7. (Zero-beta assets) Let wo be the portfolio (weights) of risky assets corresponding to the minimum-variance...

 

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7. (Zero-beta assets) Let wo be the portfolio (weights) of risky assets corresponding to the minimum-variance point in the feasible region. Let w be any other portfolio on the efficient frontier. Define ro and r to be the corresponding returns. (a) There is a formula of the form 001 = A0. Find A. [Hint: Consider the portfolios (1-)wo+aw, and consider small variations of the variance of such portfolios near = 0.] (b) Corresponding to the portfolio w there is a portfolio wz on the minimum-variance set that has zero beta with respect to w; that is, 1,z = 0. This portfolio can be expressed as w = (1-)wo+aw1. 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 Ti-Tz BiMTM-Tz), where the subscript M denotes the market portfolio and Fz 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. 7. (Zero-beta assets) Let wo be the portfolio (weights) of risky assets corresponding to the minimum-variance point in the feasible region. Let w be any other portfolio on the efficient frontier. Define ro and r to be the corresponding returns. (a) There is a formula of the form 001 = A0. Find A. [Hint: Consider the portfolios (1-)wo+aw, and consider small variations of the variance of such portfolios near = 0.] (b) Corresponding to the portfolio w there is a portfolio wz on the minimum-variance set that has zero beta with respect to w; that is, 1,z = 0. This portfolio can be expressed as w = (1-)wo+aw1. 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 Ti-Tz BiMTM-Tz), where the subscript M denotes the market portfolio and Fz 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.