casea sign reversal of the normal hedging demand occurs for suffi-ciently positive xt

This intuitive discussion suggests that we should be able toderive analytical results about the signs of the coefficients a0 andb1 in our model Indeed it is straightforward to show that whenmicro 1 s u

22 5 0 a0 5 b1 5 0 In this case the model is symmetricalpositive deviations of xt from its mean have exactly the same effect(in absolute value) as negative deviations and both myopic andhedging demand for the risky asset are zero when xt is at its meanUnfortunately we have been unable to derive comparable analyti-cal results about the signs of a0 and b1 when micro 1 s u

2 2 THORN 0However in numerical explorations we have found that with g 1and s h u 0 a0 and b1 (1 2 c ) always have the same sign as micro 1s u

22 whenever the value function is nite consistent with ourintuitive discussion of hedging demand17

Panel A in Table III reports the mean optimal allocation tostocks as a percentage of total wealth The mean allocation ispositive at all levels of g and c On average a recursive-utilityindividual with low or moderate levels of risk aversion will shortthe riskless asset in order to hold more than 100 percent of herwealth in the risky asset Large levels of relative risk aversion areneeded to keep mean stock demand below 100 percent this is amanifestation of the equity premium puzzle in our model withexogenous asset returns and endogenous portfolios

Panel B in Table III shows that average hedging demand is avery important part of total stock demand for investors whoserelative risk aversion coefficients are not close to one Averagehedging demand is calculated using (19) by setting xt 5 micro andsubtracting from the total risky-asset allocation the total alloca-

16 Kim and Omberg [1996] give a clear account of this effect [Figure 4 and pp153ndash154]

17 Kim and Omberg [1996] obtain more general analytical results for theirsimpler model with utility dened over terminal wealth


tion when g 5 1 divided by the level of relative risk aversion

a thedging(microg c ) 5 a t(microg c ) 2 a tmyopic(micro g c )

5 a t(micro g c ) 2 (1 g ) a t(micro1 c )

Average hedging demand is negative and often large for inves-tors with risk aversion coefficients of 075 for the illustrated riskaversion coefficients above one it is positive and accounts for atleast 20 percent of stock demand and often above 50 percentThus intertemporal hedging motives can easily double the equitydemand of risk-averse investors This makes it harder to explainthe equity premium puzzle with moderate levels of risk aversiona point emphasized by Campbell [1996] Brandt [1999] obtainssimilar results for the special case of power utility with riskaversion equal to 5 when he considers horizons of twenty years ormore

IV4 Optimal Consumption Behavior

Table IV and Figure II summarize the optimal consumptionpolicy Panel A in Table IV reports the exponentiated mean of theoptimal log consumption-wealth ratio Figure II is similar toFigure I but it plots Ct Wt 5 exp ct 2 wt instead of a t Both tableand gure reveal an important difference between the optimalconsumption rule and the optimal portfolio rule the optimalconsumption-wealth ratio is very sensitive to both the level of theelasticity of intertemporal substitution and the level of riskaversion while we have already noted that the optimal portfoliorule moves noticeably only with the level of risk aversion Thepattern of variation across the panel has interesting features Atlow levels of risk aversion g the optimal consumption-wealthratio decreases as the elasticity of substitution c rises (a move-ment along a row from right to left) At high levels of risk aversionon the other hand the optimal consumption-wealth ratio in-creases with c Similarly at low levels of the elasticity ofsubstitution c mdashspecically c 1mdashthe optimal ratio rises withrisk aversion g while at high levels of c it declines with g Theoptimal ratio is independent of g when c 5 1 as we already knowfrom Property 3 of the solution

This pattern of variation is also illustrated in Figures IIathrough IId where the vertical sorting of the Ct Wt curves isreversed as we move from c 5 1075 in Figure IIa to c 5 14 inFigure IIb and from g 5 075 in Figure IIc to g 5 4 in Figure IId


Figure II also shows that the sensitivity of the optimal logconsumption-wealth ratio to the state variable is modest for mostparameter values the curves for the optimal consumption policiestend to be rather at Brandt [1999] reports a similar result in anite-horizon model for the case of power utility with riskaversion of 5 Equation (20) explains this it shows that theconsumption-wealth ratio is determined only by long-run con-siderations The terms that appear on the right-hand side ofthe equation are expected discounted values of all future ex-pected returns and variances not current expected returns andvariances

TABLE IVOPTIMAL CONSUMPTION-WEALTH RATIO AND LONG-TERM EXPECTED LOG

RETURN ON WEALTH

RRA EIS

(A) Consumption-Wealth ratiosCt Wt 5 exp E[ct 2 wt] 3 100

175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027

(B) Long-term expected log return on wealthE[rpt 1 1] 3 100

175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052

Panel A reports percentage exponentiated mean optimal log consumption-wealth ratios per quarter ie100 times the exponential of E[ct 2 wt] 5 b

0 1 b1(micro 1 s u

2 2) 1 b2( s x2 1 micro2 1 micro s u

2 1 s u44) for different levels of

relative risk aversion and elasticities of intertemporal substitution Panel B reports the percentageunconditional mean of the quarterly log return on wealth These numbers are all based on the parameterestimates for the return process reported in Table II (sample period 19471ndash19954) The values on the maindiagonal correspond to the power utility case


FIG

UR

EII


To interpret the patterns in Panel A of Table IV consider rstthe right-hand column of the panel This gives the exponentiatedmean optimal log consumption-wealth ratio for an individual whois extremely reluctant to substitute consumption intertemporally( c 5 140 close to zero) Such an individual wishes to maintain aconstant expected consumption growth rate regardless of currentinvestment opportunities She can do this by consuming thelong-run average return on her portfolio with a precautionary-savings adjustment for risk But in our model both the risk andthe average return are endogenous If the investor is highlyrisk-averse as she is at the bottom of the column ( g 5 40) thenshe holds almost all her wealth in the riskless asset and earns alow return with little risk if she is close to risk-neutral as she isat the top of the column ( g 5 075) she borrows at the risklessinterest rate to earn a high but risky leveraged return Thisexplains why the mean consumption-wealth ratio is so muchhigher at the top of the column than at the bottom

To clarify this interpretation Panel B in Table IV reports theunconditional mean log portfolio return E[rpt 1 1]18 The mean logreturns in the right-hand column are close to the optimal consump-tion-wealth ratios given in the right-hand column of the upperpanel (Panel A) They are particularly close at high levels of riskaversion shown at the bottom of the tables at the top of thepanels the two variables diverge because the mean log returnreaches a maximum when the coefficient of relative risk aversiong 5 1 and starts to fall when risk aversion declines from thislevel whereas the optimal consumption-wealth ratio keeps onrising as g falls below one The investor with unit risk aversionmaximizes the conditional expectation of the log portfolio returnhence this investor must also have the highest unconditionalexpected log portfolio return The increase in the average consump-tion-wealth ratio as g falls below one is caused by the precaution-

18 We can compute the long-term or unconditional expected log return onwealth by taking unconditional expectations in Lemma 4 of Appendix 1 ie bycalculating E[Et(rpt 1 1)] which gives

E[rpt1 1] 5 rf 1 p0 1 p1Ext 1 p2Ex t2

5 rf 1 p0 1 p1micro 1 p2( s x2 1 micro2)

where p0 p1 and p2 are functions of a0 and a1 dened in Lemma 4 We can rewritep0 p1 and p2 as functions of the normalized parameters a0 and a1 by noticing thata0 5 a0 2 a1( s u

22)


ary savings effect which turns negative when c and g are on thesame side of unity as shown in equation (13)

Now consider what happens as the individual becomes morewilling to substitute intertemporally that is as c increases andwe move to the left in Panel A of the table Equation (20) helps usunderstand this If we hold xed the variance terms in (20) thederivative of ct 2 wt with respect to c is 2 [ r (1 2 r )](Et[(1 2 r )r ] middotS j 5 1

` r jrpt1 j 1 log d ) which is negative if the long-run expectedportfolio return exceeds the rate of time preference and positiveotherwise Ignoring precautionary savings effects an individualwho is willing to substitute intertemporally will have highersaving and lower current consumption than an individual who isunwilling to substitute intertemporally if the time-preference-adjusted rate of return on saving is positive but will have lowersaving and higher current consumption if the adjusted return onsaving is negative Panel A in Table IV illustrates this patternInvestors with low risk aversion g at the top of the table chooseportfolios with high average returns so a higher elasticity ofintertemporal substitution c corresponds to a lower averageconsumption-wealth ratio Highly risk-averse investors at thebottom of the table choose safe portfolios with low averagereturns so for these investors a higher c corresponds to a higheraverage consumption-wealth ratio

Our discussion so far has concentrated on the average level ofconsumption in relation to wealth We now give some intuitionabout the sensitivity of the optimal ratio to the state variable xtAlthough we have noted that the slope of the optimal consumptionpolicy is always small in absolute value relative to the interceptFigure II shows that around the mean of the state space it isnegative when c 1 and positive when c 1 Moreover itincreases in absolute value as g decreases The intertemporalsubstitution effect and the portfolio composition effect explain thispattern As xt increases in the neighborhood of its positive meanso does the expected return on wealth causing income andsubstitution effects on consumption When c 1 the substitutioneffect dominates and the investor will cut consumption to exploitfavorable investment opportunities When c 1 the income effectdominates and the investor will increase consumption because agiven quantity of wealth can sustain a greater ow of consump-tion The effect of risk aversion appears because the state variablext increases only the expected return on the risky asset not theexpected return on the riskless asset An investor with a low risk


aversion coefficient is more heavily invested in the risky assetand thus her expected portfolio return is more sensitive tochanges in xt

Finally we consider the implications of the consumption rulefor the volatility of consumption relative to past expectations andrelative to wealth Panel A in Table V reports the unconditionalstandard deviation of consumption innovations for each set ofpreferences we have considered and Panel B reports the uncondi-tional standard deviation of innovations in the log consumption-wealth ratio

Panel A in Table V shows that investors with low riskaversion have extremely volatile consumption growth for their

TABLE VVOLATILITY OF CONSUMPTION GROWTH AND VOLATILITY OF THE LOG

CONSUMPTION-WEALTH RATIO

RRA EIS

(A) Volatility of consumption growths ( D ct 1 1 2 Et[ D ct 1 1]) 3 100

175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118

(B) Volatility of the consumption-wealth ratios (ct 1 1 2 wt 1 1 2 Et[ct 1 1 2 w t 1 1]) 3 100

175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142

Panel A reports the percentage unconditional standard deviation of quarterly log consumption innova-tions for different levels of relative risk aversion and elasticities of intertemporal substitution while Panel Breports the percentage unconditional standard deviation of innovations in the quarterly log consumption-wealth ratio These numbers are all based on the parameter estimates for the return process reported in TableII (sample period 19471ndash19954) The values on the main diagonal correspond to the power utility case


consumption inherits the volatility of their portfolios Investorswith unit elasticity of substitution in consumption have constantconsumption-wealth ratios and so their consumption volatilityequals their portfolio volatility Investors with low elasticity ofintertemporal substitution have somewhat less volatile consump-tion because they react to mean-reversion in stock returns bycutting their consumption-wealth ratios when the stock marketrises A 1 percent innovation in wealth causes these investors toincrease consumption by less than 1 percent they know that a 1percent increase in consumption could not be sustained even with1 percent greater wealth because the increase in wealth isaccompanied by a decrease in expected portfolio returns Inves-tors with high elasticity of intertemporal substitution respond tothe decrease in expected returns by cutting saving so theirconsumption is more volatile than their portfolio returns Similarresults are reported by Campbell [1996] for a model with anexogenous portfolio return process

Panel B in Table V shows that investors with elasticities ofintertemporal substitution different from one have volatile con-sumption-wealth ratios because they do not consume a xedfraction of their wealth each period but a varying fraction thatchanges with the expected excess return on the risky asset Thevolatility of the consumption-wealth ratio is increasing in thedistance of the elasticity of intertemporal substitution from oneand is decreasing in risk aversion since less risk-averse investorshave riskier portfolios whose expected returns are more sensitiveto changes in investment opportunities

IV5 Portfolio Allocation and Consumption over Time

Our results can also be summarized by plotting the optimalequity allocations and consumption-wealth ratios over timeFigure III does this for preference parameters c 5 14 g 5 4 corresponding to power utility with moderate risk aversion andc 5 14 g 5 20 corresponding to a higher level of risk aversionThe upper plot of each gure shows the optimal equity allocationswhile the lower plot shows the optimal consumption-wealthratios The horizontal lines in the equity-allocation plots repre-sent 0 percent and 100 percent holdings

The gures show that stock holdings are highly volatile whilethe optimal ratio of consumption to wealth is more stable butspikes up in periods where expected returns and optimal stockholdings are unusually high The investor with lower risk aver-


FIG

UR

EII

I


sion holds on average a much larger proportion of her wealth instocks and her consumption-wealth ratio is also larger on averageand more volatile But both investors are keen stock-marketparticipants In our model investors do not face restrictions onshort sales so we allow the optimal allocation to stocks to beeither larger than 100 percent or negative Figure III shows thatin the U S postwar period both investors are long in stocksalmost always and the less risk-averse investor usually wants toshort the riskless asset and invest more than 100 percent of herwealth in the market except in periods of unusually low dividendyields such as the early 1970s and the mid-1990s

Barberis [1999] has obtained similar results for a Bayesianinvestor who maximizes power utility dened over terminalwealth and uses the log dividend-price ratio as a state variablewith a ten-year investment horizon and access to historical dataover the period 1927ndash1993 Barberisrsquo investor who is not allowedto short assets is mostly 100 percent invested in stocks BrennanSchwartz and Lagnado [1997] have studied a similar problemwith power utility of terminal wealth three state variables threeassets and frequent portfolio rebalancing They also do not allowshort sales and their optimal strategy for the period 1972ndash1992often switches between 100 percent cash and 100 percent stocksTheir optimal strategy is more volatile than Barberisrsquo or oursbecause they allow for a larger number of state variablesBrennan Schwartz and Lagnado also include long-term bonds intheir analysis but bonds do not play a major role in the optimalportfolio

IV6 The Accuracy of the Solution

The analytical solutions we present in this paper are exactonly in the limit where time is continuous and for parametervalues that imply a constant consumption-wealth ratio ( c 5 1 orconstant expected returns) For other parameter values oursolutions are only approximate One way to assess their accuracyis to compare them with solutions obtained using standardnumerical methods

In Campbell Cocco Gomes Maenhout and Viceira [1998]we have solved numerically for optimal policy functions in thecalibrated model of this paper The numerical solution discretizesthe state space and approximates the distribution for the innova-tions in the random variables using Gaussian quadrature withnine quadrature points The numerical method assumes that the


portfolio allocation rule is a pth-order polynomial in the statevariablemdashin practice a third-order polynomial is adequatemdashanduses a variant of the Newton-Raphson algorithm to optimize overthe coefficients of this polynomial

The numerical solutions we obtain are very similar to theapproximate analytical solutions except at the upper extreme ofthe state space where both the numerical consumption andportfolio allocation rules atten out Figure IV illustrates this inthe four cases we obtain when we combine c 5 17514 and g 5420 The approximate analytical solution and the numericalsolution are particularly close between the vertical lines in theplot that delimit the interval (micro 2 2 s xmicro 1 2 s x) but they do tendto diverge at the right side of the plot where the state variable xt ismore than two standard deviations above its mean The diver-gence is more serious when g 5 4 than when g 5 20 because theinvestor with g 5 4 holds a riskier portfolio with a more volatileexpected return the effect of this outweighs the greater utilitycurvature for the investor with g 5 20 For the same reason thedivergence of the approximate from the numerical solution wouldbe smaller in a model with a more stable expected return on therisky asset Full details are provided in Campbell Cocco GomesMaenhout and Viceira [1998]

V THE UTILITY COSTS OF SUBOPTIMAL PORTFOLIO CHOICE

We have shown that a long-term investor who optimallyresponds to the estimated predictability of stock returns will bothtime the stock market and use stocks to hedge against deteriora-tions in the investment opportunity set However we have not yetshown that optimal timing and hedging produce large utilitygains If the utility gains are small they might easily be out-weighed by small costs of formulating and executing the optimalpolicy

To address this issue we use our approximate analyticalmethod to solve the intertemporal optimization problem of aninvestor who follows an arbitrary portfolio rule but adjusts herconsumption optimally We then compute the investorrsquos valuefunction per unit of wealth under the suboptimal portfolio ruleand compare the unconditional expectation of this value functionwith the unconditional expectation of the value function in the


FIG

UR

EIV


unrestricted problem19 Throughout we assume that the investorknows the stochastic process driving the return on the risky assetthat is we ignore the parameter uncertainty addressed by Kandeland Stambaugh [1996] and Barberis [1999]

We consider three restricted portfolio rules The rst ruleignores the timing implied by the optimal portfolio policy and setsthe equity allocation each period to the average allocation underthe optimal rule This is a xed portfolio rule that allows forpartial hedging in the spirit of the investment strategy advocatedby Siegel [1994] Siegel argues that long-run investors should nottry to time the stock market but should buy and hold large equitypositions because these positions involve little risk at long hori-zons Siegelrsquos estimates of long-run stock market risk are lowbecause of the mean-reversion in stock returns that we havecaptured with our VAR system Thus one can interpret Siegelrsquosstrategy as a hedging strategy without market timing The secondportfolio rule is the myopic rule that times the market but ignoreshedging considerations This rule would be optimal if the covari-ance s h u were zero The third rule sets the equity allocation eachperiod to the average allocation under the myopic rule ignoringboth timing and hedging considerations

Table VI describes the optimal consumption rules implied bythe restricted portfolio rules For comparison it also includes in itsrst row the optimal consumption rule (27) under the optimalportfolio rule (26) The parameters of these rules of course dependon the exogenous parameters of the model but to save space wedo not give further details here

The top left panel of Table VII reports the unconditional meanof the value function per unit of wealth that is implied by theoptimal unrestricted consumption and portfolio rules in thecalibrated example discussed in the previous section The otherthree panels report the percentage change in the value functionwhen portfolio choice is restricted to one of the suboptimal rulesdescribed above

The table shows that suboptimal portfolio choice can causelarge losses in utility Failing to hedge intertemporally is harm-less when risk aversion g 5 1 since in this case the optimalportfolio is myopic but it can be a serious error for investors withg 1 The losses from failing to hedge increase at rst as risk

19 The implied value functions are exponentials of quadratic or linearfunctions of the state variable The results in Constantinides [1992] allow us toobtain explicit formulas for their unconditional expectations


aversion increases above one but eventually diminish as ex-tremely risk-averse investors have only very small equity posi-tions and thus have little to hedge Failing to time the marketcauses large losses for all investors except those who are ex-tremely risk-averse but extremely willing to substitute consump-tion intertemporally For all parameter values we consider thefailure to time the market causes larger utility losses than thefailure to hedge intertemporally These results conrm for avariety of investors with different levels of risk aversion andelasticity of intertemporal substitution the ndings of Balduzziand Lynch [1997b] for nite-horizon investors with isoelasticpreferences dened over wealth and relative risk aversion coeffi-cients of 2 6 and 1020 The results are also compatible with the

20 Balduzzi and Lynch nd that utility losses increase with the horizon of theinvestor Since we consider an innite horizon this helps to explain why we obtainsomewhat larger utility costs than they do for similar levels of risk aversionBalduzzi and Lynch also nd that utility costs remain substantial in the presenceof xed and proportional transaction costs

TABLE VIOPTIMAL CONSUMPTION RULES IMPLIED BY RESTRICTED PORTFOLIO RULES

Portfolio rule Optimal consumption rulegiven portfolio rule

Hedging

Timing a t 5 a0 1 a1(xt 1 s u2 2) ct 2 wt 5 b0 1 b1(xt 1 s u

2 2)1 b2(x t 1 s u

22)No-timing a t 5 a0 1 a1(micro 1 s u

2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2

ct 2 wt 5 b0nht 1 b1

nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2

ct 2 wt 5 b0nhnt 1 b1

nhnt(xt 1 s u2 2)

The second column in Table VI describes the consumption rule followed by an investor who adjustsconsumption optimally given the portfolio rule described in the rst column of the table The rst rowdescribes the optimal consumption rule implied by the unconstrained optimal portfolio rule This rule isstate-dependent and includes a hedging component Therefore the rst row of the table describes the solutionto the intertemporal optimization problem we solve in Section III The second row describes the optimalconsumption rule followed by an investor who follows a suboptimal portfolio rule consisting in allocating tostocks each period a xed fraction of her savings that equals the average allocation to stocks implied by theoptimal portfolio rule Therefore this investor ignores timing in her portfolio decisions though she allows for(imperfect) hedging The third row of the table describes the optimal consumption rule followed by an investorwho follows a myopic portfolio rule This suboptimal portfolio rule ignores hedging but it is time-dependentFinally the fourth row of the table describes the optimal portfolio rule followed by an investor who ignoresboth hedging and timing and invests in stocks each period a xed fraction of her savings that equals theaverage myopic allocation to stocks


TA

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383

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166

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82

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098

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3

Th

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pper

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Th

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rts

the

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rner

rep

orts

the

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enta

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ssin

the

valu

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nct

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wh

enth

ep

ortf

olio

rule

ism

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ic(s

eeeq

uat

ion

(21)

inte

xt)

and

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nad

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nel

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and

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nad

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ese

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esfo

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pro

cess

pres

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din

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ese

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ates

for

the

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od19

471

ndash199

54

Th

eva

lues

onth

em

ain

dia

gon

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rres

pon

dto

the

pow

eru

tili

tyca

se


ndings of Kandel and Stambaugh [1996] and Barberis [1999]that Bayesian investors experience large gains in certainty-equivalent return when they optimally respond to evidence ofpredictability in stock returns

VI CONCLUSION

One of the major objectives of modern nancial economics hasbeen to put investment advice on a scientic basis This task hasbeen accomplished for investors who have short horizons orconstant investment opportunities Unfortunately most investorshave long horizons and there is considerable evidence that theyface time-varying expected returns on risky assets Until veryrecently nancial economists have not even attempted to givesuch investors precise quantitative advice about their portfoliostrategies

Recent work on long-horizon portfolio choice has generallyignored the consumption decision and has considered portfoliochoice for investors who consume nothing until a xed terminaldate21 Our objective has been to analyze a model in whichinvestors optimize over both consumption and portfolio allocationBecause the intertemporal consumption and portfolio choiceproblem is highly intractable when expected returns are timevarying we have resorted to an analytical approximation Wehave replaced the Euler equations and budget constraint of theexact problem with approximate equations that are much easierto solve and we have explored in detail the analytical solution ofthe approximate problem

We have used Epstein-Zin-Weil recursive preferences toseparate the inuence of risk aversion and the elasticity ofintertemporal substitution on portfolio choice and consumptionWe have shown for example that portfolio choice depends on theelasticity of intertemporal substitution only indirectly throughthe effect of this elasticity on the average level of consumptionrelative to wealth

We have used our model to assess the quantitative impor-tance of intertemporal hedging demand for risky assets bylong-lived investors After calibrating the model to postwar quar-terly U S stock market data we nd that intertemporal hedging

21 Very recent exceptions to this statement include Brandt [1999] and somecases analyzed in Balduzzi and Lynch [1997a1997b]


motives can easily double the average total demand for stocks byinvestors whose coefficients of relative risk aversion exceed oneWe also nd that suboptimal myopic portfolio rules imply largeutility losses for such investors These results support the conclu-sion of other recent papers such as Barberis [1999] Balduzzi andLynch [1997a 1997b] Brandt [1999] Brennan Schwartz andLagnado [1997] and Kim and Omberg [1996] that static models ofportfolio choice can be seriously misleading Intertemporal portfo-lio choice should not remain an abstruse theoretical topic butshould be integrated into empirical research and investment practice

An important caveat is that our analysis is partial equilib-rium in nature We solve the microeconomic problem of a giveninvestor facing exogenous asset returns but we do not show howthese asset returns could be consistent with general equilibriumOne possibility is that the representative investor has differentpreferences from those assumed here perhaps the habit-formation preferences of Campbell and Cochrane [1999] that cangenerate shifts in risk aversion and hence changing risk premi-ums with a constant riskless interest rate In this setting themodel has only limited applicability since it describes the behav-ior of atypical investors whose risk aversion is constant over timeAlternatively if all investors have the preferences assumed heretheir portfolio shifts could be supported in general equilibrium byshifting asset supplies Supplies of stock would have to fall withthe risk premium to accommodate investorsrsquo desire to reduce theirstockholdings But such shifts in supplies are unlikely to beconsistent with macroeconomicdata on aggregate portfolio shares

Another caveat has to do with approximation error Ourapproximate solution is exact when the elasticity of intertemporalsubstitution is one and the time interval between consumptionand portfolio decisions is innitesimally small In a companionpaper [Campbell Cocco Gomes Maenhout and Viceira 1998] wehave checked the accuracy of the analytical approximate solutionin other cases by comparing it with a discrete-state numericalsolution in our calibrated example We have found that oursolution is generally a good approximation to the true solutionbut the approximation error does increase when the state variableis more than two standard deviations above its mean We havealso found that the numerical solution algorithm converges muchmore rapidly and reliably when we are able to provide startingvalues from the approximate solution


The approach of this paper can be applied to many relatedproblems For simplicity we have considered only a single riskyasset and a single state variable but it is straightforward toconsider multiple risky assets and state variables We can explorehorizon and rebalancing effects in the manner of Barberis [1999]and Brandt [1999] by assuming that the investor has a niterather than innite horizon and restricting the frequency atwhich she can rebalance her portfolio We can allow the risklessinterest rate to vary over time and can consider investor choicesamong indexed or nominal bonds of different maturities [Camp-bell and Viceira 1998] We can allow for time variation in thevolatility of risky asset returns and even for the presence ofexogenous labor income in the investorrsquos budget constraint [Vi-ceira 1997] We believe that in all these cases there is muchunderstanding to be gained by taking an analytical approach tothe problem

APPENDIX 1 SOME USEFUL LEMMAS

In this appendix we state as lemmas and prove nine usefulresults We use some of them to prove in Appendix 2 the mainpropositions of the paper

LEMMA 1 The conditional expectation of future values of the statevariable is a linear function of its current value while theconditional expectation of future values of the squared statevariable is a quadratic function of the current state variable

Et1 1xt 1 j 5 micro 1 f j(xt 2 micro) 1 f j 2 1h t1 1

Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h

2 1 2micro f j(1 2 f j)xt

1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1

1 2micro f j2 1h t 1 1

Proof of Lemma 1 By simple forward recursion of xt and xt2 in

(3) we have

xt 1 j 2 micro 5 f j(xt 2 micro) 1 ol 5 0

j 2 1

f l h t1 j2 l


and

(28) (xt1 j 2 micro)2 5 f 2j(xt 2 micro)2 1 ol 5 0

j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1

f 2l(xt 1 j 2 12 l 2 micro) h t 1 j 2 l

The results stated in the lemma follow from the expressionsabove after taking conditional expectations at time (t 1 1)Et 1 1 and the martingale assumption (A2) which implies thatEt 1 l 2 1h t 1 l 5 0 Et1 1h t 1 l 5 0 and Et1 1[xt1 l 2 1h t1 l] 5 0 l 1 h

LEMMA 2 The innovation in next periodrsquos squared state variableis linear in the current state variable

xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1

Proof of Lemma 2 The proof for this lemma is similar to thatfor Lemma 1 From (28) nd xt 1 1

2 by setting j 5 1 h

xt1 12 5 micro2(1 2 f )2 1 2micro f (1 2 f )xt 1 f 2xt

2 1 h t 1 12

1 [2 f (xt 2 micro) 1 2 micro] h t 1 1

Lemma 2 then follows by applying the conditional expectationsoperator Et to this expression under the martingale assumption(A2) h

LEMMA 3 The unexpected return on the risky asset and theconditional variance of the risky asset are given by

r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2

Proof of Lemma 3 This result follows trivially from (A2) and(A3) It is stated here as a lemma for completeness h

LEMMA 4 The expected portfolio return next period is quadraticin the current state variable and the unexpected portfolioreturn is linear in the current state variable

Etrpt 1 1 5 rf 1 p0 1 p1xt 1 p2xt2


and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)

Proof of Lemma 4 From (16) and our guess (i) on the optimalportfolio rule we have that

Etrpt1 1 5 a tEt[r1t1 1 2 rf] 1 rf 1 ( s u22)a t(1 2 a t)

5 (a0 1 a1xt)xt 1 rf 1 ( s u22)((a0 1 a1xt) 2 (a0 1 a1xt)2)

where the last line follows from (2) Reordering terms we get aquadratic expression in xt whose coefficients are those given in thestatement of the proposition

The expression for rpt1 1 2 Etrpt1 1 also follows from (16)and guess (i) as well as (A1)mdashconstant rf mdashand (A2)ndash(2)

(29) rpt 1 1 2 Etrpt1 1 5 a t((r1t1 1 2 rf) 2 Et[r1t 1 1 2 rf])

5 (a0 1 a1xt)ut 1 1

h

LEMMA 5 Expected optimal consumption growth over the nextperiod is quadratic in the current state variable and unex-pected consumption growth is linear in the current statevariable

Et D ct1 1 5 Etrpt1 1 1 Et[ct1 1 2 wt1 1] 21

r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2

D ct1 1 2 Et D ct 1 1 5 (a0 1 a1xt)ut 1 1 1 b1h t1 1

1 b2(2micro(1 2 f ) 1 2 f xt)h t1 1 1 b2( h t 1 12 2 s h

2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1

r1 b1(micro(1 2 f ))

1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r

Proof of Lemma 5 From (18) and (15) we can write

(30) D ct1 1 5 rpt 1 1 1 (ct 1 1 2 wt 1 1) 2 (1 r )(ct 2 wt) 1 k

Therefore

Et D ct1 1 5 Etrpt1 1 1 Et(ct1 1 2 wt1 1) 21

r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt

1 b2 micro2(1 2 f )2 1 s h2 1 2micro f (1 2 f )xt 1 f 2 2

1

rxt

2 1 k

where the second equality follows from Lemma 4 and our guess (ii)on the optimal consumption policy and the last equality followsfrom (3) and Lemmas 1 and 2 Reordering terms we get theexpression for EtD ct1 1 in the lemma as well as c0c1c2

The expression for unexpected consumption growth followsfrom (30) the expression for the unexpected portfolio returnderived in Lemma 4 and from noting that our guess (ii) on the


optimal log consumption-wealth ratio implies that

(31) ct 1 1 2 wt 1 1 2 Et(ct 1 1 2 wt 1 1) 5 b1(xt1 1 2 Etxt1 1)

1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]

where the second line follows from Lemma 2 h

LEMMA 6 The time-varying intercept in the Euler equation forportfolio returns (13) is a quadratic function of the statevariable

vpt 5 v0 1 v1xt 1 v2x t2

where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2

2 a0b1[(1 2 g ) s h u] 2 a0b2[(1 2 g )2micro(1 2 f ) s h u]

1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]


Proof of Lemma 6 From (13) (15) and (18) we have

vpt 51

2

u

cvart ( D ct 1 1 2 c rpt 1 1)

51

2

u

cEt[(D ct1 1 2 Et D ct 1 1) 2 c (rpt1 1 2 Etrpt1 1)]2

51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2

If we substitute in the bracketed expression above (29) and (31)for (rpt 1 1 2 Etrpt1 1) and (ct 1 1 2 wt 1 1) 2 Et(ct1 1 2 wt1 1) and com-pute Et under assumptions (A2) and (A3) we nd that vpt is aquadratic function of xt with the coefficients given in the state-ment of the lemma h

LEMMA 7 The parameters dening the optimal consumption rule(ii) satisfy the following three-equation system

v0 5 k 2 c log d 1 (1 2 c )rf 1 (1 2 c )a0(1 2 a0)s u

2

2

1 b0 1 21

r1 b1(micro(1 2 f )) 1 b2(micro2(1 2 f )2 1 s h

2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r

Proof of Lemma 7 This follows from the log-linearized Eulerequation for the optimal portfolio given in (12) and Lemmas 4 5


and 6 From (12) and Lemmas 4 and 6

(32) Et D ct1 1 5 c log d 1 vpt 1 c Etrpt 1 1

5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2

which is a quadratic function of the state variable But fromLemma 5 we have that Et D ct1 1 is also quadratic in xt

(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2

where c0 c1 and c2 are given in Lemma 5 Equating coefficients onthe right-hand side of (32) and (33) the lemma follows immedi-ately h

LEMMA 8 The covariance between unexpected stock returns andchanges in expected portfolio returns is linear in the statevariable

covt r1t 1 1 2 Etr1t1 1(1 2 c )(Et 1 1 2 Et) oj 5 1

`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt

where p1p2 are given in Lemma 4


Proof of Lemma 8 Lemma 4 implies that

(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`

r j(E t1 1 2 Et)xt 1 j

1 p2 oj5 1

`

r j(Et 1 1 2 Et)xt1 j2

5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f

1 2p2(xt 2 micro)r f

1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2

where the second equality follows from Lemma 2 and the thirdone follows after computing the innite summations in the secondone and reordering terms

The result stated in the lemma follows immediately fromassumptions (A2) and (A3) about the distribution of (ut1 1 h t 1 1)the expression above and the properties of the covariance operator h


LEMMA 9 The covariance between unexpected stock returns andchanges in the expected value of the intercept in the Eulerequation (13) is linear in the state variable

covt r1t1 1 2 Etr1t1 1(Et 1 1 2 Et) oj5 1

`

r jvpt1 j 5 (v1 1 2v2micro)r

1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt

where v0v1v2 are given in Lemma 6


(Et 1 1 2 Et) oj 5 1

`

r jvpt1 j 5 v1 oj 5 1

`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2

which is identical to the expression given in the proof of Lemma 8except that we have v1 and v2 instead of p1 and p2 Therefore wemust have that

(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r

1 2 r f1 2v2(xt 2 micro)

r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2

from which the lemma follows under the distributional assump-tions (A2) and (A3) h


APPENDIX 2 PROOFS OF PROPOSITIONS

Proof of Proposition 1

From (A2) Etr1t 1 1 2 rf 5 xt and from (A3) s 11t 5 s u2 Also

from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)

5 covt [r1t 1 1 2 Etr1t 1 1b1(xt 1 1 2 Etxt 1 1)

1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]

5 b1s h u 1 b2[2micro(1 2 f ) 1 2 f xt] s h u

where the second line follows from substituting guess (ii) for ct1 1 2wt1 1 the third line follows from (A2) and Lemmas 2 and 3 and thelast line follows from (A3) and the assumption of joint normality ofut1 1 and h t 1 1

Using these results we can rewrite (19) as

(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]

which is linear in xt But our guess (i) on the optimal portfoliopolicy is that a t is linear in the state variable

a t 5 a0 1 a1xt

Grouping terms in (34) we obtain a0 and a1 as stated in Proposi-tion 1 h


The proof for this proposition follows from Lemmas 6 and 7and Proposition 1 Lemma 6 denes a nonlinear equation systemfor v0v1v2 a0a1 and b0b1b2

v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2

v1 5 V21b22 1 V22a0a1 1 V23a0b2 1 V24a1b1 1 V25a1b2 1 V26b1b2

v2 5 V31a12 1 V32b2

2 1 V33a1b2

where the coefficients Vij are functions of the primitive parame-


ters of the model (both those dening the preference structure andthose dening the stochastic structure of the model) and areimmediately identiable from the statement of the system inLemma 6 For example V11 5 (1 2 g ) ( c 2 1) s u

22 and so onSimilarly Lemma 7 denes a second system for v0v1v2

a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12

where the coefficients Bij are functions of the primitive parametersof the model For example B10 5 k 2 c log d 1 (1 2 c )rf and so on

Finally Proposition 1 denes another system for a0a1 andb0b1b2

a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2

where again the coefficients Aij are also functions of the primitiveparameters of the model and are immediately identiable fromthe statement of the system in the proposition For example A10 51(2 g ) and so on

By equating the right-hand sides of the rst and secondsystem we obtain another system whose unknowns are a0a1 andb0b1b2 But the third system denes a0a1 as linear combina-tions of b1b2 Substituting this system into the one obtained bycombining the rst and second systems we obtain the equationsystem for b0b1b2 given in the proposition The coefficients L ij

relate to the coefficients Aij Bij and Vij as follows

0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h

APPENDIX 3 PROOFS OF PROPERTIES

Proof of Property 1

The approximate value function per unit of wealth obtains bydirect substitution of guess (ii) into (11) We now use equation (23)to characterize b2 This equation is

0 5 L 30 1 L 31b21 L 32b22

Substituting L ijrsquos for their values we get

(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22

This equation has two roots that we denote b21b22 A sufficient(but not necessary) condition for these roots to be real is that

L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0

This is always true when g $ 1 since (1 2 g ) 0

(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0


and all other terms in the expression for L 32L 30 are positive Wheng 1 we have L 32L 30 0 so the roots are real if

(37) L 312 2 4 L 32L 30 $ 0

To analyze the sign of the roots rewrite equation (35) as

(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22

From standard theory on quadratic equations the product of theroots is given by L ˜ 30 which is negative when g 1 and positivewhen g 1

b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1

Therefore when g 1 the roots are real and have opposite sign andwhen g 1 the roots have the same signmdashprovided that they are real

Similarly from standard theory on quadratic equations

b21 1 b22 5 2 L ˜ 31

which is always positive if c 1 g 1 f s h u 0 or c 1 g 1 f s h u 0 and always negative if c 1 1f s h u 0 or c 1 g 1 f s h u 0 since ( f 2 2 r 2 1) 0 because 0 r 1 f 1and from (36) the term in brackets in the denominator of L ˜ 31 ispositive

Therefore when g 1 and f s h u 0 both roots are positive ifc 1 and negative if c 1 so that b2(1 2 c ) 0 When g 1and f s h u 0 the same result still obtains provided that thecondition for real roots (37) holdsmdashthis condition implies thatg s u

2( f 2 2 r 2 1)2 2 (1 2 g )2 f s h u( f 2 2 r 2 1) which is sufficient toobtain the result for this case

When g 1 the roots of the equation alternate in sign If


f s h u 0 we can write the expression for the roots of equation (35) as

b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D

where A B C D are positive constantsmdashprovided that g 1 andf s h u 0mdashso choosing the positive root of the discriminantdelivers b2 0 if c 1 and b2 0 if c 1 and b2(1 2 c ) 0 Theopposite obtains if we choose the negative root If f s h u 0 we canwrite the expression for the roots of the equation as

b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D

and the same result obtains h

Proof of Property 2

To prove Property 2 we need to consider two cases the case inwhich f s h u 0 and the case in which f s h u 0

Case f s h u 0From the last part of the proof of Property 1 we have that

b2(c 2 1) 0 when we select the value of b2 associated with thepositive root of the discriminant of equation (23)

Plugging this result into the second equation of Proposition 1we obtain immediately that a1 0 when g 1 and f s h u 0 sinceall the terms in the equation are positive Also when g 5 1 thesecond term in the equation is zero so a1 5 1s u

2 0 When g 1the rst term is positive but the second is negative so we need toprove whether the sum of both terms is positive Solving for thepositive root of the discriminant in (35) and plugging the result inthe second equation in Proposition 1 we nd that

(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2

Since the denominator is always positive the sign of the slopedepends on the sign of the numerator A straightforward analysis


of the numerator shows that a couple of sufficient conditions for itto be positive are

s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u

But if the rst sufficient condition is violated the second one isimmediately veried so a1 0


b2(1 2 c ) 0 when g 1mdashprovided that the condition for realroots (37) in (35) holds Plugging this result into the secondequation of Proposition 1 we obtain immediately that a1 0since g 1 and f s h u 0 Therefore the slope of the optimalportfolio policy is always positive no matter what root we select forthe discriminant of equation (23)

When g 1 solving for the negative root of the discriminantin (35) and plugging the result in the second equation of Proposi-tion 1 we nd again (39) which is always positive when g 1 Ifwe solve for the positive root of the discriminant in (35) and weplug the result into the second equation of Proposition 1 we ndan expression similar to (39) except that the second term issubtracted A sufficient condition for this expression to be positiveis f corr (h t1 1ut1 1) f corr ( h t1 1ut1 1)3 which is always truebecause sign ( f s h u) 5 sign (f corr ( h t 1 1ut 1 1)) and corr ( h t 1 1ut1 1 ) 1 h

The Limiting Behavior of a1

Regardless of the sign of the covariance and f a1 1 ` asg 0 and a1 0 as g 1 ` To prove these results note thatfrom (39) we have that the numerator of a1 is O(g ) while thedenominator is O( g 2) Hence taking appropriate limits we obtainthe desired results h

Proof of Property 3

Part a When g 5 1 Lemma 6 in Appendix 1 implies that v0 5v1 5 v2 5 0 so the intercept term vpt in the Euler equation (12) iszero We also see this by noticing that g 5 1 implies u 5 0


Substituting g 5 1 in Proposition 1 we obtain the same myopicportfolio rule as with log utility It is straightforward to see thatthis rule maximizes the conditional expectation of the log portfolioreturn However the consumption-wealth ratio is no longerconstant unless c 5 1 as we can see from Lemma 7 in Appendix 1Therefore unit relative risk aversion implies a myopic optimalportfolio policy and a nonmyopic optimal consumption policyGiovannini and Weil [1989] emphasize this result h

Part b When c 1 the equation system in Proposition 2delivers b1 5 b2 5 0 and

b0 5 ( r (1 2 r ))(k 2 log d )

After substituting for the value of k this result simplies to

(40) b0 5 log (1 2 r )

Moreover we know from standard arguments that r 5 d There-fore it is optimal for the individual to consume each period a xedfraction of her wealth Following Giovannini and Weil [1989] wecall this optimally constant propensity to consume out of wealth amyopic consumption policy

However the agentrsquos optimal portfolio policy is not myopicThis is because from equation (41) in the proof of Property 3below we have that b2(1 2 c ) and b1(1 2 c ) are nonzero con-stants independent of c for given r Therefore when c 5 1 theterms in b1b2 in the system dening the optimal portfolio policyin Proposition 1 do not vanish and a nonmyopic portfolio policyobtains Giovannini and Weil [1989] also emphasize this result

Part c The values for b0 b1 and b2 obtain from the proof forPart b Substituting for g 5 1 into Proposition 1 we obtain a0 5 12and a1 5 1 s u

2 h

Part d With constant expected returns s h u 5 0 FromProposition 1 we obtain the same portfolio policy as in the logutility case except that g THORN 1 a0 5 1(2 g ) and a1 5 1 g s u

2 Alsosince xt is deterministic (31) implies that ct 2 wt is constant Thisis the well-known result for the optimal portfolio rule whenreturns are iid h

Proof of Property 4

A straightforward analysis of the solutions to equations(22) and (23) in Proposition 2 shows that we can write b1 and


b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )

where f1(g r ) and f1( g r ) are functions that do not depend on c After substitution in the equation system in Proposition 1 we ndthat the parameters dening the optimal portfolio rule a0a1 donot depend on c for given r However r itself is a function ofc mdashrecall that r 5 1 2 exp E[ct 2 wt] mdashso the optimal portfoliorule depends on c indirectly through r h

Proof of Property 5

To prove this result note that equation (23) in Proposition 2that determines b2 is found by equating the right-hand side of thethird equation in Lemmas 6 and 7 and substituting out a1 usingthe second equation in Proposition 1 None of these equationsdepend on micro h

HARVARD UNIVERSITY AND NATIONAL BUREAU OF ECONOMIC RESEARCH

HARVARD UNIVERSITY

REFERENCES

Balduzzi Perluigi andAnthony Lynch lsquolsquoThe Impact of Predictability and Transac-tion Costs on Portfolio Choice in a Multiperiod Settingrsquorsquo unpublished paperBoston College and New York University 1997a

Balduzzi Perluigi and Anthony Lynch lsquolsquoTransaction Costs and PredictabilitySome Utility Cost Calculationsrsquorsquo unpublished paper Boston College and NewYork University 1997b

Barberis Nicholas C lsquolsquoInvesting for the Long Run When ReturnsAre PredictablersquorsquoJournal of Finance LIV (1999) forthcoming

Brandt Michael W lsquolsquoEstimating Portfolio and Consumption Choice A ConditionalEuler Equations Approachrsquorsquo Journal of Finance LIV (1999) forthcoming

Brennan Michael J Eduardo S Schwartz and Ronald Lagnado lsquolsquoStrategic AssetAllocationrsquorsquo Journal of Economic Dynamics and Control XXI (1997) 1377ndash1403

Campbell John Y lsquolsquoStock Returns and the Term Structurersquorsquo Journal of FinancialEconomics XVIII (1987) 373ndash399 lsquolsquoIntertemporal Asset Pricing without Consumption Datarsquorsquo American Eco-nomic Review LXXXIII (1993) 487ndash512 lsquolsquoUnderstanding Risk and Returnrsquorsquo Journal of Political Economy CIV (1996)298ndash345

Campbell John Y and John H Cochrane lsquolsquoBy Force of Habit A Consumption-Based Explanation of Aggregate Stock Market Behaviorrsquorsquo Journal of PoliticalEconomy CVII (1999) 205ndash251

Campbell John Y Joao Cocco Francisco Gomes Pascal J Maenhout and Luis MViceira lsquolsquoStock Market Mean Reversion and the Optimal Equity Allocation of aLong-Lived Investorrsquorsquo unpublishedpaper Harvard University 1998


Campbell John Y and Hyeng Keun Koo lsquolsquoA Comparison of Numerical andAnalytical Approximate Solutions to an Intertemporal Consumption ChoiceProblemrsquorsquo Journal of Economic Dynamics and Control XXI (1997) 273ndash295

Campbell John Y Andrew W Lo and A Craig MacKinlay The Econometrics ofFinancial Markets (Princeton NJ Princeton University Press 1997)

Campbell John Y and N Gregory Mankiw lsquolsquoConsumption Income and InterestRates Reinterpreting the Time-Series Evidencersquorsquo in NBER MacroeconomicsAnnual 1989 (Cambridge MA MIT Press 1989)

Campbell John Y and Robert Shiller lsquolsquoThe Dividend-Price Ratio and Expecta-tions of Future Dividends and Discount Factorsrsquorsquo Review of Financial StudiesI (1988a) 195ndash227

Campbell John Y and Robert Shiller lsquolsquoStock Prices Earnings and ExpectedDividendsrsquorsquo Journal of Finance XLIII (1988b) 661ndash676

Campbell John Y and Luis M Viceira lsquolsquoConsumption and Portfolio DecisionsWhen Expected Returns Are Time Varyingrsquorsquo NBER Working Paper No 58571996

Campbell John Y and Luis M Viceira lsquolsquoWho Should Buy Long-Term BondsrsquorsquoNBER Working paper No 6801 1998

Cecchetti Steven Pok-sang Lam and Nelson C Mark lsquolsquoTesting Volatility Restric-tions on Intertemporal Marginal Rates of Substitution Implied by EulerEquations and Asset Returnsrsquorsquo Journal of Finance XLIX (1990) 123ndash152

Cochrane John and Lars P Hansen lsquolsquoAsset Pricing Explorations for Macroeconom-icsrsquorsquo in NBER Macroeconomics Annual 1992 (Cambridge MA MIT Press1992)

Constantinides George M lsquolsquoA Theory of the Nominal Term Structure of InterestRatesrsquorsquo Review of Financial Studies V (1992) 531ndash552

Cox John C and Chi-fu Huang lsquolsquoOptimal Consumption and Portfolio PoliciesWhen Asset Prices Follow a Diffusion Processrsquorsquo Journal of Economic TheoryXXXIX (1989) 33ndash83

Duffie Darrell and Lawrence Epstein lsquolsquoStochastic Differential Utilityrsquorsquo Economet-rica LX (1992) 353ndash394

Elliott Graham and James H Stock lsquolsquoInference in Time Series Regression Whenthe Order of Integration of a Regressor Is Unknownrsquorsquo Economic Theory X(1994) 672ndash700

Epstein Lawrence and Stanley Zin lsquolsquoSubstitution Risk Aversion and theTemporal Behavior of Consumption and Asset Returns A Theoretical Frame-workrsquorsquo Econometrica LVII (1989) 937ndash969

Epstein Lawrence and Stanley Zin lsquolsquoSubstitution Risk Aversion and theTemporal Behavior of Consumption and Asset Returns An Empirical Investi-gationrsquorsquo Journal of Political Economy XCIX (1991) 263ndash286

Fama Eugene and Kenneth French lsquolsquoDividend Yields and Expected StockReturnsrsquorsquo Journal of Financial Economics XXII (1988) 3ndash27

Fama Eugene and Kenneth French lsquolsquoBusiness Conditions and Expected Returnson Stocks and Bondsrsquorsquo Journal of Financial Economics XXV (1989) 23ndash49

Fischer Stanley lsquolsquoInvesting for the Short and the Long Termrsquorsquo in Zvi Bodie andJohn B Shoven eds Financial Aspects of the United States Pension System(Chicago IL The University of Chicago Press 1983)

Giovannini Alberto and Philippe Weil lsquolsquoRisk Aversion and Intertemporal Substi-tution in the Capital Asset Pricing Modelrsquorsquo NBER Working Paper No 28241989

Goldman M Barry lsquolsquoAnti-Diversication or Optimal Programmes for InfrequentlyRevised Portfoliosrsquorsquo Journal of Finance XXXIV (1979) 505ndash516

Hall Robert E lsquolsquoIntertemporal Substitution in Consumptionrsquorsquo Journal of PoliticalEconomy XCVI (1988) 339ndash357

Hamilton James D Time Series Analysis (Princeton NJ Princeton UniversityPress 1994)

Hansen Lars P and Ravi Jagannathan lsquolsquoRestrictions on Intertemporal MarginalRates of Substitution Implied by Asset ReturnsrsquorsquoJournal of Political EconomyXCIX (1991) 225ndash262

He Hua and Neil D Pearson lsquolsquoConsumption and Portfolio Policies with Incom-plete Markets and Short-Sale Constraints The Innite-Dimensional CasersquorsquoJournal of Economic Theory LIV (1991) 259ndash304


Hodrick Robert J lsquolsquoDividend Yields and Expected Stock Returns AlternativeProcedures for Inference and Measurementrsquorsquo Review of Financial Studies V(1992) 357ndash386

Ingersoll Jonathan E Jr Theory of Financial Decision Making (Totowa NJRowman and Littleeld 1987)

Kandel Shmuel and Robert Stambaugh lsquolsquoOn the Predictability of Stock ReturnsAn AssetAllocation Perspectiversquorsquo Journal of Finance LI (1996) 385ndash424

Kim Tong Suk and Edward Omberg lsquolsquoDynamic Nonmyopic Portfolio BehaviorrsquorsquoReview of Financial Studies IX (1996) 141ndash161

Kocherlakota Narayana R lsquolsquoThe Equity Premium Itrsquos Still a Puzzlersquorsquo Journal ofEconomic Literature XXXIV (1996) 42ndash71

Mehra Rajnish and Edward C Prescott lsquolsquoThe Equity Premium A PuzzlersquorsquoJournal of Monetary Economics XV (1985) 145ndash161

Merton Robert C lsquolsquoLifetime Portfolio Selection under Uncertainty The Continu-ous Time Casersquorsquo Review of Economics and Statistics LI (1969) 247ndash257 lsquolsquoOptimum Consumption and Portfolio Rules in a Continuous-Time ModelrsquorsquoJournal of Economic Theory III (1971) 373ndash413 lsquolsquoAn Intertemporal Capital Asset Pricing Modelrsquorsquo Econometrica XLI (1973)867ndash887 Continuous Time Finance (Cambridge MA Basil Blackwell 1990)

Restoy Fernando lsquolsquoOptimal Portfolio Policies under Time-Dependent ReturnsrsquorsquoBank of Spain Working Paper 9207 Bank of Spain Madrid Spain 1992

Samuelson Paul A lsquolsquoLifetime Portfolio Selection by Dynamic Stochastic Program-mingrsquorsquo Review of Economics and Statistics LI (1969) 239ndash246

Schroder Mark and Costis Skiadas lsquolsquoOptimal Consumption and Portfolio Selectionwith Stochastic Differential Utilityrsquorsquo Working Paper 226 Department of FinanceKellogg Graduate School of Management Northwestern University 1998

Siegel Jeremy J Stocks for the Long Run (Burr Ridge IL Richard D Irwin1994)

Svensson Lars E O lsquolsquoPortfolio Choice with Non-Expected Utility in ContinuousTimersquorsquo Economics Letters XXX (1989) 313ndash317

Viceira Luis M lsquolsquoOptimal Portfolio Choice for Long-Horizon Investors withNontradable Incomersquorsquo unpublished paper Harvard University 1997

Weil Philippe lsquolsquoThe Equity Premium Puzzle and the Risk-Free Rate PuzzlersquorsquoJournal of Monetary Economics XXIV (1989) 401ndash421


not hold Expected asset returns seem to vary through time sothat investment opportunities are not constant the evidence forpredictable variation in the equity premium the excess return onstock over Treasury bills is particularly strong (see Campbell[1987] Campbell and Shiller [1988a 1988b] Fama and French[1988 1989] Hodrick [1992] or the textbook treatment in Camp-bell Lo and MacKinlay [1997 Chapter 7]) Economists research-ing the equity premium puzzle nd that average excess stockreturns are too high to be consistent with a representative-investor model in which the investor has log utility (see Campbell[1996] Cecchetti Lam and Mark [1994] Cochrane and Hansen[1992] Hansen and Jagannathan [1991] Kocherlakota [1996]Mehra and Prescott [1985] or the textbook treatment in Camp-bell Lo and MacKinlay [Chapter 8])

In response to these empirical ndings several recent papershave used numerical methods to solve for optimal portfolios inmodels with realistic predictability of returns Investors aregenerally assumed to have power utility dened over wealth at asingle terminal date Different papers choose different investmenthorizons and make different assumptions about investorsrsquo abilityto rebalance their portfolios Kandel and Stambaugh [1996]consider the effects of predictability on the optimal portfolio of asingle-period Bayesian investor who takes account of parameteruncertainty while Barberis [1999] extends this work to study theoptimal portfolio of a long-horizon Bayesian investor who rebal-ances annually or not at all Brennan Schwartz and Lagnado[1997] consider a long-horizon investor who rebalances fre-quently while Balduzzi and Lynch [1997a 1997b] consider along-horizon investor who faces xed and proportional transac-tions costs which reduce the frequency of optimal rebalancing1

The results in these papers though dependent on the particularparameter values they assume illuminate the effects of predictabil-ity on portfolio choice Kim and Omberg [1996] work with asimilar framework but by assuming continuous time and zerotransactions costs are able to solve the portfolio choice problemanalytically2

A limitation of these models is that they abstract from the

1 Most of these papers like our paper work with a single state variabledriving the equity premium Only Brennan Schwartz and Lagnado [1997]consider multiple state variables

2 Kim and Omberg study the choice between a riskless asset with a constantreturn and a risky asset whose expected return follows a continuous-time AR(1)(Ornstein-Uhlenbeck) process They assume that the investor is nitely lived and
























II1 Assumptions



(1) Rpt1 1 5 a t(R1t 1 1 2 Rf) 1 Rf



(2) Etr1t1 1 2 rf 5 xt






(4) vart (ut 1 1) 5 s u2

(5) covt (ut1 1h t1 1) 5 s u h



12 g )1u u (1 2 g )








(7) Wt1 1 5 Rpt 1 1(Wt 2 Ct)



(8) 1 5 Et dCt1 1

Ct

2 1c u

Rpt1 12 (12 u )Rit1 1


(9) 1 5 Et dCt1 1

Ct

2 1 c

Rpt 1 1

u



(10) Vt 5 (1 2 d )Ct

Wt

12 1c

1 d 1 2Ct

Wt

12 1c

(Et[V t 1 112 g Rpt1 1

12 g ])1u1(1 2 c )


(11) Vt 5 (1 2 d ) 2 c (1 2 c )Ct

Wt

1(1 2 c )






0 5 u log d 2u


1

2vart

u

cD ct1 1 2 u rpt1 1















(15) D wt1 1 lt rpt 1 1 1 (1 2 (1 r ))(ct 2 wt) 1 k








(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t







(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

ER

AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

454

241

32

401

239

51

000

00

00

00

00

00

00

00

02

828

267

62

566

250

72

443

241

12

401

239

61

502

137

28

42

56

24

62

36

23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

002

200

215

32

122

210

82

90

28

22

79

27

82

580

253

42

497

247

92

454

244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

237

228

22

302

231

22

175

220

22

251

228

32

353

241

62

443

245

820

02

58

27

32

103

212

52

181

224

52

279

229

92

93

211

32

151

217

92

251

233

42

378

240

540

02

30

23

92

59

27

42

117

218

02

219

224

62

48

26

02

84

210

22

157

223

82

288

232

3

Th

epa

nel

inth

eu

pper

left

corn

erof

the

tabl

ere

port

sth

eu

nco

ndi

tion

alm

ean

ofth

eva

lue

fun

ctio

nu

nd

erth

eop

tim

alco

nsu

mpt

ion

and

por

tfol

ioru

les

Th

epa

nel

inth

eu

pper

righ

tco

rner

repo

rts

the

perc

enta

gelo

ssin

the

valu

efu

nct

ion

wh

enth

ep

ortf

olio

rule

is

xed

atth

em

ean

valu

eof

the

opti

mal

por

tfol

ioru

lean

dco

nsu

mp

tion

adju

sts

opti

mal

lyT

he

pan

elin

the

low

erle

ftco

rner

rep

orts

the

perc

enta

gelo

ssin

the

valu

efu

nct

ion

wh

enth

ep

ortf

olio

rule

ism

yop

ic(s

eeeq

uat

ion

(21)

inte

xt)

and

con

sum

ptio

nad

just

sop

tim

ally

Th

epa

nel

inth

elo

wer

righ

tco

rner

repo

rts

the

perc

enta

gelo

ssin

the

valu

efu

nct

ion

wh

enth

epo

rtfo

lio

rule

is

xed

atth

em

ean

valu

eof

the

myo

pic

rule

and

con

sum

ptio

nad

just

sop

tim

ally

Th

ese

nu

mbe

rsar

eba

sed

onth

ep

aram

eter

valu

esfo

rth

ere

turn

pro

cess

pres

ente

din

Tab

leI

Th

ese

valu

esar

ees

tim

ates

for

the

peri

od19

471

ndash199

54

Th

eva

lues

onth

em

ain

dia

gon

alco

rres

pon

dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES








































































II1 Assumptions



(1) Rpt1 1 5 a t(R1t 1 1 2 Rf) 1 Rf



(2) Etr1t1 1 2 rf 5 xt






(4) vart (ut 1 1) 5 s u2

(5) covt (ut1 1h t1 1) 5 s u h



12 g )1u u (1 2 g )








(7) Wt1 1 5 Rpt 1 1(Wt 2 Ct)



(8) 1 5 Et dCt1 1

Ct

2 1c u

Rpt1 12 (12 u )Rit1 1


(9) 1 5 Et dCt1 1

Ct

2 1 c

Rpt 1 1

u



(10) Vt 5 (1 2 d )Ct

Wt

12 1c

1 d 1 2Ct

Wt

12 1c

(Et[V t 1 112 g Rpt1 1

12 g ])1u1(1 2 c )


(11) Vt 5 (1 2 d ) 2 c (1 2 c )Ct

Wt

1(1 2 c )






0 5 u log d 2u


1

2vart

u

cD ct1 1 2 u rpt1 1















(15) D wt1 1 lt rpt 1 1 1 (1 2 (1 r ))(ct 2 wt) 1 k








(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t







(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

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FU

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TIO

NU

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TE

RN

AT

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RE

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RIC

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DP

OR

TF

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IOR

UL

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RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

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0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

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92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

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232

82

342

236

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382

240

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429

243

72

440

100

098

094

089

087

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077

076

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214

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237

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253

92

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063

054

047

044

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17

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001

15

1 21 4

1 10

1 20

1 40

17

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001

15

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110

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No

hed

gin

g0

752

284

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242

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243

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190

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222

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467

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175

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02

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02

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pow

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tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES

































































II1 Assumptions



(1) Rpt1 1 5 a t(R1t 1 1 2 Rf) 1 Rf



(2) Etr1t1 1 2 rf 5 xt






(4) vart (ut 1 1) 5 s u2

(5) covt (ut1 1h t1 1) 5 s u h



12 g )1u u (1 2 g )








(7) Wt1 1 5 Rpt 1 1(Wt 2 Ct)



(8) 1 5 Et dCt1 1

Ct

2 1c u

Rpt1 12 (12 u )Rit1 1


(9) 1 5 Et dCt1 1

Ct

2 1 c

Rpt 1 1

u



(10) Vt 5 (1 2 d )Ct

Wt

12 1c

1 d 1 2Ct

Wt

12 1c

(Et[V t 1 112 g Rpt1 1

12 g ])1u1(1 2 c )


(11) Vt 5 (1 2 d ) 2 c (1 2 c )Ct

Wt

1(1 2 c )






0 5 u log d 2u


1

2vart

u

cD ct1 1 2 u rpt1 1















(15) D wt1 1 lt rpt 1 1 1 (1 2 (1 r ))(ct 2 wt) 1 k








(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t







(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

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NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

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LIO

RU

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ISU

NR

ES

TR

ICT

ED

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RC

EN

TA

GE

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VA

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AT

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UL

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RR

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Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

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0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

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92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

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232

82

342

236

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240

82

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243

72

440

100

098

094

089

087

081

077

076

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214

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92

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237

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486

253

92

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200

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068

063

054

047

044

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17

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001

15

1 21 4

1 10

1 20

1 40

17

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001

15

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110

120

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No

hed

gin

g0

752

284

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02

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242

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82

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190

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175

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02

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pow

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tili

tyca

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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



























































II1 Assumptions



(1) Rpt1 1 5 a t(R1t 1 1 2 Rf) 1 Rf



(2) Etr1t1 1 2 rf 5 xt






(4) vart (ut 1 1) 5 s u2

(5) covt (ut1 1h t1 1) 5 s u h



12 g )1u u (1 2 g )








(7) Wt1 1 5 Rpt 1 1(Wt 2 Ct)



(8) 1 5 Et dCt1 1

Ct

2 1c u

Rpt1 12 (12 u )Rit1 1


(9) 1 5 Et dCt1 1

Ct

2 1 c

Rpt 1 1

u



(10) Vt 5 (1 2 d )Ct

Wt

12 1c

1 d 1 2Ct

Wt

12 1c

(Et[V t 1 112 g Rpt1 1

12 g ])1u1(1 2 c )


(11) Vt 5 (1 2 d ) 2 c (1 2 c )Ct

Wt

1(1 2 c )






0 5 u log d 2u


1

2vart

u

cD ct1 1 2 u rpt1 1















(15) D wt1 1 lt rpt 1 1 1 (1 2 (1 r ))(ct 2 wt) 1 k








(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t







(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

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AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

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214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

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23

82

50

28

02

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220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

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239

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000

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828

267

62

566

250

72

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12

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239

61

502

137

28

42

56

24

62

36

23

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92

681

259

42

522

249

02

448

242

52

418

241

42

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200

215

32

122

210

82

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28

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79

27

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253

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247

92

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244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

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367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

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228

22

302

231

22

175

220

22

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820

02

58

27

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52

181

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52

279

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92

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32

151

217

92

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42

378

240

540

02

30

23

92

59

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218

02

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62

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02

84

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22

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3

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epa

nel

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pper

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pow

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tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES





















































II1 Assumptions



(1) Rpt1 1 5 a t(R1t 1 1 2 Rf) 1 Rf



(2) Etr1t1 1 2 rf 5 xt






(4) vart (ut 1 1) 5 s u2

(5) covt (ut1 1h t1 1) 5 s u h



12 g )1u u (1 2 g )








(7) Wt1 1 5 Rpt 1 1(Wt 2 Ct)



(8) 1 5 Et dCt1 1

Ct

2 1c u

Rpt1 12 (12 u )Rit1 1


(9) 1 5 Et dCt1 1

Ct

2 1 c

Rpt 1 1

u



(10) Vt 5 (1 2 d )Ct

Wt

12 1c

1 d 1 2Ct

Wt

12 1c

(Et[V t 1 112 g Rpt1 1

12 g ])1u1(1 2 c )


(11) Vt 5 (1 2 d ) 2 c (1 2 c )Ct

Wt

1(1 2 c )






0 5 u log d 2u


1

2vart

u

cD ct1 1 2 u rpt1 1















(15) D wt1 1 lt rpt 1 1 1 (1 2 (1 r ))(ct 2 wt) 1 k








(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t







(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

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FU

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OP

TIM

AL

PO

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ES

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EN

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VA

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Tim

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EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

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0018

59

935

653

577

497

462

452

447

283

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257

22

525

246

32

432

242

22

417

150

652

508

429

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368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

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240

82

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243

72

440

100

098

094

089

087

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077

076

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214

52

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92

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237

42

486

253

92

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200

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063

054

047

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464

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62

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23

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220

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52

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17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

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23

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21

21

72

16

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52

925

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61

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137

28

42

56

24

62

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92

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249

02

448

242

52

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200

215

32

122

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82

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253

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247

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244

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436

243

34

002

190

219

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204

220

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216

222

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222

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367

238

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242

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246

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467

247

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104

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62

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22

175

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02

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27

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42

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02

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3

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epa

nel

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eu

pper

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lues

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pow

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tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES























































(7) Wt1 1 5 Rpt 1 1(Wt 2 Ct)



(8) 1 5 Et dCt1 1

Ct

2 1c u

Rpt1 12 (12 u )Rit1 1


(9) 1 5 Et dCt1 1

Ct

2 1 c

Rpt 1 1

u



(10) Vt 5 (1 2 d )Ct

Wt

12 1c

1 d 1 2Ct

Wt

12 1c

(Et[V t 1 112 g Rpt1 1

12 g ])1u1(1 2 c )


(11) Vt 5 (1 2 d ) 2 c (1 2 c )Ct

Wt

1(1 2 c )






0 5 u log d 2u


1

2vart

u

cD ct1 1 2 u rpt1 1















(15) D wt1 1 lt rpt 1 1 1 (1 2 (1 r ))(ct 2 wt) 1 k








(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t







(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

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ED

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RC

EN

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GE

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NU

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DP

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Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

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0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

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241

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239

51

000

00

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02

828

267

62

566

250

72

443

241

12

401

239

61

502

137

28

42

56

24

62

36

23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

002

200

215

32

122

210

82

90

28

22

79

27

82

580

253

42

497

247

92

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244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

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22

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231

22

175

220

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820

02

58

27

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52

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52

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217

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42

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02

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92

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3

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nel

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eu

pper

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ain

dia

gon

alco

rres

pon

dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



















































(10) Vt 5 (1 2 d )Ct

Wt

12 1c

1 d 1 2Ct

Wt

12 1c

(Et[V t 1 112 g Rpt1 1

12 g ])1u1(1 2 c )


(11) Vt 5 (1 2 d ) 2 c (1 2 c )Ct

Wt

1(1 2 c )






0 5 u log d 2u


1

2vart

u

cD ct1 1 2 u rpt1 1















(15) D wt1 1 lt rpt 1 1 1 (1 2 (1 r ))(ct 2 wt) 1 k








(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t







(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

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FU

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SN

oti

min

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IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

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244

32

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242

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59

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577

497

462

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447

283

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257

22

525

246

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242

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417

150

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368

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346

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586

251

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244

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241

32

409

200

383

345

319

308

294

287

285

284

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92

511

247

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242

92

415

241

12

409

400

166

165

166

166

166

166

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166

232

82

342

236

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382

240

82

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243

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440

100

098

094

089

087

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077

076

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214

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237

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253

92

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200

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047

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23

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52

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17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

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72

16

21

52

925

274

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253

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239

61

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137

28

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56

24

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92

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249

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242

52

418

241

42

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200

215

32

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210

82

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27

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253

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247

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244

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436

243

34

002

190

219

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220

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3

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tim

ates

for

the

peri

od19

471

ndash199

54

Th

eva

lues

onth

em

ain

dia

gon

alco

rres

pon

dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES






























































(15) D wt1 1 lt rpt 1 1 1 (1 2 (1 r ))(ct 2 wt) 1 k








(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t







(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

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17

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15

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110

120

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gin

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148

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767

386

075

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927

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17

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001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

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110

120

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No

hed

gin

g0

752

284

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32

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243

34

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190

219

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3

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54

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tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES
























































(17) D wt1 1 lt a t(r1t1 1 2 rf) 1 rf 1 1 21

r(ct 2 wt)

1 k 11

2a t(1 2 a t) s 11t







(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

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NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

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ISU

NR

ES

TR

ICT

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PE

RC

EN

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INT

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VA

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FU

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NU

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TE

RN

AT

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DP

OR

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UL

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RR

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Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

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0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

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240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

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27

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214

12

181

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62

620

400

074

067

058

052

041

032

028

026

23

82

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220

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366

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52

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17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

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23

22

21

21

72

16

21

52

925

274

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253

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000

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828

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250

72

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239

61

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137

28

42

56

24

62

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92

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259

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249

02

448

242

52

418

241

42

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200

215

32

122

210

82

90

28

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79

27

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253

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247

92

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244

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436

243

34

002

190

219

32

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220

92

216

222

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222

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367

238

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175

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02

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3

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epa

nel

inth

eu

pper

left

corn

erof

the

tabl

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port

sth

eu

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tion

alm

ean

ofth

eva

lue

fun

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nu

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eop

tim

alco

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and

por

tfol

ioru

les

Th

epa

nel

inth

eu

pper

righ

tco

rner

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the

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enta

gelo

ssin

the

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ion

wh

enth

ep

ortf

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is

xed

atth

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ean

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the

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nsu

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tion

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sts

opti

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lyT

he

pan

elin

the

low

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rep

orts

the

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enta

gelo

ssin

the

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nct

ion

wh

enth

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ism

yop

ic(s

eeeq

uat

ion

(21)

inte

xt)

and

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sum

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nad

just

sop

tim

ally

Th

epa

nel

inth

elo

wer

righ

tco

rner

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wh

enth

epo

rtfo

lio

rule

is

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ean

valu

eof

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rule

and

con

sum

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nad

just

sop

tim

ally

Th

ese

nu

mbe

rsar

eba

sed

onth

ep

aram

eter

valu

esfo

rth

ere

turn

pro

cess

pres

ente

din

Tab

leI

Th

ese

valu

esar

ees

tim

ates

for

the

peri

od19

471

ndash199

54

Th

eva

lues

onth

em

ain

dia

gon

alco

rres

pon

dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES






















































(18) D ct 1 1 5 (ct1 1 2 wt1 1) 2 (ct 2 wt) 1 D wt1 1


s 1ct 5 covt (r1t 1 1 D ct 1 1)


5 s 1c 2 wt 1 a t s 11t




5 a ts 11t


Etr1t 1 1 2 rf 11

2s 11t 5

u

c(s 1c 2 wt 1 a t s 11t) 1 (1 2 u ) a t s 11t


(19) a t 51

g

Etr1t1 1 2 rf 1 12 s 11t

s 11t2

1

1 2 c

g 2 1

g

s 1c 2 wt

s 11t






ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

ER

AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

454

241

32

401

239

51

000

00

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02

828

267

62

566

250

72

443

241

12

401

239

61

502

137

28

42

56

24

62

36

23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

002

200

215

32

122

210

82

90

28

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79

27

82

580

253

42

497

247

92

454

244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

237

228

22

302

231

22

175

220

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02

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52

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52

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42

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02

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92

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218

02

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3

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epa

nel

inth

eu

pper

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the

tabl

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port

sth

eu

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ndi

tion

alm

ean

ofth

eva

lue

fun

ctio

nu

nd

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eop

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and

por

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epa

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righ

tco

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gelo

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wh

enth

ep

ortf

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is

xed

atth

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the

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nsu

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tion

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sts

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he

pan

elin

the

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erle

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rep

orts

the

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the

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nct

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wh

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(21)

inte

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ally

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ese

nu

mbe

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onth

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aram

eter

valu

esfo

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turn

pro

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pres

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din

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leI

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ese

valu

esar

ees

tim

ates

for

the

peri

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471

ndash199

54

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eva

lues

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em

ain

dia

gon

alco

rres

pon

dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES





















































ct 1 1 2 wt1 1 5 Et 1 1 oj5 1

`

r j(rpt1 11 j 2 D ct1 11 j) 1r k

1 2 r


(20) ct 1 1 2 wt 1 1 5 (1 2 c )Et 1 1 oj5 1

`

r jrpt 1 11 j 2 Et 1 1 oj5 1

`

r jvpt 1 j

1r

1 2 r(k 2 c log d )






(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

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AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

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595

253

32

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241

32

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239

51

000

00

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828

267

62

566

250

72

443

241

12

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239

61

502

137

28

42

56

24

62

36

23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

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200

215

32

122

210

82

90

28

22

79

27

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580

253

42

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247

92

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244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

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228

22

302

231

22

175

220

22

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32

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820

02

58

27

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52

181

224

52

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92

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32

151

217

92

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42

378

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540

02

30

23

92

59

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218

02

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62

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22

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3

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epa

nel

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eu

pper

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tabl

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ean

ofth

eva

lue

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(21)

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valu

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ese

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esar

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for

the

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54

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eva

lues

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the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES





















































(i) a t 5 a0 1 a1xt










a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

ER

AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

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241

32

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239

51

000

00

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828

267

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250

72

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12

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239

61

502

137

28

42

56

24

62

36

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22

92

681

259

42

522

249

02

448

242

52

418

241

42

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200

215

32

122

210

82

90

28

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27

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253

42

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247

92

454

244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

237

228

22

302

231

22

175

220

22

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228

32

353

241

62

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245

820

02

58

27

32

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52

181

224

52

279

229

92

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32

151

217

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42

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02

30

23

92

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218

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22

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3

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epa

nel

inth

eu

pper

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ean

ofth

eva

lue

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nu

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(21)

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lues

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pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES




















































a0 51

2 g2

b1

1 2 c

g 2 1

g

s h u

s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2micro(1 2 f )

a1 51

g s u2

2b2

1 2 c

g 2 1

g

s h u

s u2

2 f






(21) 0 5 L 10 1 L 11b0 1 L 12b1 1 L 13b12 1 L 14b2

1 L 15b22 1 L 16b1b2

(22) 0 5 L 20 1 L 21b1 1 L 22b2 1 L 23b22 1 L 24b1b2

(23) 0 5 L 30 1 L 31b2 1 L 32b22





2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

ER

AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

454

241

32

401

239

51

000

00

00

00

00

00

00

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02

828

267

62

566

250

72

443

241

12

401

239

61

502

137

28

42

56

24

62

36

23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

002

200

215

32

122

210

82

90

28

22

79

27

82

580

253

42

497

247

92

454

244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

237

228

22

302

231

22

175

220

22

251

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32

353

241

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820

02

58

27

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52

181

224

52

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32

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42

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540

02

30

23

92

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218

02

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22

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3

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epa

nel

inth

eu

pper

left

corn

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tabl

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tion

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ean

ofth

eva

lue

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nu

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pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES





















































2 1 (1 2 g )( f 2 2 (1 r ))4 f s h u 2 (1 2 g )4 f 2s h2 $ 0












(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

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PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

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AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

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IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

454

241

32

401

239

51

000

00

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02

828

267

62

566

250

72

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241

12

401

239

61

502

137

28

42

56

24

62

36

23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

002

200

215

32

122

210

82

90

28

22

79

27

82

580

253

42

497

247

92

454

244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

237

228

22

302

231

22

175

220

22

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820

02

58

27

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52

181

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52

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02

30

23

92

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epa

nel

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pper

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ofth

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lue

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pow

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tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES






















































(24) Vt 5 expb0 2 c log (1 2 d )

1 2 c1

b1

1 2 cxt 1

b2

1 2 cxt

2





















2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

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gin

g0

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54

148

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767

386

075

545

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312

927

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59

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577

497

462

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447

283

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525

246

32

432

242

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417

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508

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368

353

348

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586

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32

409

200

383

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319

308

294

287

285

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92

511

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12

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166

165

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100

098

094

089

087

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077

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se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES
































































2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

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12

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110

120

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gin

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54

148

38

767

386

075

545

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312

927

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59

935

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577

497

462

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447

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22

525

246

32

432

242

22

417

150

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508

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368

353

348

346

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586

251

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244

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32

409

200

383

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319

308

294

287

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92

511

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92

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12

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166

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82

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100

098

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089

087

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077

076

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92

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1 21 4

1 10

1 20

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284

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190

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3

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pper

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tion

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ean

ofth

eva

lue

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pow

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tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES























































2
















(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

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075

545

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927

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59

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577

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462

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525

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32

432

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368

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319

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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



























































(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

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17

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15

12

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110

120

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17

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15

12

14

110

120

140

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gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

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59

935

653

577

497

462

452

447

283

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687

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22

525

246

32

432

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22

417

150

652

508

429

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368

353

348

346

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586

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52

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32

409

200

383

345

319

308

294

287

285

284

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92

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12

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165

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82

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100

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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



















































(25)r1t 1 1 2 rf

dt1 1 2 pt 1 15

u 0

b 01

u 1

b 1(dt 2 pt) 1

e 1t1 1

e 2t 1 1



2 5 u 12V 22 s u












r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES






















































r1t 1 1 2 rf

dt1 1 2 pt1 15

0173(0066)

2 0146(0073)

1

0047(0020)0957

(0022)

(dt 2 pt) 1e 1t1 1

e 2t1 1

V 5

5296E 2 3 2 4290E 2 3(0540E 2 3) (0522E 2 3)

2 4290E 2 3 6397E 2 3(0522E 2 3) (0653E 2 3)

R2 500280910

(B) Derived model

r1t 1 1 2 rf 5 xt 1 ut 1 1

xt1 1 5 1250E 2 2 1

(0005)0957 (xt 2 micro) 1 h t1 1

(0022)

s u2 s u h

s u h s h2 5

5296E 2 3 2 0203E 2 3(0540E 2 3) (0090E 2 3)

2 0203E 2 3 0014E 2 3(0090E 2 3) (0012E 2 3)

rf 5 0071E 2 2 s x2s u

2 5 3215E 2 2 corr ( h u) 5 2 0737






(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

ER

AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

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242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

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569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

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tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES





















































(26) a t 5 a0 1 a1(xt 1 s u

22)

and


22))2


22) 1 b2( s u44) b1 5 b1 2











RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

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FU

NC

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NW

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NT

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OP

TIM

AL

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RT

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NR

ES

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RC

EN

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GE

LO

SS

INT

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VA

LU

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TIO

NU

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AL

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RN

AT

IVE

RE

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RIC

TE

DP

OR

TF

OL

IOR

UL

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RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

454

241

32

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239

51

000

00

00

00

00

00

00

00

02

828

267

62

566

250

72

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401

239

61

502

137

28

42

56

24

62

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23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

002

200

215

32

122

210

82

90

28

22

79

27

82

580

253

42

497

247

92

454

244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

237

228

22

302

231

22

175

220

22

251

228

32

353

241

62

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245

820

02

58

27

32

103

212

52

181

224

52

279

229

92

93

211

32

151

217

92

251

233

42

378

240

540

02

30

23

92

59

27

42

117

218

02

219

224

62

48

26

02

84

210

22

157

223

82

288

232

3

Th

epa

nel

inth

eu

pper

left

corn

erof

the

tabl

ere

port

sth

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nco

ndi

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ean

ofth

eva

lue

fun

ctio

nu

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eop

tim

alco

nsu

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and

por

tfol

ioru

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Th

epa

nel

inth

eu

pper

righ

tco

rner

repo

rts

the

perc

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the

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nct

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wh

enth

ep

ortf

olio

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is

xed

atth

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ean

valu

eof

the

opti

mal

por

tfol

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lean

dco

nsu

mp

tion

adju

sts

opti

mal

lyT

he

pan

elin

the

low

erle

ftco

rner

rep

orts

the

perc

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the

valu

efu

nct

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wh

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ep

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olio

rule

ism

yop

ic(s

eeeq

uat

ion

(21)

inte

xt)

and

con

sum

ptio

nad

just

sop

tim

ally

Th

epa

nel

inth

elo

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righ

tco

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the

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just

sop

tim

ally

Th

ese

nu

mbe

rsar

eba

sed

onth

ep

aram

eter

valu

esfo

rth

ere

turn

pro

cess

pres

ente

din

Tab

leI

Th

ese

valu

esar

ees

tim

ates

for

the

peri

od19

471

ndash199

54

Th

eva

lues

onth

em

ain

dia

gon

alco

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pon

dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES























































RRA EIS


175 100 115 12 14 110 120 140075 2 386 2 294 2 234 2 210 2 182 2 167 2 162 2 160100 00 00 00 00 00 00 00 00150 271 235 205 192 175 166 163 162200 338 307 279 266 248 239 236 235400 299 298 297 296 295 294 294 294

1000 160 172 185 192 204 212 214 2162000 88 97 108 114 124 131 133 1344000 46 52 58 62 68 73 74 75

(B) Slope a1

175 100 115 12 14 110 120 140075 2225 2251 2273 2283 2296 2304 2306 2307100 1888 1888 1888 1888 1888 1888 1888 1888150 1455 1444 1433 1429 1422 1418 1417 1417200 1185 1175 1166 1161 1155 1151 1150 1150400 681 681 680 680 680 679 679 679

1000 299 303 307 309 313 315 316 3162000 155 158 161 163 166 168 168 1684000 79 81 83 84 86 87 87 87





22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

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AG

EM

EA

NV

AL

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AL

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ED

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A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

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244

32

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242

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0018

59

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653

577

497

462

452

447

283

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687

257

22

525

246

32

432

242

22

417

150

652

508

429

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368

353

348

346

267

42

586

251

52

483

244

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241

32

409

200

383

345

319

308

294

287

285

284

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92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

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166

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232

82

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236

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100

098

094

089

087

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077

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17

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001

15

1 21 4

1 10

1 20

1 40

17

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15

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No

hed

gin

g0

752

284

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92

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243

34

002

190

219

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12

467

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3

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epa

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eu

pper

left

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sth

eu

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tion

alm

ean

ofth

eva

lue

fun

ctio

nu

nd

erth

eop

tim

alco

nsu

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and

por

tfol

ioru

les

Th

epa

nel

inth

eu

pper

righ

tco

rner

repo

rts

the

perc

enta

gelo

ssin

the

valu

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nct

ion

wh

enth

ep

ortf

olio

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is

xed

atth

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the

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nsu

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tion

adju

sts

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he

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elin

the

low

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ftco

rner

rep

orts

the

perc

enta

gelo

ssin

the

valu

efu

nct

ion

wh

enth

ep

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olio

rule

ism

yop

ic(s

eeeq

uat

ion

(21)

inte

xt)

and

con

sum

ptio

nad

just

sop

tim

ally

Th

epa

nel

inth

elo

wer

righ

tco

rner

repo

rts

the

perc

enta

gelo

ssin

the

valu

efu

nct

ion

wh

enth

epo

rtfo

lio

rule

is

xed

atth

em

ean

valu

eof

the

myo

pic

rule

and

con

sum

ptio

nad

just

sop

tim

ally

Th

ese

nu

mbe

rsar

eba

sed

onth

ep

aram

eter

valu

esfo

rth

ere

turn

pro

cess

pres

ente

din

Tab

leI

Th

ese

valu

esar

ees

tim

ates

for

the

peri

od19

471

ndash199

54

Th

eva

lues

onth

em

ain

dia

gon

alco

rres

pon

dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES




















































22 1 s u22




RRA EIS


2 2)] 3 100

175 100 115 12 14 110 120 140075 2984 3116 3209 3247 3296 3322 3331 3334100 2860 2860 2860 2860 2860 2860 2860 2860150 2474 2421 2376 2355 2327 2314 2309 2307200 2132 2086 2044 2025 1997 1983 1978 1976400 1331 1327 1327 1326 1324 1323 1323 1323

1000 613 631 651 661 677 689 693 6952000 322 336 352 361 375 384 388 3894000 165 174 184 189 198 204 206 207


175 100 115 12 14 110 120 140075 2 278 2 224 2 188 2 174 2 157 2 148 2 145 2 144100 00 00 00 00 00 00 00 00150 229 213 198 191 181 176 174 174200 329 315 301 294 284 279 277 276400 463 462 461 461 460 460 460 460

1000 534 547 560 567 578 585 587 5882000 556 575 594 603 619 628 631 6324000 566 588 611 621 639 649 653 655



FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

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TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

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RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

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464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

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241

32

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239

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000

00

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828

267

62

566

250

72

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12

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239

61

502

137

28

42

56

24

62

36

23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

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200

215

32

122

210

82

90

28

22

79

27

82

580

253

42

497

247

92

454

244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

237

228

22

302

231

22

175

220

22

251

228

32

353

241

62

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245

820

02

58

27

32

103

212

52

181

224

52

279

229

92

93

211

32

151

217

92

251

233

42

378

240

540

02

30

23

92

59

27

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218

02

219

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62

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26

02

84

210

22

157

223

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3

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epa

nel

inth

eu

pper

left

corn

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tabl

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tion

alm

ean

ofth

eva

lue

fun

ctio

nu

nd

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tim

alco

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and

por

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epa

nel

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eu

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righ

tco

rner

repo

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the

perc

enta

gelo

ssin

the

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ion

wh

enth

ep

ortf

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rule

is

xed

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(21)

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tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































FIG

UR

EI







































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



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110

120

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gin

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54

148

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767

386

075

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927

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59

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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES























































































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

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NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

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AL

TE

RN

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RE

ST

RIC

TE

DP

OR

TF

OL

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UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

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257

22

525

246

32

432

242

22

417

150

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508

429

401

368

353

348

346

267

42

586

251

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244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

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52

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222

92

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063

054

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044

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12

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032

028

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28

02

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220

32

366

247

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17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

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for

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tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES
















































































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

ER

AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

92

569

200

081

075

068

063

054

047

044

042

27

52

95

214

12

181

229

02

464

255

62

620

400

074

067

058

052

041

032

028

026

23

82

50

28

02

108

220

32

366

247

52

557

17

51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

595

253

32

454

241

32

401

239

51

000

00

00

00

00

00

00

00

02

828

267

62

566

250

72

443

241

12

401

239

61

502

137

28

42

56

24

62

36

23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

002

200

215

32

122

210

82

90

28

22

79

27

82

580

253

42

497

247

92

454

244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

237

228

22

302

231

22

175

220

22

251

228

32

353

241

62

443

245

820

02

58

27

32

103

212

52

181

224

52

279

229

92

93

211

32

151

217

92

251

233

42

378

240

540

02

30

23

92

59

27

42

117

218

02

219

224

62

48

26

02

84

210

22

157

223

82

288

232

3

Th

epa

nel

inth

eu

pper

left

corn

erof

the

tabl

ere

port

sth

eu

nco

ndi

tion

alm

ean

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eva

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and

por

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(21)

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ally

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and

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sum

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ese

nu

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tim

ates

for

the

peri

od19

471

ndash199

54

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eva

lues

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dto

the

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eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES







































































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



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alm

ean

ofth

eva

lue

fun

ctio

nu

nd

erth

eop

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alco

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and

por

tfol

ioru

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nel

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eu

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is

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the

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nsu

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tion

adju

sts

opti

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lyT

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pan

elin

the

low

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ftco

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rep

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the

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enta

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ssin

the

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efu

nct

ion

wh

enth

ep

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olio

rule

ism

yop

ic(s

eeeq

uat

ion

(21)

inte

xt)

and

con

sum

ptio

nad

just

sop

tim

ally

Th

epa

nel

inth

elo

wer

righ

tco

rner

repo

rts

the

perc

enta

gelo

ssin

the

valu

efu

nct

ion

wh

enth

epo

rtfo

lio

rule

is

xed

atth

em

ean

valu

eof

the

myo

pic

rule

and

con

sum

ptio

nad

just

sop

tim

ally

Th

ese

nu

mbe

rsar

eba

sed

onth

ep

aram

eter

valu

esfo

rth

ere

turn

pro

cess

pres

ente

din

Tab

leI

Th

ese

valu

esar

ees

tim

ates

for

the

peri

od19

471

ndash199

54

Th

eva

lues

onth

em

ain

dia

gon

alco

rres

pon

dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES




























































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



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BL

EV

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AG

EM

EA

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VA

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SN

oti

min

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17

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001

15

12

14

110

120

140

17

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001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

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0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

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420

241

32

409

200

383

345

319

308

294

287

285

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92

511

247

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454

242

92

415

241

12

409

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166

165

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100

098

094

089

087

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077

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1 20

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No

hed

gin

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752

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190

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02

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tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES




















































RETURN ON WEALTH

RRA EIS


175 100 115 12 14 110 120 140075 058 153 246 292 362 404 418 425100 078 153 229 266 322 355 366 372150 103 153 204 230 267 290 297 301200 119 153 188 205 231 247 252 254400 152 153 155 156 157 158 158 158

100 179 153 128 115 094 082 078 076200 190 153 117 098 070 052 046 044400 196 153 111 090 057 037 030 027


175 100 115 12 14 110 120 140075 379 377 374 373 372 371 371 371100 384 384 384 384 384 384 384 384150 372 370 369 368 367 367 366 366200 348 346 343 342 340 340 339 339400 257 257 256 256 256 256 256 256

100 137 140 143 144 146 148 149 149200 078 081 083 085 087 089 089 089400 045 046 048 049 050 051 052 052


0 1 b1(micro 1 s u





FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































FIG

UR

EII








22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

ER

AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

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ese

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tim

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for

the

peri

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54

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dto

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tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES
























































22)












RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



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17

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12

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120

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12

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110

120

140

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gin

g0

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54

148

38

767

386

075

545

385

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927

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32

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59

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462

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283

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257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

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087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

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200

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063

054

047

044

042

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52

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214

12

181

229

02

464

255

62

620

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052

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026

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50

28

02

108

220

32

366

247

52

557

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51

001

15

1 21 4

1 10

1 20

1 40

17

51

001

15

12

14

110

120

140

No

hed

gin

g0

752

284

29

12

44

23

22

21

21

72

16

21

52

925

274

42

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253

32

454

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32

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239

51

000

00

00

00

00

00

00

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02

828

267

62

566

250

72

443

241

12

401

239

61

502

137

28

42

56

24

62

36

23

12

29

22

92

681

259

42

522

249

02

448

242

52

418

241

42

002

200

215

32

122

210

82

90

28

22

79

27

82

580

253

42

497

247

92

454

244

02

436

243

34

002

190

219

32

204

220

92

216

222

12

222

222

32

367

238

22

407

242

02

444

246

12

467

247

010

02

104

212

42

162

218

62

237

228

22

302

231

22

175

220

22

251

228

32

353

241

62

443

245

820

02

58

27

32

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212

52

181

224

52

279

229

92

93

211

32

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217

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251

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42

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240

540

02

30

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59

27

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218

02

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ese

valu

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ees

tim

ates

for

the

peri

od19

471

ndash199

54

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eva

lues

onth

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dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES




























































RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



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valu

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din

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tim

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for

the

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tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES























































RRA EIS


175 100 115 12 14 110 120 140075 3712 3117 2741 2612 2471 2413 2397 2390100 3344 2747 2349 2208 2055 1991 1974 1966150 2743 2232 1863 1728 1583 1526 1511 1505200 2307 1884 1562 1441 1312 1265 1254 1249400 1393 1163 969 893 816 795 792 792

100 631 542 461 429 399 397 400 402200 329 287 247 231 217 219 221 222400 168 148 128 120 114 115 117 118


175 100 115 12 14 110 120 140075 897 000 672 949 1312 1502 1561 1590100 777 000 607 865 1203 1386 1443 1471150 611 000 513 738 1047 1215 1269 1295200 504 000 445 648 930 1090 1141 1167400 294 000 292 438 655 785 828 849

100 130 000 144 222 346 427 455 469200 067 000 078 122 194 241 258 266400 034 000 041 064 103 128 138 142









FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

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EV

IIP

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CE

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NC

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PO

RT

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LIO

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TR

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GE

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075

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319

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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES
























































FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

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the

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se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































FIG

UR

EII

I














FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

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AL

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EN

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17

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001

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12

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120

140

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54

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38

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312

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59

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525

246

32

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242

22

417

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508

429

401

368

353

348

346

267

42

586

251

52

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244

22

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241

32

409

200

383

345

319

308

294

287

285

284

255

92

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247

32

454

242

92

415

241

12

409

400

166

165

166

166

166

166

166

166

232

82

342

236

82

382

240

82

429

243

72

440

100

098

094

089

087

081

077

076

075

214

52

173

222

92

270

237

42

486

253

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214

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1 20

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17

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15

12

14

110

120

140

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22

21

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42

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249

02

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241

42

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32

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82

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27

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253

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247

92

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244

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243

34

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32

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220

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216

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12

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222

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238

22

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242

02

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246

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247

010

02

104

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42

162

218

62

237

228

22

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231

22

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02

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27

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02

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peri

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471

ndash199

54

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eva

lues

onth

em

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dia

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dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES






























































FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



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the

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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES
























































FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

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PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

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FU

NC

TIO

NU

ND

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AL

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A

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17

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15

12

14

110

120

140

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12

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110

120

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gin

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312

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348

346

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383

345

319

308

294

287

285

284

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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































FIG

UR

EIV













Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



TA

BL

EV

IIP

ER

CE

NT

AG

EM

EA

NV

AL

UE

FU

NC

TIO

NW

HE

NT

HE

OP

TIM

AL

PO

RT

FO

LIO

RU

LE

ISU

NR

ES

TR

ICT

ED

A

ND

PE

RC

EN

TA

GE

LO

SS

INT

HE

VA

LU

E

FU

NC

TIO

NU

ND

ER

AL

TE

RN

AT

IVE

RE

ST

RIC

TE

DP

OR

TF

OL

IOR

UL

ES

RR

A

Tim

ing

EI

SN

oti

min

gE

IS

17

51

001

15

12

14

110

120

140

17

51

001

15

12

14

110

120

140

Hed

gin

g0

7554

54

148

38

767

386

075

545

385

312

927

275

32

611

255

32

480

244

32

431

242

51

0018

59

935

653

577

497

462

452

447

283

42

687

257

22

525

246

32

432

242

22

417

150

652

508

429

401

368

353

348

346

267

42

586

251

52

483

244

22

420

241

32

409

200

383

345

319

308

294

287

285

284

255

92

511

247

32

454

242

92

415

241

12

409

400

166

165

166

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ates

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the

peri

od19

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54

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tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES





























































Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



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249

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253

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247

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244

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243

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242

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246

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247

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42

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218

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22

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231

22

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245

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92

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02

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59

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218

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Th

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valu

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rth

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pro

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pres

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din

Tab

leI

Th

ese

valu

esar

ees

tim

ates

for

the

peri

od19

471

ndash199

54

Th

eva

lues

onth

em

ain

dia

gon

alco

rres

pon

dto

the

pow

eru

tili

tyca

se



VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES






















































Hedging


2 2)1 b2(x t 1 s u


2 2) c t 2 wt 5 b0hnt 1 b1

hnt (xt 1 s u2 2)

No-hedging

Timingat 5

xt 1 s u2 2

g s u2


nh t(xt 1 s u2 2)

1 b2nh t(xt 1 s u

2 2)

No-Timingat 5

micro 1 s u2 2

g s u2


nhnt(xt 1 s u2 2)



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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































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VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



















































VI CONCLUSION
















Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



























































Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES























































Et 1 1xt 1 j2 5 micro2(1 2 f j)2 1

1 2 f 2( j 2 1)

1 2 f 2s h


1 f 2 jxt21 f 2( j 2 1)h t1 1

2 1 2 f 11 2( j 2 1)(xt 2 micro) h t1 1



(3) we have


j 2 1

f l h t1 j2 l


and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































and


j2 1

f 2l h t1 j2 l2

1 2 f ol 5 0

j 2 1




xt 1 12 2 Etxt1 1

2 5 ( h t1 12 2 s h

2) 1 (2micro(1 2 f ) 1 2 f xt) h t1 1




2 1 h t 1 12




r1t 1 1 2 Etr1t 1 1 5 ut1 1

s 11t 5 s u2





and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































and

rpt 1 1 2 Etrpt1 1 5 (a0 1 a1xt)ut1 1

where

p0 5 a0(1 2 a0)( s u22)

p1 5 a0 1 a1(1 2 2a0)( s u22)

p2 5 a1 2 a12(s u

22)







5 (a0 1 a1xt)ut 1 1

h



r(ct 2 wt) 1 k

5 c0 1 c1xt 1 c2xt2



2)


where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































where

c0 5 rf 1 a0(1 2 a0)s u

2

21 k 1 b0 1 2

1


1 b2(micro2(1 2 f )2 1 s h2)

c1 5 a0 1 a1(1 2 2a0)s u

2

21 b1 f 2

1

r1 b2(2micro f (1 2 f ))

c2 5 a1 2 a12

s u2

21 b2 f 2 2

1

r



Therefore


r(ct 2 wt) 1 k

5 p0 1 p1xt 1 p2xt2 1 rf

1 b0 1 21

r1 b1 Etxt 1 1 2

1

rxt 1 b2 Etxt 1 1

2 21

rxt

2 1 k

5 a0(1 2 a0)s u

2

21 a0 1 a1

s u2

22 a0a1s u

2 xt 1 a1 1 2 a1

s u2

2xt

2

1 rf 1 b0 1 21

r1 b1 micro(1 2 f ) 1 f 2

1

rxt


1

rxt

2 1 k






1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

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1 b2(xt1 12 2 Etxt 1 1

2 )

5 b1h t1 1 1 b2[( h t1 12 2 s h

2)

1 (2micro(1 2 f ) 1 2 f xt) h t 1 1]




where

v0 5 a02 (1 2 g )( c 2 1)

1

2s u

2 1 b12

1 2 g

c 2 1

1

2s h

2

1 b22

1 2 g

c 2 1(s h

2 1 2micro2(1 2 f )2) s h2


1 b1b2

1 2 g

c 2 12micro(1 2 f ) s h

2

v1 5 b22

1 2 g

c 2 14micro f (1 2 f ) s h

2 1 a0a1[(1 2 g )( c 2 1) s u2]

2 a0b2[(1 2 g )2 f s h u] 2 a1b1[(1 2 g ) s h u]

2 a1b2[(1 2 g )2micro(1 2 f )s h u]

1 b1b2

1 2 g

c 2 12 f s h

2

v2 5 a12 (1 2 g )( c 2 1)

1

2s u

2 1 b22

1 2 g

c 2 12 f 2 s h

2

2 a1b2[(1 2 g )2 f s h u]



vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



















































vpt 51

2

u


51

2

u


51

2

u

cEt[(1 2 c )(rpt 1 1 2 Etrpt 1 1)

1 (ct 1 1 2 wt 1 1) 2 Et(ct 1 1 2 wt 1 1)]2




2

2

1 b0 1 21


2)

v1 5 a0(1 2 c ) 1 (1 2 c )a1(1 2 2a0)s u

2

21 b1 f 2

1

r

1 b2(2micro f (1 2 f ))

v2 5 a1(1 2 c ) 2 (1 2 c )a12

s u2

21 b2 f 2 2

1

r





5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES




















































5 c log d 1 c rf 1 v0 1 c a0(1 2 a0)s u

2

2

1 v1 1 c a0 1 a1

s u2

22 a0a1s u

2 xt

1 v2 1 c a1 1 2 a1

s u2

2xt

2


(33) Et D ct1 1 5 c0 1 c1xt 1 c2xt2




`

r jrpt1 11 j

5 (1 2 c ) ( p1 1 2p2micro)r

1 2 r f2 2p2micro

r f

1 2 r f 2s h u

1 (1 2 c ) 2p2

r f

1 2 r f 2s h u xt




(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



















































(Et 1 1 2 Et) oj 5 1

`

r jrpt1 j1 1 5 p1 oj 5 1

`


1 p2 oj5 1

`


5 p1h t1 1 oj 5 1

`

r jf j 2 1 1p2s h

2

1 2 f 2

middot oj 5 1

`

r j( f 2 j 2 f 2( j2 1))

1 p2h t 1 12 o

j5 1

`

r j f 2( j2 1)

1 2p2h t1 1(xt 2 micro)f 2 1 oj 5 1

`

r jf 2j

1 p2h t1 12micro f 2 1 oj 5 1

`

r jf j

5 p1

r

1 2 r f


1 2 r f 2

1 2p2micror

1 2 r fh t1 1

1 p2

r

1 2 r f 2h t 1 1

2 2 p2

r

1 2 r f 2s h

2






`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES




















































`


1 2 r f

2 2v2micror f

1 2 r f 2s h u

1 2v2

r f

1 2 r f 2s h u xt



(Et 1 1 2 Et) oj 5 1

`


`

r j(E t 1 1 2 Et)xt 1 j

1 v2 oj 5 1

`

r j(E t 1 1 2 Et)xt1 j2


(Et1 1 2 Et) oj5 1

`

r jvpt 1 j 5 v1

r


r f

1 2 r f 2

1 2v2micror

1 2 r fh t1 1

1 v2

r

1 2 r f 2h t1 1

2 2 v2

r

1 2 r f 2s h

2






from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES





















































from guess (ii)

s 1c 2 wt 5 covt (r1t 1 1ct 1 1 2 wt 1 1)


1 b2(xt 1 12 2 Etx t1 1

2 )]

5 covt [ut 1 1b1h t1 1 1 b2( h t 1 12 2 s h

2)

1 b2(2micro(1 2 f ) 1 2 f xt) h t 1 1]




(34) a t 51

g

xt

s u2

11

2 g2

1 2 g

g ( c 2 1)

s h u

s u2

b1 1 b2[2micro(1 2 f ) 1 2 f xt]


a t 5 a0 1 a1xt




v0 5 V11a02 1 V12b1

2 1 V13b22 1 V14a0b1 1 V15a0b2 1 V16b1b2


v2 5 V31a12 1 V32b2

2 1 V33a1b2





a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES




















































a0a1 and b0b1b2

v0 5 B10 1 B11b0 1 B12b1 1 B13b2 1 B14a0 1 B15a02

v1 5 B21a0 1 B22b1 1 B23b2 1 B24a1 1 B25a0a1

v2 5 B31a1 1 B32b2 1 B33a12



a0 5 A10 1 A11b1 1 A12b2

a1 5 A20 1 A21b2




0 5 2 B10 1 (V11 2 B15)A102 2 B14A10 1 ( 2 B11)b0

1 [2(V11 2 B15)A10A11 2 B12 1 V14A10 2 B14A11]]b1

1 [(V11 2 B15)A112 1 V12 1 V14A11]b1

2

1 [2(V11 2 B15)A10A12 2 B13 1 V15A10 2 B14A12]b2

1 [(V11 2 B15)A122 1 V13 1 V15A12]b2

2

1 [2(V11 2 B15)A11A12 1 V14 A12 1 V15A11 1 V16]b1b2

0 5 2 B21A10 1 (V22 2 B25)A10A20 2 B24A20


1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































1 [ 2 B21A11 2 B22 1 (V22 2 B25)A11A20 1 V24A20]b1

1 [ 2 B21A12 2 B23 1 (V22 2 B25)(A10A21 1 A12A20)

1 V23 A10 1 V25A20 2 B24A21]b2

1 [V21 1 (V22 2 B25)A12A21 1 V23A12 1 V25A21]b22

1 [(V22 2 B25)A11A21 1 V23A11 1 V24 A21 1 V26]b1b2

0 5 (V31 2 B33)A202 2 B31A20

1 [2(V31 2 B33)A20A21

2 B31A21 2 B32 1 V33A20]b2

1 [(V31 2 B33)A212 1 V321 V33 A21]b2

2 h


Proof of Property 1


0 5 L 30 1 L 31b21 L 32b22


(35) 0 5( c 2 1)

2 g s u2

2(1 2 g )2 f s h u

g s u2

1 f 2 21

rb2

12(1 2 g ) f 2[ s h u

2 1 g ( s u2 s h

2 2 s h u2 )]

( c 2 1) g s u2

b22


L 32L 30 0

ie

(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

g 2s u4

0


(36) s u2 s h

2 2 s h u2 5 s u

2 s h2(1 2 corr (uh )2) $ 0



(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



















































(37) L 312 2 4 L 32L 30 $ 0


(38) 0 5(c 2 1)2

4(1 2 g ) f 2[s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

2( c 2 1) s h u

f [ s h u2 1 g ( s u

2 s h2 2 s h u

2 )]

1( c 2 1) g s u

2( f 2 2 1r )

2(1 2 g ) f 2[ s h u2 1 g ( s u

2 s h2 2 s h u

2 ])b2 1 b2

2

or

0 5 L ˜ 30 1 L ˜ 31b2 1 b22


b21 middot b22 5 L ˜ 30

0 if g 1

0 if g 1



b21 1 b22 5 2 L ˜ 31







b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



















































b2 5 (c 2 1)2 (A 1 B) 6 Icirc (A 1 B)2 1 C

2 D


b2 5 ( c 2 1)( 2 A 1 B) 6 Icirc ( 2 A 1 B)2 1 C

2 D


Proof of Property 2






(39) a1 54 g f 2s h

2 s u2 2 2 g f s h us u

2( f 2 2 1r ) 1 Icirc A

4 g f 2s u2[ s h u

2 1 g ( s h2 s u

2 2 s h u2 )]

where

A 5 2 g f s h us u2 f 2 2

1

r

2

1 16 g (1 2 g ) f 3s h u3 s h

2 f 2 21

r

2 16 g (1 2 g ) f 4s h u2 s u

2 s h2




s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES



















































s h2

2 f 2

(1 r 2 f 2)$ 2 f s h u

and

s h2

f 2

(1 r 2 f 2) 2 f s h u







Proof of Property 3





b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES




















































b0 5 ( r (1 2 r ))(k 2 log d )


(40) b0 5 log (1 2 r )




2 h



Proof of Property 4



b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES


















































b2 as

(41)

b1 5 ( c 2 1)f1( g r )

b2 5 (c 2 1)f2( g r )


Proof of Property 5



HARVARD UNIVERSITY

REFERENCES









































































































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CONSUMPTION AND PORTFOLIO DECISIONS WHEN ...pages.stern.nyu.edu/~dbackus/GE_asset_pricing...ity on portfolio choice. Kim and Omberg [1996] work with a similar framework but, by assuming