Financial Economics - Block I: Consumption based asset pricing

Course requirements What is Finance? Course outline

Financial EconomicsBlock I: Consumption based asset pricing

Peter Kondor

Winter 2009


Part

Course overview


Outline of Part 0

Course requirements

What is Finance?

Course outline


Contact

Office 414 Nador u. 9.Phone 2206E-mail [email protected] Hours by appointment


Course requirements

1. Presentation of a paper (20%)• chosen from a menu• 30 minutes• What is the main idea?• Which are the main ingredients?• How does it work?• How does it relate to the material of this course?


2. 2 Problem sets (10% each)• in groups of ≤ 4• on website: new problems are added on each week• 1st is due to 3rd February, beginning of class

3. Final exam (40% or 60%)


• Optional:

4. 8-10-page essay (20%) by the last class of the course• choose a topic, pick a research question• find the paper which is closest to this question and many

related papers• write detailed literature review (more on the most relevant

paper, less on the others)• if you have ideas on how to address the question, write

about it.

• If you skip the essay, final exam counts for 60%• As the deadline of the essay is the last class, you have to

decide to do the essay before you write the final exam.


Basic notions

• What is an asset?• Price vs returns?• stocks/bonds/ contingent claims (derivatives)?• portfolio?• mutual fund/hedge fund?


What is Finance?• Structure of financial markets


• in the language of economics: Asset prices ensure thatsavings equal investment

• allocation of consumption and investment across statesand time

• risk-sharing among individuals• think of: insurance, investments in your pension, saving on

your deposit• has to match: firms production possibilities and decisions• if allocation works well: efficient production


• First part of the course:• Suppose financial markets work smoothly: Can we build a

model of portfolio choice, which matches the factsquantitatively?

• no frictions: representative agent’s decision on optimalconsumption/investment

• in focus: relationship between aggregate consumption dataand returns of assets

• Second part of the course:• What can go wrong on financial markets?• frictions: asymmetric information, search frictions, capital

constraints, bubbles• ”changing liquidity”


Course outline

I Consumption based asset pricing1. The Euler-equation and classic issues in Finance2. Factor pricing models and the cross-section3. Time series: predictability and the equity premium


II Asset pricing and liquidity4. Asset pricing under asymmetric information: static set-up5. Sequential trading and herding6. Optimal dynamic trading with informational advantage7. Search in finance8. Capital constraints and limits to arbitrage9. Delegated portfolio management

10. Bubbles and crashes11. Aggregate liquidity

The problem Interpretations Classic issues in Finance Consumption based model in practice Alternative approaches

Part I

The Euler equation and the classic issues inFinance


Outline of Part I

The problem

Interpretations

Classic issues in Finance

Consumption based model in practice

Alternative approaches


The problem of the representative investor

• The problem: how much to consume/ how much to invest

maxξ

u(ct ) + βEt [u(ct+1)]

s.t .ct = et −ptξ

ct+1 = et+1 + xt+1ξ

• First-order condition / Euler-equation

ptu′(ct ) = Et [βu′(ct+1)xt+1]

pt = Et [βu′(ct+1)

u′(ct )xt+1]


Interpretations of the Euler condition

• rewrite Euler as

pt = Et [mt+1xt+1]

mt+1 = βu′(ct+1)

u′(ct )

• or p = E [mx ]

• m is the pricing kernel• prices any claim (as we will see)• for example: risk free rate• xt+1 = Rt ,pt = 1

Rt =1

Et [mt+1]



• m also determines the stochastic discount factor• suppose an asset paying the random dt in each t• problem in infinite time

max{ξj}∞

j=t

Et

[∞

∑j=0

βju(ct+j

)]ct = et −ξtpt

ct+j = dt+j

t+j−1

∑k=t

ξk + et+j −ξt+jpt+j for all j > 0


• from first order condition

pt+j = Et

(∞

∑k=t+j

βk−t u′ (ck+1)

u′ (ck )dk+1

)= Et

(∞

∑k=t+j

mt ,k+1dk+1

)

• Note that mt ,k+1 works as a discount factor (but it isstochastic, hence the name)

• check that this implies

pt = Et (mt+1 (pt+1 + dt+1))

• same condition as before


• m also determines the state price density• consider S states indexed by s = 1,2... and corresponding

Arrow-Debreu securities• if the price of the A-D security is pc(s) in state s, then the

price of our asset with pay-off x(s) must be

p =S

∑s=1

pc(s)x(s)

• ”Happy-meal theorem” (Cochrane)


• or

p =S

∑s=1

π (s)pc (s)

π (s)x (s) =

S

∑s=1

π (s)m (s)x (s) = E (mx)

• π(s)m(s) is the state-price density• transformed state-price


• m also determines risk-neutral probabilities• define risk neutral probabilities

π∗ (s)≡ m (s)

E (m)π (s) = Rf m (s)π (s)


p =1

Rf

S

∑s=1

π∗ (s)x (s) =

1Rf E∗ (x)

• where we took expectations under the new probabilities• adjust probabilities to take into account the ”importance” of

the event• basis of risk-neutral pricing


Example: Risk-neutral derivative pricing by binomialtrees

• Used extensively on Wall Street• Idea: all assets are priced by the same pricing kernel, i.e.

same risk-neutral probabilities• you know the value of a derivative conditional on the value

of the underlying• ⇒ the same risk neutral probabilities which price the

underlying must price the derivative• guess the return distribution of the underlying, calculate

implied risk-neutral probabilities, calculate value of thederivative


• Suppose the risk free rate is Rf

• Suppose the current price of a stock is p0 and and inperiod 1 it will be either p1 = xu or p1 = xd < xu

• Price a derivative contract which pays you max(x1−K ,0)where K is a prespecified constant and xd < K < xu

• This is a European call option


• Instead calculate the risk-neutral probabilities(π(u),1−π(u)) which is consistent with the price of thestock from

p =π(u)xu + (1−π(u))xd

Rf

• the price of the call option is

pc =π(u)(xu−K )

Rf


• by increasing the number of nodes in the tree, one canapproximate arbitrary distribution for the price of theunderlying

• starting from the last nodes backward, a computer caneasily solve for the risk-neurtral probabilities and price thederivative

• works for any complicated derivative contract


Classic issues in Finance: The risk free rate

• suppose no uncertainty in consumption growth• suppose CRRA utility:

u(c) =c1−γ

1− γ

• if no uncertainty in consumption growth thenRf = 1/E(m) = 1/Et [

1β

(ct+1ct

)−γ ] implies

Rf =1β

(ct+1

ct)γ

• Risk free rate increases in...


• suppose consumption is lognormally distributed andr ft ≡ lnRf

t , δ ≡− lnβ

• using

E (ez) = eE(z)+ var(z)2

• gives

Rft =

[e−δ e−γEt (∆ lnct+1)+

γ2var(∆ lnct+1)2

]−1


• taking logarithm

r ft = δ + γEt (∆ lnct+1)− γ2var (∆ lnct+1)

2• similar comparative statics plus precautionary savings• can do the argument backwards: consumption growth

varies with the risk free rate• particulars of the CRRA: γ substitution across time, across

states, precautionary savings


Classic issues: Risk correction

• Let us turn to risky assets• Remember

cov(m,x)≡ E(mx)−E(m)E(x)

• Thus, p = E(mx) implies

p = E(m)E(x) + cov(m,x) =E(x)

Rf + cov(m,x)

• present value + risk adjustment


•

p =E(x)

Rf + cov(βu′(ct+1)

u′(ct ),xt+1)

• which asset has a high price?• why? (think about insurance)• why covariance and not variance?


• in terms of returns:• 1 = E(mR i) implies

1 = E(m)E(R i) + cov(m,R i)

E(R i)−Rf = −Rf cov(m,R i)

E(R i)−Rf = −cov(u′(ct+1),R i

t+1)

E [u′(ct+1)]

• expected returns and expected excess returns are centralin finance

• we say ”riskier securities trade for larger expected return”• what ”riskier” means is an important issue


Classic issues: Expected Return-Beta Representation

• write the previous equation as

E(R i) = Rf +cov(m,R i)

var(m)(−var(m)

E(m))

E(R i) = Rf + βi ,mλm

• expected return=risk free rate+quantity of risk x price ofrisk

• β : regression coefficient of asset return on discount factor,asset specific,

• λ is market specific• one-factor model• how to get λ?


• assuming CRRA and taking Taylor approximation (for smallvariance, in the problem set)

•

E(R i) = Rf + βi ,∆cλ∆c

λ∆c = γvar(∆c)

• expected return should increase linearly in consumptiongrowth

• λ depends on risk aversion and volatility of consumption• returns are larger if risk aversion is larger, or environment

is more volatile


Classic issues: idiosyncratic risk does not effect prices

• large variance, large return?• only risk correlated with m gets compensation: if

cov(m,x) = 0 then p = E(x)Rf

• idea: add a little (ξ ) to your portfolio, it has onlysecond-order effect on your consumption stream

σ2(c + ξx) = σ

2(c) + 2ξcov(c,x) + ξ2σ

2(x)

• decompose payoff as

x = proj(x |m) + ε

• no compensation for ε part as E(mε) = 0 by definition• the compensation for x is the same as the compensation

for x = proj(x |m)


Classic issues: Mean-variance frontier

• from

1 = E(

mR i)

= E (m)E(

R i)

+ ρm,R i σ

(R i)

σ (m)

E(

R i)

= Rf −ρm,R iσ(m)

E(m)σ(R i)

• observing that∥∥ρm,R i

∥∥≤ 1 gives∥∥∥E(R i)−Rf∥∥∥≤ σ

(R i)

σ(m)

E(m)

• This is the mean-variance frontier


The Mean-variance frontier


The Mean-variance frontier

1. means and variances of asset returns must lie in awedge-shaped region. The boundary is themean-variance frontier. Maximum return for given level ofvariance.

2. all returns are perfectly correlated with m, ( |ρm,R i |= 1 ).Higher part, maximally risky assets, highest compensation,lower part, best insurance assets, lowest compensation.


3. Any mean-variance efficient return carries all pricinginformation. Given a mean-variance efficient return and therisk free rate, we can find a discount factor that prices allassets and vice-versa (in the problem set)

m = a + bRmv

Rmv = d + em

4. all frontier returns are perfectly correlated with each other,we can synthesize/span any frontier return from two suchreturns.

Rmv = Rf + a(Rm−Rf )


5. there is a single-beta representation using anymean-variance efficient return. Thus, in the beta, expectedreturn space, all assets are along a line. (in the problemset)

E(R i) = Rf + βi ,mv (E(Rmv )−Rf )

6. we can decompose returns into a priced/systematiccomponent and into a residual/idiosyncratic component.The priced part is perfectly correlated with the discountfactor, the residual part generates no expected return.Assets inside the frontier are not worse than assets on thefrontier. You would not want to put your whole portfolio inone ”inefficient” asset, but you might put some wealth insuch assets.


Consumption based model in practice

• let us take the Euler-condition seriously• suppose that the representative agent solves our problem

with utility

u(c) =c1−γ

1− γ

• then m = β (ct+1ct

)γ thus any excess return has to obey

0 = Et [β (ct+1

ct)−γRe

t+1]

• orEt [Re

t+1] =−Rf cov(β (ct+1

ct)−γ ,Re

t+1]


• we have data on aggregate consumption and we have dataon returns. We can calculate the covariance on the lefthand side

• it should work for any asset (stocks, bonds, options)• Does returns line up with any γ?


Something is wrong


Something is wrong

• there is positive correlation• put pricing errors are same order of magnitude as the

spread across portfolios


Alternative approaches

1. Different utility functions• nonseparabilities (stock of durable goods, past

consumption affects marginal utility)• micro data on individual consumption (cross-sectional

variance of income)• (more on this in part 3)


2. Factor pricing models• connect pricing kernel directly to factors which we can

measure better• (more on this in the next part)


3. General equilibrium models• we could close our economy• let us model the decision rule ct = f (yt , it ...)• we can connect pricing kernel to other variables• perhaps, consumption data is bad• The simplest version is the Lucas’s asset-pricing model

(Lucas, 1978)


The Lucas-tree model

• endowment economy:• large number of identical agents with log preferences• only durable good: a set of identical trees (”Lucas-tree”),

one for each person in the economy• at the beginning of each period tree gives a stochastic,

perishable fruit dt (aggregate risk)• fruit follows a Markov process: dt = ds,s = 1, ...S• one share per tree is traded, each agent starts with one tree• representative agent is fine• bonds are in zero-net supply


• problem with ξ shares and ψ bonds

V (ws) = maxξ ,ψ

u (cs) + βE(V(w ′s′))

w ′s′ = ξ (ds′ + ps′) + ψRs

ws = cs + ξps + ψ

• or

V (ws) = maxξ ,ψ

u (ws− (ξps + ψ))+βE (V (x (ds′ + ps′) + ψRs))


• The first order conditions are

u′ (cs)ps = βE(

V ′w ′s (ds′ + ps′))

u′ (cs) = βE(

V ′w ′sRs

)• and the envelope condition is

V ′ws(ws) = u′ (cs)

• (in problem set: verify that value function is logarithmic)


• giving

u′ (cs)ps = βE(u′ (cs′)(ds′ + ps′)

)u′ (cs) = βE

(u′ (cs′)Rs

)• with logarithmic utility and market clearing cs = ds

ps =β

1−βds

• note that price dividend ratio is constant across states


• In finance, it is more common to use the Mehra-Prescott(1985) version (this is the famous equity premium puzzlepaper)

• modify that growth rate of dt follows a markov process• that is dt

dt−1changes across λs,s = 1, ...,S and utility is

CRRA


u′ (dt )pt = βE(u′ (dt+1)(dt+1 + pt+1)

)pt

dt= βE

((dt

dt+1

)γ(dt+1

dt+

pt+1

dt

))pt

dt= βE

((dt

dt+1

)γ dt+1

dt

(1 +

pt+1

dt+1

))pt

dt= β

S

∑s=1

πss′ (λs′)1−γ

(1 +

pt+1

dt+1

)pds = β

S

∑s=1

πss′ (λs′)1−γ (1 + pds′)

• solve for price-dividend ratios in different states


• from price-dividend ratios returns are

Rss′ =ps′ + ds′

ps==

ds′

ds

ps′/ds′ + ds′

ps/ds=

λs′pds′ + 1

pds

• why does this version fit better for calibration?


• In these versions production is not influenced byinvestment

• endowment economy• more sophisticated general equlibrium models include full

blown production functions• ”Production based asset pricing”• Jermann (1998), Boldrin, Chirstiano, and Fisher (2001)

Theory of factor models One factor model: the CAPM A multi-factor model: the ICAPM Tests of factor models: CAPM and ICAPM

Part II

Factor pricing models and the cross-section


Outline of Part II

Theory of factor models

One factor model: the CAPM

A multi-factor model: the ICAPM

Tests of factor models: CAPM and ICAPM


Factor Pricing Models

• consumption might not be enough to explain thecross-section of asset prices

• now we try from the other end• suppose we find which factors do explain the cross-section• under what condition this will be a meaningful exercise

1. Theory2. empirics


Theory of factor models

• Factor pricing models replace pricing kernel (marginalutility growth) with an additive factor structure

mt+1 = a + b′ft+1

βu′(ct )

u′(ct )= a + b′ft+1


Which factors to pick

• Think of pricing kernel as state-prices• we need variables which proxy ”bad states” when agents

value consumption more• returns on broad-based portfolios, interest rates, growth in

GDP, investment, etc.• arriving ”news” about high income in the future means the

state is better today• any variable which forecasts returns, macroeconomic

variables• for many empirical studies, this seems sufficient motivation


• problem: very little discipline• if economic theory does not restrict ”fishing”, we will end

up with factors fit well in one sample, but does not help inother sample

• we do not have so many samples!• In theory, we should

1. GE model: real investment today results in real outputtomorrow

2. determinants of consumption from exogenous variables:ct = f (yt , it ...)

3. determine one list of factors4. prove that the relationship is linear5. all factor models should be derived from consumption

based models


Capital Asset Pricing Model - CAPM

• most widely use and most famous asset pricing model• discount factor linearly determined by the returns on

”wealth portfolio”m = a + bRW

• in the classic case large market index is proxy e.g. S&P500• often written as

E(R i)−Rf = βi ,Rw [E(Rw )−Rf ]

≈ cov(R i ,RM)

var(RM)[E(RM)−Rf ]


Derivation I: consumption in one-period, exponentialutility, normal distributions

• u(c) =−e−αc

• problem

maxc

E(u(c)) = maxc

[−e−αE(c)+(α2/2)var(c)

]c = y f Rf + y’R

W = y f + y’1

• gives

y = Σ−1E(R)−Rf

α


• orE(R)−Rf = αΣy = αcov(R,RW )

• total risky portfolio is y’R, Σy covariance with investor’srisky portfolio, and overall portfolio

• representative investor implies overall portfolio is marketportfolio

• market portfolio is wealth portfolio if all wealth can beinvested in stock market

• applying to market return: market price of risk depends onrisk aversion

E(RW )−Rf = αvar(RW )

• express alpha, substitute back to get the CAPM equation


Derivation II: infinite periods, quadratic utility

• i.i.d returns• first start with general utility function. Bellman:

V (Wt ) = maxc

u(ct ) + βEt [V (Wt+1)]

s.t .Wt+1 = RWt+1(Wt −ct )

RWt = w’R

w’1 = 1



u′(ct ) = βEt [V ′(Wt+1)R i ]

• or, using the envelope theorem for u′(ct ) = V ′(Wt )

mt+1 = βV ′(Wt + 1)

u′(ct )= β

V ′(Wt + 1)

V ′(Wt )


• easy to check that if the value function is quadraticV (Wt+1) =−ν/2(Wt+1−W ∗)2 then

mt+1 =−βνRW

t+1(Wt −ct )−W ∗

u′(ct )=

=βνW ∗

u′(ct )+−βν(Wt −ct )

u′(ct )RW

t+1 = at + btRWt+1

• given the information in period t , at ,bt are constants• CAPM• only if value function is quadratic


• turns out that the assumption u(ct ) =−12(ct −c∗)2 implies

a quadratic value function• guess that the value function has the previous form• show that the optimal consumption is

ct =c∗−βν(E(RW

t+1)W ∗− (E([RWt+1]2)Wt ))

1 + βνE([RWt+1]2)

• linear in Wt

• substitute in for value function and validate that the guesswas right

• we established that quadratic utility with i.i.d returnsimplies CAPM

• where did we use the i.i.d returns?


Derivation III: any utility with one state variable,linearized

• if only state variable is Wt and there is a solution thenct = g(Wt ) and

mt+1 = βV ′(Wt+1)

u′(ct )= β

V ′(RWt+1(Wt −g(Wt )))

u′(g(Wt ))

= G(RWt+1)≈ at + btRW

t+1

• you might want to do the Taylor-expansion around theconditional mean of the factor

mt+1 ≈G(E(RWt+1)) + G′(E(RW

t+1))(RWt+1−E(RW

t+1))


A multi-factor model: the ICAPM• how to validate a multifactor model?• suppose that marginal utility of wealth depends on other

variable(s) zt

• then we can write ct = g(Wt ,zt ) and

mt+1 = βV ′(Wt+1,zt+1)

u′(ct )= β

V ′(RWt+1(Wt −g(Wt ,zt )),zt+1)

u′(g(Wt ,zt ))

= G(RWt+1,zt+1)≈ at + bW

t RWt+1 + bz

t zt+1

• any variables which might affect the marginal utility ofwealth can be included

• Fama: ”fishing license”• not that much, e.g. if you argue that a state variable is

there to forecast investment-opportunity sets, they betterdo


Empirical evidence on the cross-section I: the CAPM

• Early tests (Litner 1965)• calculate betas of individual stocks βi = cov(RM ,Ri )

var(RM )

• regress betas to average returns• not too much success: too much dispersion, slope too flat,

crosses the y-axis at an implausible risk-free rate

• problem (Miller-Scholes 1972) betas are measured witherrors

• portfolios have lower residual variance + individual stockschange as the business changes: portfolio betas are bettermeasured

• variance lower, you see better the differences of expectedreturns among portfolios with different characteristics


• Fama-MacBeth(1973)• found individual betas• grouping stocks into portfolios according to their betas (for

the whole period)• regress cross-sectional regression of returns on betas in

each year• slope estimates: average slope coefficients over the whole

period• standard error of estimates: standard error across years

(with appropriate weights)• every strategy with high average return turned out to have

high betas• every strategy without a high beta would not have high

average returns


Test of the CAPM


Empirical evidence on the cross-section II: Multi-factormodels

• as we saw, you do not need too much to have multiplefactors

• if anything else then the aggregate market influences themarginal value of a dollar: you are there


• Since Fama-MacBeth, almost all asset pricing tests followthe same steps:

1. find a characteristic that might be associated with averagereturns

2. sort stocks into portfolios based on characteristics, checkwhether there is spread in average returns (”worry hereabout measurement, survival bias, fishing bias and all theother things that can ruin a pretty picture out of sample”)

3. compute betas for the portfolios, check whether averagereturn spread is accounted for by the spread in betas

4. if not: anomaly, consider multiple betas

• Econometrics would suggest: use characteristics asinstruments instead of portfolio grouping. Then you couldinvestigate individual portfolios.


• Fama-French (1993,1996) 3 factors• value stocks:

• market values are small compared to accountant’s bookvalue (essentially track past investments)

• large average returns• opposite: growth stocks

• large/small stocks• if CAPM were right, value stocks and small stocks would

have high betas• form 25 portfolios based on these two characteristics:

Fama-French 25 portfolios


Fama-French 25 portfolios and the CAPM






• once in each year sort stocks into two size groups (S/B)and three B/M groups (H,M,L)

• in each month calculate factor SMB by calculating theaverage return of small minus average return of big groupfor each period

• calculate factor HML by calculating the average return ofhigh B/M minus average return of low B/M group for eachperiod

• form a 2x3 (3x3, 5x5) matrix by characteristics• regress each portfolio on market, HML, and SMB


Fama-French 25 and the three factor model


What is the theory behind F-F?

• To make sense of the results, we should understand thereal, macroeconomic, aggregate, nondiversifiable risk thatis proxied by the returns of the HML and SMB portfolios

• F-F argues ”distress risk”: value firms experienced a seriesof bad news, near financial distress, in liquidity crunchthese firms would do badly, and these are the times whenit hurts the investor the most. Thus, investors requireexcess returns to hold these stocks.

• not much evidence that the HML portfolio would beassociated with other measures of distress(Lakonishok-Shleifer-Vishny, 1993)

• Heaton and Lucas (1997): typical stock holder is aproprietor of a small business, the same events which hurtvalue firms hurt them as well, they need excess returns


• you might think of the result as characterizing thecomovement of returns:

• just the fact that small firms have higher returns does notimply that small prices of small firms move together

• But they do.• finding three portfolios which characterize all the

comovement, implies that they describe returns• otherwise you would have arbitrage (APT).

• 3-factor model explains other strategies not explained byCAPM: other price multiples (P/E) and five-year salesgrowth


Macroeconomic factors

• test whether stock performance during badmacroeconomic times determines average returns

• Jagannathan and Wang (1996): labor income• Chen, Roll, Ross(1986): industrial production, inflation• Cochrane (1996): investment growth• calculating betas (covariance) with these macro variables• expected returns line up• value and size portfolios are not that well explained as F-F• but much easier to motivate• Lettau and Ludvigson(2001): scaling consumption with a

proxy for total wealth, conditional CCAPM, works as well asF-F‘


Reversal and momentum

• reversal strategy:• from a portfolio of losers and winners based on years -5 to

-1• sell winners buy losers• excess return not explained by CAPM• but explained by F-F (HML picks it up)


Reversal and momentum

• momentum strategy:• from a portfolio of losers and winners based on the last six

month (or year)• buy the winners, sell the losers• excess return not explained the CAPM• neither by F-F (momentum stocks are negatively correlated

with value stocks)• related to small autocorrelation on short-horizont• small predictability times large past return = large expected

return• might be associated with large transaction cost• widely popular• could be a forth risk-factor but it is very hard to see what

should be the risk


Performance of the F-F factors and momentum

-200

-100

0

100

200

300

400

500

600

700

800

900

192701

192811

193009

193207

193405

193603

193801

193911

194109

194307

194505

194703

194901

195011

195209

195407

195605

195803

196001

196111

196309

196507

196705

196903

197101

197211

197409

197607

197805

198003

198201

198311

198509

198707

198905

199103

199301

199411

199609

199807

200005

200203

200401

200511

200709

mkt

smb

hml

momentum

Documents

Financial Economics - Block I: Consumption based asset pricing