320 SECTION THREE calculated. Suppose that you are offered the chance to play the following game. You start by investing $100. Then two coins are flipped. For each head that comes up your starting balance will be increased by 20 percent, and for each tail that comes up your starting balance will be reduced by 10 percent. Clearly there are four equally likely outcomes: FIGURE 3.15 Historical returns on major asset classes, 1926–1998. Rate of return, percent Number of years 0Ϫ10 10 Average return, percent Standard deviation, percent 3.8 3.2 Treasury bills Rate of return, percent Number of years 0Ϫ10Ϫ20Ϫ30Ϫ40 10 20 30 40 50 13.2 20.3 Common stocks Rate of return, percent Number of years 0Ϫ10 10 20 3.2 4.5 Inflation Rate of return, percent Number of years 0Ϫ10 10 20 30 40 5.7 9.2 Treasury bonds 0 1 2 3 50 45 40 35 30 25 20 15 10 5 0 4 5 6 7 9 8 0 5 30 25 20 15 10 35 40 50 45 0 5 10 15 20 25 Source: Stocks, Bonds, Bills and Inflation® 1999 Yearbook, © 1999 Ibbotson Associates, Inc. Based on copyrighted works by Ibbotson and Sinquefield. All Rights Reserved. Used with permission. Introduction to Risk, Return, and the Opportunity Cost of Capital 321 • Head + head: You make 20 + 20 = 40% • Head + tail: You make 20 – 10 = 10% • Tail + head: You make –10 + 20 = 10% • Tail + tail: You make –10 – 10 = –20% There is a chance of 1 in 4, or .25, that you will make 40 percent; a chance of 2 in 4, or .5, that you will make 10 percent; and a chance of 1 in 4, or .25, that you will lose 20 percent. The game’s expected return is therefore a weighted average of the possible out- comes: Expected return = probability-weighted average of possible outcomes = (.25 × 40) + (.5 × 10) + (.25 × –20) = +10% If you play the game a very large number of times, your average return should be 10 percent. Table 3.10 shows how to calculate the variance and standard deviation of the returns on your game. Column 1 shows the four equally likely outcomes. In column 2 we cal- culate the difference between each possible outcome and the expected outcome. You can see that at best the return could be 30 percent higher than expected; at worst it could be 30 percent lower. These deviations in column 2 illustrate the spread of possible returns. But if we want a measure of this spread, it is no use just averaging the deviations in column 2—the av- erage is always going to be zero. To get around this problem, we square the deviations in column 2 before averaging them. These squared deviations are shown in column 3. The variance is the average of these squared deviations and therefore is a natural meas- ure of dispersion: Variance = average of squared deviations around the average = 1,800 = 450 4 When we squared the deviations from the expected return, we changed the units of measurement from percentages to percentages squared. Our last step is to get back to percentages by taking the square root of the variance. This is the standard deviation: Standard deviation = square root of variance = √ 450 = 21% Because standard deviation is simply the square root of variance, it too is a natural measure of risk. If the outcome of the game had been certain, the standard deviation would have been zero because there would then be no deviations from the expected TABLE 3.10 The coin-toss game; calculating variance and standard deviation (1) (2) (3) Percent Rate of Return Deviation from Expected Return Squared Deviation +40 +30 900 +10 0 0 +10 0 0 –20 –30 900 Variance = average of squared deviations = 1,800/4 = 450 Standard deviation = square root of variance = √ 450 = 21.2, about 21% 322 SECTION THREE outcome. The actual standard deviation is positive because we don’t know what will happen. Now think of a second game. It is the same as the first except that each head means a 35 percent gain and each tail means a 25 percent loss. Again there are four equally likely outcomes: • Head + head: You gain 70% • Head + tail: You gain 10% • Tail + head: You gain 10% • Tail + tail: You lose 50% For this game, the expected return is 10 percent, the same as that of the first game, but it is more risky. For example, in the first game, the worst possible outcome is a loss of 20 percent, which is 30 percent worse than the expected outcome. In the second game the downside is a loss of 50 percent, or 60 percent below the expected return. This in- creased spread of outcomes shows up in the standard deviation, which is double that of the first game, 42 percent versus 21 percent. By this measure the second game is twice as risky as the first. A NOTE ON CALCULATING VARIANCE When we calculated variance in Table 3.10 we recorded separately each of the four pos- sible outcomes. An alternative would have been to recognize that in two of the cases the outcomes were the same. Thus there was a 50 percent chance of a 10 percent return from the game, a 25 percent chance of a 40 percent return, and a 25 percent chance of a –20 percent return. We can calculate variance by weighting each squared deviation by the probability and then summing the results. Table 9.3 confirms that this method gives the same answer. ᭤ Self-Test 3 Calculate the variance and standard deviation of this second coin-tossing game in the same formats as Tables 3.10 and 3.11. MEASURING THE VARIATION IN STOCK RETURNS When estimating the spread of possible outcomes from investing in the stock market, most financial analysts start by assuming that the spread of returns in the past is a rea- TABLE 3.11 The coin-toss game; calculating variance and standard deviation when there are different probabilities of each outcome (1) (2) (3) (4) Percent Rate Probability Deviation from Probability × of Return of Return Expected Return Squared Deviation +40 .25 +30 .25 × 900 = 225 +10 .50 0 .50 × 0 = 0 –20 .25 –30 .25 × 900 = 225 Variance = sum of squared deviations weighted by probabilities = 225 + 0 + 225 = 450 Standard deviation = square root of variance = √ 450 = 21.2, about 21% Introduction to Risk, Return, and the Opportunity Cost of Capital 323 sonable indication of what could happen in the future. Therefore, they calculate the standard deviation of past returns. To illustrate, suppose that you were presented with the data for stock market returns shown in Table 3.12. The average return over the 5 years from 1994 to 1998 was 24.75 percent. This is just the sum of the returns over the 5 years divided by 5 (123.75/5 = 24.75 percent). Column 2 in Table 3.12 shows the difference between each year’s return and the av- erage return. For example, in 1994 the return of 1.31 percent on common stocks was below the 5-year average by 23.44 percent (1.31 – 24.75 = –23.44 percent). In column 3 we square these deviations from the average. The variance is then the average of these squared deviations: Variance = average of squared deviations = 801.84 = 160.37 5 Since standard deviation is the square root of the variance, Standard deviation = square root of variance = √ 160.37 = 12.66% It is difficult to measure the risk of securities on the basis of just five past outcomes. Therefore, Table 3.13 lists the annual standard deviations for our three portfolios of securities over the period 1926–1998. As expected, Treasury bills were the least variable security, and common stocks were the most variable. Treasury bonds hold the middle ground. TABLE 3.12 The average return and standard deviation of stock market returns, 1994–1998 Deviation from Year Rate of Return Average Return Squared Deviation 1994 1.31 –23.44 549.43 1995 37.43 12.68 160.78 1996 23.07 –1.68 2.82 1997 33.36 8.61 74.13 1998 28.58 3.83 14.67 Total 123.75 801.84 Average rate of return = 123.75/5 = 24.75 Variance = average of squared deviations = 801.84/5 = 160.37 Standard deviation = square root of variance = 12.66% Source: Stocks, Bonds, Bills and Inflation 1999 Yearbook, Chicago: R. G. Ibbotson Associates, 1999. TABLE 3.13 Standard deviation of rates of return, 1926–1998 Portfolio Standard Deviation, % Treasury bills 3.2 Long-term government bonds 9.2 Common stocks 20.3 Source: Computed from data in Ibbotson Associates, Stocks, Bonds, Bills and Inflation 1999 Yearbook (Chicago, 1999). 324 SECTION THREE Of course, there is no reason to believe that the market’s variability should stay the same over many years. Indeed many people believe that in recent years the stock mar- ket has become more volatile due to irresponsible speculation by . . . (fill in here the name of your preferred guilty party). Figure 3.16 provides a chart of the volatility of the United States stock market for each year from 1926 to 1998. 6 You can see that there are periods of unusually high variability, but there is no long-term upward trend. Risk and Diversification DIVERSIFICATION We can calculate our measures of variability equally well for individual securities and portfolios of securities. Of course, the level of variability over 73 years is less interest- ing for specific companies than for the market portfolio because it is a rare company that faces the same business risks today as it did in 1926. Table 3.14 presents estimated standard deviations for 10 well-known common stocks for a recent 5-year period. 7 Do these standard deviations look high to you? They should. Remember that the market portfolio’s standard deviation was about 20 percent over the entire 1926–1998 period. Of our individual stocks only Exxon had a standard deviation of less than 20 percent. Most stocks are substantially more variable than the market portfolio; only a handful are less variable. This raises an important question: The market portfolio is made up of individual stocks, so why isn’t its variability equal to the average variability of its components? The answer is that diversification reduces variability. 6 We converted the monthly variance to an annual variance by multiplying by 12. In other words, the variance of annual returns is 12 times that of monthly returns. The longer you hold a security, the more risk you have to bear. 7 We pointed out earlier that five annual observations are insufficient to give a reliable estimate of variability. Therefore, these estimates are derived from 60 monthly rates of return and then the monthly variance is mul- tiplied by 12. FIGURE 3.16 Stock market volatility, 1926–1998. Annualized standard deviation of monthly returns, percent ’26 ’30 ’34 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 ’38 ’42 ’46 ’50 ’54 ’58 Year ’62 ’66 ’70 ’74 ’78 ’82 ’86 ’90 ’94 ’98 DIVERSIFICATION Strategy designed to reduce risk by spreading the portfolio across many investments. Introduction to Risk, Return, and the Opportunity Cost of Capital 325 Selling umbrellas is a risky business; you may make a killing when it rains but you are likely to lose your shirt in a heat wave. Selling ice cream is no safer; you do well in the heat wave but business is poor in the rain. Suppose, however, that you invest in both an umbrella shop and an ice cream shop. By diversifying your investment across the two businesses you make an average level of profit come rain or shine. ASSET VERSUS PORTFOLIO RISK The history of returns on different asset classes provides compelling evidence of a risk–return trade-off and suggests that the variability of the rates of return on each asset class is a useful measure of risk. However, volatility of returns can be a misleading measure of risk for an individual asset held as part of a portfolio. To see why, consider the following example. Suppose there are three equally likely outcomes, or scenarios, for the economy: a re- cession, normal growth, and a boom. An investment in an auto stock will have a rate of return of –8 percent in a recession, 5 percent in a normal period, and 18 percent in a boom. Auto firms are cyclical: They do well when the economy does well. In contrast, gold firms are often said to be countercyclical, meaning that they do well when other firms do poorly. Suppose that stock in a gold mining firm will provide a rate of return of 20 percent in a recession, 3 percent in a normal period, and –20 percent in a boom. These assumptions are summarized in Table 3.15. It appears that gold is the more volatile investment. The difference in return across the boom and bust scenarios is 40 percent (–20 percent in a boom versus +20 percent in a recession), compared to a spread of only 26 percent for the auto stock. In fact, we can confirm the higher volatility by measuring the variance or standard deviation of re- turns of the two assets. The calculations are set out in Table 3.16. Since all three scenarios are equally likely, the expected return on each stock is Portfolio diversification works because prices of different stocks do not move exactly together. Statisticians make the same point when they say that stock price changes are less than perfectly correlated. Diversification works best when the returns are negatively correlated, as is the case for our umbrella and ice cream businesses. When one business does well, the other does badly. Unfortunately, in practice, stocks that are negatively correlated are as rare as pecan pie in Budapest. TABLE 3.14 Standard deviations for selected common stocks, July 1994–June 1999 Stock Standard Deviation, % Biogen 46.6 Compaq 46.7 Delta Airlines 26.9 Exxon 16.0 Ford Motor Co. 24.9 MCI WorldCom 34.4 Merck 24.5 Microsoft 34.0 PepsiCo 26.5 Xerox 27.3 326 SECTION THREE simply the average of the three possible outcomes. 8 For the auto stock the expected re- turn is 5 percent; for the gold stock it is 1 percent. The variance is the average of the squared deviations from the expected return, and the standard deviation is the square root of the variance. ᭤ Self-Test 4 Suppose the probabilities of the recession or boom are .30, while the probability of a normal period is .40. Would you expect the variance of returns on these two investments to be higher or lower? Why? Confirm by calculating the standard deviation of the auto stock. The gold mining stock offers a lower expected rate of return than the auto stock, and more volatility—a loser on both counts, right? Would anyone be willing to hold gold mining stocks in an investment portfolio? The answer is a resounding yes. To see why, suppose you do believe that gold is a lousy asset, and therefore hold your entire portfolio in the auto stock. Your expected return is 5 percent and your standard TABLE 3.16S Expected return and volatility for two stocks Auto Stock Gold Stock Deviation from Deviation from Rate of Expected Squared Rate of Expected Squared Scenario Return, % Return, % Deviation Return, % Return, % Deviation Recession –8 –13 169 +20 +19 361 Normal +5 0 0 +3 +2 4 Boom +18 +13 169 –20 –21 441 Expected return 1 (–8 + 5 + 18) = 5% 1 (+20 + 3 – 20) = 1% 33 Variance a 1 (169 + 0 + 169) = 112.7 1 (361 + 4 + 441) = 268.7 33 Standard deviation √ 112.7 = 10.6% √ 268.7 = 16.4% (= √ variance) a Variance = average of squared deviations from the expected value. TABLE 3.15 Rate of return assumptions for two stocks Rate of Return, % Scenario Probability Auto Stock Gold Stock Recession 1/3 –8 +20 Normal 1/3 +5 +3 Boom 1/3 +18 –20 8 If the probabilities were not equal, we would need to weight each outcome by its probability in calculating the expected outcome and the variance. Introduction to Risk, Return, and the Opportunity Cost of Capital 327 deviation is 10.6 percent. We’ll compare that portfolio to a partially diversified one, in- vested 75 percent in autos and 25 percent in gold. For example, if you have a $10,000 portfolio, you could put $7,500 in autos and $2,500 in gold. First, we need to calculate the return on this portfolio in each scenario. The portfo- lio return is the weighted average of returns on the individual assets with weights equal to the proportion of the portfolio invested in each asset. For a portfolio formed from only two assets, Portfolio rate = ( fraction of portfolio ؋ rate of return ) of return in first asset on first asset + ( fraction of portfolio ؋ rate of return ) in second asset on second asset For example, autos have a weight of .75 and a rate of return of –8 percent in the reces- sion, and gold has a weight of .25 and a return of 20 percent in a recession. Therefore, the portfolio return in the recession is the following weighted average: 9 Portfolio return in recession = [.75 × (–8%)] + [.25 × 20%] = –1% Table 3.17 expands Table 3.15 to include the portfolio of the auto stock and the gold mining stock. The expected returns and volatility measures are summarized at the bot- tom of the table. The surprising finding is this: When you shift funds from the auto stock to the more volatile gold mining stock, your portfolio variability actually de- creases. In fact, the volatility of the auto-plus-gold stock portfolio is considerably less than the volatility of either stock separately. This is the payoff to diversification. We can understand this more clearly by focusing on asset returns in the two extreme scenarios, boom and recession. In the boom, when auto stocks do best, the poor return on gold reduces the performance of the overall portfolio. However, when auto stocks are stalling in a recession, gold shines, providing a substantial positive return that boosts TABLE 3.17 Rates of return for two stocks and a portfolio Rate of Return, % Portfolio Scenario Probability Auto Stock Gold Stock Return, % a Recession 1/3 –8 +20 –1.0% Normal 1/3 +5 +3 +4.5 Boom 1/3 +18 –20 +8.5 Expected return 5% 1% 4% Variance 112.7 268.7 15.2 Standard deviation 10.6% 16.4% 3.9% a Portfolio return = (.75 × auto stock return) + (.25 × gold stock return). 9 Let’s confirm this. Suppose you invest $7,500 in autos and $2,500 in gold. If the recession hits, the rate of return on autos will be –8 percent, and the value of the auto investment will fall by 8 percent to $6,900. The rate of return on gold will be 20 percent, and the value of the gold investment will rise 20 percent to $3,000. The value of the total portfolio falls from its original value of $10,000 to $6,900 + $3,000 = $9,900, which is a rate of return of –1 percent. This matches the rate of return given by the formula for the weighted average. 328 SECTION THREE portfolio performance. The gold stock offsets the swings in the performance of the auto stock, reducing the best-case return but improving the worst-case return. The inverse relationship between the returns on the two stocks means that the addition of the gold mining stock to an all-auto portfolio stabilizes returns. A gold stock is really a negative-risk asset to an investor starting with an all-auto portfolio. Adding it to the portfolio reduces the volatility of returns. The incremental risk of the gold stock (that is, the change in overall risk when gold is added to the port- folio) is negative despite the fact that gold returns are highly volatile. In general, the incremental risk of a stock depends on whether its returns tend to vary with or against the returns of the other assets in the portfolio. Incremental risk does not just depend on a stock’s volatility. If returns do not move closely with those of the rest of the portfolio, the stock will reduce the volatility of portfolio returns. We can summarize as follows: ᭤ EXAMPLE 1 Merck and Ford Motor Let’s look at a more realistic example of the effect of diversification. Figure 3.17a shows the monthly returns of Merck stock from 1994 to 1999. The average monthly re- turn was 3.1 percent but you can see that there was considerable variation around that average. The standard deviation of monthly returns was 7.1 percent. As a rule of thumb, in roughly one-third of the months the return is likely to be more than one standard de- viation above or below the average return. 10 The figure shows that the return did indeed differ by more than 7.1 percent from the average on about a third of the occasions. Figure 3.17b shows the monthly returns of Ford Motor. The average monthly return on Ford was 2.3 percent and the standard deviation was 7.2 percent, about the same as that of Merck. Again you can see that in about a third of the cases the return differed from the average by more than one standard deviation. An investment in either Merck or Ford would have been very variable. But the for- tunes of the two stocks were not perfectly related. 11 There were many occasions when a 1. Investors care about the expected return and risk of their portfolio of assets. The risk of the overall portfolio can be measured by the volatility of returns, that is, the variance or standard deviation. 2. The standard deviation of the returns of an individual security measures how risky that security would be if held in isolation. But an investor who holds a portfolio of securities is interested only in how each security affects the risk of the entire portfolio. The contribution of a security to the risk of the portfolio depends on how the security’s returns vary with the investor’s other holdings. Thus a security that is risky if held in isolation may nevertheless serve to reduce the variability of the portfolio, as long as its returns vary inversely with those of the rest of the portfolio. 10 For any normal distribution, approximately one-third of the observations lie more than one standard devi- ation above or below the average. Over short intervals stock returns are roughly normally distributed. 11 Statisticians calculate a correlation coefficient as a measure of how closely two series move together. If Ford’s and Merck’s stock moved in perfect lockstep, the correlation coefficient between the returns would be 1.0. If their returns were completely unrelated, the correlation would be zero. The actual correlation between the returns on Ford and Merck was .03. In other words, the returns were almost completely unrelated. Introduction to Risk, Return, and the Opportunity Cost of Capital 329 decline in the value of one stock was canceled by a rise in the price of the other. Be- cause the two stocks did not move in exact lockstep, there was an opportunity to reduce variability by spreading one’s investment between them. For example, Figure 3.17c FIGURE 3.17 The variability of a portfolio with equal holdings in Merck and Ford Motor would have been only 70 percent of the variability of the individual stocks. Ϫ20 Ϫ15 Ϫ10 Ϫ5 0 5 10 15 20 25 30 Merck return, percent Ϫ25 Ϫ30 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 Ϫ20 Ϫ15 Ϫ10 Ϫ5 0 5 10 15 20 25 30 Ϫ25 Ϫ30 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 Ϫ20 Ϫ15 Ϫ10 Ϫ5 0 5 10 15 20 25 30 Ϫ25 Ϫ30 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 Ford Motor return, percentPortfolio return, percent (a) (b) (c) [...]... 2 3 4 5 6 7 8 9 85 76. 5 69 62 .5 56. 5 56 61.5 67 .5 61 56 51 46 41.5 38 41.5 45.5 50 45.5 59.5 53.5 49 47 51.5 56. 5 62 60 54 149 164 180 198 217 1 96 177 160 160 80 80 80 79.5 80 80 80 80.5 72.5 * Number of shares outstanding Source: We are indebted to Professor Mary M Cutler for providing these data 8 Stock Market History a What was the average rate of return on large U.S common stocks from 19 26 to 1998?... sequence of keystrokes, assuming that the opportunity cost of capital is 7 percent: Hewlett-Packard HP-10B –350,000 16, 000 16, 000 466 ,000 7 CFj CFj CFj CFj I/YR {NPV} Sharpe EL-733A –350,000 16, 000 16, 000 466 ,000 7 NPV Texas Instruments BA II Plus CFi CF CFi 2nd CFi CFi i –350,000 16, 000 16, 000 466 ,000 –350,000 16, 000 16, 000 466 ,000 CFj CFj CFj CFj {IRR/YR} Sharpe EL-733A –350,000 16, 000 16, 000 466 ,000... build the office block (Notice that C0 is negative, reflecting the fact that it is a cash outflow.) 3 46 SECTION FOUR FIGURE 4.1 Cash flows and their present values for office block project Final cash flow of $ 466 ,000 is the sum of the rental income in Year 3 plus the forecasted sales price for the building $ 466 ,000 $450,000 $ 16, 000 $ 16, 000 $ 16, 000 0 1 2 3 Present value 16, 000 ϭ 14,953 1.07 16, 000 ϭ 13,975... of return is the discount rate at which NPV equals zero.3 2 If the opportunity cost of capital is less than the project rate of return, then the NPV of your project is positive If the cost of capital is greater than the project rate of return, then NPV is negative Thus the rate of return rule and the NPV rule are equivalent 60 40 Rate of return ϭ 14.3% 20 0 Ϫ20 Ϫ40 NPV profile 60 Ϫ80 0 4 8 12 16. .. + 16, 000 + 16, 000 + 466 ,000 The IRR is the discount rate at which these cash flows would have zero NPV Thus NPV = –$350,000 + $ 16, 000 $ 16, 000 $ 466 ,000 + + =0 1 + IRR (1 + IRR)2 (1 + IRR)3 There is no simple general method for solving this equation You have to rely on a little trial and error Let us arbitrarily try a zero discount rate This gives an NPV of $148,000: NPV = –$350,000 + $ 16, 000 $ 16, 000 $ 466 ,000... present values in 19 86 for the Channel Tunnel The investment apparently had a small positive NPV of £251 million (figures in millions of pounds) Year Cash Flow PV at 13 Percent 19 86 1987 1988 1989 1990 1991 1992 1993 1994 1995 19 96 1997 1998 1999 2000 2001 2002 2003 2004 2005 20 06 2007 2008 2009 2010 Total –£457 –4 76 –497 –522 –551 –584 61 9 211 489 455 502 530 544 63 6 594 68 9 729 7 96 859 923 983 1,050... 14,953 1.07 16, 000 ϭ 13,975 1.072 466 ,000 ϭ 380,395 1.073 409,323 C1 C2 C3 + + 1+r (1 + r)2 (1 + r)3 $ 16, 000 $ 16, 000 $ 466 ,000 = –$350,000 + + + = $59,323 1.07 (1.07)2 (1.07)3 NPV = C0 + Let’s check that the owners of this project really are better off Suppose you put up $350,000 of your own money, commit to build the office building, and sign a lease that will bring $ 16, 000 a year for 3 years Now you... project in each year can be calculated as follows: Book Value Start of Year ($ thousands) Net Income during Year ($ thousands) Book Value, End of Year ($ thousands) Book Rate of Return = Income/Book Value at Start of Year 90 60 30 50 – 30 = 20 50 – 30 = 20 50 – 30 = 20 60 30 0 20/90 = 222 = 22.2% 20 /60 = 333 = 33.3% 20/30 = 66 7 = 66 .7% We have already seen that cash flows and accounting income may... rate of 50 percent In this case NPV is –$194,000: NPV = –$350,000 + SEE BOX $ 16, 000 + $ 16, 000 + $ 466 ,000 = –$194,000 1.50 (1.50)2 (1.50)3 NPV is now negative So the IRR must lie somewhere between zero and 50 percent In Figure 4.3 we have plotted the net present values for a range of discount rates You can see that a discount rate of 12. 96 percent gives an NPV of zero Therefore, the IRR is 12. 96 percent... rate of return is 1 + nominal return 1.108 –1= – 1 = 065 , or 6. 5% 1 + inflation rate 1.04 2 The risk premium on stocks is the average return in excess of Treasury bills This was 14.7 – 5.2 = 9.5% The maturity premium is the average return on Treasury bonds minus the return on Treasury bills It was 6. 4 – 5.2 = 1.2% 3 Rate of Return +70% +10 +10 –50 Deviation Squared Deviation +60 % 0 0 60 3 ,60 0 0 0 3 ,60 0 . 1,000* 1 85 56 59.5 149 80 2 76. 5 51 53.5 164 80 3 69 46 49 180 80 4 62 .5 41.5 47 198 79.5 5 56. 5 38 51.5 217 80 6 56 41.5 56. 5 1 96 80 7 61 .5 45.5 62 177 80 8 67 .5 50 60 160 80.5 9 61 45.5 54 160 72.5 *. % Biogen 46. 6 Compaq 46. 7 Delta Airlines 26. 9 Exxon 16. 0 Ford Motor Co. 24.9 MCI WorldCom 34.4 Merck 24.5 Microsoft 34.0 PepsiCo 26. 5 Xerox 27.3 3 26 SECTION THREE simply the average of the three. deviation of stock market returns, 1994–1998 Deviation from Year Rate of Return Average Return Squared Deviation 1994 1.31 –23.44 549.43 1995 37.43 12 .68 160 .78 19 96 23.07 –1 .68 2.82 1997 33. 36 8 .61