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Solutions to Chapter 10 Introduction to Risk, Return, and the Opportunity Cost of Capital capital gain + dividend ($44 ? $40) + $2 = = 0. 15 = 15. 0% initial share price $40 1. Rate of return = Dividend yield = dividend/initial share price = $2/$40 = 0. 05 = 5% Capital gains yield = capital gain/initial share price = $4/$40 = 0. 10 = 10% 2. Dividend yield = $2/$40 = 0. 05 = 5% The dividend yield is unaffected; it is based on the initial price, not the final price. Capital gain = $36 ??? $40 = ? $4 Capital gains yield = ???$4/$40 = ???0. 10 = ???10% capital gain + dividend ($38 ? 40) + $2 = = 0% initial share price $40 1 + nominal rate of return 1+ 0 ? 1 = ? 1 = ? 0. 0385 = ? 3. 85% 1 + inflation rate 1 + 0. 04 3. a. Rate of return = Real rate of return = Rate of return = b. ($40 ? $40) + $2 = 0. 05 = 5% $40 1 + nominal rate of return 1. 05 ? 1 = ? 1 = 0. 0096 = 0. 96% 1 + inflation rate 1. 04 Real rate of return = Rate of return = c. ($42 ? $40) + $2 = 0. 10 = 10% $40 1 + nominal rate of return 1. 10 ? 1 = ? 1 = 0. 0577 = 5. 77% 1 + inflation rate 1. 04 Real rate of return = 10-1 4. Real rate of return = 1 + nominal rate of return ? 1 1 + inflation rate 1. 95 ? = 0. 0833 = 8. 33% 1. 80 Costaguana: Real return = U. S. : Real return = 1. 12 ? 1 = 0. 0980 = 9. 80% 1. 02 The U. S. provides the higher real rate of return despite the lower nominal rate of return. Notice that the approximate relationship between real and nominal rates of return is valid only for low rates: real rate of return ? nominal rate of return ??? inflation rate This approximation incorrectly suggests that the Costaguanan real rate was higher than the U. S. real rate. 5. We use the following relationship: Real rate of return = 1 + nominal rate of return ? 1 1 + inflation rate
Asset class Treasury bills Treasury bonds Common stocks Nominal rate of return 4. 0% 5. 3% 11. 7% Inflation rate 3. 0% 3. 0% 3. 0% Real rate of return 0. 97% 2. 23% 8. 45% 6. The nominal interest rate cannot be negative. If it were, investors would choose to hold cash (which pays a rate of return equal to zero) rather than buy a Treasury bill providing a negative return. On the other hand, the real expected rate of return is negative if the inflation rate exceeds the nominal return. 7. Average price of Quarter stocks in market 1 902. 50 2 866. 67 3 888. 33 4 876. 67 Index (using DJIA method) 100. 00 96. 03 98. 43 97. 4 Total market value of stocks 628,880 608,260 607,760 569,100 Index (using S&P method) 100. 00 96. 72 96. 64 90. 49 10-2 8. a. b. c. For the period 1900-2004, Average rate of return = 11. 7% (See Table 10-1) For the period 1900-2004, Average risk premium = 7. 6% (See Table 10-1) For the period 1900-2004, Standard deviation of returns = 20. 0%. (See Table 10-5) 9. a. Year 2000 2001 2002 2003 2004 b. c. Stock market return -10. 89 -10. 97 -20. 86 31. 64 12. 62 T-bill return 5. 89 3. 83 1. 65 1. 02 1. 20 Average Risk premium -16. 78 -14. 80 -22. 51 30. 62 11. 42 -2. 41 Deviation from mean -14. 37 -12. 39 -20. 10 33. 03 13. 3 Squared deviation 206. 4969 153. 5121 404. 0100 1,090. 9809 191. 2689 409. 2538 The average risk premium was: -2. 41% The variance (the average squared deviation from the mean) was 409. 2538 (without correcting for the lost degree of freedom). Therefore: standard deviation = 409. 2538 = 20. 23% 10. In early 2004, the Dow was substantially more than three times its 1990 level. Therefore, in 2004, a 40-point movement was far less significant in percentage terms than it was in 1990. We would expect to see more 40-point days in 2004 even if market risk as measured by percentage returns is no higher than it was in 1990. 1. Investors would not have invested in bonds during the period 1977-1981 if they had expected to earn negative average returns. Unanticipated events must have led to these results. For example, inflation and nominal interest rates during this period rose to levels not seen for decades. These increases, which resulted in large capital losses on long-term bonds, were almost certainly unanticipated by investors who bought those bonds in prior years. The results for this period demonstrate the perils of attempting to measure ‘normal’ maturity (or risk) premiums from historical data.
While experience over long periods may be a reasonable guide to normal premiums, the realized premium over short periods may contain little information about expectations of future premiums. 10-3 12. If investors become less willing to bear investment risk, they will require a higher risk premium to compensate them for holding risky assets. Security prices of risky investments will fall until the expected rates of return on those securities rise to the now-higher required rates of return. 13. Based on the historical risk premium of the S&P 500 (7. 6 percent) and the current level of the risk-free rate (about 3. percent), one would predict an expected rate of return of 11. 1 percent. If the stock has the same systematic risk, it also should provide this expected return. Therefore, the stock price equals the present value of cash flows for a one-year horizon. P0 = $2 + $50 = $46. 80 1. 111 14. Boom: $5 + ($195 ? $80) = 150. 00% $80 $2 + ($100 ? $80) = 27. 50% $80 $0 + ($0 ? $80) = ? 100. 00% $80 Normal: Recession: r= 150 + 27. 50 + (? 100) = 25. 83% 3 1 1 1 ? (150 ? 25. 83) 2 + ? (27. 50 ? 25. 83) 2 + ? (? 100 ? 25. 83) 2 = 10,418. 06 3 3 3 Variance = Standard deviation = variance = 102. 07% 15.
The bankruptcy lawyer does well when the rest of the economy is floundering, but does poorly when the rest of the economy is flourishing and the number of bankruptcies is down. Therefore, the Leaning Tower of Pita is a risk-reducing investment. When the economy does well and the lawyer’s bankruptcy business suffers, the stock return is excellent, thereby stabilizing total income. 10-4 16. Boom: $0 + ($18 ? $25) = ? 28. 00% $25 $1 + ($26 ? $25) = 8. 00% $25 $3 + ($34 ? $25) = 48. 00% $25 Normal: Recession: r= (? 28) + 8 + 48 = 9. 33% 3 1 1 1 ? (? 28 ? 9. 33) 2 + ? (8 ? 9. 33) 2 + ? (48 ? 9. 33) 2 = 963. 6 3 3 3 Variance = Standard deviation = variance = 31. 04% Portfolio Rate of Return Boom: (? 28 + 150)/2 = 61. 00% Normal: (8 + 27. 5)/2 = 17. 75% Recession: (48 ???100)/2 = ???26. 0% Expected return = 17. 58% Standard deviation = 35. 52% 17. a. Interest rates tend to fall at the outset of a recession and rise during boom periods. Because bond prices move inversely with interest rates, bonds provide higher returns during recessions when interest rates fall. rstock = [0. 2 ? (? 5%)] + (0. 6 ? 15%) + (0. 2 ? 25%) = 13. 0% rbonds = (0. 2 ? 14%) + (0. 6 ? 8%) + (0. 2 ? 4%) = 8. 4% Variance(stocks) = [0. ? (? 5? 13)2] + [0. 6 ? (15? 13)2] + [0. 2 ? (25 ??? 13)2] = 96 Standard deviation = 96 = 9. 80% Variance(bonds) = [0. 2 ? (14? 8. 4)2] + [0. 6 ? (8? 8. 4)2] + [0. 2 ? (4? 8. 4)2] = 10. 24 Standard deviation = 10. 24 = 3. 20% b. c. Stocks have both higher expected return and higher volatility. More risk averse investors will choose bonds, while those who are less risk averse might choose stocks. 10-5 18. a. Recession Normal economy Boom (? 5% ? 0. 6) + (14% ? 0. 4) = 2. 6% (15% ? 0. 6) +(8% ? 0. 4) = 12. 2% (25% ? 0. 6) + (4% ? 0. 4) = 16. 6% b. Expected return = (0. 2 ? 2. 6%) + (0. 6 ? 2. 2%) + (0. 2 ? 16. 6%) = 11. 16% Variance = [0. 2 ? (2. 6 ??? 11. 16)2] + [0. 6 ? (12. 2 ??? 11. 16)2] + [0. 2 ? (16. 6 ??? 11. 16)2] = 21. 22 Standard deviation = c. 21. 22 = 4. 61% The investment opportunities have these characteristics: Stocks Bonds Portfolio Mean Return 13. 00% 8. 40% 11. 16% Standard Deviation 9. 80% 3. 20% 4. 61% The best choice depends on the degree of your aversion to risk. Nevertheless, we suspect most people would choose the portfolio over stocks since the portfolio has almost the same return with much lower volatility. This is the advantage of diversification. 19.
If we use historical averages to compute the “normal” risk premium, then our estimate of “normal” returns and “normal” risk premiums will fall when we include a year with a negative market return. This makes sense if we believe that each additional year of data reveals new information about the “normal” behavior of the market portfolio. We should update our beliefs as additional observations about the market become available. 20. Risk reduction is most pronounced when the stock returns vary against each other. When one firm does poorly, the other will tend to do well, thereby stabilizing the return of the overall portfolio. 1. a. General Steel ought to have greater sensitivity to broad market movements. Steel production is more sensitive to changes in the economy than is food consumption. Club Med sells a luxury good (expensive vacations) while General Cinema sells movies, which are less sensitive to changes in the economy. Club Med will have greater market risk. b. 10-6 22. a. The expected rate of return on the stock is 4 percent. The standard deviation is 24 percent. Because the stock offers a risk premium of zero (its expected return is the same as the expected return for Treasury bills), it must have no market risk.
All the risk must be diversifiable, and therefore of no concern to investors. b. 23. Sassafras is not a risky investment for a diversified investor. Its return is better when the economy enters a recession. Therefore, the company risk offsets the risk of the rest of the portfolio. Sassafras is a portfolio stabilizer despite the fact that there is a 90 percent chance of loss. Compare Sassafras to purchasing an insurance policy. Most of the time, you lose money on your insurance policy. But the policy pays off big if you suffer losses elsewhere — for example, if your house burns down.
For this reason, we view insurance as a risk-reducing hedge, not as speculation. Similarly, Sassafras may be viewed as analogous to an insurance policy on the rest of your portfolio since it tends to yield higher returns when the rest of the economy fares poorly. In contrast, the Leaning Tower of Pita has returns that are positively correlated with the rest of the economy. It does best in a boom and goes out of business in a recession. For this reason, Leaning Tower would be a risky investment for a diversified investor since it increases exposure to the macroeconomic or market risk to which the investor is already exposed. 10-7