SMChap008 - Chapter 08 Portfolio Theory and the Capital...

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Chapter 08 - Portfolio Theory and the Capital Asset Pricing Model CHAPTER 8 Portfolio Theory and the Capital Asset Pricing Model Answers to Problem Sets 1. a. (.5 x 0%) + (.5 x 14%) = 7%. b. With Perfect Positive Correlation: Portfolio variance = [(.5) 2 x (28) 2 ] + [(.5) 2 x (26) 2 ] + 2 (.5 x .5 x 1 x 28 x 26) = 729 Standard deviation = the square root of 729 = 27%. With Perfect Negative Correlation: Portfolio variance = [(.5) 2 x (28) 2 ] + [(.5) 2 x (26) 2 ] + 2 (.5 x .5 x -1 x 28 x 26) = 1 Standard deviation = the square root of 1 = 1%. With no correlation: Portfolio variance = [(.5) 2 x (28) 2 ] + [(.5) 2 x (26) 2 ] + 2 (.5 x .5 x 0 x 28 x 26) = 365 Standard deviation = the square root of 365 = 19.1%. c. See Figure 1 below. d. No, measure risk by beta, not by standard deviation. 8-1
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Chapter 08 - Portfolio Theory and the Capital Asset Pricing Model Est. Time: 06- 10 2. a. Portfolio A (higher expected return, same risk) b. Cannot say (depends on investor’s attitude -toward risk) c. Portfolio F (lower risk, same expected return) Est. Time: 01 - 05 3. The long-term risk premium for securities as shown in Chapter 7 is 7.3%, and the long-term standard deviation for security returns is 20.0%. Therefore the Sharpe ratio = 7.3/20.0 = .365. Est. Time: 01 - 05 4. a. Figure 8.11b: Diversification reduces risk (e.g., a mixture of portfolios A and B would have less risk than the average of A and B). b. Those along line A in Figure 8.11a c. See Figure 2 below. Est. Time: 01 - 05 5. a. See Figure 3 below. 8-2
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Chapter 08 - Portfolio Theory and the Capital Asset Pricing Model b. A, D, G; A is inefficient because an investor could get a higher return with a lower standard deviation (less risk) if he or she invested in Portfolio B instead. D is inefficient because an investor could earn a slightly higher return at the same risk level if he or she invested in Portfolio E instead. G is inefficient because an investor could earn the same return with less risk if he or she invested in Portfolio F instead. c. To calculate the Sharpe ratio for each portfolio, subtract the risk free rate of 12% from each portfolio’s return and divide that by the standard deviation. The chart below shows the Sharpe ratio for each portfolio. F has the highest Sharpe ratio. Portfolio A B C D E F G H Return, r 10 12.5 15 16 17 18 18 20 Standard Deviation 23 21 25 29 29 32 35 45 Sharpe Ratio: ( r − r f )/Standard Deviation - 8.7% 2.4% 12.0% 13.8% 17.2% 18.8% 17.1% 17.8% d. If the maximum standard deviation is 25, then we must find the portfolio with the highest return whose standard deviation is no greater than 25. This is found at 15% in Portfolio C. e. Put 25/32 of your money in F and lend 7/32 at 12%. This fraction of portfolio F will have a standard deviation of only 25% since all of portfolio F has a standard deviation of 32%. The expected return on this strategy = 7/32 X 12% + 25/32 X 18% = 16.7%. If you could borrow without limit, you would achieve as high an expected return as you’d like, with correspondingly high risk, of course.
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