Mean-variance portfolio optimization. An investor with a capital of $100,000 con
ID: 3232250 • Letter: M
Question
Mean-variance portfolio optimization. An investor with a capital of $100,000 considers two risky stocks and a fixed-rate investment (e.g. riskless bank account). Let X_1 and X_2 denote the annual random returns of these stocks (per share), respectively. We are given that E(X_1] = 6, Var (X_1) = 55, E[X_2] = 4, Var (X_2) = 28, and rho(X_1, X_2) = -0.3. Moreover, the current price of the first stock is $60 per share and that of the second stock is $48 per share. The riskless return rate is 3.6% per year. The portfolio will consist of s_1 shares of the first stock, s_2 shares of the second stock and the remaining money, say B_0, invested at the fixed rate. Therefore, the annual random return on such a portfolio, R, is given by R = s_1X_1 + s_2X_2 + (0.036)B_0, under the constraints 60s_1 + 48s_2 + B_0 = 100,000 s_1 greaterthanorequalto 0, s_2 greaterthanorequalto 0, B_0 greaterthanorequalto 0 (no short selling or borrowing). (a) If the investor decide to buy 600 shares of the first stock and 800 shares of the second stock (and invest the rest in the bank), compute the mean and variance of the annual return of this portfolio. (b) Now assume that the investor's target return rate is 7%. In other words, the investor would be fine with an average return of $7,000 (or more) out of a total investment of $100,000. Using Calculus, determine the optimal values of the portfolio (s_1, s_2 and B_0) to minimize the variance of the return. Then, convert the decimal expressions to integers and rearrange the allocations to satisfy the constraints.Explanation / Answer
Amount invested in bank account = 100000 – (600x60) - (800x48) = 25600
R = 600x1 + 800x2 + (25600x0.036) = 600x1 + 800x2 + 912.6
So, E(R) = 600E(x1) + 800E(x2) + E(912.6) = (600x6) + (800x4) + (912.6)
= 7712.6 ANSWER 1
Now, given (X1, X2) = - 0.3, Cov(X1, X2) = - 0.3(65x28) = - 12.7984.
V(R) = V(600x1 + 800x2) [variance of a constant is zero and return from bank account is independent of X1 and X2]
So, V(R) = 6002V(x1) + 8002V(x2) + 2x600x800 Cov(X1, X2)
= 29033536 ANSWER 2