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



6 Appendix

6.1 Proof of Lemma 1

Investor i derives her optimal portfolio of date τ -claims from


maxE[ui(xi)]   s.t.   E[xiφ(Dτ)] = w0τ.

w0τ is the investor’s endowment reserved for buying claims on Dτ . φ(Dτ )is
the stochastic discount factor, i.e.
φ(Dτ) = Φ0 exp(-rf τ). The FOC for
xi is (λi denotes the Lagrange-multiplier of the budget constraint)


u'i(Xi) = λiφ(Dτ) ; Dτ.


Differentiate the log of this equation with respect to ln Dτ . This yields


d ln x

ni (Xi)        = ПМ (Dτ) ; Dτ                    (8)

d ln Dτ

Since d ln xi/d ln Dτ = (dxi∕dDτ)/αi (Dτ) and ]∑i dxi∕dDτ = 1, aggregating
equation (8) across all investors yields


1    =    αi ( Dτ )

ПМ ( Dτ )   ʌ ni (Xi )


(9)


Differentiate equation (9) with respect to Dτ . This yields


nM( Dτ ) = V ni( xi ) dxi α ( D ) -V 1 α ( D )     (10)

[ПМ(Dτ)]2 b [ni(Xi)]2 dDτα(T) ≥-η ηi(Xi)αi(t).     (0)

The first term on the right hand side of equation (10) can be rewritten using
(8) as


ni ( Xi ) d ln Xi

[ni(Xi)]2 dln Dτ


[αi(Dτ)]2


1 v ni ( Xi ) ∣^ D D }MM ( Dτ ) ^∣ 2
nM ( Dτ ) V ni (Xi ) I- i( τ ) ni (Xi ) -


20




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