Let $w_{0}$ be the portfolio (weights) of risky assets corresponding to the minimum-variance point in the feasible region. Let $\mathbf{w}_{1}$ be any other portfolio on the efficient frontier. Define $r_{0}$ and $r_{1}$ to be the corresponding returns.

(a) There is a formula of the form $\sigma_{01}=A \sigma_{0}^{2}$. Find $A$. [Hint: Consider the portfolios $(1-\alpha) \mathbf{w}_{0}+\alpha \mathbf{w}_{1}$, and consider small variations of the variance of such portfolios near $\alpha=0$.]

(b) Corresponding to the portfolio $\mathbf{w}_{1}$ there is a portfolio $\mathbf{w}_{z}$ on the minimum-variance set that has zero beta with respect to $\mathbf{w}_{1}$; that is, $\sigma_{1,2}=0$. This portfolio can be expressed as $\mathbf{w}_{z}=(1-\alpha) \mathbf{w}_{0}+\alpha \mathbf{w}_{1}$. Find the proper value of $\alpha$.

(c) Show the relation of the three portfolios on a diagram that includes the feasible region.

(d) If there is no risk-free asset, it can be shown that other assets can be priced according to the formula

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

where the subscript $M$ denotes the market portfolio and $\bar{r}_{z}$ is the expected rate of return on the portfolio that has zero beta with the market portfolio. Suppose that the expected returns on the market and the zero-beta portfolio are $15 %$ and $9 %$, respectively. Suppose that a stock $i$ has a correlation coefficient with the market of.5. Assume also that the standard deviation of the returns of the market and stock $i$ are $15 %$ and $5 %$, respectively. Find the expected return of stock $i$.