Suppose an investor uses the quadratic utility function $U(x)=$ $x-\frac{1}{2} c x^{2}$. Suppose there are $n$ risky assets and one risk-free asset with total return $R$. Let $R_{M}$ be the total return on the optimal portfolio of risky assets. Show that the expected retum of any asset $i$ is given by the formula

\[\bar{R}_{i}-R=\beta_{i}\left(\bar{R}_{M}-R\right)\]

where $\beta_{i}=\operatorname{cov}\left(R_{M}, R_{i}\right) / \sigma_{M}^{2}$.