Risk and Rates of Return 1. Suppose you won the lottery and had two options: (1) receiving $0.5 million or (2) taking a gamble in which at the flip of a coin you receive $1 million if a head comes up but receive zero if a tail comes up. a) What is the expected value of the gamble? b) Would you take the sure $0.5 million or the gamble? c) If you chose the sure $0.5 million, would that indicate that you are a risk averter or a risk seeker? d) Suppose the payoff was actually $0.5 million — that that was the only choice. You now face the choice of investing it in a U.S. Treasury bond that will return $537,500 at the end of a year or a common stock that has a 50-50 chance of being worthless or worth $1, 150,000 at the end of the year. i. The expected profit on the T-bond investment is $37,500. What is the expected dollar profit on the stock investment? ii. The expected rate of return on the T-bond investment is 7.5%. What is the expected rate of return on the stock investment? 2.
Stocks A and B have the following historical returns: Year 2007 2008 2009 2010 2011
Stock A’s Returns, rA
Stock B’s Returns, r B
(18.00%) 33.00 15.00 (0.50) 27.00
(14.50%) 21.80 30.50 (7.60) 26.30
a) b)
Calculate the average rate of return for each stock during t he period 2004 through 2008. Assume that someone held a portfolio consisting of 50% of Stock A and 50% of Stock B.What would the realized rate of return on the portfolio have been each year? Whatwould the average return on the portfolio have been during this period? c) Calculate the standard deviation of returns for each stock and for the portfolio. d) Calculate the coefficient of variation for each stock and for the portfolio. e) Assuming you are a risk-averse investor, would you prefer to hold Stock A, Stock B, orthe portfolio? Why?
3.
A stock’s returns have the following distribution:
Calculate the stock’s expected return, standard deviation, and coefficient of variation.
I.Stocks X and Y have the following probability distributions of expected future returns:
Calculate the expected returns, standard deviation, and coefficient of variation of Stocks X and Y. Is it possible that most investors will regard Stock Y as being less risky than Stock X? Explain. II.Calculate the portfolio return, ̂ ,for the following portfolio: Security A B C D
4.
Expected Return, 12.5% 3.0% 9.8% 10.5%
Investment $400,000 600,000 1,000,000 2,000,000
An individual has $35,000 invested in a stock with a beta of 0.8 andanother $40,000 invested in a stock with a beta of 1.4. If these are the only two investmentsin her portfolio, what is her portfolio’s beta?
5. Assume that the risk-free rate is 6% and the expected returnon the market is 13%. What is the required rate of return on a stock with a beta of 0.7? 6. Assume that the risk-free rate is 5% and themarket risk premium is 6%. What is the expected return for the overall stock market? Whatis the required rate of return on a stock with a beta of 1.2? 7. Consider the following information for threestocks, Stocks X, Y, and Z. The returns on the three stocks are positively correlated, but theyare not perfectly correlated. (That is, each of the correlation coefficients is between 0 and 1.)
Fund Q has one-third of its funds invested in each of the three stocks. The risk-free rate is 5.5%, and the market is in equilibrium. (That is, required returns equal expected returns.) a) What is the market risk premium (rM – rRF)? b) What is the beta of Fund Q? c) What is the expected return of Fund Q? d) Would you expect the standard deviation of Fund Q to be less than 15%, equal to 15%,or greater than 15%? Explain.
