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



=E


№+/


Ds,
ST.ds


dSs     Ds

h+ + J Ss-ds



τ

I dμv ( Ss ) - E
t


τ

У v (Ss)
t



By equation (5), μv(St) = ηt,DηS,DσD + rf and ΣS(St) = ηS,DσD. Since
Vt = αtSt , we obtain ηtS,D = ηtV,D and ΣV (St) = ΣS(St) = ηtV,DσD . Hence
we can rewrite the covariance as


τ

f J ηV,DσDdWs
t


VD s,D , Φ,D V,D 1 3 DdW
ηs   ∂Ds+ ηs   ∂Ds σDD Ds   s


j E ( ʃ vγrDs,D + „ФD V,D 1 σ4 D ry,D
J
ɪs  ∂Ds + ηs  ∂Ds jDDDsηs


Dt ds,


since by Ito’s Lemma the stochastic part of v(St) is given by


Dt


Φ,D V,D 2

d ( ηt  η, σD + rf ) σD Dt dWt.

∂Dt

Theelasticities ηV,D and ηs,d are positive. Hence, Cov (CERt,τv () — rf Dt) <
[>] 0 if aggregate RRA is declining [increasing] and ηsV,D is non-increasing
[non-declining] in
Ds . The latter condition is equivalent to the condition that
the instantaneous volatility of the return index, Σ
V (Ss), is not increasing
[not declining] because Σ
V(Ss) = ηV,DσD.                               


6.5 Proof of Lemma 2

Let X denote the aggregate supply of claims. Then equation (9) yields for
ηi(xi)=γi,i = 1...n

1    = αi ( X )

ηM ( X )         γi

Differentiating with respect to X yields

ηM( X ) = ^ αi( X )

[пм(X)]2 = V Yi '


23




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