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

Hence ηM (X) → 0 if a_- (X) → 0, ∀ i. The first order condition for an
optimal portfolio of claims is

χ-^γ = λ- ∙ φ(X); ∀(i,X).

Hence

— ■ --1 1

xi- = a-(X) = λ- ^γi [φ(X)]^-^γi/X,
X

so that a_i(X)e(0, 1).

Differentiating ln(αi(X)) with respect to ln(X) yields

a_- ( X )
a- ( X )

= ⅛¹ ПМ ( X ) - 1]

X γi

(15)

As shown by Benninga and Mayshar (2000), η'M(X) < 0 and for X → ∞,
an → 1, so that ηM(X) → γ_n. Hence, for X → ∞, d_-(X) → 0,∀i, and
^ηM⁽^x) → 0.

Now consider X → 0. Then η'M(X) → 0 if a-(X) → 0, i = 2, ..,n, since
ɪʌ αi(X) = 0. From the first order condition, optimal risk sharing implies

'∙ (⅛)^- ■

Y 1
^{γ γ}i
x₁

X-

X ²

a- ( X ) f λi ∖ ^γi x^γi

X = U1) X²

As shown by Benninga and Mayshar (2000), α1(X) → 1forX → 0 so that
ηM (x) → γ1 and x1 → X. Hence the last equation yields for X → 0

αi ( X )
X

→ (^λ- ) ^γi X ~^γ1i 2 ,i = 2 ,..,n.
λ1

This term goes to zero for X → 0ifγ1 > 2γi,i =2, .., n. Then, by equation
(1), a- ( X ) → 0 for X → 0, i = 2 ,...,n. Hence η'M ( X ) → 0 for X → 0. ■