Need help with question 2 - 5. This course is Mathematics of Finance.

423 Mathematics of Finance, Fall 2016. Homework 2 Due: Thursday 6 October 2016, NO LATER than 1pm. You may hand in your work during the class or put your solution in the dropbox on my office door. Please include the SECTION NUMBER in your work. The maximum number of points you can receive for this homework is 50. 1. In anticipation of the new iPhone and Galaxy Note, you decide to invest some of your $100,000 savings in the Apple and Samsung stocks. The current stock prices are $112.71 for Apple and $1550.00 for Samsung. To decide how much to invest in each stock, you ask your trusted investment advisor, Rupert, how he sees the stock prices in one moth. He tells you that the following 4 scenarios are equally likely to happen: (i) Both stocks will go up by 10%; (ii) Both will go down by 10%; (iii) Apple will go up by 10% while Samsung will go down by 5%; or (iv) Samsung will go up 10% while Apple price doesn't change. (a) (5 points) Suppose that you buy 500 shares of apple and 20 shares of Samsung and keep the remainder of your money in your current account (which pays zero interest rate). What is the mean and variance of your portfolio return? (b) (5 points) Suppose that you decide to spend all of your wealth in the two stocks. How many shares of each stock should you buy (or short-sell) such that your portfolio return has the minimum possible variance? 2. (a) (3 points) After examining years of daily data on hundreds of US equities, you discover the following relationship between the monthly returns of 3 stocks, A, B, and C: 1 3 1 RA = RB + RC . 2 4 400 What should the effective annual risk free rate be? Why? (Hint: Find a risk-free portfolio of A, B, and C.) (b) (5 points) Let R1 and R2 be the daily return of two equities, and let 1 , 2 , 12 and 22 be the mean and variance of the returns. Find a relationship between R1 and R2 if the returns are perfectly correlated. Determine the risk free rate. (Hint: Recall that two random variables R1 and R2 are perfectly (anti-)correlated if and only if R1 = a R2 + b, for some constants a and b.) (c) (2 points) How would your answers to part (b) change if the stocks are perfectly anti-correlated? 3. One of the most common criterion for measuring portfolio return is the Sharpe-Ratio of a portfolio (named after the economist William F. Sharpe). It is defined as the ratio of the mean of portfolio return divided by its standard deviation , i.e. SR = . Note that the denominator is the standard q 2. deviation, i.e. Consider a portfolio of two assets, with mean 1 and 2 , variance 12 and 22 , and zero correlation (i.e 12 = 0). Denote the portfolio weight of the first asset as 1 such that the weight of the second asset is 1 1 . (a) (2 points) Find a formula for SR as a function of 1 . (b) (3 points) Find the portfolio that maximizes the Sharpe-Ratio. Show your calculations in detail. (Hint: Find the value of 1 that maximizes the function SR (1 ) in the previous part.) (c) (2 points) Is the portfolio you found in part (b) an efficient portfolio? Explain. 1 4. Suppose that you are managing the only investment firm in a country with only two companies: National Auto Co. (NAC) and National Oil Inc. (NOI). From historical data, you make the following estimates for the moments of the stocks annualized return: NAC = 9%, NOI = 15%, NAC = 5%, NOI = 11%, NAC,NOI = 0.1. Also, assume that all the investors in the country are Mean-Variance investors. (a) (5 points) A client asked you for a portfolio that offers two times the return of the minimum variance portfolio. Find the portfolio weight of each stock in such a portfolio. (b) (5 points) The effective annual risk free rate is 2%. What portfolio of only the two stocks should you be offering to your clients? Explain your answer. 5. You are interested in investing in the following 4 sectors of the economy: (1) Agricultural, (2) Manufacturing, (3) Energy, and (4) Financial. Your trusted investment firm offers 4 ETFs (Exchange Traded Funds) that closely mimic the performance of each sector. You are also given the following estimates for the mean, variance and correlation of annualized returns of the ETFs. ET F Agricultural 0.08 0.08 Manufacturing 0.12 0.10 Energy 0.15 0.13 Financial 0.06 0.1 Manufacturing Energy Financial Agricultural 0.5 0.3 0.0 Manufacturing 0.4 0.1 Energy 0.0 Hint: When answering the following questions, you may refer to formulas presented during the class. Your are also allowed to workout matrix manipulations (e.g. matrix inversion and multiplication) by a calculator or softwares such as MATLAB or MATHEMATICA. (a) (2 points) What is the minimum variance portfolio of the 4 ETFs? (b) (3 points) Find the parametric formula for the efficient frontier, if only investing in the ETFs is allowed. (c) (3 points) Which efficient portfolio provides a return 4 times that of the minimum variance portfolio? (d) (3 points) Assume that the effective rate of interest is 3% per annum, find the mutual fund in the one fund theorem. Find the efficient frontier when investing risk-free is allowed in addition to investing in the ETFs. (e) (2 points) Which efficient frontier is superior (in M-V sense)? The one you found in part (b), or the one you found in part (d)? Explain your answer. 2