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

6.6 The Price of the Market Portfolio

The price of the market portfolio at date t, St ,is

St = £ exp(rf (t - s))E[DsΦ_t,_s∖D_t) + exp(-rf h)E(dpDt+ eΦ_t,_t+_h∖D_t)
s=t

= A ^£ exp(r, (t - s)) ^£ Dtв

exp[(1 - δ_tV(s - t)){μ₁D - δ_iβσD,/2)]

N

+A exp(-rf h) £ Dt^{(1 -δi})β_i dp^{1 -δi}

exp[(1 - δi)dh{μD - [(1 - δi)d - 1]⅛/2)]
with

A^-1 ≡ Y D-^δitB_t,t

Bis ≡ βi dp^-δiexp[-δ_zd(t + h - s){μD - [δ_zd + 1]σD/2)]

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