Example 5.10 (3 securities) Consider three securities with expected returns, standard devia- tions of returns and correlations between returns H1 = 0.10, 01 = 0.28, P12 = P21 = -0.10, H2 = 0.15, 09 = 0.24, p23 = pa2 = 0.20, Ha=0.20, 03 = 0.25, P31 = pia = 0.25. We arrange the wi's into a one-row matrix m and I's into a one-row matrix u, m=[ 0.10 0.15 0.20 ], u= [1 1 1). Next we compute the entries cy = pujarny of the covariance matrix C, and find the inverse matrix to C, 0.0784 -0.0067 0.0175 13.954 2.544 -4.396 -0.0067 0.0576 0.0120 2.544 18.548 -4.274 0.0175 0.0120 0.0625 -4.396 -4.274 18.051 5. Portfolio Management 111 UC- uCw y From Proposition 5.9 we can compute the weights in the minimum variance portfolio. Since uc-[ 12.102 16.818 9.382 ], MC T = 38,302, we obtain [ 0.316 0.439 0.245 ). The expected return and standard deviation of this portfolio are My em" = 0,146, VwCWT 0.162 The minimum variance line can be computed using Proposition 5.10. To this end we compute uc-1*12.102 16.818 9.382 ), mc-1 [ 0.898 2.1822.530 ] uc-'7.38.302, mcm 0.923, uc'm" mc 475.609. Substituting these into the formula for w in Proposition 5.10, we obtain the weights in the portfolio with minimum variance among all portfolios with ex- pected return hv: w [ 1.578 8.6144 0.845 2.7694 -1.422 +11.3844ev ]. The standard deviation of this portfolio is o =Vww. 0.237 - 2.8854 +9.8504. Assume the same hypothesis of Example 5.10 at Page 110 of the TEXTBOOK. (a) (10 pts) Given risk/standard deviation 7 = 1.5, find the corresponding expected return on the efficient frontier. (b) (15 pts) In addtion to the three risky securities therein, assume now a risk-free security with return R = 0.10 is available. Find the expected return im and standard deviation Om for the market portfolio, and derive a linear equation for the capital market line in the expected return-risk plane. Example 5.10 (3 securities) Consider three securities with expected returns, standard devia- tions of returns and correlations between returns H1 = 0.10, 01 = 0.28, P12 = P21 = -0.10, H2 = 0.15, 09 = 0.24, p23 = pa2 = 0.20, Ha=0.20, 03 = 0.25, P31 = pia = 0.25. We arrange the wi's into a one-row matrix m and I's into a one-row matrix u, m=[ 0.10 0.15 0.20 ], u= [1 1 1). Next we compute the entries cy = pujarny of the covariance matrix C, and find the inverse matrix to C, 0.0784 -0.0067 0.0175 13.954 2.544 -4.396 -0.0067 0.0576 0.0120 2.544 18.548 -4.274 0.0175 0.0120 0.0625 -4.396 -4.274 18.051 5. Portfolio Management 111 UC- uCw y From Proposition 5.9 we can compute the weights in the minimum variance portfolio. Since uc-[ 12.102 16.818 9.382 ], MC T = 38,302, we obtain [ 0.316 0.439 0.245 ). The expected return and standard deviation of this portfolio are My em" = 0,146, VwCWT 0.162 The minimum variance line can be computed using Proposition 5.10. To this end we compute uc-1*12.102 16.818 9.382 ), mc-1 [ 0.898 2.1822.530 ] uc-'7.38.302, mcm 0.923, uc'm" mc 475.609. Substituting these into the formula for w in Proposition 5.10, we obtain the weights in the portfolio with minimum variance among all portfolios with ex- pected return hv: w [ 1.578 8.6144 0.845 2.7694 -1.422 +11.3844ev ]. The standard deviation of this portfolio is o =Vww. 0.237 - 2.8854 +9.8504. Assume the same hypothesis of Example 5.10 at Page 110 of the TEXTBOOK. (a) (10 pts) Given risk/standard deviation 7 = 1.5, find the corresponding expected return on the efficient frontier. (b) (15 pts) In addtion to the three risky securities therein, assume now a risk-free security with return R = 0.10 is available. Find the expected return im and standard deviation Om for the market portfolio, and derive a linear equation for the capital market line in the expected return-risk plane