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,τ 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 D_τ

Since d ln xi/d ln D_τ = (dx_i∕dD_τ)/α_i (D_τ) and ]∑_i dx_i∕dD_τ = 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 dDτ^α(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 ^{( X}i ) ∣^ D D _}MM ⁽ Dτ ) ^∣ 2
nM ( Dτ ) V ni (Xi ) I- i^{( τ} ) ni (Xi ) -

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