# ch7_cool2 - Homework 2 with solutions Class 9#1 An investor...

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Homework 2 – with solutions Class 9 #1 An investor is considering adding another investment to a portfolio. To achieve the maximum diversification benefits, the investor should add, if possible, an investment that has which of the following correlation coefficients with the other investments in the portfolio? a. –1.0 b. -0.5 c. 0.0 d. 1.0 Solution: Case a. (-1.0). #2 A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are: Expected Return Standard Deviation Stock fund (S) 15% 32% Bond fund (B) 9 23 The correlation between the fund returns is 0.15. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal risky portfolio? Investment opportunity set for stocks and bonds min var B S CAL 0 2 4 6 8 10 12 14 16 18 01 02 03 04 0 Standard Deviation (%)

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Solution: The graph approximates the points: #3 If you were to use only the two risky funds and still require an expected return of 12%, what would be the investment proportions of your portfolio? Compare the standard deviation to that of the optimal portfolio in the previous problem. What do you conclude? Solution: Using only the stock and bond funds to achieve a mean of 12% we solve: 12 = 15w S + 9(1 w S ) = 9 + 6w S w S = 0.5 Investing 50% in stocks and 50% in bonds yields a mean of 12% and standard deviation of: σ P = [(0.50 2 × 1024) + (0.50 2 × 529) + (2 × 0.50 × 0.50 × 110.4)] 1/2 = 21.06% The efficient portfolio with a mean of 12% has a standard deviation of only 20.61%. Using the CAL reduces the SD by 45 basis points. #4 Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate. The characteristics of the two stocks are as follows: Stock Expected Return Standard Deviation A 8% 40% B 13 60 Correlation = -1. Could the equilibrium risk-free rate be greater than 10%? (Hint: Can a particular stock portfolio be substituted for the risk-free asset?) Solution: Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio in equilibrium will always be the risk-free rate. To find the proportions of this portfolio [with w A invested in Stock A and w B = (1 – w A ) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to: σ P = Abs[w A σ A w B σ B ] E(r) σ Minimum Variance Portfolio 10.89% 19.94% Tangency Portfolio 13.25% 24.57%
0 = 40 w A 60(1 – w A ) w A = 0.60 The expected rate of return on this risk-free portfolio is: E(r) = (0.60 × 8%) + (0.40 × 13%) = 10.0% Therefore, the risk-free rate must also be 10.0%.

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ch7_cool2 - Homework 2 with solutions Class 9#1 An investor...

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