Let Y 1 , Y 2 , Y 1 , Y 2 , be independent

Let $Y_1,Y_2,\dots$ be independent and identically distributed (iid) non-negative random variables with mean one. For $t$ greater than or equal to one, define $X_t=Y_1{Y}_2\cdots Y_t$.

(a) Show that $E{X}_t=1$ for all $t$.

(b) Let $a=E\sqrt{Y_t}$. Show that $0<a\le 1$. Under what conditions does $a=1$ ?

(c) $X_t$ is a non-negative martingale, so its limit as $t$ increases exists and is finite almost surely. Call this limit $X_{\infty }$. A result of Kakutani, applied to finance by Martin (2012), can be stated as follows in this example. If $a=1,X_{\infty }=1$. Otherwise, $X_{\infty }=0$. Explain intuitively how it is possible to have both $E{X}_t=1$ for all $t$ and $X_{\infty }=0$.

(d) Consider an asset with iid gross returns $(1+R_1),(1+R_2),\dots$ and an iid stochastic discount factor $M_1,M_2,\dots$ Define the risk-adjusted value of the asset at time $t$ as $V_t=M_1{M}_2\dots M_t(1+R_1)(1+R_2)\dots (1+R_t)$. Define the limiting risk-adjusted value in the infinite future as $V_{\infty }$. Show that there is a unique asset and SDF for which $V_{\infty }=1$. Otherwise, $V_{\infty }=0$. For what asset and SDF do we have $V_{\infty }=1$ ?

(e) For any other asset, explain intuitively how it is possible to have $E{V}_t=1$ for all $t$ despite the fact that $V_{\infty }=0$. Illustrate your mechanism by reference to the properties of the gross return series, and then by reference to the properties of the SDF series. Discuss the economic difference between these two cases.