The xt random variable thus moves slowly (depending on ϕ), with a tendency to return to its mean value. Lastly, mean reversion is captured by assuming cov(ηt+1, ut+1)=σηu<0, which translates, as per below, into a statement about risky return autocorrelations:

0>σηu=cov(u˜t+1,η˜t+1)=covt[(r˜t+1Etr˜t+1),(x˜t+1x¯ϕ(xtx¯))]=covt(r˜t+1,x˜t+1)=covt(r˜t+1,Etr˜t+2rf+σu22)=covt(r˜t+1,r˜t+2u˜t+2rf+σu22)=covt(r˜t+1,r˜t+2)

image

a high return today reduces expected returns next period. Thus,

vart(r˜t+1+r˜t+2)=2vart(r˜t+1)+2covt(r˜t+1,r˜t+2)<2vart(rt+1)

in contrast to the independence case. More generally, for all horizons k,

vart(r˜t+1+r˜t+2++r˜t

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