Consider

4. Consider n risky securities with expected returns R. ....R. Let i be the column vector of expected returns. Let R, be the risk-free rate, let e be an nx 1 vectors of ones, and q = F - Rre be the vector of expected excess returns. Let V denote the covariance matrix. For any portfolio Ip, let qp = q? Xdenote the expected return of portfolio Ip, let o = x V xp denote the variance of the return of portfolio Ip, and let Sp = 9p/0, denote the Sharpe ratio of portfolio Ip. Finally, for any two portfolios Ip and Is, let Ops = XTVX, denote the covariance between the returns of these portfolios. Let v-te Te = ef y-le and v-9 Iq=eTy-la You should recognize from your class notes that te is the minimum variance portfolio, and X, is the portfolio with maximum Sharpe ratio. Assume that a te. (a) Assume qe > 0, find qe and qq. (b) The Two-Fund Theorem states that we can express any efficient portfolio of risky securities as a convex combination of two efficient portfolios. Consider the port- folio 2. 94 9pm. + 9p lep 94-9e 9q-qe Verify that the expected excess return of xp is qp. (c) Find o ego (d) Find o? = Dee and on = 099- (e) Use part (c) and (d) to find o for portfolio Ip of part (b). (f) Let kronikoje, show that o2 = 0% +R(qq qe)? 4. Consider n risky securities with expected returns R. ....R. Let i be the column vector of expected returns. Let R, be the risk-free rate, let e be an nx 1 vectors of ones, and q = F - Rre be the vector of expected excess returns. Let V denote the covariance matrix. For any portfolio Ip, let qp = q? Xdenote the expected return of portfolio Ip, let o = x V xp denote the variance of the return of portfolio Ip, and let Sp = 9p/0, denote the Sharpe ratio of portfolio Ip. Finally, for any two portfolios Ip and Is, let Ops = XTVX, denote the covariance between the returns of these portfolios. Let v-te Te = ef y-le and v-9 Iq=eTy-la You should recognize from your class notes that te is the minimum variance portfolio, and X, is the portfolio with maximum Sharpe ratio. Assume that a te. (a) Assume qe > 0, find qe and qq. (b) The Two-Fund Theorem states that we can express any efficient portfolio of risky securities as a convex combination of two efficient portfolios. Consider the port- folio 2. 94 9pm. + 9p lep 94-9e 9q-qe Verify that the expected excess return of xp is qp. (c) Find o ego (d) Find o? = Dee and on = 099- (e) Use part (c) and (d) to find o for portfolio Ip of part (b). (f) Let kronikoje, show that o2 = 0% +R(qq qe)