(Finance) Portfolio optimization question.
Q. 1 (Numerical portfolio optimization, 57 pts). In portfolio optimization with more than two risky assets, one uses matrix notation as a handy tool for the computations and the analysis of the efficient frontier. After all the results are formulated in matrix form, the numerical calculations are yet to be carried out, which is the purpose of this exercise. In case you need to compute the inverse of a matrix, you are welcome to do it with software such as MATLAB or WolframAlpha. However, I strongly encourage you to do it by hand using some linear algebra tricks discussed in the lectures. Consider an investor who plans to choose a portfolio of four stocks. From market data, the expected returns of these stocks are estimated as p1 = -0.2, H2 = 0.1, 143 = 0.2, H4 = 0.3. The estimated values of the corresponding standard deviations of the returns are 01 = 0.2, 02 = 0.1, 03 = 0.2, 04 = 0.3. Based on statistical evidence, it is assumed that the only correlated pairs of stocks are {1,3}, {1,4} and {2,4} with respective correlation coefficients P13 = 0.3, P14 = 0.1, P24 = -0.1 for the returns. a. (4) What is the expected return vector and the covariance matrix of these five stocks? b. (4) What are the weights of the minimum variance portfolio (MVP)? Does the MVP have shortselling? c. (4) What are the expected return and standard deviation of the MVP? d. (4) What are the weights of the portfolio admitting the minimum variance among all portfolios with expected return u = 2? e. (4) What are the weights of the portfolio admitting the minimum variance among all portfolios with expected return u = -3? f. (5) Using two-fund theorem, describe the minimum variance line in terms of the two portfolios computed in parts d and e. g. (5) Describe the risk-expected return curve of the portfolios on the minimum vari- ance line. Draw the curve. Now, in addition to these stocks, a riskless bond is available in the market with return r= 0.05. h. (4) Why does there exist a market portfolio? i. (4) What are the weights of the market portfolio? j. (4) What are the expected return and standard deviation of the market portfolio? k. (5) Describe the capital market line. 1. (5) What are the weights of the portfolio on the efficient frontier whose expected return is some fixed u E [r, +co)? Note that the weight vector you are asked to compute is in R5. m. (5) Consider the set W of all weight vectors of the portfolios on the efficient frontier. Give a representation of W of the form W= {(wi(s), . . . , W5(s)) R | >0}, for some functions w1, ..., W5 of a real variable such that w5(0) = 1. Q. 1 (Numerical portfolio optimization, 57 pts). In portfolio optimization with more than two risky assets, one uses matrix notation as a handy tool for the computations and the analysis of the efficient frontier. After all the results are formulated in matrix form, the numerical calculations are yet to be carried out, which is the purpose of this exercise. In case you need to compute the inverse of a matrix, you are welcome to do it with software such as MATLAB or WolframAlpha. However, I strongly encourage you to do it by hand using some linear algebra tricks discussed in the lectures. Consider an investor who plans to choose a portfolio of four stocks. From market data, the expected returns of these stocks are estimated as p1 = -0.2, H2 = 0.1, 143 = 0.2, H4 = 0.3. The estimated values of the corresponding standard deviations of the returns are 01 = 0.2, 02 = 0.1, 03 = 0.2, 04 = 0.3. Based on statistical evidence, it is assumed that the only correlated pairs of stocks are {1,3}, {1,4} and {2,4} with respective correlation coefficients P13 = 0.3, P14 = 0.1, P24 = -0.1 for the returns. a. (4) What is the expected return vector and the covariance matrix of these five stocks? b. (4) What are the weights of the minimum variance portfolio (MVP)? Does the MVP have shortselling? c. (4) What are the expected return and standard deviation of the MVP? d. (4) What are the weights of the portfolio admitting the minimum variance among all portfolios with expected return u = 2? e. (4) What are the weights of the portfolio admitting the minimum variance among all portfolios with expected return u = -3? f. (5) Using two-fund theorem, describe the minimum variance line in terms of the two portfolios computed in parts d and e. g. (5) Describe the risk-expected return curve of the portfolios on the minimum vari- ance line. Draw the curve. Now, in addition to these stocks, a riskless bond is available in the market with return r= 0.05. h. (4) Why does there exist a market portfolio? i. (4) What are the weights of the market portfolio? j. (4) What are the expected return and standard deviation of the market portfolio? k. (5) Describe the capital market line. 1. (5) What are the weights of the portfolio on the efficient frontier whose expected return is some fixed u E [r, +co)? Note that the weight vector you are asked to compute is in R5. m. (5) Consider the set W of all weight vectors of the portfolios on the efficient frontier. Give a representation of W of the form W= {(wi(s), . . . , W5(s)) R | >0}, for some functions w1, ..., W5 of a real variable such that w5(0) = 1
