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(Dτ) = 0)

v O ( D ) ( —___1__~ V αi ( Dτ ) D

1

ηi ( Xi )


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

=Dj ∑
i


d ln αi

d ln Dτ


1

ηi ( Xi )


пм ( Dτ )
Dτ


1

ηi ( Xi )


12

" J αi( Dτ )


The last equation follows from d ln αi∕d ln Dτ = d ln xi/d ln Dτ 1 and equa-
tion (8) which implies
d ln αi∕d ln Dτ = ηM()[1 i(xi) 1 M(Dτ)].

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 =
Φ
0,s/Φ0,t. Because of the martingale property Φ0,t = E0,sDt), the elas-
ticity of the (forward) stochastic discount factor Φ
t,s with respect to the
dividend
Dt is given by

ln Φt,s = ln Φo,s
ln Dt     ln Ds


e(∂^ Φ ts

ln Ds


Df)) = ηΦ D + E ( ηφ D φts Dt),


since ln Ds = 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

S,D
ηt ,


1+


ʃ exp (rf (s t)) E (dsΦt,s (ηφD + E (ηsφ,DΦt>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

S,D

ηt   =1 -J

t


covP ( Dsφ ,D IDt ) ds


exp (rf (s t)) St ,


where covP(.) is the covariance under the equivalent martingale measure P7.
Thus, if
ηφ,D is constant, then ηS,D П Dt = 1. Declining aggregate RRA,
d∂Djr- < 0, implies covP(Dsφ,DDt) < 0 and, hence, ηS,D > 1. Increasing
aggregate RRA,
'DDD, > 0, implies ηS,D < 1.                            ■

21



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