Ec2723, Fall 2013: Assignment 3

Due in class, Tuesday December 3

1. A representative agent has Epstein-Zin preferences, where � is the discount factor, risk aversion, the elasticity of intertemporal substitution, and � = (1� )(1�1= ). De�ne� = exp(�r�); where r� is the pure rate of time preference. Suppose the consumption growthlog(Ct+1=Ct) of the representative agent is i.i.d. Recall that, under the budget constraintand regularity conditions, the Euler equation is given by

1 = Et

24(exp(�r�)�Ct+1Ct

�� 1

)� �1

1 +Rw;t+1

�1��(1 +Ri;t+1)

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where 1 +Rw;t+1 is the gross simple return on the portfolio of all invested wealth.

a) Consider an asset paying a dividend stream Dt = C�t for some constant �. Iteratingthe Euler equation forward and imposing a standard no-bubble condition, show that thepresent value formula for the asset price Pt is

Pt = Et

1Xj=1

exp(�r��j)�Ct+jCt

�� � �

1

1 +Rw;t+j

�1��C�t+j ;

where (1 +Rw;t+j) is the j-period gross simple return on the wealth portfolio.

b) Following a �guess and verify� approach, we suppose that the consumption-wealthratio C=W is a constant. Denote exp(�) = 1 + C=W . Express 1 + Rw;t+j only in terms ofconsumption and the constant �.

For the random variable X = log(Ct+j=Ct), de�ne the cumulant-generating function

c(#) = log E exp(#X) ;

for all # for which the expectations are �nite. (Note: Do not confuse the preference parameter� with the argument # of the cumulant-generating function.) Use c(�) when necessary in thefollowing steps.

c) Based on your answers in parts a) and b), derive the log dividend-price ratio d=p =log(1 +Dt=Pt) in terms of � and model parameters r�, �, , , and �.

d) Solve for the constant � in terms of model parameters r�, �, , and . (Hint: considerthe case with � = 1.)

e) Based on your answers in part c) and d), derive the log riskfree rate rf , in terms ofmodel parameters r�, �, , and . (Hint: consider the case with � = 0.)

f) Finally, derive the log of the expected gross return, er = log(1 + ERi;t+1), as well asthe log risk premium rp = er � rf , in terms of model parameters r�, �, , , and �. (Hint:We know that the price-dividend ratio is a constant.)

g) Discuss which of your answers to previous parts are a¤ected by the use of Epstein-Zinpreferences rather than power utility.

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2. Consider the following model for the stochastic discount factor, Mt+1:

mt+1 � log(Mt+1);

�mt+1 = xt + ��t+1;

xt+1 = �xt + �t+1;

where �t+1 is randomly drawn from one of two distributions. With probability �, state 1occurs at time t+ 1 and �t+1 = �1;t+1, where �1;t+1 � N (0; �21); with probability 1� �, state2 occurs at time t+ 1 and �t+1 = �2;t+1, where �2;t+1 � N (0; �22).a) Is the stochastic discount factorMt+1 lognormally distributed, conditional on informa-

tion available at time t? Is it lognormally distributed, conditional on information availableat time t and knowledge of the state (1 or 2) that occurs at time t+ 1?

b) Use the formula for the one-period zero-coupon bond yield,

1 + Y1t =1

EtMt+1

;

to solve for the one-period bond yield in this economy. Show that the log yield, y1t �log(1 + Y1t), is linear in the state variable xt.

c) Use the recursive equation for bond prices,

Pnt = Et[Pn�1;t+1Mt+1];

where Pnt is the price of an n-period zero-coupon bond at time t, to show that all log bondyields are linear in xt. Derive an expression for the slope coe¢ cient relating n times the logbond yield of maturity n to the state variable xt. What is the standard name for modelswith log bond yields linear in state variables? What makes this model di¤erent from othermodels you have seen with this property?

d) Which of the following phenomena are displayed by this model? Explain.

(i) Time-varying risk premia in the term structure of interest rates.

(ii) Changing volatility of interest rates.

(iii) Excess kurtosis of interest rate movements.

(iv) Imperfect conditional correlation of bond returns.

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3. Consider an in�nite-lived investor with Epstein-Zin preferences allocating a portfoliobetween a risky asset, with log return rt+1, and a short-term safe asset, with log returnrf;t+1. The risky asset return is lognormally distributed, with a constant risk premium overthe short-term safe asset. If the asset returns and the investor�s consumption are jointlyconditionally lognormal, Epstein and Zin have shown that the �rst-order conditions for theinvestor�s optimization problem can be written as

Etrt+1 � rf;t+1 +�2t2= �

Covt(rt+1;�ct+1)

+ (1� �)Covt(rt+1;rp;t+1);

where �2t is the conditional variance of the risky asset return, �ct+1 is the change in theinvestor�s log consumption, and rp;t+1 is the log return on the investor�s portfolio. Theparameters de�ning preferences are the coe¢ cient of relative risk aversion , the elasticityof intertemporal substitution in consumption , and � = (1� )=(1� 1= ):Using a loglinear approximation to the investor�s intertemporal budget constraint, the

innovation in log consumption can be written as

ct+1 � Etct+1 = rp;t+1 � Etrp;t+1 + (1� )(Et+1 � Et)1Xj=1

�jrp;t+1+j;

where � � 1� expfE(ct � wt)g is a parameter of loglinearization.a) Give an intuitive explanation of this expression for the innovation in log consumption.

b) Use this expression to rewrite the investor�s �rst-order condition so that consumptiondoes not appear. Show that the parameter drops out of the �rst-order condition when itis rewritten in this way.

c) Write �t for the share of the risky asset in the portfolio. Derive an equation for �t.[Hint: Relate the conditional covariance Covt(rt+1;rp;t+1) to �t and �2t .] Show that �t = �,a constant, and show that � has two components that can be interpreted as myopic demandand intertemporal hedging demand respectively.

d) A real perpetuity or consol bond pays one real dollar each period forever. Using aloglinear approximation, the return on this bond can be written as

rc;t+1 = rf;t+1 + �c � (Et+1 � Et)1Xj=1

�jcrf;t+1+j;

where �c captures a constant risk premium on the consol, �c � 1� expfE(�pc;t)g, and pc;t isthe log price of the consol including its current coupon. Use this expression to show that ifthe risky asset is a consol, then an investor who is in�nitely risk-averse and in�nitely averseto intertemporal substitution will invest all her wealth in the consol. Explain the economicintuition behind this result, and explain why it requires in�nite aversion to intertemporalsubstitution.

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4. Consider the risksharing problem of two in�nitely lived agents, who receive randomshares of a �xed endowment e. Each period there are two states of the economy. In state1, agent 1 receives e=2 + k and agent 2 receives e=2 � k, while in state 2 agent 1 receivese=2 � k and agent 2 receives e=2 + k. The conditional probabilities of the two states areconstant and equal. Each agent maximizes a discounted sum of expected period utilities,with time discount factor �. Period utility is given by

u(c) = ec� �ec2;where ec � c� e=2 and c is the level of consumption during the period.

Following Alvarez and Jermann (Econometrica 2000) assume that the only punishmentfor default is permanent exclusion from the �nancial market.

a) Calculate an agent�s expected utility under perfect risksharing.

b) Calculate an agent�s expected utility under autarchy, if the state is initially bad andif the state is initially good.

c) Derive a condition under which no risksharing is possible.

d) Show that this condition is equivalent to �high implied interest rates�in autarchy asde�ned by Alvarez and Jermann.

e) Characterize the conditions under which partial risksharing, but not full risksharing,is possible.

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5. Following Athanasoulis and Shiller (Review of Financial Studies 2000), consider asingle-period model with two agents. Agent 1 has income x1 = c+ v + ", while agent 2 hasincome x2 = c+ v � ". The term c is deterministic, while v and " are normally distributedshocks with mean zero. The variance of the common income shock v is 1, while the varianceof the relative income shock " is a. Agent 1 has constant absolute risk aversion coe¢ cient 1, while agent 2 has constant absolute risk aversion coe¢ cient 2.

A single futures contract exists in this economy. The payo¤ on the contract is �1v+ �2",and it costs p payable next period. Normalize the variance of the payo¤ to 1. (Note: thisimposes restrictions on �1 and �2.) The contract is in zero net supply.

a) Given the price p, solve for the quantities of the futures contract q1 and q2 that eachagent wishes to hold.

b) Solve for the price p that must hold in equilibrium given that the futures contract isin zero net supply.

c) Write down an expression for the maximized utility of each agent. Add the maximizedutility of agent 1 to the maximized utility of agent 2 to get an expression for social welfare.

d) Now assume that 1 = 2. Show that a futures contract designer who wishes tomaximize social welfare will set �1 = 0. Interpret this result.

e) Now assume that 1 = 0 while 2 > 0. Show that the welfare-maximizing contractdesigner will set �2 = ��1. Interpret this result.f) Now assume that 2 = 0 while 1 > 0. Show that the welfare-maximizing contract

designer will set �2 = �1. Interpret this result.

g) What assumptions would be needed for the welfare-maximizing contract to have �2 =0? Explain.

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6. This question asks you to use Generalized Method of Moments (GMM) to analyze thetwo-beta asset pricing model of Campbell and Vuolteenaho (2004). It is not necessary touse a powerful programming language or statistics package to work with the data. MicrosoftExcel is su¢ cient (but Matlab would make life easier). If you plan to use Excel, �rst readthe help �les on matrix functions (MMULT, MDETERM, MMULT, TRANSPOSE), how tode�ne matrixes with CTRL+SHIFT+ENTER, etc.

Download the �le problem set 3 data.xls from the course web site. The �le has thefollowing columns: <date>, <excess return on the market over the risk-free asset, RMt �Rft>, <expected log excess market return known at time t � 1, Et�1(reMt)>, <minus themarket�s discount-rate news, �NDR;t>, <the market�s cash-�ow news, NCF;t>, <net risk-free rate (for quarter t, known at t-1), Rft>, and <net simple test-asset returns on 9 of the25 Fama-French ME-BE/ME portfolios, Ri;t�1>.

a) Estimate the parameters of a linear stochastic discount factor, speci�ed as

Mt = a+ b(�NDR;t) + cNCF;t

using GMM. Use ten moment conditions (one for the risk-free asset and nine for the stockportfolios) of the form 0 = E[Mt(1 + Rit) � 1]. Use the identity weighting matrix W =I. Clearly write down the expression that you are minimizing. What are the estimatedparameter values?

b) Next, generate and report the covariance matrix of the parameter estimates. Useformulas in Cochrane, Asset Pricing, that take into account the fact that you used a pre-speci�ed (identity) weighting matrix. Clearly write out the formulas you use in your answer.Also report the z-statistics and p-values for the hypothesis that the coe¢ cient is zero foreach coe¢ cient individually.

c) Suppose a slightly modi�ed version of the �rst-order condition in the paper �Bad Beta,Good Beta�holds:

E(Rit �Rft) = Cov[(Rit �Rft); NCF;t] + Cov[(Rit �Rft);�NDR;t]:

What restriction does this �rst-order condition impose on the parameters of the linear SDFspeci�cation? (Use the fact that the means of both NDR;t and NCF;t are zero.) Test thisrestriction. Is it rejected?

d) Again, suppose that the �rst-order condition in part c) holds. Based on the estimatedparameter vector [a; b; c]0 and its covariance matrix you produced above, give an estimate of and the standard error of this estimate. (Use the delta method.)

e) Finally, use your S estimate to produce the asymptotically optimal weighting matrixspeci�ed as W = S�1. Use this weighting matrix to get optimal parameter estimates andtheir covariance matrix. Compare the parameter estimates and covariance matrix to thoseproduced in part b). Perform the J-test of overidentifying restrictions. What is the p-valueof the J-test?

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