During each of $N$ time periods a venture capitalist, Vicky, is presented with an investment opportunity for which the rate of return for that period is a random variable; the rates of return in successive periods are independent identically distributed random variables with distributions concentrated on $[-1, \infty)$. Thus, if $x_{n}$ is Vicky's capital at period $n$, then $x_{n+1}=\left(1-p_{n}\right) x_{n}+p_{n} x_{n}\left(1+R_{n}\right)$, where $p_{n} \in[0,1]$ is the proportion of her capital she chooses to invest at period $n$, and $R_{n}$ is the rate of return for period $n$. Vicky desires to maximize her expected yield over $N$ periods, where the yield is defined as $\left(\frac{x_{N}}{x_{0}}\right)^{\frac{1}{N}}-1$, and $x_{0}$ and $x_{N}$ are respectively her initial and final capital.

(a) Express the problem of finding an optimal policy in a dynamic programming framework.

(b) Show that in each time period, the optimal strategy can be expressed in terms of the quantity $p^{*}$ which solves the optimization problem $\max _{p} \mathbb{E}\left(1+p R_{1}\right)^{1 / N}$. Show that $p^{*}>0$ if $\mathbb{E} R_{1}>0$. [Do not calculate $p^{*}$ explicitly.]

(c) Compare her optimal policy with the policy which maximizes her expected final capital $x_{N}$.