EQUILIBRIUM, SDF AND THE EULER EQUATIONS ∗

9.7 EQUILIBRIUM, SDF AND THE EULER EQUATIONS ∗

We have seen, throughout Chapters 5, 6 and 7, the importance of the rational expectations hypothesis as a concept of equilibrium for determining asset prices. In Chapter 3, we have also used the maximization of the expected utility of consumption to determine a rationality leading to a pricing mechanism we have called the SDF (stochastic discount factor). In other words, while in rational expectations we have an asset price determined by:

Current price =

E ∗ {Future Prices}

1+R f

where E ∗ denotes expectation with respect to a ‘subjective’ probability (in J. Muth, 1961 words) which we called the risk-neutral probability and R f is the risk-free rate. In the SDF framework, we had:

Current price = E ∗ Future Prices

In this section, we extend the two-period framework used in Chapter 3 to multiple periods. To do so, we shall use Euler’s equation, providing the condition for an equilibrium based on a rationality of expected utility of consumption. Let an investor maximizing the expected utility of consumption:

where u(c t+j ) is the utility of consumption at time t + j, T is the final time and

G (W T ) is the terminal wealth state at time T . At time t, the change rate in the wealth is:

WW

c t+j and therefore c +R t t+j −W t −1 t =q t t −R t t+j =

q t+j We insert this last expression in the utility to be maximized:

Application of Euler’s equation, a necessary condition for value maximization, yields:

t+j

W t+j

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and therefore we have the following ‘equilibrium’: ∂ V t

= q W = constant or =E ρ

q t+j W t+j In other words, the marginal utility of wealth increments (savings) equals the dis-

t+j t+j t+j

W t + j−1

counted inflation-adjusted marginal utilities of consumption. If wealth is invested in a portfolio of assets such that:

N t+j and therefore, ∂ u (c t + j−1 )

N t+j since at time t−1, the future price at time t is random, we have:

where M t is the kernel, or the stochastic discount factor, expressing the ‘con- sumption impatience’. This equation can also be written as follows:

1+R t = ;1=E M t p t → 1 = E {M (1 + R t t }

t −1

p t −1

which is the standard form of the SDF equation.

Example: The risk-free rate

If p t is a bond worth $1 at time t, then for a risk-free discount rate:

= E {M t } (1) and therefore E {M t }=

1+R f 1+R f This leads to: M t

and finally to p t −1 =E

1+R f where E ∗ t is a modified (subjective) probability distribution.

(c

t + j−1

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Example: Risk premium and the CAPM beta

For a particular risky asset, the CAPM provides a linear discount mechanism which is:

M t +1 =a t +b t R M,t +1

In other words, for a given stock, whose rate of return is 1 + R t +1 =p t +1 / p t , we have:

1 cov (M t +1 , 1+R t +1 )

1 = E {M t +1 (1 + R t +1 )} → E(1 + R t +1 )= −

E (M t +1 ) and therefore, upon introducing the linear SDF, we have:

E (M t +1 ) After we insert the linear model for the kernel we have:

E (M t +1 )

E (1 + R t +1 ) = (1 + R f,t ) [1 − cov (M t +1 , 1+R t +1 )] and

E (1 + R t +1 ) = (1 + R f,t +1 )[1 − cov(a + bR M,t +1 , 1+R t +1 )] which is reduced to:

cov(R M,t +1 −R f,t +1 , R −R )

var(R

E (R t +1 −R f,t +1 )=βE t (R M,t +1 −R f,t +1 )

However, the hypothesis that the kernel is linear may be limiting. Recent studies have suggested that we use a quadratic measurement of risk with a kernel given by:

t +1 =a t +b t R M,t +1 +c t R M,t +1

In this case, the skewness of the distribution also enters into the determination of the value of the stock.