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Chapter 6 Analysis of Risk and Return CHAPTER 6 ANALYSIS OF RISK AND RETURN ANSWERS TO QUESTIONS: 1. a. Risk refers to the chance for some unfavorable event to occur. In finance, risk is the possibility that actual returns or cash flows from an investment will be less than expected. b. Probability distributions define the percentage chance of occurrence of each of one or more possible outcomes. c. The standard deviation is a statistical measure of the dispersion of a variable about the mean. It is an absolute measure of risk and is measured as the square root of the weighted average squared deviations of individual observations from the mean. d. The required rate of return of an investment is the level of return investors demand, given the risk of the investment. e. The coefficient of variation is a measure of relative risk. It is defined as the ratio of the standard deviation to the mean of some variable. f. A portfolio is efficient if, for a given standard deviation, there is no other portfolio with a higher expected return, or, for a given expected return, there is no other portfolio with a lower standard deviation. g. The efficient frontier consists of the set of efficient portfolios h. The capital market line is a line joining the risk-free rate and the market portfolio. It indicates the risk and expected returns that can be obtained by investing various proportions of one’s wealth in the risk-free security and the market portfolio of (risky) securities. i. Beta is a measure of the systematic risk of an asset or security. It is defined as the ratio of the covariance of returns for some asset or security j and the market portfolio m to the variance of returns on the market portfolio. j. CAPM is the Capital Asset Pricing Model, a theory which describes the relationship between risk and required return for securities and other assets. k. The correlation coefficient is a relative statistical measure of the degree to which two series of numbers, such as the returns from two securities, tend to move or vary together. l. A portfolio is a collection of two or more assets or securities. m. The characteristic line is a regression line relating the periodic holding period returns for a specific security to the periodic holding period returns on the market portfolio. n. The security market line defines the relationship between systematic risk and the required returns for individual securities. o. The covariance is an absolute statistical measure of the degree to which two series of numbers tend to move or vary together. p. Systematic risk is that portion of the variability in a security's return that is caused by factors affecting the market as a whole. It is also called nondiversifiable risk. q. Unsystematic risk is risk that is unique to a firm. It is also called diversifiable risk. 2. The probability distribution of a security whose returns are known with certainty is a single vertical line with a height equal to 1.0 (100%) at the level of the known return. 3. "Risk-free" U.S. Government bonds have virtually no risk of default, but they are exposed to interest rate risk -- the chance that interest rates will rise, causing bond prices to fall and investors to experience either a real or an "opportunity" loss. 4. With increased inflation expectations, required returns on bonds would tend to increase and bond prices would decline. 5. The coefficient of variation and the standard deviation of a security's return will give the same relative measure of risk when compared to the risk of another security when both securities have the same level of expected returns. 6. Diversification can reduce the risk of a portfolio of assets below that of the weighted average risk of the individual assets because when assets are combined together in a group or portfolio, extreme returns in one direction from one asset may be offset by extreme returns in the other direction from another asset. 7. The primary variables that influence the risk of a portfolio of assets are the risk of the individual assets in that portfolio, the relative proportion of each asset in the portfolio, and the correlation between the returns of the assets in the portfolio. 8. Systematic risk refers to that portion of the variability of an individual security's return that is caused by factors affecting the market as a whole. It is usually measured by the security's "beta" value. Unsystematic risk is risk that is unique to the firm. It is the variability of a security's returns that is caused by such factors as management capabilities and decisions, strikes, availability of supplies, unique effects of government regulation, the firm's operating and financial leverage, and the effects of foreign competition. An investor who owns a well-diversified portfolio of individual securities will be exposed to very little unsystematic risk. Hence this risk dimension can be ignored in many cases. 9. An increase in the expected future inflation rate increases the investors’ required rates of return on all securities, including common stocks. An increase in the required rate of return causes stock prices to decline as investors become unwilling to purchase these securities at existing prices. 10. A beta of 1.4 indicates that Amrep has more than an average level of systematic risk (an average level being indicated by a beta of 1.0). An investor would expect Amrep common stock to have 40 percent more variability in its systematic returns than a stock with a beta of 1.0. 11. A security's beta value is computed as the slope of a regression line between holding period returns on the market portfolio and the individual security's holding period returns. It is specifically defined as the ratio of the covariance of the security's holding period returns with market holding period returns to the variance of the market’s holding period returns. 12. The beta concept can be used to estimate the cost of equity capital for a firm, assuming that investors tend to hold well-diversified portfolios of securities, making systematic risk the appropriate risk measure. Because beta must be estimated from historical data, it is important to remember that historically estimated betas do not capture any future change in the firm's systematic risk characteristics. Hence, if a firm is about to merge or significantly change its product line makeup, then the historical beta may not be an accurate reflection of expected future systematic risk. When computing the cost of equity using the beta concept, an analyst must estimate the expected risk-free rate and the expected market rate of return. These estimates are subject to differences in judgment between analysts. 13. a. C, D, E, and F b. F c. C 14. The term structure of interest rates is the pattern of interest rate yields for debt securities that are similar in all respects except for their time remaining to maturity. 15. The risk structure of interest rates is the pattern of interest rate yields for debt securities that are similar in all respects except for their risk. 16. Risk in a financial sense is defined as variability of returns. The greater the variability of a security's returns over time, the higher is its risk. 17. In general, the higher a security's risk, the greater is its expected return. 18. Required rates of return on all securities include a default-risk-free rate of return, composed of a real rate of return and a purchasing power loss premium. Government and corporate bonds also have a maturity risk premium, reflecting the term structure of interest rates prevailing in the capital markets. Corporate bonds and equities have a default risk premium to reflect the possibility that interest and/or principal will not be paid when due. Some corporations issue various classes of debt and preferred stock securities. For the more junior debt securities of a firm, that is, the bonds with the least protection in the case of default, such as subordinated debentures, investors demand a seniority risk premium. Within individual security classes, such as common stock or corporate bonds, one observes significant differences in required rates of return between firms. These differences reflect the business and financial risk facing that firm. Finally, some securities, such as the stock of closely held firms or small corporate bond offerings, reflect a marketability risk premium. 19. Downward sloping yield curves usually indicate that investors expect interest rates (and inflation rates) to decline in the future. Upward sloping yield curves reflect both future interest rate expectations but the liquidity premium required for long-term securities. 20. There is virtually no risk of default on U.S. Government bonds, because the U.S. government can always print more money (although it would be worthless). In contrast, IBM bonds do carry some default risk, albeit a small risk. 21. No. The cost of equity capital for a firm must always be greater than the cost of debt capital to that same firm, because equity capital has the lowest priority of payment in case of default, has no fixed dividends, and has no known maturity date. Sometimes, at very high interest rates it might appear that the cost of equity is lower than the cost of debt for a firm, but this illusion simply reflects errors in the process of calculating the cost of equity. SOLUTIONS TO PROBLEMS: 1. a. E(RX) = .1(-10%) + .2(10%) + .4(15%) + .2(20%) + .1(40%) = 15% E(RY) = .2(2%) + .2(7%) + .3(12%) + .2(15%) + .1(16%) = 10% b. X = [(-10-15)2.1 + (10-15)2.2 + (15-15)2.4 + (20-15)2.2 + (40-15)2.1].5 = 11.62% Y = [(2-10)2.2 + (7-10)2.2 + (12-10)2.3 + (15-10)2.2 + (16-10)2.1].5 = 4.94% c. Stock X is riskier because it has a higher standard deviation of returns than Y. The coefficient of variation of returns is also higher for X (0.77) than for Y (0.49). 2. a. Expected Return = r = 0.10(0%) + 0.20(5%) + 0.40(8%) + 0.20(12%) + 0.10(15%) = 8.1% b. Variance of Returns = σ2 = (0.10)(0 – 8.1)2 + (0.20)(5 – 8.1)2 + (0.40)(8 – 8.1)2 + (0.20)(12 – 8.1)2 + (0.10)(15 – 8.1)2 = 6.561 + 1.922 + 0.004 + 3.362 + 5.041 = 16.890 σ = (16.890)1/2 = 4.11% c. Calculate the Coefficient of Variation (CV) to decide which is better. CV1 = 4.11/8.1 = 0.5074 CV2 = 15/12 = 1.25 Investment #1 has 0.51 units of risk per unit of return versus Investment #2’s 1.25 units of risk per unit of return. Given the information, based on risk and return, Investment #1 is more attractive. 3. a. The coefficient of variation of returns for Cornhusker's stock is 0.75 (15%/20%). The coefficient of variation of returns for Mustang's stock is 0.90 (9%/10%). Therefore, according to the coefficient of variation criterion (which considers total risk -- both systematic and unsystematic), Mustang's stock is riskier. b. Looking only at systematic risk (i.e., assuming that investors are well-diversified), Cornhusker appears to be riskier. 4. r = (0.50)(10%) + (0.50)(2%) = 6% σ2 = (0.5)2(20)2 + (0.5)2(0)2 + 2(.5)(.5)(20)(0)(0) = 100 σ = 10% Beta = (0.5)(1.00) + (0.5)(0) = 0.5 5. a. (i) E(R) = .4(.10) + .6(.07) = .082 or 8.2% (ii) p = [(.4)2(.08)2 + (.6)2(.04)2 + 2(.4)(.6)(.5)(.08)(.04)].5= .0487 or 4.87% b. (i) E(R) = .7(.10) + .3(.07) = .091 or 9.1% (ii) p = [(.7)2(.08)2 + (.3)2(.04)2 + 2(.7)(.3)(.5)(.08)(.04)].5= .0629 or 6.29% c. Expected returns remain the same. They do not depend on the correlation of returns among securities. (i) xy = +1; p = .056 or 5.6% (by substituting +1 for + 0.5 in part 6. a. (ii) above. (ii) xy = 0; p = .04 or 4% (iii) xy = -.7; p = .023 or 2.3% 6. a. From a portfolio risk perspective, Xerox would be considered the riskier stock because it has a larger beta (1.2) than Kodak (0.98). When considering each stock individually, a measure of total risk (adjusted for relative magnitudes of expected returns) is appropriate. This measure would be the coefficient of variation. The coefficient of variation for Xerox is 0.30 (4.5%/15%). The coefficient of variation for Kodak is 0.32 (3.8%/12%), indicating that Kodak is the riskier of the two securities. b. E(R) = .35(15%) + .65(12%) = 13.05% c. p = [(.35)2(4.5%)2 + (.65)2(3.8%)2 + 2(.35)(.65)(.6)(4.5%)(3.8%)].5 = 3.64% d. p = .35(1.2) + .65(.98) = 1.06 7. a. k = 5 + 1.65(8) = 18.2% b. k = 5 + 0.80(8) = 11.4% c. k = 5 + 1.05(8) = 13.4% d. k = 5 + 1.85(8) = 19.8% 8. a. kj = .06 + .094(1.5) = .201 or 20.1% b. kj = .08 + .094(1.5) = .221 or 22.1% 9.a. Percentage Rate of Return = [(100,000 - 92,500)/92,500] x 100% = 8.11% b. Percentage Rate of Return = [(1,000 - 975 + 70)/975] x 100% = 9.74% c. Risk premium = Expected return - Risk-free return = 9.74% - 8.11% = 1.63% 10. a. rf = 3% real return + 7% inflation premium = 10% b. 12% - 10% = 2% maturity risk premium between a 10-year bond and a 1-year bond. c. Default risk premium = 13% - 12% = 1% d. Seniority risk premium = 16% - 13% = 3% 11. Expected return = r = (0.10)(-5) + (0.25)(5) + (0.35)(10) + (0.20)(15) + (0.10)(20) = 9.25% Required return = k = 5 + (1.2)(8.5) = 15.2% r < k, therefore, do not invest. 12. a. E(Rp) = .3(14%) + .7(16%) = 15.4% b. i. With perfect positive correlation, the risk of the portfolio is the weighted average risk of the individual securities, or: p = .3(5%) + .7(9%) = 7.8% ii. When the correlation is -0.2, the portfolio risk can be computed as: p = [(.3)2(5)2 + (.7)2(9)2 + 2(.3)(.7)(-0.2)(5)(9)].5 = 6.18% 13. a. r = (0.6)(11) + (0.4)(16) = 13% b. σ2 = (0.6)2(22)2 + (0.4)2(30)2 + 2(0.6)(0.4)(22)(30)(0.4) = 174.24 + 144 + 126.72 = 444.96 σ = 21.09% c. Use the Coefficient of Variation to decide. CV1 = 21.09/13 = 1.62 CV2 = 30/15 = 2.00 Choose Portfolio #1 as it has only 1.62 units of risk per unit of return versus Portfolio #2’s 2 units of risk per unit of return. 14. Expected return = 14%; Required return = 7 + .8(15 - 7) = 13.4% z = (13.4% - 14%)/ 8% = -0.075 From Table V, p(z < - 0.075)  47% 15. a. E(Rp) = .4(.13) + .6(.17) = .154 or 15.4% b. p = [.42(8)2 + .62(14)2 + 2(.4)(.6)(.3)(8)(14)].5 = 9.85% c. Jones: E(R) = 13%; Required return = 8% + .9(15% - 8%) = 14.3% Bush: E(R) = 17%; Required return = 8% + 1.2(15% - 8%) = 16.4% Therefore, the Bush stock is the better buy because the expected return exceeds the required return. 16. E(RA) = .25(10%) + .5(12%) + .25(14%) = 12% E(RB) = .3(13%) + .35(16%) + .35(19%) = 16.15% E(RC) = .4(14%) + .3(18%) + .3(22%) = 17.6% E(Rp) = .4(12%) + .3(16.15%) + .3(17.6%) = 14.93% A = [(10-12)2.25 + (12-12)2.5 + (14-12)2.25].5 = 1.41% B = 2.41% C = 3.32% p = [(.4)2(1.41)2 + (.3)2(2.41)2 + (.3)2(3.32)2 + 2(.4)(.3)(.7)(1.41)(2.41) + 2(.4)(.3)(.6)(1.41)(3.32) + 2(.3)(.3)(.85)(2.41)(3.32)].5 = 2.07% 17. a. z = (0% - 20%)/12% = -1.67 From Table V, p(z < -1.67) = 4.75% b. z = (6% - 20%)/12% = -1.17 From Table V, p(z < -1.17) = 12.1% 18. wx + wy = 1 wy = 1 - wx E(Rp) = wx E(Rx) + wy E(Ry) wx = [E(Rp) - E(Ry)]/[E(Rx) - E(Ry)] = (20% - 26%)/(15% - 26%) = 0.545 or 54.5% wy = 1 - 0.545 = 0.455 or 45.5% p = 0.545(0.94) + 0.455(1.33) = 1.12 19. Expected return = r = 10/30 = 33.3% Standard deviation of returns = 5/30 = 16.7% Z = (10 – 33.3)/16.7 = -1.40 The probability that a return of less than 10% will be realized = 1- 0.9192 = 0.0808 20. a. Security Beta P 0.8(10)(8)/(8)2 = 1.00 Q 0.6(20)(8)/(8)2 = 1.50 R 0.4(15)(8)/(8)2 = 0.75 b. Security Expected Return Beta Required Return P 12% 1.00 7% + 8.8 (1.00) = 15.8% Q 18 1.50 7% + 8.8 (1.50) = 20.2% R 15 0.75 7% + 8.8 (0.75) = 13.6% Only security R appears to be an attractive investment at this time. 21. p = [(0.5)2 (4)2 + (0.5)2 (6)2 + 2(0.5)(0.5)(-1.0)(4)(6)].5 = 1.0% 22. a. ke = 8% + 1.5(8%) = 20% b. z = (20% - 24%)/12% = -0.33 From Table V, p(z > -0.33) = 1.0 - 0.3707 = 0.6293 or 62.93% 23. Inflation premium = 5% - 2% = 3% Maturity risk premium = 8% - 5% = 3% Default risk premium (Forester) = 14% - 8% = 6% Seniority risk premium (Forester common stock) = 18% - 14% = 4% Investors may require a higher rate of return on Brown’s common stock compared to Forester’s because Brown may have greater business and/or financial risk than Forester. Also, there could be a difference in the marketability (liquidity) risk premiums of the two stocks. 24. kj = 4% + 8.8%(1.5) = 17.2% Expected return = ($60 - $50)/$50 = 20% A required rate of return of 17.2% implies that the stock must appreciate to $50(1.172) = $58.60 by the end of the year. The probability that the stock is overvalued is equal to the probability that the stock price will be less than $58.60 at the end of the year. Given that the stock price is normally distributed with a mean of $60 and a standard deviation of $2.50, then: p(Stock price < $58.60) = p(z < ($58.60 - $60)/$2.5) = p(z < -0.56) From Table V, p(z < -0.56) = 0.2877 or 29% 25. a. p = .2 (0.7) + .4 (1.3) + .4 (1.1) = 1.10 b. 1.00 = (.2 + x)(0.7) + (.4 - x)(1.3) + .4(1.1) x = 0.16667 or 0.16667($100,000) = $16,667 wA = 0.36667; wB = 0.23333; wc = 0.40 c. kj = 5 + 8.8j Texaco: kj = 5 + 8.8(0.70) = 11.16% General Electric: kj = 5 + 8.8(1.3) = 16.44% Whirlpool: kj = 5 + 8.8(1.1) = 14.68% Original portfolio: E(Rp) = .2(11.16) + .4(16.44) + .4(14.68) = 14.68% Revised portfolio: E(Rp) = .36667(11.16) + .23333(16.44) + .4(14.68) = 13.80% 26. a. kj = 8 + 8.6  j b. No picture provided. SML will have a y intercept of 8 percent and the same slope as that shown in Figure 6.16. c. Before: kj = 6 + 8.6 ( 1.65) = 20.2% After: kj = 8 + 8.6 ( 1.65) = 22.2% 27. Required return = 6% + 0.9(7.4%) = 12.66% Expected return: $15 = $25 (PVIFk, 5) ke = 10.76% The stock appears to be overvalued because the expected return is less than the required return. z = (12.66 - 10.76)/2 = 0.95 p (z > 0.95) = 17.11% Hence the probability that the stock is overvalued is: 100% - 17.11% = 82.89% 28. No recommended solution.  65