7. (Zero-beta assets) Let wo be the portfolio (weights) of risky assets corresponding the minimum-variance point in the feasible region Let w, be any other portfolio on the efficient frontier Define and r to be the corresponding returns.

(a) There is a formula of the form = A Find A. [Hint Consider the portfolios (I-a) wo+aw, and consider small variations of the variance of such portfolios near a = 0]

(b) Corresponding to the portfolio w, there is a portfolio w, on the minimum-variance set that has zero beta with respect to wi; that is, = 0. This portfolio can be expressed as w = (1-a)wo+aw Find the proper value of a

(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 F-F BIM (FM-F:) where the subscript M denotes the market portfolio and F 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. Sup- pose that a stock 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