6. In this question, let k = {0, 1, 2, ...,8,9} be given by k = 6 Consider two assets, Asset A and Asset B, with expected returns of 8% pa and 10% pa and standard deviations of 12% pa and 20% pa respectively. The correlations coefficient between the returns yield by the two securities is given by p= 0.1k (a) Using mean-variance portfolio theory, prove that the efficient frontier becomes a straight line in the presence of a risk-free asset. [Hint: What is the mean and variance of return of a portfolio which is a combination of an arbitrary risky portfolio and the risk-free asset?] (b) If only Assets A and B are available, calculate the equation of the efficient frontier in expected return-standard deviation space. (c) A third Asset, Asset C, is risk-free and has an certain return of 6% pa. A Lagrangian function is to be used to calculate the equation of the new efficient frontier. Write down, but do not solve, the five simultaneous equations that result from this procedure. (d) Use your simultaneous equations to derive the relationship between IA and Xb, the holdings of Assets A and B, on the new efficient frontier. Hence derive the equation of the new efficient frontier in E - o space. (e) Determine the minimum variance portfolio associated with an expected return of: i. 8%. ii. 4%. Compute the variances of return of each portfolio. (f) Show that the portfolio obtained in part (e) (ii) is inefficient. Ex plain why any portfolio with return less than the riskfree return is inefficient. 6. In this question, let k = {0, 1, 2, ...,8,9} be given by k = 6 Consider two assets, Asset A and Asset B, with expected returns of 8% pa and 10% pa and standard deviations of 12% pa and 20% pa respectively. The correlations coefficient between the returns yield by the two securities is given by p= 0.1k (a) Using mean-variance portfolio theory, prove that the efficient frontier becomes a straight line in the presence of a risk-free asset. [Hint: What is the mean and variance of return of a portfolio which is a combination of an arbitrary risky portfolio and the risk-free asset?] (b) If only Assets A and B are available, calculate the equation of the efficient frontier in expected return-standard deviation space. (c) A third Asset, Asset C, is risk-free and has an certain return of 6% pa. A Lagrangian function is to be used to calculate the equation of the new efficient frontier. Write down, but do not solve, the five simultaneous equations that result from this procedure. (d) Use your simultaneous equations to derive the relationship between IA and Xb, the holdings of Assets A and B, on the new efficient frontier. Hence derive the equation of the new efficient frontier in E - o space. (e) Determine the minimum variance portfolio associated with an expected return of: i. 8%. ii. 4%. Compute the variances of return of each portfolio. (f) Show that the portfolio obtained in part (e) (ii) is inefficient. Ex plain why any portfolio with return less than the riskfree return is inefficient
