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Derek Chau

MGTB09H3 – Principles of Finance  Fall 2010  Additional self­study exercises topic 6 – Risk Return Analysis       answers at the end     Questions   Question 1 (Risk and Return) Given the following information on a portfolio of 3 stocks held by Asif Thakor: oftRetMMeteieir Economy Probability Roeturn Reotfurn Rotfurn Boo.30.300.41.35 Normal 0.400.200.20.45 Bust 0.25 0.00 -0.35 -0.95 a. If Asif’s portfolio is invested 30% in Mei Wa, 50% in Man-Lei, and 20% in Masair, what is the portfolio expected return? The variance? The standard deviation? b. If the expected T-Bill rate is 5%, what is the expected risk premium on the portfolio? c. If the expected inflation rate is 2%, what is the expected real return on the portfolio? What is the expected real risk premium on the portfolio? d. What is the required rate of return on the portfolio, if the market risk premium is 7%, and portfolio beta is 1.6? Question 2 (Risk and Return) Following is the historical cash dividend and price data on Aaron Kam & Yeon Kim Systems (AYS) and Nhi La & Eric Leung Enterprises (NEE) stocks: Aaron Kam & Yeon Kim Systems Nhi La & Eric Leung Enterprises CnYsr. CnYsr. Year Dividend Price YearividendPrice 1993 $--- $40.00 1993 $--- $15.00 199.43.0199--22.00 199.38.5199.18.50 199.48.0199.14.00 199.44.0199.28.50 a. Calculate the yearly rate of returns for AYS and NEE stocks, and a portfolio comprised of 40% invested in AYS stock and 60% invested in NEE stock. b. Calculate the mean rate of returns and standard deviations for AYS stock, NEE stock and the portfolio. Suppose the risk free rate is 4%. Calculate the Sharpe ratios of both stocks and that of the portfolio. What is your conclusion based on these Sharpe ratios?   1    Question 3 (Risk and Return) The expected returns for two firms, A and B are as follows: State of Economy Probability Return of Firm A Return of Firm B 1 0.1 -0.05 -0.10 2 0.40.10 0.15 3 0.30.25 0.10 4 0.20.30 0.18 Firm A has total investment in assets of $75,000,000; three times that of Firm B. Assume that a new firm, C, is formed through a merger between Firms A and B. The share of A and B in the portfolio represented by Firm C is based on the ratio of their total assets prior to the merger. a. Calculate the expected return and standard deviation of Firm A and B before the merger. b. Calculate the expected return and standard deviation of Firm C. c. Based on risk/return, which firm will, you invest in? And why? (Assume the risk free rate is 4%.)   Question 4 (Risk and Return) Lisa Harvey & Fanny Lam Ltd. (LFL) and Yousuf Talal & Jessica Wu Inc. (YJI) stocks have the following historical dividend and price data: StFcI Year Dividend Year-end Pric e DividendYeaP rined 0 --- $22.50 --- $43.75 1 $2.00 16.00 $3.40 35.50 2 3 2.420.23.951.75 4 2.617.24.044.50 5 2.918.74.245.25 a. Calculate the realized rate of return for each stock in each year. Then assume that someone held a portfolio that was half LFL stock half YJI stock. What was the realized rate of return on the portfolio in each year from year 1 through year 5? What are the average returns for each stock and for the portfolio? b. Calculate the standard deviation of returns for each stock and for the portfolio. c. Based on the extent to which the portfolio has a lower risk than the stocks held individually, would you guess that the correlation coefficient between returns on the two stocks is closer to 0.9, 0.0, or –0.9? d. If you add more stocks at random to the portfolio, what is the most accurate statement of what happens to σ p 1. σp remains constant. 2. σp declines to approximately 15% 3. σp declines to zero if enough stocks are included. 2    Solutions   Answer 1 (Risk and Return) a. First, calculate the expected portfolio return in each of the three possible future states of the economy. BoEo(r : ) = 0.3(0.3) + 0.5(0.45) + 0.20(1.35) = 0.585 p Normal:E(r p) = 0.3(0.2) + 0.5(0.25) + 0.2(0.45) = 0.275 BuEs(:r p) = 0.3(0) + 0.5(-0.35) + 0.20(-0.95) = -0.365 Next, calculate the portfolio expected return, taking the probabilities of the future states of the economy into account: E(R p) = 0.35(0.585) + 0.4(0.275) + 0.25(-0.365) = 0.2235 Note that we can also calculate Ep by first calculating the expected return for the three individual stocks and then taking a weighted sum (using the portfolio weights) to calculate the portfolio expected return. σ 20.35(0.585-0.2235) 2+0.40(0.275-0.2235) +0.25(-0.365-0.2235) = 0.1334 p σ p = 0.3652 b. RP p= E(R
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