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

The second term on the right hand side of equation (10) can be rewritten
as (since ∑_i α'_i(D_τ) = 0)

^v O ( D ) ( —___¹__~ ^{V α}i ( Dτ ) D

²-α i( ^τ) ∖(x(Xi)   ηM(Dτ)J   Dτ^ O (Dτ) '

The last equation follows from d ln α_i∕d ln D_τ = d ln xi/d ln D_τ — 1 and equa-
tion (8) which implies d ln α_i∕d ln D_τ = ηM(Dτ)[1 /η_i(x_i) — 1 /ηM(D_τ)].

Multiplying equation (10) by ηM(Dτ) proves Lemma 1.

6.2 Proof of Proposition 1

For 0 ≤ t ≤ s the forward stochastic discount factor is defined by Φt,s =
Φ₀,s/Φ₀,t. Because of the martingale property Φ₀,_t = E(Φ₀,_s∣D_t), the elas-
ticity of the (forward) stochastic discount factor Φt,s with respect to the
dividend Dt is given by

_since ∂ ln D_s = ₁ d ^φ,D ≡   ^d l∏^φ0,s

^since ∂ ln Dt = ¹ and ^ηs   ≡   ∂ ln Ds ■

Differentiating the logarithm of equation (4) with respect to ln Dt yields
after some manipulation

ʃ∞ exp (—rf (s — t)) E (d_sΦt,s (—η^φD + E (η_s^φ,DΦ_t>s∣ Dt)) ∣ Dt) ds

St

The forward stochastic discount factor defines an equivalent martingale mea-
sure P , i.e. P (A)= _A Φt,sdP , where P is the physical or objective proba-
bility measure. Hence the previous equation yields

where covP(.) is the covariance under the equivalent martingale measure P⁷.
Thus, if η^φ,D is constant, then ηS^,D ≡ ∂П D_t = 1. Declining aggregate RRA,
^d∂Djr- < 0, implies covP(D_s,η^φ,D∣D_t) < 0 and, hence, ηS^,D > 1. Increasing
aggregate RRA, 'DDD, > 0, implies ηS^,D < 1.                            ■

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