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



6.3 Proof of Proposition 2

We know that for constant aggregate RRA

Var


Vτ Vτ

ln —

к  Vt


Dt


D D D-

= Var I ln —
к
Dt


t<τ.


By Proposition 1, for declining aggregate RRA ηtΦ,D, the elasticity ηtS,D > 1
so that the conditional variance of asset returns is higher than the (condi-
tional) variance of the dividend process, i.e.

Var


(l,V


Dt


Dτ

> Var I ln —

Dt


t<τ.


(11)


Consider now the unconditional variance (i.e. θ =0):

Var


(I- S)


Var(E(lnVτ |Dt) -lnVt)+E(Var(lnVτ |Dt))(12)


with

E(lnVτ |Dt)-lnVt


- 1∑V (Ss)2)


ds


We need to show that Var


is greater than


Var


(∣-3


Var(lnDτ |Dt) .


(14)


From (11) it follows that the second term on the right hand side of equa-
tion (12) exceeds
Var ɑn D-^. As the first term on the right hand side of
equation (12) is also positive, we are done. The proof is the same for the
variance conditional on
Dθ ; 0 < θ < t.

6.4 Proof of Proposition 3

Since by definition CERt- f- (dSs/ Ss) + f- ( Ds/ Ss rf ) ds, and the
riskless rate
rf is assumed constant, the covariance is given by

Cov ( CERt-v (S- ) rf Dt) = Cov


---
ddSs DDs D

J Ss- + J S-ds, J dμv(Ss)



22




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