Problem 1: Markowitz Portfolio (Risky Assets) cov([r 1, r 2, r 3 ] T ) = V =

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1 Homework II Financial Mathematics and Economics Professor: Paul J. Atzberger Due: Monday, October 3rd Please turn all homeworks into my mailbox in Amos Eaton Hall by 5:00pm. Problem 1: Markowitz Portfolio (Risky Assets) Consider three risky assets with returns r 1, r 2, and r 3. Assume that the covariance is given by cov([r 1, r 2, r 3 ] T ) = V = (1) and that the mean returns are µ 1 = 1, µ 2 = 2, and µ 3 = 4. a) Use the method of Lagrange multipliers to derive the weights w p of the portfolio with mean return µ p having the least variance σ 2 p. Be sure to show each step for full credit. b) Using the general formula you derived in (a) find the weights w p for the portfolio with mean return µ p = 3, also compute the variance σp 2 of the return. (Hint: Use Gaussian elimination to compute the vectors x = V 1 µ and y = V 1 1.) c) Find the weights w p, mean return µ p, and variance σp 2 of the minimum variance portfolio p of the assets. (Hint: First determine µ p then the weights.) 1

2 Problem 2: Markowitz Portfolio (Risky Assets) Consider three assets with returns r 1, r 2, and r 3. Assume that the covariance is given by V = (2) and that the mean returns are µ 1 = 2, µ 2 = 4, and µ 3 = 10. a) Find the weights w p for a portfolio with mean return µ p = 6 having the least variance σ 2 p. (Hint: Use Gaussian elimination to compute the vectors x = V 1 µ and y = V 1 1.) b) Find the weights w p, mean return µ p, and variance σp 2 of the minimum variance portfolio p of the assets. c) Is the portfolio p you found in (a) efficient? d) Find the frontier portfolio q which has zero covariance with p. e) What is the mean and variance of the return of q? f) Is the portfolio q efficient? 2

3 Problem 3: Markowitz Portfolio (Risky Assets + Risk-Free Asset) Consider three risky assets with returns r 1, r 2, and r 3. Also take into consideration that there is a risk-free asset with return r 0 = Assume that the covariance is given by V = (3) and that the mean returns are µ 1 = 0.1, µ 2 = 0.05, and µ 3 = Answer the following equations being sure to show all steps carefully for full credit. a) Use the method of Lagrange multipliers to find the weights w p for a portfolio with mean return µ p = 1 and the least variance σ 2 p. Be sure to show all steps for full credit. b) Find the weights w p, mean return µ p, and variance σp 2 of the minimum variance portfolio p of the assets. c) Is the portfolio p you found in (a) efficient? Problem 4: Markowitz Portfolio (Risky Assets + Risk-Free Asset) Consider three risky assets with returns r 1, r 2, and r 3. Also take into consideration that there is a risk-free asset with return r 0 =

4 Assume that the covariance is given by V = (4) and that the mean returns are µ 1 = 2.75, µ 2 = 1.5, and µ 3 = 1.5. Answer the following equations being sure to show all steps carefully for full credit. a) Use the method of Lagrange multipliers to find the weights w p for a portfolio with mean return µ p = 1 and the least variance σ 2 p. Be sure to show all steps for full credit. b) Find the weights w p, mean return µ p, and variance σp 2 of the minimum variance portfolio p of the assets. c) Is the portfolio p you found in (a) efficient? Problem 5: Numerical Computation Consider n risky assets with returns r i. Assume that the covariance has the following structure σ i,j = 0.05(1 + (j + i)/n)e i j (5) and that the mean returns are µ i = 0.1(1 + i/n). a) Use the general solution of the Markowitz problem determined by 4

5 the method of Lagrange multipliers in class to compute the weights w p for the frontier portfolio p with minimum variance for n = 5 assets. Also compute the return s mean µ p and variance σp 2. Write a matlab script to perform these calculations. Be sure to submit as an appendix to the homework your matlab script file. b) Find the weights w p (k) using a matlab script for the frontier portfolios having mean returns µ p (k) = ( k)µ p where k = 1, 2, Use the weights to compute the variance of the returns σp 2 (k). Plot the results in a (σ,µ) plot using matlab, as we did in class. Include in the plot (σ p, µ p ) from the previous problem. c) Compute the weights w q and the variance σq 2 of the return for the frontier portfolio q with mean return µ q = 0.5µ p. Include this data point on your plot. Use the plot to estimate the return you can achieve for the other portfolio having this same variance σq 2. Is the portfolio q efficient? Which would you invest in? Problem 6: Probability Theory. This problem emphasizes the importance of using probability theory to carefully make deductions. Suppose a game show host presents you with three doors behind one of which is a new car. You are then asked to choose one of the three doors. The host then opens one of the remaining two doors you did not choose revealing a donkey. You are then asked if you wish to keep your original choice or chose the only other remaining door. What should you do? Does it matter if you keep your chosen door or make a change? a) Formulate a probability space (Ω, E, P) for the problem. 5

6 b) Compute the probability of winning if you keep your original choice. c) Compute the probability of winning if you change your choice after the donkey is revealed. d) Which probability is greater? Which strategy is preferred? Problem 7: Probability Theory An elevator makes serial runs up and down a 5 story building without changing direction except at the top or bottom. Answer the following questions being sure to show the steps of your calculation carefully. a) A person enters the elevator on the 4 th floor. What is the probability it is going up? (Hint: The probability of going up is not 50% - 50%, think about how the direction relates to the state of the elevator when the request button is pressed. Use the probability space: Ω = {(a, b) : a {1, 2, 3, 4, 5}, b { 1, 1}} \ {(5, 1), (1, 1)} E = all subsets P {(a 0, b 0 )} = 1 Ω In words, the sample space consists of all permissible elevator states, where a represents the floor number and b represents the direction. Since the button can be pressed at any time, all states of the elevator are equally probable.) 6

7 b) A person enters the elevator on the 3 rd floor. What is the probability it is going down? c) Suppose there are two elevators the first as above and the second broken with a malfunction in which it randomly moves up or down. You and a friend decide to race by both entering the elevators at the same time on the 3 rd floor. If you enter the malfunctioning one, what is the probability that you reach the ground floor before your friend? (Hint: You may assume that the elevators move between the floors at the same speed and that when the elevator is at the top or botton floor it moves up or down deterministically one floor.) d) You and a friend again decide to race both randomly entering the elevators at the same time on the 3 rd floor, but in this case which elevator is malfunctioning is unknown. What is the probability you win the race? Extra Credit: e) You and a friend yet again decide to race both entering the elevators at the same time on the 3 rd floor, but this time both elevators are now malfunctioning. What is the probability you win the race to the ground floor? What is the probability to win the race to the top floor? Is the probability different or the same? Why? 7