A single-period market consists of $n$ assets whose prices at time $t$ are denoted by $S_{t}=\left(S_{t}^{1}, \ldots, S_{t}^{n}\right)^{T}, t=0,1$, and a riskless bank account bearing interest rate $r$. The value of $S_{0}$ is given, and $S_{1} \sim N(\mu, V)$. An investor with utility $U(x)=-\exp (-\gamma x)$ wishes to choose a portfolio of the available assets so as to maximize the expected utility of her wealth at time 1. Find her optimal investment.

What is the market portfolio for this problem? What is the beta of asset $i$ ? Derive the Capital Asset Pricing Model, that

Excess return of asset $i=$ Excess return of market portfolio $\times \beta_{i}$.

The Sharpe ratio of a portfolio $\theta$ is defined to be the excess return of the portfolio $\theta$ divided by the standard deviation of the portfolio $\theta$. If $\rho_{i}$ is the correlation of the return on asset $i$ with the return on the market portfolio, prove that

Sharpe ratio of asset $i=$ Sharpe ratio of market portfolio $\times \rho_{i}$.