L4: Static Portfolio Choice 1 Static Portfolio Choices (L4) Following topics are covered: –Mean and Variance as Choice Criteria – an example –Insurance

L4: Static Portfolio Choice 1

Static Portfolio Choices (L4)

• Following topics are covered:– Mean and Variance as Choice Criteria – an example

– Insurance

• Optimal insurance with loading

• Optimal coinsurance

• Optimality of deductible insurance

– Optimal Investment Portfolio

• Portfolio of single risky and risk-free assets

• The effect of background risk

• Portfolio of multiple risky assets and risk-free assets

• Materials covered in Chapter 5, CWS

• Efficient frontier

Materials from CWS3&5 and EGS3&4


Using Mean and Variance as Choice Criteria: An Example

• If the distribution of return offered by assets is jointly normal, then we can maximize expected simply by selecting the best combination of mean and variance.

• Assuming the return on an asset is normal distributed with mean E and variance σ2, we can write the utility function as: U=U(R, E, σ)

• Then the expected utility is:

• We want to show that the marginal rate of substitution between return and risk is positive and that the indifference curves are convex.

dRERfRUUE );;()()(


Positive Slope in Indifference Curve

• Converting R to a standard normal variable, Z=(R-E)/σ.

– then f(R;E;σ)=(1/σ)f(Z;0;1)

• An indifference curve is defined as the locus of points where the change in expected utility is zero – dE(U)=0.

• The denominator must be positive because U’(.) must be positive. The numerator is negative (why?)

dZZfZEUUE )1;0;()()(

0)1;0;()('

)1;0;()('

dZZfZEU

dZZZfZEU

d

dE


Means in Managing Risks

• Insurance– Risky assets with a full insurance is like investing in a risk-free asset

– Partial insurance is like a combo of risk free assets (full insurance) and risky assets

• Diversified portfolio– Risk-free asset and single risky asset

– Multiple risky assets

– Risk-free asset and multiple risky assets


Static Portfolio Choice I: Insurance• Maximize an agent’s utility when there is costly or costless hedging

contract available • The case of actuarially fairly priced insurance• Assuming loss follows a distribution of x (where x>=0); premium = P;

Indemnity schedule = I(x)• Insurer reimburses policyholder for the full value of any loss, I(x)=x• When the premium is actuarially fair, P=EI(x)=E(x)• Suppose the insured is risk averse, how much is he willing to pay for the

insurance?– With insurance, his expected utility is u(E(x))– Without insurance, his expected utility is E(u(x))=u(E(x)-П)– Insurance increases the certainty equivalent by П

• As a result, risk-averse agents would take full insurance when insurance prices are actuarially fair

• When the insurance has loading, optimal insurance amount is determined by the transaction cost and risk reduction.


Insurance Loading

• Suppose the chance of the ship being sunk is ½.

• Insurer’s loading is 10% of the actual value of the policy

• Suppose the amount of insurance purchased is I, P(I)=(I/2)+0.1(I/2)=0.55I

• A square-root utility function

• Questions– What is the expected wealth?

– What is the expected utility?

– What is maximum utility?


Optimal Coinsurance• Definition: I(x)=βx

• Insurance pricing rule:– P(β)=(1+λ)EI(x)= βP0, where P0 =(1+ λ)Ex

• Random final wealth = w0- βP0-(1- β)x))1(()(max 00 xPwEuH

)](')[()(

)(' 0 yuPxEyEu

H

0)]('')[()(

)('' 202

2

yuPxE

yEuH

EU would be hump-shaped w. r. t. β. Thus β* is determined by:

)](')[()(

)(' 0 yuPxEyEu

H

=0


Mossin’s Theorem• Full insurance (β*=1) is optimal at an actuarially fair price,

λ=0, while partial coverage (β*<1) is optimal if the premium includes a positive loading, λ>0.– An intuition is that full insurance may be still optimal if the degree of

risk aversion of the policyholder is sufficiently high.

– This is not correct given that risk aversion is a second-order phenomenon. For a very small level of risk, individual behavior toward risk approaches risk neutrality. For risk neutral policyholder, any saving in transaction cost is good.

– Also may have corner solution of β*=0 when λ>0, occurring λ≥λ* where

– Undiversified risk is an alternative form of transaction cost (page 52))('

)(',cov(*

0

0

xwExEu

xwux


Comparative Statics in Coinsurance Problem

• The effect of risk aversion

• Proposition 3.2: Consider two utility function u1 and u2 that are increasing and concave, and suppose that u1 is more risk averse than u2 in the sense of Arrow and Pratt (Equation 1.7, page 11). Then, β1*>β2*– Sounds natural



• Effect of initial wealth:

• Proposition 3.3: An increase in initial wealth will

(i) decrease the optimal rate of coinsurance β* if u exhibits decreasing absolute risk aversion (proof see page 54 and the next slide)

(ii) increase the optimal rate of coinsurance β* if u exhibits increasing absolute risk aversion

(iii) cause no change in optimal rate of coinsurance β* if u exhibits constant absolute risk aversion

w

H

)(' *

w *

The sign of is same as


Proof of Proposition 3.3

• The sign of әβ*/ әw is same as әH’(β)/ әw

)]('')[()('

0 yuPxEw

Hyy

where P0=(1+λ)E(x), y=w0-βP0-(1-β)x, and H’(β)=E[(x-P0)u’(y)] DARA implies -u’ is more concave than u.

Note that )]('')[()]('[

0 yuPxEyuE

yy

)]('')[( 0 yuPxE yy >0 at β= β*. Thus proved.



• Effect of loading

• Proposition 3.4: An increase in the premium loading factor λ will cause β* to

(i) decrease if u exhibits constant or increasing absolute risk aversion

(ii) either increase or decrease if u exhibits decreasing absolute risk aversion

Look at *

*)('H

To evaluate


Deductible Insurance

• Deductible provide the best compromise between the willingness to cover the risk and the limitation of the insurance deadweight cost.

• Suppose a risk-averse policyholder selects an insurance contract (P, I(.)) with P=(1+λ)EI(x) and I(x) nodecreasing and I(x)≥0 for all x. Then the optimal contract contains a straight deductbile D; that is I(x)=max(0, x-D)


Static Portfolio Choice II: Diversification

• Investors who consume their entire wealth at the end of the current period• The case containing a risk-free asset and a risky asset• The risk-free rate of return of the bond is r. the return of the stock is a

random variable x• Initial wealth w0

• α is invested in stock• Ending Portfolio Value

=(w0- α)(1+r)+ α(1+x)=w0(1+r)+a(x-r)=w+ αy


Optimal Investment in Risky Assets)(maxarg* ywEua

Assume that H=Eu(w+αy). H’(α)=E[u’(w+ αy )*y]; H’’(α)=E(u’’*y2)≤0 The optimal α* follows H’(α*)=E[u’(w+ α*y )*y] If α*=0, then H’(0)=E[u’(w)*y]=u’(w)*y=0. As u’(w)>0 (increasing utility function), to have H’(0)=0, y=0. This leads to the first part of proposition 4.1 (p66): The optimal investment in the risky asset is positive iff the expected excess return is positive. Note, this is very similar to the optimal coinsurance problem in Ch3 (p50). Investing in risky asset α*>0 is equivalent to taking coinsurance β*<1

Other results in proposition 4.1 should also follow.

What can we learn from the above condition?


Further Thoughts• As long as there is a positive excess return y, investors should

invest in the stock market– Participation puzzle

• Under constant relative risk aversion (CRRA), the demand for stocks is proportional to wealth: a*=kw. More specifically, we have

– Equity premium puzzle• Assuming a reasonable level of risk aversion lead to unreasonable shares

of investment in common stocks• Using historical data, μ=6%, and σ=16%. If R=2, the investment in equity

is 117%. Evan when R=10, equity investment would be 23%. (EGS page 66)

• Mehra and Prescott (85)

)(

1*22 wRuw yy

y


The Effect of Background Risk

)(maxarg~

** ywEv

)(maxarg~~

** ywEu

One way to explain the surprisingly large demand for stocks in the theoretical model is to recognize that there are other sources of risk on final wealth than the riskiness of assets returns.

We want to compare α** with α*, the demand for risky asset when there is no background risk.

Assuming v(z)=Eu(z+ε), we have

We just need to check if v is more concave than u, utility function corresponding to y without background risk.

)(maxarg~

* ywEu


v and u

• To show v is more concave than u, we need show

)('

)(''

)('

)(''

)('

)(''

zu

zu

zEu

zEu

zv

zv

for all ε such that Eε=0. The above condition is equivalent to showing Eh(z, ε)≤0, where h(z, ε)=u’’(z+ε)u’(z)-u’’(z)u’(z+ε) A necessary condition is h is concave in ε for all z.

I.e., )('''

)(''''

)('

)(''

zu

zu

zu

zu for all z.


Conditions regarding Background Risk

Consider the following three statements:

1. any zero-mean background risk reduces the demand for other independent risk

2. for all z, -u’’’’(z)/u’’’(z)>=-u’’(z)/u’(z)

3. absolute risk aversion is decreasing and convex.

Condition 2 is necessary for condition 1 under the assumption that u’’’ is positive. Condition 3 is sufficient for condition 1 and 2.

Power utility function satisfies (3).


Portfolio of Risky Assets

• Two assets following the same distributions of x1 and x2 that are independently and identically distributed

• Perform expected utility maximization

• If two assets are i.i.d., holding a balanced portfolio is optimal– Home bias


Diversification in Mean-Variance Model

• There are n risky assets, indexed by i = 1, 2, …, n

• The return of asset i is denoted by xi, whose mean is µi and covariance between returns of assets I and j is σij

• Risk free asset whose return r= x0


Diversification in Expected-Utility Model

n

iii

n

iii

n

ii xxaxxaaxz

100

10 )(1)1(1

A person maximizes the certainty equivalent of final wealth Ez-1/2Avar[z].

n

iii xaxEz

100 )(1

n

i

n

jijji aazVar

1 1

][

Differentiating the certainty equivalent wealth wrt ai and setting it to 0:

n

jijji aAx

1

*0 0

In a matrix form, *0 Aaxi


Interpretation

The investment in risky asset i: )(1

* 01 x

Aa

The investment in risk-free asset:

n

jjaa

1

*0 1*

When returns are independently distributed, we have:

ii

ii

x

Aa

0* 1

• Investment Implication: all investors, whatever their attitude to risk, should purchase the same portfolio of risky assets.

• Two-fund Separation

Frontier Portfolios

• A portfolio is a frontier portfolio if it has the minimum variance among portfolios that have the same expected rate of return. A portfolio p is a frontier portfolio if and only if wp, the N-vector portfolio weights of p, is the solution to the quadratic program:


VwwT

w 2

1min

}{

11

][

T

pT

w

rEews.t.,

See HL, page 63-65.

Frontier in Mean-Variance Space


See HL, page 66 (3.11.2a)

Minimum Variance Portfolio

• Minimum variance portfolio (mvp): expected return=A/C; var=1/C

• The covariance of the minimum variance portfolio and any portfolio (not only those on the frontier) is always equal to the variance of the rate of return on the minimum variance portfolio


Efficient Portfolios

• Those frontier portfolios which have expected rates of return strictly higher than that of the minimum variance portfolios

• Inefficient portfolios

• Any combination of efficient portfolios will be an efficient portfolio.

• For any portfolio p on the frontier, expect for mvp, there exists a unique frontier portfolio, denoted by zc(p), which has a zero covariance with p. (HL, page 70)


Portfolios with Risk Free Assets


VwwT

w 2

1min

}{

][)11( pfTT rErwew s.t.

Solution see HL, page 76-80.

How does it differ from EGS’s derivations?


Materials covered in Ch5, CWS

• Two-asset portfolios– Minimum variance portfolio

– Minimum variance opportunity set

– Efficient set

• Efficent set with one risky and one risk-free asset – page 126

• Many assets– CML


Technical Note on Comparative Statics• Comparative statics: how the equilibrium condition changes when an

exogenous variable changes

Assume: g=g(y,α). y is endogenous, i.e., y=y(α), α is exogenous.

Deriving comparative statics is to see the sign of y

.

The FOC of g is 0),(),(

yg

y

ygy

The SOC of g is 0),(

2

2

y

yg if g is concave.

If g is concave, the optimal y* must satisfy the condition 0),( yg y

When α changes, it will affect y and thus the equilibrium condtion differentiate 0),( yg y in both sides, we have

yy

y

g

gdy

*

sign )()*

( ygsigndy

=>we need to look at

),(yg y for *dy


Technical Note on Differentiation

• Leibnitz’s Rule

dxxf

d

daaf

d

dbbfdxxf

d

d b

a

b

a

)(

)(

)(

)(

),()()),((

)()),((),(

Special case, if b(θ)=b and a(θ)=a, then only last term remains.


Example

Assumptions: (1) x=end of period value before tax, which follows a distribution of f(x); (2) y = amount promised to debt holders (3) interest rate = 0 (4) τ = tax rate (5) k=proportional bankruptcy cost = k*x The firm picks up appropriate debt payment level y which maximize the firm’s value. Derive the comparative statics: ∂y*/∂ τ (adapted from Bradley, Jarrell and Kim, 1984, JF 39(3), pp. 857-878)


Exercises

• EGS, 3.2; 3.3; 4.2

• Set up a model for one of your ongoing projects– Provide a non-technical (i.e., no reference please) introduction of the

paper

– Identify the primary tradeoff in your story

– Setup the model

Documents

L4: Static Portfolio Choice 1 Static Portfolio Choices (L4) Following topics are covered: –Mean and Variance as Choice Criteria – an example –Insurance