Risk and Returns

Editor's Notes

• #17: We can use the standard deviation (􏰦, pronounced “sigma”) to quantify the tightness of the probability distribution.7 The smaller the standard deviation, the tighter the probability distribution and, accordingly, the lower the risk.
• #20: If we assume that the probability distribution of returns for the Norman Company is normal, 68% of the possible outcomes would have a return ranging between 13.59 and 16.41% for asset A and between 9.34 and 20.66% for asset B; 95% of the possible return outcomes would range between 12.18 and 17.82% for asset A and between 3.68 and 26.32% for asset B; and 99% of the possible return outcomes would range between 10.77 and 19.23% for asset A and between 􏰂1.98 and 31.98% for asset B. The greater risk of asset B is clearly reflected in its much wider range of possible returns for each level of confidence (68%, 95%, etc.).
• #21: When the standard deviations (from Table 5.5) and the expected returns (from Table 5.4) for assets A and B are substituted into Equation 5.4, the coefficients of variation for A and B are 0.094 (1.41%􏰇15%) and 0.377 (5.66%􏰇15%), respectively. Asset B has the higher coefficient of variation and is therefore more risky than asset A—which we already know from the standard deviation. (Because both assets have the same expected return, the coefficient of variation has not provided any new information.)
• #22: Judging solely on the basis of their standard deviations, the firm would prefer asset C, which has a lower standard deviation than asset D (9% versus 10%). However, management would be making a serious error in choosing asset C over asset D, because the dispersion—the risk—of the asset, as reflected in the coeffi- cient of variation, is lower for D (0.50) than for C (0.75). Clearly, using the coef- ficient of variation to compare asset risk is effective because it also considers the relative size, or expected return, of the assets.
• #30: To reduce overall risk, it is best to combine, or add to the portfolio, assets that have a negative (or a low positive) correlation. Combining negatively correlated assets can reduce the overall variability of returns. Figure 5.6 shows that a portfolio containing the negatively correlated assets F and G, both of which have the same expected return, 􏰕k, also has that same return 􏰕k but has less risk (variability) than either of the individual assets. Even if assets are not negatively correlated, the lower the positive correlation between them, the lower the resulting risk. Some assets are uncorrelated—that is, there is no interaction between their returns. Combining uncorrelated assets can reduce risk, not so effectively as com- bining negatively correlated assets, but more effectively than combining positively correlated assets.
• #31: The assets therefore have equal return and equal risk. The return patterns of assets X and Y are perfectly negatively correlated. They move in exactly opposite directions over time. The returns of assets X and Z are per- fectly positively correlated. They move in precisely the same direction. The risk in this portfolio, as reflected by its standard deviation, is reduced to 0%, whereas the expected return remains at 12%. Thus the combination results in the complete elimination of risk. Whenever assets are perfectly negatively correlated, an optimal combination (similar to the 50–50 mix in the case of assets X and Y) exists for which the resulting standard deviation will equal 0.
• #32: Three possible correlations—perfect positive, uncorrelated, and perfect nega- tive—illustrate the effect of correlation on the diversification of risk and return. Table 5.9 summarizes the impact of correlation on the range of return and risk for various two-asset portfolio combinations. The table shows that as we move from perfect positive correlation to uncorrelated assets to perfect negative corre- lation, the ability to reduce risk is improved. Note that in no case will a portfolio of assets be riskier than the riskiest asset included in the portfolio. In all cases, the return will range between the 6% return of R and the 8% return of S. The risk, on the other hand, ranges between the individual risks of R and S (from 3% to 8%) in the case of perfect positive correlation, from below 3% (the risk of R) and greater than 0% to 8% (the risk of S) in the uncorrelated case, and between 0% and 8% (the risk of S) in the perfectly negatively correlated case.
• #33: note that as the correlation becomes less positive and more negative (moving from the top of the figure down), the ability to reduce risk improves.
• #34: Figure 5.8 depicts the behavior of the total portfolio risk (y axis) as more securities are added (x axis). With the addition of securities, the total portfolio risk declines, as a result of the effects of diversification, and tends to approach a lower limit. Research has shown that, on average, most of the risk-reduction benefits of diver- sification can be gained by forming portfolios containing 15 to 20 randomly selected securities.17
• #35: Diversifiable risk (sometimes called unsystematic risk) represents the portion of an asset’s risk that is associated with random causes that can be eliminated through diversification. It is attributable to firm-specific events, such as strikes, lawsuits, regulatory actions, and loss of a key account. Nondiversifiable risk (also called systematic risk) is attributable to market factors that affect all firms; it can- not be eliminated through diversification. (It is the shareholder-specific market risk described in Table 5.1.) Factors such as war, inflation, international inci- dents, and political events account for nondiversifiable risk. Because any investor can create a portfolio of assets that will eliminate virtu- ally all diversifiable risk, the only relevant risk is nondiversifiable risk. Any investor or firm therefore must be concerned solely with nondiversifiable risk. The measurement of nondiversifiable risk is thus of primary importance in select- ing assets with the most desired risk–return characteristics.
• #38: Because any investor can create a portfolio of assets that will eliminate virtu- ally all diversifiable risk, the only relevant risk is nondiversifiable risk. Any investor or firm therefore must be concerned solely with nondiversifiable risk. The measurement of nondiversifiable risk is thus of primary importance in select- ing assets with the most desired risk–return characteristics.
• #40: The tendency of a stock to move with the market is measured by its beta coefficient, b. Ideally, when estimating a stock’s beta, we would like to have a crystal ball that tells us how the stock is going to move relative to the overall stock market in the future. But since we can’t look into the future, we often use historical data and assume that the stock’s historical beta will give us a reasonable estimate of how the stock will move relative to the market in the future.
• #41: Note that the horizontal (x) axis measures the historical market returns and that the vertical (y) axis mea- sures the individual asset’s historical returns. The first step in deriving beta involves plotting the coordinates for the market return and asset returns from various points in time. Such annual “market return–asset return” coordinates are shown for asset S only for the years 1996 through 2003. For example, in 2003, asset S’s return was 20 percent when the market return was 10 percent. By use of statistical techniques, the “characteristic line” that best explains the relationship between the asset return and the market return coordinates is fit to the data points.18 The slope of this line is beta.
• #42: The return of a stock that is half as respon- sive as the market (b 􏰀 .5) is expected to change by 1/2 percent for each 1 percent change in the return of the market portfolio. A stock that is twice as responsive as the market (b 􏰀 2.0) is expected to experience a 2 percent change in its return for each 1 percent change in the return of the market portfolio.
• #44: Portfolio betas are interpreted in the same way as the betas of individual assets. They indicate the degree of responsiveness of the portfolio’s return to changes in the market return. For example, when the market return increases by 10 percent, a portfolio with a beta of .75 will experience a 7.5 percent increase in its return (.75 􏰄 10%); a portfolio with a beta of 1.25 will experience a 12.5 per- cent increase in its return (1.25 􏰄 10%). Clearly, a portfolio containing mostly low-beta assets will have a low beta, and one containing mostly high-beta assets will have a high beta.
• #45: The preceding section demonstrated that under the CAPM theory, beta is the most appropriate measure of a stock’s relevant risk. The next issue is this: For a given level of risk as measured by beta, what rate of return is required to compensate investors for bearing that risk? risk-free of interest, RF, which is the required return on a risk-free asset, typically a 3-month U.S. Treasury bill (T-bill), a short-term IOU issued by the U.S. Treasury
• #46: The market risk premium, RPM, shows the premium that investors require for bearing the risk of an average stock. The size of this premium depends on how risky investors think the stock market is and on their degree of risk aversion.
• #49: the higher the beta, the higher the required return, and the lower the beta, the lower the required return.