Sauder School of Business COMM 371 Marked Homework 2 • This homework can be done in groups of 2-5 and it has to be typed. • Submit a hard copy (with detailed explanations) in person and an Excel file (so that your calculations can be checked) on Connect. • The deadline for both submissions is 18:00 on Tuesday, December 2nd. • Use the Excel file from Connect (COMM371_W2014_Marked_HW2_Data.xlsx, see below) and include all your calculations, answers, and detailed explanations. Use existing tabs. • Add names and UIDs of all the group members on the tab GROUP INFO. • For the hard copy, print tabs: GROUP INFO, PROBLEM 1 ANSWERS, and PROBLEM 2 ANSWERS • Hand in the hard copy at the end of the class or bring it to the instructor’s office. • The Excel file has to be uploaded on Connect by one group member only. Problem 1 Download the file COMM371_W2014_Marked_HW2_Data.xlsx from the course web page. It contains the prices, at the monthly frequency, of two Exchange Traded Funds (a) TLT (a long-term bond fund) and (b) IWM (a small cap stock fund) from August 1, 2002 to September 2, 2014. The purpose of this problem is to illustrate the concepts of risk-return tradeoff and diversification. 1 (a) [5 marks] Using the monthly price data, compute the monthly returns on the two funds. (b) [7 marks] Compute estimates of the expected values, the standard deviations and the correlation of the monthly returns for the two funds. (c) [3 marks] Obtain estimates of the annualized expected values and standard deviations of the funds’ returns. To do this, multiply the monthly expected return by 12 and the monthly standard deviation by √ 12. Note that the correlation is not affected by the frequency. (d) [3 marks] Assume that the annual risk-free rate is 1.4%. Using the annualized expected returns and standard deviations, compute the Sharpe ratio for the two funds. Which fund offers the best reward-to-variability ratio? (e) [2 marks] Based on the correlation computed in part (b), do you expect to achieve any risk reduction by forming a portfolio that combines the two funds? (f) [10 marks] Suppose you want to create a portfolio that invests in the two funds. Create a list of possible weights for TLT starting from 0 and ending at 1 with increments of 0.05. For each weight on the list, compute the annualized expected return, standard deviation, and Sharpe ratio of the corresponding portfolio. For which weight on the list can you achieve the highest Sharpe ratio? Draw the expected return-standard deviation frontier and indicate the position of the risk-free asset and the portfolio with the highest Sharpe ratio. (g) [10 marks] Use the formulas given in class to find the weights of the optimal tangent portfolio obtained by combining the two funds. How do they compare with the weights obtained in part (f)? (h) [10 marks] Suppose you want to invest in the risk-free asset and the two funds. If you want to achieve the best reward-to-variability ratio and an expected return of 10%, what are the three weights of your portfolio? What is the standard deviation of the return on that portfolio? 2 Problem 2 Assume that the price of stock XYZ can be described by a binomial model. In every period, the price of stock XYZ may increase by a factor u or decrease by a factor d. The up shock is u = 1.20, the down shock is d = .95, and the risk-free rate is r = 0.02 per period. The probability of an up shock is 0.5. The current price (t = 0) of the ABC stock is S0 = $100. This means that the evolution of the stock price for the next two periods is described by t = 0 t = 1 t = 2 144 120 100 114 95 90.25 (a) [3 marks] Consider a European call option that expires in two periods with strike price X = $110. Find the payoffs of the European call option at maturity. (b) [10 marks] Using a replication argument, compute the price of the European call option at t = 0. (c) [3 marks] Consider a European put option that expires in two periods with strike price X = $110. Find the payoffs of the European put option at maturity. (d) [10 marks] Using a replication argument, compute the price of the European put option at t = 0. (e) [6 marks] Use your answers in parts (b) and (d) to verify that the Put-Call Parity holds. (Do not use continuous compounding to compute the present value of the strike price). (f) [9 marks] Compute the expected value and the standard deviation of the stock return over the two periods. (g) [9 marks] Compute the expected value and the standard deviation of the call option return over the two periods. 3 Hints: For parts (f) and (g), assume that the two periods are independent. This implies that, at time t = 2, the probabilities for the events [Stock Price = 144], [Stock Price = 114], and [Stock Price = 90.25] are 0.5×0.5 = 0.25, 0.5×0.5+0.5×0.5 = 0.5, and 0.5×0.5 = 0.25, respectively. 4