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 = –.0291 = –2.91% = = 05 = 5% Real rate = − = − = 0194 = 1.94% c Rate of return = = 10 = 10% Real rate = − = − = 0680 = 6.80% 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 Gov’t bonds Common stocks Nominal Return 4.7% 6.5 11.7 Inflation 3.2% 3.2 3.2 Real Rate 1.45% 3.20 8.24 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 Average price of Quarter stocks in market 902.50 866.67 Index (using DJIA method) 100.00 96.03 10-2 Copyright © 2006 McGraw-Hill Ryerson Limited Total market value of stocks 628,880 608,260 Index (using S&P method) 100.00 96.72 888.33 876.67 98.43 97.14 607,760 569,100 96.64 90.49 Quarter Sep 2003 Dec 2003 Mar 2004 Average rate of return Standard deviation of return Tanzania Brewerie s, Quarterly Rates of Return Tanzani a Equal weighted Cigarett Tanzania e portfolio Tea Compan Dahac Packers y Simba o 1/6 of each stock TOL -0.0317 -0.1667 -0.0164 -0.019 0.04 -0.0397 0.16 0.0118 0.0145 0.0962 0.025 -0.0172 0.1857 0.0175 -0.0131 -0.0605 -0.006 -0.008 0.0635 0.0109 0.1635 -0.1333 -0.0423 -0.0143 -0.0102 0.062 0.0512 -0.0093 0.0284 0.1081 0.0405 0.0325 The simple average of the individual stocks’ standard deviation, is 0692 or 6.92% The standard deviation of the equal-weighted portfolio, shown in the table, is 3.25% 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 quarter, the squared difference between the quarter’s’s return and the average rate of return for all quarters is calculated The squared deviations are summed and divided by (the number of quarterly returns s minus 1) This gives the sample variance The sample standard deviation is the square root of the sample variance 2003 2004 2005 2006 2007 average Std Dev TSX 26.61 14.58 24.39 17.01 9.83 18.48 6.95 T-Bill Long Bond 2.93 8.06 2.24 8.46 2.65 15.05 4.01 3.22 4.28 3.30 3.22 7.62 0.88 4.85 10-3 Copyright © 2006 McGraw-Hill Ryerson Limited TSX risk Long bond risk premium premium 23.68 5.13 12.34 6.22 21.74 12.40 13.00 -0.79 5.55 -0.98 15.26 4.40 7.43 5.56 b The average TSX risk premium was 15.26 % The average long bond risk premium was 4.4% for these five years These results are largely due to the very good performance of the TSX in 2003 to 2007 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)] × [(23.68 – 15.26)2 + (12.34 – 15.26)2 + (21.74 – 15.26)2 + (13.0 – 15.26)2 + (5.55 – 15.26)2 = 55.20 Standard deviation of TSX risk premium = = 7.43% 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 7.43% standard deviation and the Long Bond risk premium has a 5.56% standard deviation There is more variation in the TSX risk premium because there is more variation in the TSX return than for the Long Bond portfolio 10 In 2007, the S&P/TSX was more than four 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 10-4 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 = 0.3 × (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% + ì 48% = 5.2% 10-5 Copyright â 2006 McGraw-Hill Ryerson Limited Variance = × (−28 – 5.2)2 + × (8 – 5.2)2 + × (48 – 5.2)2 = 700.96 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 (−5% × 6) + (14% × 4) = 2.6% (15% × 6) + ( 8% × 4) = 12.2% (25% × 6) + ( 4% × 4) = 16.6% 10-6 Copyright © 2006 McGraw-Hill Ryerson Limited b Expected return = × 2.6% + × 12.2% + × 16.6% = 11.16% 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 10.7 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 10-7 Copyright © 2006 McGraw-Hill Ryerson Limited 21 22 23 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 b Exotic World Tours Agency sells a luxury good (expensive vacations) while General Cinema sells movies, which are less sensitive to changes in the economy Exotic World Tours Agency will have greater market risk a Expected return = × (-20%) + × 30% = 5% Standard deviation = [ × (-20% - 5%)2 + × (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% 10-8 Copyright © 2006 McGraw-Hill Ryerson Limited c 25 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 somewhat reduces total variability a The following table contains the annual rates of return, the five-year average rate of return and the standard deviation of the rates of return for each index and the portfolio with one-third in each of the indexes: 2003 2004 2005 2006 2007 average Std Dev TSX 26.61 14.58 24.39 17.01 9.83 T-Bill Long Bond Portfolio 2.93 8.06 12.53 2.24 8.46 8.43 2.65 15.05 14.03 4.01 3.22 8.08 4.28 3.30 5.80 18.48 6.95 3.22 0.88 7.62 4.85 9.77 3.40 b The table summarizes the calculations from (a): Average Standard return (%) deviation (%) TSX 300 18.48 6.95 Long Bond 7.62 4.85 Treasury Bill 3.22 0.88 Portfolio 9.77 3.40 The average standard deviation of the three securities is 4.23% = (6.95+4.85+0.88)/3, higher than the portfolio standard deviation of 3.40%, 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) 26 The correlation coefficients between the quarterly rates of return on Tanzania Breweries and each of the stocks are as follows: Tanzania Breweries TOL Tanzania Tea Packers 10-9 Copyright © 2006 McGraw-Hill Ryerson Limited Tanzania Cigarette Company Simba Dahaco Correlation with TB 1.0000 (0.6009) 0.1694 (0.1927) (0.9678) 0.7990 As expected, the correlation of Tanzania Breweries with itself is The stock offering the best diversification benefit is Simba Its return is most negatively correlated with Tanzania Breweries’ rate of return 27 Internet: Arithmetic Average 1928-2007 1967-2007 1997-2007 Risk Premium 1928-2007 1967-2007 1997-2007 Stocks 11.69% 11.98% 9.39% T.Bills 3.91% 6.05% 4.13% T.Bonds 5.26% 7.66% 6.71% Stocks - T.Bills Stocks - T.Bonds 7.78% 6.42% 5.94% 4.33% 5.26% 2.68% a From the above tables, the overall risk premium is bigger when using the Treasury Bill as the risk free security than using Treasury Bonds as the risk free security This makes sense: Treasury Bills are less risky than Treasury Bonds, making the difference in risk between Treasury Bills and the market index bigger than the difference in risk between Treasury Bonds and the market index b The risk premium becomes smaller over time 28 Internet: TD Canadian Index Fund, http://www.globefund.com/servlet/Page/document/v5/data/fund? style=na_eq&id=18353&gf_uid=globeandmail.gf.03428539934 3yr risk: 15.17 TD Precious Metals http://www.globefund.com/servlet/Page/document/v5/data/fund? style=na_eq&id=18350&gf_uid=globeandmail.gf.03428539934 3yr risk: 31.66 TD Energy http://www.globefund.com/servlet/Page/document/v5/data/fund? style=na_eq&id=18345&gf_uid=globeandmail.gf.03428539934 3yr risk: 25.16 10-10 Copyright © 2006 McGraw-Hill Ryerson Limited TD Entert & Communications GIF II http://www.globefund.com/servlet/Page/document/v5/data/fund? style=globe_eq&id=52905&gf_uid=globeandmail.gf.03428539934 3yr risk: 17.89 TD Health Sciences http://www.globefund.com/servlet/Page/document/v5/data/fund? style=na_eq&id=25995&gf_uid=globeandmail.gf.03428539934 yr risk: 11.17 Except for TD Health Sciences, the other sectors all have higher risk than the index fund It indicates some sectors have risk above Index, some below Index 29 Standard & Poor's Expected results: Students have experience calculating rates of return for companies They will see differences in dividend and capital gains yields One thing to note: The S&P database provides a rolling years worth of stock prices At the time this data was retrieved, there were not December closing prices If another month had been selected, five years of data would have been available December Closing Price 82.55 71.98 80.55 80.43 Dividend 1.15 1.52 1.52 1.48 2004 2005 2006 2007 14.64 7.72 7.51 6.73 Microsoft 2004 (MSFT) 2005 2006 2007 26.72 26.15 29.86 35.6 Company Year Magna 2004 (MGA) 2005 2006 2007 Ford (F) Dividend Yield Capital Gains Yield Rate of return 0.0184 0.0211 0.0184 -0.1280 0.1191 -0.0015 -0.1096 0.1402 0.0169 0.25 0.4 0.4 0.0171 0.0518 0.0533 -0.4727 -0.0272 -0.1039 -0.4556 0.0246 -0.0506 0.43 0.39 0.34 3.32 0.0146 0.0130 0.1112 -0.0213 0.1419 0.1922 -0.0067 0.1549 0.3034 10-11 Copyright © 2006 McGraw-Hill Ryerson Limited 30 Expected results: Students will see diversification in action Return GOOG Return BAC Return Portfolio Return 22.330 (0.163) 359.36 (0.103) 24.170 (0.309) (0.299) 0.166 26.690 (0.022) 400.52 (0.135) 35.00 4.460 (0.071) 27.290 0.061 463.29 (0.022) 31.140 (0.002) 4.800 (0.002) 25.720 (0.065) 473.75 (0.100) 32.90 0.378 59.240 (0.173) 4.810 (0.293) 27.510 (0.029) 526.42 (0.101) 23.87 (0.298) (0.179) May08 71.600 (0.041) 6.800 (0.177) 28.320 (0.007) 585.80 0.020 34.01 (0.094) (0.060) Apr08 74.630 8.260 0.444 28.520 0.005 574.29 0.304 37.54 (0.010) 0.155 Mar08 72.150 (0.014) 5.720 (0.124) 28.380 0.043 440.47 (0.065) 37.910 (0.046) (0.041) Feb08 73.210 (0.071) 6.530 (0.017) 27.200 (0.166) 471.180 (0.165) 39.74 (0.100) (0.104) Jan08 78.800 (0.020) 6.640 (0.013) 32.600 (0.084) 564.30 (0.184) 44.150 0.070 (0.046) Dec07 80.430 (0.046) 6.730 (0.104) 35.600 0.060 691.48 (0.002) 41.26 (0.106) (0.040) Nov07 84.270 (0.111) 7.510 (0.153) 33.600 (0.087) 693.00 (0.020) 46.13 (0.045) (0.083) Oct07 94.760 (0.016) 8.870 0.045 36.810 0.249 707.00 0.246 48.28 (0.040) 0.097 Sep07 96.310 0.077 8.490 0.087 29.460 0.025 567.27 0.101 50.27 (0.008) 0.056 Aug07 89.450 0.020 7.810 (0.082) 28.730 (0.009) 515.250 0.010 50.68 0.069 0.002 Jul07 87.710 8.510 (0.097) 28.990 (0.016) 510.00 (0.024) 47.42 (0.030) (0.041) Jun07 90.990 9.420 0.129 29.470 (0.040) 522.70 0.050 48.89 (0.036) 0.025 MGA Return F Return Oct08 33.670 (0.342) 2.190 (0.579) Sep08 51.190 (0.106) 5.200 Aug08 57.270 (0.031) Jul08 59.100 Jun08 0.034 (0.036) 0.021 MSFT 10-12 Copyright © 2006 McGraw-Hill Ryerson Limited 0.124 (0.053) 0.005 (0.023) 0.042 May07 89.150 0.126 8.340 0.037 30.690 0.025 497.90 0.056 50.71 (0.004) 0.048 Apr07 79.150 0.054 8.040 0.019 29.940 0.074 471.380 0.029 50.90 (0.002) 0.035 Mar07 75.110 0.021 7.890 (0.003) 27.870 (0.011) 458.16 0.019 51.02 0.004 0.006 Feb07 73.570 (0.058) 7.910 (0.027) 28.170 (0.087) 449.45 (0.104) 50.83 (0.033) (0.062) Jan07 78.120 (0.030) 8.130 0.083 30.860 0.033 501.50 0.089 52.58 (0.015) 0.032 Dec06 80.550 0.048 7.510 (0.076) 29.860 0.017 460.48 (0.050) 53.39 (0.009) (0.014) Nov06 76.860 0.028 8.130 (0.018) 29.360 0.023 484.810 0.018 53.85 (0.000) 0.010 Oct06 74.800 Return Portfolio Return 8.280 MGA Return mean (0.028) stdev 0.093 F 476.39 28.710 Return MSFT (0.034) 0.182 Return GOOG 53.87 Return BAC (0.007) (0.006) (0.025) (0.020) 0.084 0.116 0.129 0.091 Mean standard deviation of the standard deviations of all five stocks: 0.121 - calculated as average of the individual stocks’ standard deviations Portfolio standard deviation: 0.091 - calculated using the monthly rates of return of the portfolio Notice that the portfolio has lower monthly standard deviation than the simple average of the individual stocks’ standard deviations This is due to the fact that the stock returns are less than perfectly correlated Diversification reduces portfolio standard deviation 31 a b See solution of problem 30 Correlation MGA F MSFT GOOG BAC MGA 1.000 0.692 0.483 0.441 0.550 F 0.692 1.000 0.326 0.529 0.551 10-13 Copyright © 2006 McGraw-Hill Ryerson Limited c MSFT 0.483 0.326 1.000 0.657 0.098 GOOG 0.441 0.529 0.657 1.000 (0.001) BAC 0.550 0.551 0.098 (0.001) 1.000 MGA and F, MSFT and GOOG are pairs of closely related industries, more correlated than other pairs MGA: Magna International – makes car parts F: Ford Motor Company – designs and assembles cars MSFT: Microsoft – makes computer software GOOG: Goggle – runs internet search business BAC: Bank of America 10-14 Copyright © 2006 McGraw-Hill Ryerson Limited ... 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. .. 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... 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