For two assets, let their 1 -year simple returns, \(R_{1}, R_{2}\), be independent and identically distributed as follows

\[P\left[R_{i}=right]=p, \quad P\left[R_{i}=-right]=1-p\]

for some \(r>0\) and \(0

\[R_{P}=w R_{1}+(1-w) R_{2}\]

Starting with unit initial wealth \(W_{0}=1\), the wealth after 1 year is \[W_{1}=W_{0} \times\left(1+R_{P}ight)=1+w R_{1}+(1-w) R_{2}\]
Find the optimal allocation to maximize expected utility of future wealth, \(E\left[U\left(W_{1}ight)ight]\), via the following steps:

(a) Write the explicit expression for \(E\left[U\left(W_{1}ight)ight]\) in terms of \(p, r, w\)
\[\begin{aligned}E\left[U\left(W_{1}ight)ight] & =p^{2} U(1+w r+(1-w) r) \\& +p(1-p) U(\ldots)+(1-p) p U(\ldots) \\& +(1-p)^{2} U(1-w r-(1-w) r)\end{aligned}\]

(b) Using the chain rule, calculate \(d E\left[U\left(W_{1}ight)ight] / d w\) as a function of \(w, r, p, U^{\prime}(\cdot)\) and set it to zero.

(c) Since \(U^{\prime \prime}<0, U^{\prime}(\cdot)\) is a strictly decreasing function, it implies \(U^{\prime}\left(x_{1}ight)=U^{\prime}\left(x_{2}ight)\) if and only if \(x_{1}=x_{2}\). Use this result to find \(w^{*}\) that makes \(d E\left[U\left(W_{1}ight)ight] / d w\) equal zero.

(d) Since \(U^{\prime \prime}(\cdot)<0\), show that \(E\left[U\left(W_{1}ight)ight]\) is maximized at \(w^{*}\) by showing \(d^{2} E\left[U\left(W_{1}ight)ight] / d w^{2}\) evaluated at \(w^{*}\) is \(\leq 0\).

(e) What are the optimal weights \(\left(w^{*}, 1-w^{*}ight)\) ?