Solutions to Chapter 10 Introduction to Risk, Return, and the Opportunity Cost of Capital return = = = 15 = 15% dividend yield = dividend / initial price = 2/40 = 05 = 5% capital gains yield = capital gains / initial price = 4/40 = 10 = 10% dividend yield = 2/40 = 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 = –.10 = – 10% a Rate of return = = Real rate = − = − b Rate of return =0 = –.0476 = –4.76% = = 05 = 5% Real rate = − = − = c Rate of return = = 10 = 10% Real rate = − = − = 0476 = 4.76% 10-1 Copyright © 2006 McGraw-Hill Ryerson Limited Real return = − Costaguana: Real return = − = 0833 = 8.33% Canada: Real return = − = 1067 = 10.67% Canada provides the higher real return despite the lower nominal return Notice that the approximation real rate ≈ nominal rate – inflation rate would incorrectly suggest that the Costaguanan real rate was higher than the Canadian real rate The approximation is valid only for low rates We use the relationship (with all rates expressed as decimals) that: Real rate = − Asset class Treasury bills Treasury bonds Common stocks Nominal Return 4.7% 6.4 11.4 Inflation 3% 3 Real Rate 1.65% 3.3 8.16 The nominal interest rate cannot be negative If it were, investors would choose to hold cash (which pays a return of zero) rather than buy a 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 10-2 Copyright © 2006 McGraw-Hill Ryerson Limited Average price of % change Equal stocks in in average Weighted Week market stock price Index 85.90 100 85.00 -1.05% 98.95 84.80 -0.24% 98.72 85.70 1.06% 99.77 88.60 3.38% 103.14 86.00 -2.93% 100.12 85.20 -0.93% 99.19 83.60 -1.88% 97.32 78.60 -5.98% 91.50 % % % change Total change change in in Equal Market in total Value Value Weighted value of market Weighted Weighted Indexa stocks value Indexb Index 571,400 100 -1.05% 552,950 -3.23% 96.77 -3.23% -0.24% 540,300 -2.29% 94.56 -2.29% 1.06% 536,900 -0.63% 93.96 -0.63% 3.38% 552,050 2.82% 96.61 2.82% -2.93% 546,550 -1.00% 95.65 -1.00% -0.93% 553,900 1.34% 96.94 1.34% -1.88% 548,750 -0.93% 96.04 -0.93% -5.98% 511,050 -6.87% 89.44 -6.87% Notes: a The value of the portfolio for the equal weighted index is the simple average of the prices of the stocks in the index For any week, the value of the index is the current simple average price of the stocks in the index divided by the average stock price for the first week of the index, multiplied by an arbitrary starting value for the index Here we start the index at 100 A second way to calculate the new value of the index is to multiply the previous value by plus the percentage change in average stock price for that week You can see this by noting that the weekly percentage change in the simple average stock price is identical to the percentage change in the equal weighted index NOTE: the values above have been calculated with Excel and more decimal places were used than are shown If you use the rounded values in the table, your answers will differ slightly than the ones above b The market value index tracks the value of a portfolio that holds shares in each firm in proportion to the number of outstanding shares The composition of the value-weighted portfolio is identical to that of the entire market Therefore, the new value of the index equals the previous index value increased (or decreased) by the percentage change in the total value of stocks in the market The market value index can also be calculated as the current market value of the stocks in the index divided by the market value of the stocks in the index for the first week of the index, multiplied by an arbitrary starting value for the index Again we start the index at 100 Note that the weekly percentage change in the total market value of the stocks is identical to the percentage change in the value weighted index NOTE: The values above have been calculated with Excel and more decimal places were used than are shown If you use the rounded values in the table, your answers will differ slightly than the ones above 10-3 Copyright © 2006 McGraw-Hill Ryerson Limited Week Average rate of return Standard deviation of return Tonsil Weekly Rates of Return on Stocks Prochnik Krosno Exbud Kable Average -0.1000 -0.0980 -0.0942 -0.0960 -0.0088 0.0982 0.0976 -0.0963 -0.0893 -0.0980 -0.0978 -0.0843 0.0921 0.0964 0.0989 -0.0900 -0.1008 -0.0841 -0.0408 0.0957 0.0971 0.0973 -0.0323 -0.1000 0.1007 0.0976 0.1000 0.0960 -0.0968 -0.0969 -0.0960 0.0000 0.0000 0.0000 -0.0062 0.0063 0.0000 0.0000 0.0063 -0.0994 -0.0379 -0.0365 -0.0278 0.0035 0.0167 0.0390 0.0149 -0.0771 -0.0372 -0.0215 -0.0085 0.0131 -0.0116 -0.0132 0.0888 0.0973 0.0906 0.0967 0.0357 0.0381 The average of the individual stocks’ standard deviation, is 0818 or 8.18% The standard deviation of the equal-weighted portfolio, shown in the table is 3.81% This is striking evidence of the benefits of diversification Note: Since the question works with observed data, the sample standard deviations are calculated Thus for each stock the average rate of return is calculated Then, for each week, the squared difference between the week’s return and the average rate of return for all weeks is calculated The squared deviations are summed and dividend by (the number of weeks minus 1) This gives the sample variance The sample standard deviation is the square root of the sample variance a 2000 2001 2002 2003 2004 average Std Dev TSX 7.41 -12.57 -12.44 26.72 11.53 4.13 16.80 T-Bill Long 5.63 4.14 2.55 2.93 2.28 3.50 1.39 Bond 13.64 3.92 10.09 8.06 8.82 TSX risk premium 1.78 -16.71 -14.99 23.79 9.25 Long bond risk premium 8.01 -0.22 7.54 5.13 6.54 8.91 3.51 0.63 17.01 5.40 3.33 10-4 Copyright © 2006 McGraw-Hill Ryerson Limited b The average TSX risk premium was 0.63 % The average long bond risk premium was 5.4% for these five years These results are largely due to the very poor performance of the TSX in 2001 and 2002 No investor expected to lose 12% each year on their stock portfolio! c A fast way to calculate standard deviation of a sample of data is using a spreadsheet, such as Excel In Excel, use the STDEV function Alternatively, the standard deviation can be calculated by hand First, calculate the sample variance, then take the square root The sample variance is the sum of the squared deviations from the mean, divided by the number of observations minus We illustrate with the TSX risk premium: Variance of TSX risk premium = [1/(5-1)] × [(1.78 - 63)2 + (-16.71 - 63)2 + (-14.99 - 63)2 + (23.79 - 63)2 + (9.25 - 63)2 = 289.21 Standard deviation of TSX risk premium = = 17.01% We would expect that the risk premium standard deviation would be higher for the TSX than for the Long Bond portfolio This is what we find: the TSX risk premium has a 14.2% standard deviation and the Long Bond risk premium has a 3.33% standard deviation There is a lot more variation in the TSX risk premium because there is a lot more variation in the TSX return than for the Long Bond portfolio 10 In early 2000, the Dow was more than three times its 1990 level Therefore a 40point movement was far less significant in percentage terms than in 1990 We would expect to see more 40-point days even if market risk as measured by percentage returns is no higher than in 1990 11 Investors would not have invested in bonds 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 surely unanticipated by investors who bought those bonds in prior years The results from 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 12 If investors become less willing to bear investment risk, they will require a higher risk premium for holding risky assets Security prices will fall until the expected rates of return on those securities rise to the now-higher required rates of return 10-5 Copyright © 2006 McGraw-Hill Ryerson Limited 13 Based on the historical risk premium of the TSX (7.0 percent), and the current level of the risk-free rate (about 2.75 percent), one would predict an expected rate of return of 9.75 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 + 50 P0 = 1.0975 = $47.38 14 Boom = 122.22% Normal = 13.33% Recession = –100% Expected return = × 122.22 + × 13.33 + × (−100)= 23.33% Variance = × (122.22 − 23.33)2 + × (13.33−23.33)2 + × (−100−23.33)2 = 6025.8 Standard deviation = = 77.63% 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 Tower of Pita is a good hedge When the economy does well and the lawyer’s bankruptcy business suffers, the stock return is excellent, thereby stabilizing total income The owner of the gambling casino probably does well when the economy is flourishing and less well when it is doing poorly For the casino owner, holding Tower of Pita stock will not stabilize total income as much as it does for the bankruptcy lawyer 16 Rate of Return Boom = –28% Normal = Recession = 48% 8% Expected return = × (−28%) + × 8% + × 48% = 5.2% Variance = × (−28 – 5.2)2 + × (8 – 5.2)2 + ì (48 5.2)2 = 700.96 10-6 Copyright â 2006 McGraw-Hill Ryerson Limited Standard deviation = = 26.5% Portfolio Rate of Return Boom Normal Recession (−28 + 122.22)/2 = 47.11% (8 + 13.33)/2 = 10.665% (48 –100)/2 = –26.0% Expected return = × 47.11% + × 10.665% + × (-26.0%) = 14.27% Variance = × (47.11 – 14.27)2 + × (10.665 – 14.27)2 + × (-26.0 – 14.27)2 = 654.4 Standard deviation = = 25.6% Standard deviation is lower than for either firm individually because the variations in the returns of the two firms serve to offset each other When one firm does poorly, the other does well, which reduces the risk of the combination of the two 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 will provide higher returns during recessions when interest rates fall b rstock = × (−5%) + × 15% + × 25% = 13% rbonds = × 14% + × 8% + × 4% = 8.4% Variance(stocks) = × (−5−13)2 + × (15−13)2 + × (25 – 13)2 = 96 Standard deviation = = 9.80% Variance(bonds) = × (14−8.4)2 + × (8−8.4)2 + × (4−8.4)2 = 10.24 Standard deviation = = 3.20% 18 c Stocks have higher expected return and higher volatility More risk averse investors will choose bonds, while others will choose stocks a Recession Normal Boom b Expected return = × 2.6% + × 12.2% + × 16.6% = 11.16% (−5% × 6) + (14% × 4) = 2.6% (15% × 6) + ( 8% × 4) = 12.2% (25% × 6) + ( 4% ì 4) = 16.6% 10-7 Copyright â 2006 McGraw-Hill Ryerson Limited Variance = × (2.6 – 11.16)2 + × (12.2 – 11.16)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 13.0% 8.4 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 it gives almost the same return with much lower volatility This is the advantage of diversification d To calculate the correlation coefficient, rearrange the formula for the portfolio standard deviation as we did in Check Point 9.6 Correlation between bond and stock returns = (σp2 – xs2 σs2 – xb2 σb2) / ( xs xb σs σb) = (.04612 – 62× 0982 – 42 × 0322) / ( × × × 098 × 032) = -.995 The stocks and bonds are almost perfectly negatively correlated 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” behaviour 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 well, thereby stabilizing the return of the overall portfolio By contrast stock returns that move together provide no risk reduction If stock returns are independent, some risk reduction (variability reduction) occurs but it is less than if the stock returns vary against each other 21 a General Steel ought to have more sensitivity to broad market movements Steel production is more sensitive to changes in the economy than is food consumption 10-8 Copyright © 2006 McGraw-Hill Ryerson Limited 22 23 b 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 a Expected return = × (-20%) + × 30% = 5% Standard deviation = [ × (-20% - 5%) + × (30% - 5%)2]1/2 = 25% The expected rate of return on the stock is percent The standard deviation is 25 percent b Because the stock offers a risk premium of zero (its expected return is the same as for Treasury bills), it must have no market risk All the risk must be diversifiable, and therefore of no concern to investors Sassafras is not a risky investment to 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 It 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 will lose money on your insurance policy But the policy will pay 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 is faring 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 to a diversified investor since it increases exposure to the macroeconomic or market risk to which the investor is already exposed 24 a Portfolio expected return = × 9% + × 8% = 8.3% Portfolio standard deviation = [.32 × 22 +.72 × 252 + × × × × × 25]1/2 = 196 = 19.6% b With correlation of 7, the portfolio standard deviation is = [.32 × 22 +.72 × 252 + × × × × × 25]1/2 = 221 = 22.1% c The higher is the correlation between two variables, the less potential for diversification In (a), with correlation of only 2, the portfolio standard deviation is less than the standard deviation of return of either of the two stocks in the portfolio However, with the higher correlation of 7, the stocks’ return move more closely together and forming a portfolio only 10-9 Copyright © 2006 McGraw-Hill Ryerson Limited somewhat reduces total variability 25 a and b Average return (%) 4.13 8.91 3.50 5.51 TSX 300 Long Bond Treasury Bill Portfolio Standard deviation (%) 16.8 3.51 1.39 6.01 The average standard deviation of the three securities is 7.23% = (16.8 +3.51+1.39)/3, higher than the portfolio standard deviation of 6.01%, showing the benefit of diversification If there were no benefits from diversification, the portfolio standard deviation would simply be the average of the standard deviations of each of the securities in the portfolio, weighted by their portfolio weights (here the weights are each 1/3) TSX T-Bill Long Bond Portfolio 2000 7.41 5.63 13.64 8.89 2001 -12.57 4.14 3.92 -1.50 2002 -12.44 2.55 10.09 0.07 2003 26.72 2.93 8.06 12.57 2004 11.53 2.28 8.82 7.54 Average 4.13 3.50 8.91 5.51 Std Dev 16.80 1.39 3.51 6.01 26 The correlation coefficients between the weekly rates of return on Tonsil and each of the stocks are as follows: Tonsil Prochnik Krosno Exbud Kable 0.93 0.46 -0.87 0.30 Correlation with Tonsil As expected, the correlation of Tonsil with itself is The stock offering the best diversification benefit is Exbud Its return is negatively correlated with Tonsil's rate of 10-10 Copyright © 2006 McGraw-Hill Ryerson Limited return 27 Internet: Rates of Return on Canadian Financial Securities Unfortunately, the Society of Canadian Actuaries no longer gives non-members access to their publications This internet problem can no longer be done We will see if the Society will allow us to post their report on the Online Learning Centre for the textbook 28 Internet: Rates of Return on US Financial Securities Unfortunately, the free sample data on this website has changed as we write this solution However, rates of return on US stocks, bonds and bills can be found in an Excel spreadsheet put together by Professor Aswath Damodaran Look for http://pages.stern.nyu.edu/~adamodar/New_Home_Page/data.html and download the data for "Historical Returns on Stocks, Bonds and Bills - United States" Expected results: Students should see that the risk premium measured relative to the treasury bill is larger than when the long-term government bond is used as the risk-free security They can look for patterns over time 29 Internet: Mutual Funds Expected results: Students may find that the standard deviation of the index fund is lower than "aggressive growth" funds but higher than bond funds or balanced fund 30 Standard & Poor's Expected results: Students have experience calculating rates of return for companies They will see differences in dividend and capital gains yields 31 Standard & Poor's Expected results: Students will see diversification in action 10-11 Copyright © 2006 McGraw-Hill Ryerson Limited ... value of the portfolio for the equal weighted index is the simple average of the prices of the stocks in the index For any week, the value of the index is the current simple average price of the... correlation of only 2, the portfolio standard deviation is less than the standard deviation of return of either of the two stocks in the portfolio However, with the higher correlation of 7, the... average of the individual stocks’ standard deviation, is 0818 or 8.18% The standard deviation of the equal-weighted portfolio, shown in the table is 3.81% This is striking evidence of the benefits of