topic8

F500: Empirical Finance

Oliver [email protected]

March 11, 2014

Oliver Linton [email protected] () F500: Empirical Finance March 11, 2014 1 / 44

Topic Eight: Intertemporal Equilibrium Pricing

Outline:

1 The Stochastic Discount Factor2 The Consumption Capital Asset Pricing Model3 The Equity Premium Puzzle4 Explanations for the Puzzle5 Other approaches


The Stochastic Discount Factor

In the standard framework, investors have a concave, positively sloped,time-invariant utility function for consumption, and a constant rate of timepreference . They invest in risky assets fRi ,tgni=1 and consume theproceeds over time fCtgInvestors choose investment/consumption to maximize the discountedexpected utility of lifetime consumption

Vt = Et

"

j=0

jU (Ct+j )

#= U (Ct ) +

cont . valuez }| {EtVt+1

subject to a budget constraint

Wt+1 = (Wt Ct )N

i=1wit (1+ Ri ,t+1)


First order condition for each risky asset i , the so-called Euler equation

U 0 (Ct ) = Et(1+ Ri ,t+1)U 0 (Ct+1)

,

where Et means expectational on information at time t.Dening

Mt+1 = U 0(Ct+1)U 0(Ct )

and rearranging the series of rst-order conditions (i = 1, . . . , n)

1 = Et [(1+ Ri ,t+1)Mt+1]


The random variable Mt > 0 is called the stochastic discount factororpricing kernel. It is the (random) ratio of marginal utilities between eachinvestmentdate-state and realized returndate-state, weighted bypure time preference.Pricing formula for any asset

Pt = Et [Mt+1Xt+1] ,

where Xt+1 is the cash ow in period t + 1 (e.g., Pt+1 +Dt+1)


Relationship can be derived more generally from non-arbitrage assumption:There does not exist a negative-cost portfolio with a uniformlynon-negative payo.

Given non-arbitrage, by the separating hyperplane theorem there exists astrictly positive random variable Mt+1 such that for all i

Et [(1+ Rit+1)Mt+1] = 1

Note that the stochastic discount factor of any investor is an allowablenon-arbitrage pricing kernel.


Girsanovs TheoremIn mathematics, a kernel is the positive weighting function inside of aweighted integral, e.g., the probability weights in the expectation integralE [X ] = xP(x)Replace the true probability weights in the basic pricing expectation withhypotheticalprobability weights (nite state notation)

P(x) Mt (x)P(x),

and taking expectations under these transformed probabilities gives

Et [Xt+1Mt+1] = E t [Xt+1]

All assets have the same expected return under the transformedprobabilities. This new hypothetical "probabilitymeasure is called theequivalent martingale measureor the risk neutral measure. It is very usefulfor empirical derivatives pricing (not covered in this course).


The Consumption Capital Asset Pricing Model Again

It is often convenient to work with the unconditional expectation

1 = E [(1+ Rit )Mt ]

= E [(1+ Rit )]E [Mt ] + cov (Rit ,Mt )

Let R0t denote an asset such that

cov(R0t ,Mt ) = 0

(zero beta or risk free asset). Then

E [Mt ] =1

E [1+ R0t ]= E

U 0(Ct )U 0(Ct1)


Then substituting in and rearranging we obtain for any asset i

E [Rit R0t ] = cov(Rit ,Mt ) E (1+ R0t ) = covRit ,

u0(Ct )u0(Ct1)

An asset whose covariance with Mt is small tends to have low returnswhen the investors marginal utility of consumption is high ie whenconsumption is low. Require a large risk premium to hold it.


Suppose there is an asset Rmt that pays o exactly Mt then

E [Rmt R0t ] = var(Mt ) E (1+ R0t )

Therefore,

E [Rit R0t ] = imE [Rmt R0t ]

im =cov(Rit ,Rmt )var(Rmt )

This pricing model is called the consumption CAPM.This pricing model applies conditionally or unconditionally

Es [Rit R0t ] = covs (Rit ,Rmt )vars (Rmt ) Es [Rmt R0t ], all s < t


Need to specify U(.) in order to estimate ic from consumption data.Simple, elegant utility function is the CRRA class with risk aversionparameter

U (Ct ) =C 1t 11

Calculating the stochastic discount factor gives

Mt+1 = Ct+1Ct

mt+1 = logMt+1 = log gt+1 ; gt+1 = log (Ct+1/Ct )

1+ Rft =1

Ct+1Ct


Note that the CCAPM model has:

cross-sectional predictions (relative risk premia are proportional toconsumption betas),

time-series predictions (expected returns vary with expectedconsumption growth rates, etc.),

joint time-series/cross-sectional predictions.

The standard CAPM only has cross-sectional predictions.


It is straightforward to test the consumption CAPM. Hansen andSingleton (1982) GMM conditional moment restriction

Et

"(1+ Rit+1)

Ct+1Ct

1#= 0

Do not need to specify dynamics for returns or consumption exceptstationarity on consumption growth.Convert to unconditional moment restriction and do GMM


Let with Xt denoting all the data and = (,)

g(Xt , ) =

2640B@ 1+ R1,t+1...1+ Rn,t+1

1CA Ct+1Ct

1

375 instrumentsz}|{Zt 2 RpThen

E [g(Xt , )] = 0.

Estimate the parameters by the Generalized Method of Moments(GMM) using p > 2 sample moments

GT () =1T

T

t=1g(Xt , ).

This is nonlinear in .


Test whether overidentifying restrictions (p > 2) hold using the J-test.

jjGT (b)jjWoptThis is asymptotically chi squared under the null hypothesis that themoments are correct.Empirically it performs very poorly, at least in its standard formulation.The empirical failure of the consumption CAPM is among the mostimportant anomalies of asset pricing theory.


The Equity Premium Puzzle

We make an assumption that log consumption growth and log equitymarket return are jointly normal(can hold a little more generally in anapproximate sense like the Campbell log linearization) and that utility isCRRA.Then letting rit be logarithmic returns and ct = logCt we have the linear(in parameters) equation

0 = Et [ri ,t+1] + log Et [ct+1] +122i +

22c 2ic,

whereic = covt (ri ,t+1 Et (ri ,t+1), ct+1 Et (ct+1))

2i = vart (ri ,t+1) and 2c = vart (ct+1)


The risk-free rate is determined endogenously in the model (setting2i = ic = 0)

Et [rft ] = log + g 22c2

where g is the mean growth rate of consumptionFor any other asset i we have

Et [ri ,t+1 rf ,t+1] = ic 2i

2 ic

This is a pricing equation for the risk premium in terms of covariation withconsumption growth


The Equity Premium Puzzle. Empirically, ic is very small relative tothe observed premium of equities over xed income securities, hence thisimplies a very high coe cient of risk aversion .The Risk Free Rate Puzzle.If is set high enough to explain observedequity risk premia, it is too high (given average consumption growth) toexplain observed risk-free returns! The rate of pure time preference isdriven below zero.


Mehra and Prescott (1985).

"Historically the average return on equity has far exceededthe average return on short-term virtually default-free debt. Overthe ninety-year period 1889-1978 the average real annual yield onthe Standard and Poor 500 Index was seven percent, while theaverage yield on short-term debt was less than one percent. Thequestion addressed in this paper is whether this large dierentialin average yields can be accounted for by models that abstractfrom transactions costs, liquidity constraints and other frictionsabsent in the Arrow-Debreu set-up. Our nding is that it cannotbe, at least not for the class of economies considered. "


Explanations for the Puzzle

A large number of explanations for the puzzle have been proposed. Theseinclude:

a contention that the equity premium does not exist: that the puzzleis a statistical illusion

modications to the assumed preferences of investors, and

imperfections in the model of risk aversion.


Statistical Illusion

The most basic explanation is that there is no puzzle to explain: that thereis no equity premium. Essentially, we dont have enough statistical powerto distinguish the equity premium from zero.Sample selection bias: US equity market is the most intensively studies inequity market research. Not coincidentally, it had the best equity marketperformance in the 20th century; others (e.g. Russia, Germany, andChina) produced a gross return of zero due to bankruptcy events.Low number of data points: the period 19002005 provides only 105independent years which is not a large number of years statistically.Sample period choice: returns of equities (and relative returns) varygreatly depending on which points are included. Using data starting fromthe top of the market in 1929 or starting from the bottom of the market in1932 (leading to estimates of equity premium of 1% lower per year), orending at the top in 2000 (vs. bottom in 2002) or top in 2007 (vs. bottomin 2009 or beyond) completely change the overall conclusion.


Market frictions

If some or all assets cannot be sold short by some or all investors, then thestochastic discount factor equation is much weaker:

E [(1+ Rit )Mt ] 1

The inequality-version of the stochastic discount factor does not aggregateacross investors. Hence aggregate consumption is not directly relevant.


Separating Risk Aversion and Intertemporal Substitution

The standard multiperiod von Neumann-Morgenstern utility function iselegant but may not provide an accurate representation.

Multiperiod von Neumann-Morgenstern utility has a single parameter (therisk aversion parameter) which governs both the elasticity of intertemporalsubstitution and Arrow-Pratt risk aversion.


Elasticity of intertemporal substitution

Consider an investor who consumes/saves in period zero and consumes inperiod one with no risk. Suppose that the risk-free interest rate is Rf .The elasticity of intertemporal substitution is dened as the percentagechange in optimal consumption growth for a percentage change in therisk-free interest rate:

EIS =%(C1/C0)%(Rf )

In the CRRA case it is easy to show EIS = 1 .

Note that EIS is an intertemporal concept with no connection to risk.


Arrow-Pratt Risk Aversion

Consider the family of risky investments x() = pi + 2z , where pi is aconstant and z is a unit-variance, zero-mean random variable. Note thatfor any > 0 the mean/variance ratio of this investment equals pi.Let C denote a riskless consumption level and U(C ) a vN-M utilityfunction. The absolute risk aversion of U(C ) is the value of pi whichleaves the investor approximately indierent for small values of

ARA = fpi s.t. lim!0

E [U(C + x ())] = U(C )g

It is easy to show that

ARA = U00(C )

2U 0(C ).

It is not di cult to show that if ARA is constant for all C thenU(C ) = exp(C ) for some .


The relative risk aversion is the ARA divided by the level of consumption:

RRA = ARA/C

It is not di cult to show that if RRA is constant for all C then (choosinga convenient scaling for the utility function) U(C ) = 11C

1 which isthe CRRA utility function.

Given CRRA then RRA = for all C .

Note that RRA is a pure risk concept with no intertemporal componentwhereas EIR is a pure intertemporal concept with no risk component. Inthe multiperiod vN-M framework they are inextricably linked together.


The Epstein-Zin-Weil "utility" functionSeparates EIS and RRA, it is dened recursively by

Ut =(1 )C

1

t + EthU1t+1

i 1

1

where is discount factor, is coe cient of relative risk aversion is theelasticity of intertemporal substitution

=1 1 1

The rst-order conditions are more complex than in the vN-M case, butone useful series of rst-order conditions is

E t

24(Ct+1Ct

1) 1(1+ Rm,t+1)

1(1+ Ri ,t+1)

35 = 1where Rm,t is the return on the market portfolio.


Assuming that consumption and the return on the market portfolio arejointly lognormal, and substituting gives

r f ,t+1 = log + 12 2m

222c +

1Et [ct+1]

Et [ri ,t+1] rf ,t+1 = ic + (1 ) im

2i

2

Consumption betas and market portfolio betas both aect asset riskpremia.


Eliminate Consumption. Applying Campbells log-linearisation from aprevious lecture

Et [ri ,t+1] rf ,t+1 = im + ( 1) ih 2i

2

ih = covt

ri ,t+1,

j=1

j fEt+1rm,t+1+j Et rm,t+1+jg!

Risk premia depend on market betas and on "changing opportunity setbetas.Covariation with news about future returns to the market aects riskpremia.This two-beta representation also can be expanded into a k-beta model ifthe market return is generated by a k-vector VAR process


Vector AutoregressionsLet

Xt = (rmt , ct , ipt , yieldt , etc ., ...)

where the rst element is the market return and the other are observablevariates interacting with consumption via a linear VAR process

Xt+1 = AXt + et+1,

where A = (aij ) is a parameter matrix and et+1 is an error vector withE (et+1jXt , . . .) = 0. Let e|1 = (1, 0, . . . , 0), then

rm,t+1 = e|1Xt+1 = e

|1AXt + e

|1et+1

We can forecast the future by

EtXt+1 = AXt =) Et rm,t+1 = e|1AXt

EtXt+j = AjXt =) Et rm,t+j = e|1AjXtOliver Linton [email protected] () F500: Empirical Finance March 11, 2014 30 / 44

The factor betas are nonlinear combinations of the VAR coe cients andthe extra-market factors are the VAR innovations. Let

ik = cov(ri ,t+1, ek ,t+1)

Inserting this into the above equation gives:

E t [ri ,t+1] r f ,t+1 = 2i

2+ i1 + ( 1)

K

k=1

kik

which (except for the log expectation adjustment) is identical to themulti-beta pricing models tested previously


Other Asset Pricing ApproachesHabits models. Dierence and Ratio models. Dierence model(Constantinides (1990))

Ut = Et

j=0

j(Ct+j Xt+j )1 1

1

Ratio model (Abel (1990))

Ut = Et

j=0

j(Ct+j/Xt+j )

1 11

Habit Xt , for example Xt some level of previous consumption. Givesadditional exibility, but not very plausible.Hyperbolic discounting (Laibson (1996))

U(Ct ) + Et

j=0

jU (Ct+j )


Lettau and Ludvigson (2001,2004)

Standard dynamic optimization with wealth W . Let rw be the log of netreturn on aggregate wealth. By linearization and solving forward obtain

ct wt =

i=1

iwEt (rw ,t+i ct+i )

LL assume that wealth is composed of asset holdings A and human capitalH and that H is related to labor income Yt in a specic wayApproximating the nonstationary component of human capital byaggregate labour income, they obtain

caytz }| {ct aat y yt

=

i=1

iwEt ((1 )ra,t+i + yt+i ct+i )


The prediction is that consumption, asset values and income arecointegrated. Their residual summarizes expectations of futurereturns on the market portfolio.

Using U.S. quarterly stock market data, they nd that uctuations inthe consumptionwealth ratio (cay) are strong predictors of both realstock returns and excess returns over a Treasury bill rate.

They nd that this variable is a better forecaster of future returns atshort and intermediate horizons than is the dividend yield, thedividend payout ratio, and several other popular forecasting variables.


Long run risks model Bansal and Yaron (2004)

Epstein Zin preferences

E t

24(Ct+1Ct

1) 1(1+ Ra,t+1)

1(1+ Ri ,t+1)

35 = 1where Ra is the gross return on an asset that delivers aggregateconsumption as its dividend each period (like, but not equal to, the marketportfolio). In logs with gt+1 = logCt+1/Ct and zt = log(Pt/Ct ), whereP is the price level

ra,t+1 = 0 + 1zt+1 zt + gt

mt+1 = log gt+1 + ( 1)ra,t+1


They specify dynamics for consumption and dividend growth rates

gt+1 = + xt + tt+1

gd ,t+1 = d + xt + dtut+1

where the unobserved state variables (x is the "Long Run Risks")

xt+1 = xt + etet+1

2t+1 = 2 + 1(

2t 2) + wwt+1

innovations et+1,wt+1, t+1, ut+1 are standard normal and iid andmutually independent. The parameter 1


Captures the idea that news about growth rates and economicuncertainty (i.e., consumption volatility) alters perceptions regardinglong-term expected growth rates and economic uncertainty

Asset prices will be fairly sensitive to small growth rate andconsumption volatility news.

Log linearizing, they obtain


Innovation to the pricing kernel

mt+1 Etmt+1 = m,consumption shockz }| {

tt+1 m,eLRR shockz }| {tet+1 m,ww

shock to volz}|{wt+1

Equity premium

Et (rm,t+1 rf ,t+1) + 12vart (rm,t+1) = m,em,e2t + m,wm,w

2w

vart (rm,t+1) = (2m,e + 2d )

2t +

2m,w

2w

Consumption shock is not pricedRisk return relationship

Et (rm,t+1 rf ,t+1) = 0 + 1vart (rm,t+1)


Bansal et al. show thatI The model is capable of justifying the observed magnitudes of theequity premium, the risk-free rate, and the volatility of the marketreturn and the dividend-yield.

I It captures the volatility feedback eect, that is, the negativecorrelation between return news and return volatility news.

I As in the data, dividend yields predict future returns and the volatilityof returns is time-varying.

I At plausible values for the preference parameters (IES and RRA), areduction in economic uncertainty or better long-run growth prospectsleads to a rise in the wealthconsumption and the pricedividend ratios.There is a signicant negative correlation between pricedividend ratiosand consumption volatility.

I They show that about half of the variability in equity prices is due touctuations in expected growth rates, and the remainder is due touctuations in the cost of capital.


Is there an equity premium puzzle in the USA?3-month tbill rate annualized


S&P500 daily total return annualized


Annualized moments of daily returns, log returns, and tbills 1954-2012

med IQR/1.3 (1)(1950-) log returns 0.0309 0.0504 0.0675 0.0484 0.02933(1950-) returns 0.0832 0.1160 0.1549 0.1114 0.02883(1954-) tbill 0.0475 0.0466 0.0019 0.0016 0.99940(1954-) Risk premium 0.0254 0.0631 0.1568 0.1136 0.02524

The correlation between returns and log returns is 0.9997, but see howdierent the mean and standard deviations are over this periodThe correlation between returns and the tbill rate is -0.014 so almostinsignicant.


Bloomberg. Bonds Beat Stocks in Earth-ShatteringReversal: Chart ofDay


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