Return Predictability and Stock Market Crashes in a Simple Rational Expectations Model

shown by Wachter (2002), in a complete market an investor with time-
additive utility over consumption allocates wealth as if she saves for every
consumption date separately. Thus, the pricing of date τ1 -claims can be an-
alyzed independently of that of τ2-claims subject to the usual no-arbitrage
requirement. At any date t, each agent can trade claims on aggregate con-
sumption of date τ, τ > t.Letxi(Dτ) denote the consumption of agent i
at date τ as a function of aggregate consumption D_τ and α_i (D_τ) = x_i∕D_τ
her share of consumption. Then _i αi (Dτ) = 1 for every level of aggregate
consumption. As shown by Benninga and Mayshar (2000), in equilibrium
aggregate RRA ηM (Dτ) is related to the investors’ RRA by the harmonic
mean,

1/ПМ(Dτ) = E (1 /η.(xi)) αi(Dτ).                 (1)

i

ηi(xi) is agent i’s RRA given her consumption xi. In order to find out
whether aggregate RRA declines in aggregate consumption Dτ , we differen-
tiate equation (1) with respect to Dτ. As shown in the appendix, we obtain
the following result.

Lemma 1 The growth rate of aggregate RRA is

^ ni⁽ xi ) [^α⅛⁽ Dτ )
ʌ ηi (Xi ) I. ηi (Xi )

The lemma shows that in equilibrium the growth rate of aggregate RRA is
the difference between two terms, the first being the sum of weighted growth
rates of individual RRA, the second being a pseudo-variance of the inverse
individual levels of RRA. If every investor has a positive share of consump-
tion α_i(D_τ), then, apart from [ηM(D_τ)]2/D_τ, the last term in equation (2)
has the properties of a variance term. The more heterogeneous agents are in
their preferences and, hence, in their equilibrium levels of RRA, the higher is
this pseudo-variance. It depends on the equilibrium allocation of consump-
tion since the shares αi(Dτ) and the individual RRA ηi(Xi) are endogenous.

Hence, as already shown by Benninga and Mayshar (2000), aggregate RRA
is declining if every agent has constant RRA, i.e. if η^l_i(x_i) = 0. Now
suppose η'_i(x_i) = 0. Then the first term on the right hand side of equa-
tion (2) multiplies the individual RRA growth rate η'_i(x_i)/η_i(x_i) by zi ≡

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