unwillingness to hold stocks even in the lace of a large equity premium. Bonomo and Garcia (1993) ohlain similar results in a consumption-based model with loss aversion.

In related work, Epstein and /in (1990) have developed a parametric version of the choice theory of Yaari (1987). Their specification for period utility displays first-order risk aversion-the risk premium required to induce an investor to take a small gamble is proportional to the standard deviation of the gamble rather than the variance as in standard theory. This feature increases the risk premia predicted by the model, but in a calibration exercise in the style of Mehra and Prcscott (1985), Epstein and Zin find that they can fit only about one third of the historical equity premium.

Another strand of the literature alters the specification of discounting in (8.4.20). Ainslie (1992) and I.ocwcnstein and Prelec (1992) have argued that experimental evidence suggests not geometric discounting but hyperbolic discounting: The discount factor for horizon К is not 5h but a function of the form (1 + S\K)-S/S, where both $\ and S% arc positive as in the standard theory. This functional form implies that a lower discount rale is used for periods further in the future. Laibson (1996) argues that hyperbolic iscounting is well approximated by a utility specification

tAO + ySE,

(8.4.22)

where the additional parameter /3 < 1 implies greater discounting over \c nekt period than between any periods further in the future.

Hyperbolic discounting leads to lime-inconsistent choices: Because the iscount rate between any two dates shifts as the dates draw nearerv the ptimal plan for those dates changes over time even if no new information arrives. The implications for consumption and portfolio choice depend tn the way in which this time-consistency problem is resolved. Laibson (1996) derives the Eulcr equations for consumption choice assuming lhat t ic individual chooses each periods consumption in that period without

3eing able to constrain future consumption choices. Interestingly, he shows at with hyperbolic discounting the elasticity of intertemporal substitution less than the reciprocal of the coefficient of relative risk aversion even vtjhen the period utility function has the power form.

8.5 Conclusion

Financial economists have not yet produced a generally accepted model of the stochastic discount factor. Nonetheless substantial progress has been made. Wc know that the stochastic discount factor must be extremely volatile

if it is lo explain the cross-sectional pattern of asset returns. We also know that the conditional expectation of the stochastic discount factor must be comparatively stable in order to explain the stability of the riskless real interest rate. These properties put severe restrictions on the kinds of asset pricing models that can be considered.

There is increasing interest in the idea that risk aversion may vary over lime with the slate of the economy, lime-varying risk aversion can explain the huge body of evidence that excess returns on slocks and other risky assets are predictable. One mechanism that can produce time-varying risk aversion is habit formation in the utility function of a representative agent. Bui it is also possible lhat investors appear lo have time-varying risk aversion because they trade on the basis of irrational expectations, or that time-varying risk aversion arises from the interactions of heterogeneous agents. Grossman and Zhou (1996), for example, present a model in which two agents with different risk-aversion coefficients trade with each other. One of die agents has an exogenous lower bound on wealth, and the resulting equilibrium has a time-varying price of risk. This is likely lo be an active area for future research.

Problems-Chapter 8

8.1 Prove lhat the benchmark portfolio has the properties (PI) through (P5) stated on pages 298 and 300 of this chapter.

8.2 Consider an economy with a representative agent who has power utility with coefficient of relative risk aversion y. The agent receives a nor.storablc endowment. The process for the log endowment, or cquivalently the log of consumption o, is

ДГ/+1 = / + 0До + ,-н.

where die coefficient <p may be either positive or negative. U<p is positive then endowment fluctuations are highly persistent; if it is negative then they have an important transitory component.

8.2.1 Assume thai consumption and asset returns are jointly log-normal, with constant variances and covariances.

i. Use die representative agents Eulcr equations lo show lhat the expected log return on any asset is a linear function of the expected growth rate of the endowment. What is the slope coefficient in this relationship?

ii. Use the representative agents Eulcr equations to show that the difference between the expected log return on any asset and the log riskfree interest rale, plus one-half the own variance of

inn 14ш/нлш и/шиипиш iMllllflS

11к- log asscl return (call litis sum the premium on the asset), is proportional lo the conditional covariance of the log asset return with consumplion growlh. What is the slope coefficient in this relationship?

8.2.2 To a dose approximation, ihe unexpected return on any asscl / can he written as

/./+! - >t.H I - iH

£рл<..

+1+/

i +1

2 P ru+i+j Lj--t

2 pru+i+j

where d,is die dividend pair! on asset i at time /. This approximation was developed as (7.1.2Г>) in Chapter 7.

i. Use this expression to calculate the unexpected return on an equity which pays aggregate consumption as its dividend.

ii. Use this expression lo calculate the unexpected return on a real consul bond which has a fixed real dividend each period.

8.2.3

i. (:.il( ul.ile the equity premium and the consul bond premium.

ii. Show thai the bond premium has (he opposite sign (оф and is proportional to the square of у. Give an economic interpretation of ibis result.

iii. Show that the equity premium is always larger than the bond premium, and the difference between them is proportional to у. Give an economic interpretation of this result.

iv. Relate your discussion lo the empirical literature on the cq-uilv premium puzzle.

8.3 Consider a two-period world with a continuum of consumers. Each consumer has a random endowment in the second period and consumes only in the second period. In ihe first period, securities are traded but no money changes hands until the second period. ЛИ consumers have log utility over second-period consumption.

8.3.1 Suppose lhat all consumers endowments are the same. They arc in with probability 1/2 and (1- a)m with probability 1/2, where 0 < и < I. Suppose dial a claim to the second-period aggregate endowment is traded and lhat il costs (>m either slate, payable in the second period. (.otnpulc the equilibrium price ji and the expected return on the claim.

Problems

8.3.2 Now suppose that in the second period, with probability all consumers receive m\ with probability 1/2, a fraction (1-6) of consumers receive m and a fraction 6 receive (1-a/b)m. In the first period, all consumers face the same probability of being in the : latter group, but no insurance markets exist through which they can hedge this risk. Compute the expected return on the claim defined above. Is it higher or lower than before? Is it bounded byj a function of a and m?

8.3.3 Relate your answer to the recent empirical literature on the determination of stock returns in representative-agent models. To what extent do your results in parts 8.3.1 and 8.3.2 depend on the details of the model, and to what extent might they hold more generally?

Note: This problem is based on Mankiw (1986).

Derivative Pricing Models

Tl-lK PRICING OK OPTIONS, warrants, and other derivative scent ides-financial securities whose payoffs depend on the prices of other securities-is one of the great successes of modern financial economics. Based on the well-known Law of One Price or no-arbitrage condition, the option pricing models of Black and Scholes (1973) and Mcrton (1973b) gained an almost immediate acceptance among academics and investment professionals that is unparalleled in the history of economic science.1

The fundamental insight of the Black-Scholes and Mcrton models is that under certain conditions an options payoff can be exactly replicated by a particular dynamic investment strategy involving only the underlying stock and riskless debt. This particular strategy may be constructed to be self-financing, i.e., requiring no cash infusions except at the start and allowing no cash withdrawals until the option expires; since the strategy replicates the options payoff al expiration, the initial cost of this self-financing investment strategy must be identical lo the options price, otherwise an arbitrage opportunity will arise. This noarbilrage condition yields not only the options price but also the means to replicate ihe option synthetically-via the dynamic investment strategy of stocks and riskless debt-if it does not exist.

This method of pricing options has since been used to price literally hundreds of other types of derivative securities, some considerably more complex than a simple option. In the majority of these cases, the pricing formula can only be expressed implicitly as the solution of a parabolic partial differential equation (PDE) with boundary conditions and initial values determined by the contractual terms of each security. To obtain actual prices, the PDE must be solved numerically, which might have been problematic in 1973 when Black and Scholes and Mcrton first published iheir papers but is now commonplace llianks to the breakthroughs in computer

Sec Bernstein (IW2, (;h,i)lci II) for .1 lively шиши of the iniellei 1u.1l history ol the Black-Scholes/Mcrton option pricing formula.