Consider the power utility defined by the function \(F(S)=\frac{1}{\gamma} S^{\gamma}\), for \(\gamma \leq 1\). There are available \(n\) assets, each of which follows geometric Brownian motion, including a risk-free asset with rate of return \(r\). A portfolio of these leads to a process for the portfolio as \(\mathrm{d} P=\mu P \mathrm{~d} t+\sigma \mathrm{d} z\).

(a) Show that the gamma investor will seek to maximize \(\mu+\frac{1}{2}(\gamma-1) \sigma^{2}\).

(b) Find a pricing formula for \(\mu_{i}-r\) in terms of \(\operatorname{cov}\left(P_{\mathrm{opt}}, S_{i}\right)\), where \(P_{\mathrm{opt}}\) is the optimal portfolio of the gamma investor.