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Chapter 07 - Introduction to Risk and Return CHAPTER 7 Introduction to Risk and Return The values shown in the solutions may be rounded for display purposes. However, the answers were derived using a spreadsheet without any intermediate rounding. Answers to Problem Sets 1. Expected payoff = (.10 × $500) + (.50 × $100) + (.40 × $0) = $100 Rates of return: ($500 – 100) / $100 = 400% ($100 – 100) / $100 = 0% ($0 – 100) / $100 = –100% Expected rate of return = (.10 × 400%) + (.50 × 0%) + (.40 × –100%) Expected rate of return = 0% Variance = .10(400% – 0)2 + .50(0% – 0)2 + .40(–100% – 0)2 Variance = 20,000 Standard deviation = 20,000.5 Standard deviation = 141.42% Est. Time: 01 – 05 2. a. Average nominal return = (.172 + .010 + .161 + .331 + .127) / 5 Average nominal return = .1602, or 16.02% Variance = [(.172 – .1602)2 + (.010 – .1602)2 + (.161 – .1602)2 + (.331 – .1602)2 + (.127 – .1602)2] / 5 Variance = .010595 Standard deviation = .010595.5 Standard deviation = .1029, or 10.29% b. Average real return = {[(1.172 / 1.015) – 1] + [(1.010 / 1.030) – 1] + [(1.161 / 1.017) – 1] + [(1.331 / 1.015) – 1] + [(1.127 / 1.008) – 1]} / 5 Average real return = .1412, or 14.12% Est. Time: 01 – 05 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return 3. Ms. Sauros: Average return = [.249 + (–.009) + .186 + .421 + .152] / 5 Average return = .1998, or 19.98% Variance = [(.249 – .1998)2 + (–.009 – .1998)2 + (.186 – .1998)2 + (.421 – .1998)2 + (.152 – .1998)2] / 5 Variance = .019485 Standard deviation = .0194 Standard deviation = .1396, or 13.96% S&P 500: Average return = (.172 + .010 + .161 + .331 +.127) / 5 Average return = .1602, or 16.02% Variance = [(.172 – .1602)2 + (.010 – .1602)2 + (.161 – .1602)2 + (.331 – .1602)2 + (.127 – .1602)2 Variance = .010595 Standard deviation = .010595.5 Standard deviation = .1029, or 10.29% Est. Time: 01 – 05 4. a. False. Investors prefer diversified portfolios because diversification reduces variability and therefore reduces risk. However, the diversification of an individual company does not necessarily make it less risky. b. True. Stocks must be less than perfectly positively correlated in order to obtain diversification benefits. c. False. The risk eliminated by diversification is called specific risk, or the risk surrounding an individual company or industry. Market risk will still exist in a fully diversified portfolio. d. False. It is true that the greatest benefit to diversification occurs when stocks are uncorrelated. However, most stocks tend to move in the same direction. There are still benefits to diversification any time stocks are less than perfectly positively correlated. e. False. The contribution to portfolio risk depends on the relationship of the stock to the market as a whole. Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return f. True. Market risk is the risk that relates to a diversified portfolio. g. True. The only risk inherent in a diversified portfolio is market risk. h. False. An undiversified portfolio with a beta of 2 is twice as risky as the market portfolio. Part of that risk will be market risk and part will be specific risk. Est. Time: 01 – 05 5. Perfect negative correlation does the most to reduce risks because the stocks always move in opposite directions. When one goes up, the other goes down, and vice versa. Est. Time: 01 – 05 6. Est. Time: 01 – 05 7. a. σp = 1.3 × 20% = 26% b. σp = 0 × 20% = 0% c. βp = 15% / 20% = .75 d. βp = Less than 1 βp = 20% / 20% = 1, This would be the beta if the portfolio were diversified. Since the portfolio is non-diversified, the beta must be less than 1 because part of the portfolio’s risk is specific, or unique, risk. Est. Time: 01 – 05 8. βp = {(5 × 1.2) + [(10 – 5) × 1.4)]} / 10 βp = 1.3 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return Beta measures systematic risk which cannot be eliminated by diversification. Est. Time: 01 – 05 9. The beta of each stock is given by the slope of the line, or the rise divided by the run. The run is the range of the market returns while the rise is the range of the stock returns. BetaA = (0 – 20) / (–10 – 10) = 1 BetaB = (–20 – 20) / (–10 – 10) = 2 BetaC = (–30 – 0) / (–10 – 10) = 1.5 BetaD = (15 – 15) / (–10 – 10) = 0 BetaE = (10 – (–10) / (–10 – 10) = –1 Est. Time: 01 – 05 10. a. r = [(1 + R) / (1 + h)] – 1 r1929 = {[1 + (–.145)] / (1 + .002)} – 1 = –.1467, or –14.67% r1930 = {[1 + (–.283)] / [1 + (–.060)]} – 1 = –.2372, or –23.72% r1931 = {[1 + (–.439)] / [1 + (–.095)]} – 1 = –.3801, or –38.01% r1932 = {[1 + (–.099)] / [1 + (–.103)]} – 1 = .0045 or .45% r1933 = [(1 + .573) / (1 + .005)] – 1 = .5652, or 56.52% b. Average real return = [–.1433 + (–.2372) + (–.3801) + .0045 + .5652] / 5 Average real return = –.0382, or –3.82% c. Risk premium1929 = –.145 – .048 = –.1930, or –19.30% Risk premium1930 = –.283 – .024 = –.3070, or –30.70% Risk premium1931 = –.439 – .011 = –.4500, or –45.00% Risk premium1932 = –.099 – .010 = –.1090, or –10.90% Risk premium1933 = .573 – .003 = .5700, or 57.00% d. Average risk premium = [–.1930 + (–.3070) + (–.4500) + (–.1090) + .5700] /5 Average risk premium = –0978, or –9.78% Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return e. σRisk premium = {[–.1930 – (–.0978)]2 + [–.3070 – (–.0978)]2 + [–.4500 – (– .0978)]2 + [–.1090 – (–.0978)]2 + [.5700 – (–.0978)]2]} / 5 σRisk premium = .1246, or 12.46% Est. Time: 06 – 10 11. a. A long-term United States government bond is always considered to be safe in terms of the dollars received. However, the price of the bond fluctuates as interest rates change, and the rate at which coupon payments received can be invested also changes as interest rates change. And, of course, the payments are all in nominal dollars, so inflation risk must also be considered. b. It is true that stocks offer higher long-run rates of return than do bonds, but it is also true that stocks have a higher standard deviation of return. So, which investment is preferable depends on the amount of risk one is willing to tolerate. This is a complicated issue and depends on numerous factors, one of which is the investment time horizon. If the investor has a short time horizon, then stocks are generally not preferred. c. Unfortunately, 10 years is not generally considered a sufficient amount of time for estimating average rates of return. Thus, using either a 5- or 10year average is likely to be misleading. Est. Time: 06 – 10 12. The risk to Hippique shareholders depends on the market risk, or beta, of the investment in the black stallion. The information given in the problem suggests that the horse has very high unique risk, but we have no information regarding the horse’s market risk. So, the best estimate is that this horse has a market risk about equal to that of other racehorses, and thus this investment is not a particularly risky one for Hippique shareholders. Est. Time: 01 – 05 13. In the context of a well-diversified portfolio, the only risk characteristic of a single security that matters is the security’s contribution to the overall portfolio risk. This contribution is measured by beta. Lonesome Gulch is the safer investment for a diversified investor because its beta of .10 is lower than the beta of Amalgamated Copper of .66. For a diversified investor, the standard deviations are irrelevant. Est. Time: 01 – 05 14. a. σP2 = .602 × .102 + .402 × .202 + 2(.60 × .40 × 1 × .10 × .20) Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return σP2 = .0196 b. σP2 = .602 × .102 + .402 × .202 + 2(.60 × .40 × .50 × .10 × .20) σP2 = .0148 c. σP2 = .602 × .102 + .402 × .202 + 2(.60 × .40 × 0 × .10 × .20) σP2= .0100 Est. Time: 06 – 10 15. a. Refer to Figure 7.13 in the text. With 100 securities, the box is 100 by 100. The variance terms are the diagonal terms, and thus there are 100 variance terms. The rest are the covariance terms. Because the box has (100 × 100) terms altogether, the number of covariance terms is: Number of covariance terms = 1002 – 100 = 9,900 Half of these terms (i.e., 4,950) are different. b. Once again, it is easiest to think of this in terms of Figure 7.13. With 50 stocks, all with the same standard deviation (.30), the same weight in the portfolio (.02), and all pairs having the same correlation coefficient (.40), the portfolio variance is: σ2 = 50(.02)2(.30)2 + [(50)2 – 50](.02)2(.40)(.30)2 = .03708 σ = .193, or 19.3% c. For a fully diversified portfolio, portfolio variance equals the average covariance: σ2 = (.30)(.30)(.40) = .036 σ = .190, or 19.0% Est. Time: 06 – 10 16. a. Refer to Figure 7.13 in the text. For each different portfolio, the relative weight of each share is (1 / number of shares (n) in the portfolio), the standard deviation of each share is .40, and the correlation between pairs is .30. Thus, for each portfolio, the diagonal terms are the same, and the off-diagonal terms are the same. There are n diagonal terms and (n2 – n) off-diagonal terms. In general, we have: Variance = n(1 / n)2(.4)2 + (n2 – n)(1 / n)2(.3)(.4)(.4) Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return For one share: Variance = 1(1)2(.4)2 + 0 = .160000 For two shares: Variance = 2(.5)2(.4)2 + 2(.5)2(.3)(.4)(.4) = .104000 The results are summarized in the second and third columns of the table in part (c) below. b. The underlying market risk that cannot be diversified away is the second term in the formula for variance above: Underlying market risk = (n2 – n)(1 / n)2(.3)(.4)(.4) As n increases, [(n2 – n)(1 / n)2] = [(n – 1) / n] becomes close to 1, so that the underlying market risk is: [(.3)(.4)(.4)] = .048. c. This is the same as Part (a), except that all of the off-diagonal terms are now equal to zero. The results are summarized in the fourth and fifth columns of the table below. (Part a) No. of Shares 1 2 3 4 5 6 7 8 9 10 Variance .160000 .104000 .085333 .076000 .070400 .066667 .064000 .062000 .060444 .059200 (Part a) Standard Deviation .400 .322 .292 .276 .265 .258 .253 .249 .246 .243 (Part c) Variance .160000 .080000 .053333 .040000 .032000 .026667 .022857 .020000 .017778 .016000 (Part c) Standard Deviation .400 .283 .231 .200 .179 .163 .151 .141 .133 .126 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return Graphs for Part (a): Portfolio Standard Deviation Portfolio Variance 0 .5 Standard Deviation 0 .2 Variance 0 .1 5 0 .1 0 .0 5 0 0 .4 0 .3 0 .2 0 .1 0 0 2 4 6 8 10 12 0 2 N u m b e r o f S e c u ritie s 4 6 8 10 12 N u m b e r o f S e c u ritie s Graphs for Part (c): Portfolio Standard Deviation Portfolio Variance 0 .5 Standard Deviation 0 .2 Variance 0 .1 5 0 .1 0 .0 5 0 0 .4 0 .3 0 .2 0 .1 0 0 2 4 6 8 N u m b e r o f S e c u ritie s 10 12 0 2 4 6 8 10 12 N u m b e r o f S e c u ritie s Est. Time: 16 – 20 17. The table below uses the format of Figure 7.13 in the text in order to calculate the portfolio variance. The portfolio variance is the sum of all the entries in the matrix. Portfolio variance is .03178. BHP BP Fiat Heineken Korea Electric Nestlé Sony Tata Motors BHP .0006126 .0003781 .0005062 .0000893 .0002841 -.0000090 .0002636 .0006050 BP .0003781 .0013231 .0007832 .0002051 .0003290 .0000529 .0008359 .0005157 Korea Fiat Heineken Electric .0005062 .0000893 .0002841 .0007832 .0002051 .0003290 .0028971 .0002063 .0003558 .0002063 .0005085 .0001334 .0003558 .0001334 .0012102 -.0000653 .0001203 .0000042 .0013274 .0004677 .0003120 .0008420 .0002866 .0002211 Nestlé -.0000090 .0000529 -.0000653 .0001203 .0000042 .0001470 .0001563 .0000474 Sony .0002636 .0008359 .0013274 .0004677 .0003120 .0001563 .0031416 .0005206 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Tata .0006050 .0005157 .0008420 .0002866 .0002211 .0000474 .0005206 .0023900 Chapter 07 - Introduction to Risk and Return Est. Time: 21 – 25 18. “Safest” means lowest risk; in a portfolio context, this means lowest variance of return. Half of the portfolio is invested in British Petroleum stock (BP), and half of the portfolio must be invested in one of the other securities listed. Thus, we calculate the portfolio variance for seven different portfolios to see which is the lowest (see table below). The safest attainable portfolio is comprised of British Petroleum stock (BP) and Nestlé. Company with BP: Variance Variance BHP = .52 × .19802 + .52 × .29102 + 2 × .5 × .5 × .42 × .1980 × .2910 = .04307 Fiat Chrysler = .52 × .43062 + .52 × .29102 + 2 × .5 × .5 × .40 × .4306 × .2910 = .09259 Heineken = .52 × .18042 + .52 × .29102 + 2 × .5 × .5 × .25 × .1804 × .2910 = .03587 Korea Electric = .52 × .27832 + .52 × .29102 + 2 × .5 × .5 × .26 × .2783 × .2910 = .05106 Nestlé = .52 × .09702 + .52 × .29102 + 2 × .5 × .5 × .12 × .0970 × .2910 = .02522 Sony = .52 × .44842 + .52 × .29102 + 2 × .5 × .5 × .41 × .4484 × .2910 = .09819 Tata Motors = .52 × .39112 + .52 × .29102 + 2 × .5 × .5 × .29 × 39.11 × .2910 = .07591 Est. Time: 11 – 15 19. a-1. Change in stock’s rate of return = .05 × –.25 = –.0125, or –1.25% a-2. Change in stock’s rate of return = –.05 × –.25 = .0125, or 1.25% b. “Safest” implies lowest risk. Assuming the well-diversified portfolio is invested in typical securities, the portfolio beta is approximately one. The largest reduction in beta is achieved by investing the $20,000 in a stock with the negative beta. Est. Time: 06 – 10 20. E(r)P = .12 = .1wa + .15(1 – wa); wa = .60; (1 – wa) = .40 σp = (.602 × .202 + .402 × .402 + 2 × .60 × .40 × .50 × .20 × .40).5 σp = .2433, or 24.33% Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return Est. Time: 06 – 10 21. a. In general: σP = (x1212 + x2222 + 2x1x21212).5 Thus: P = (.52 × .35802 + .52 × .21002 + 2 × .5 × .5 × .30 × .3580 × .2100).5 P = .2331, or 23.31% b. We can think of this in terms of Figure 7.13 in the text, with three securities. One of these securities, T-bills, has zero risk and, hence, zero standard deviation. Thus: P = [(1/3)2 × .35802 + (1/3)2 × .21002 + 2 × (1/3) × (1/3) × .30 × .3580 × .2100].5 P = .1554, or 15.54% Another way to think of this portfolio is that it is comprised of one-third T-Bills and two-thirds a portfolio which is half Bank of America and half Starbucks. Because the risk of T-bills is zero, the portfolio standard deviation is two-thirds of the standard deviation computed in Part (a) above: P = (2/3) × 23.31% = 15.54% c. With 50% margin, the investor invests twice as much money in the portfolio as he had to begin with. Thus, the risk is twice that found in Part (a) when the investor is investing only his own money: P = 2 23.31% = 46.62% Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return d. With 100 stocks, the portfolio is well diversified, and hence the portfolio standard deviation depends almost entirely on the average covariance of the securities in the portfolio (measured by beta) and on the standard deviation of the market portfolio. Thus, for a portfolio made up of 100 stocks each with Bank of America’s beta of 1.57, the portfolio standard deviation is approximately: P = 1.57 23.0% = 36.11% For stocks like Starbucks, the approximate standard deviation is: P = .83 23.0% = 19.09% Est. Time: 11 – 15 22. For a two-security portfolio, the formula for portfolio risk is: σP2= x1212 + x2222 + 2x1x21212 If security one is Treasury bills and security two is the market portfolio, then 1 is zero, and 2 is 20%. Therefore: σP2 = x2222 = x22 × .202 σP = .20x2 Portfolio expected return = x1(.06) + x2(.06 + .085) Portfolio expected return = .06x1 + .145x2 Portfolio x1 x2 1 2 3 4 5 6 1.0 .8 .6 .4 .2 .0 .0 .2 .4 .6 .8 1.0 Expected Return .060 .077 .094 .111 .128 .145 Standard Deviation .000 .040 .080 .120 .160 .200 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return Portfolio Risk & Return .16 Expected Return .14 .12 .10 .08 .06 .04 .02 .00 .00 .05 .10 .15 .20 .25 Standard Deviation Est. Time: 11 – 15 23. The matrix below displays the variance for each of the eight stocks along the diagonal and each of the covariances in the off-diagonal cells: Variances/Covariances BHP BP Fiat Heineken Korea Nestlè Sony Tata σs,mkt BHP 0.0392040 0.0241996 0.0323983 0.0057151 0.0181841 -0.0005762 0.0168688 0.0387189 0.0218391 BP 0.0241996 0.0846810 0.0501218 0.0131241 0.0210562 0.0033872 0.0534986 0.0330049 0.0353842 Fiat 0.0323983 0.0501218 0.1854164 0.0132056 0.0227688 -0.0041768 0.0849557 0.0538905 0.0548225 Heineken 0.0057151 0.0131241 0.0132056 0.0325442 0.0085349 0.0076995 0.0299298 0.0183442 0.0161372 Korea 0.0181841 0.0210562 0.0227688 0.0085349 0.0774509 0.0002700 0.0199664 0.0141496 0.0227976 Nestlè -0.0005762 0.0033872 -0.0041768 0.0076995 0.0002700 0.0094090 0.0100038 0.0030349 0.0036314 Sony 0.0168688 0.0534986 0.0849557 0.0299298 0.0199664 0.0100038 0.2010626 0.0333202 0.0562007 Tata 0.0387189 0.0330049 0.0538905 0.0183442 0.0141496 0.0030349 0.0333202 0.1529592 0.0434278 The covariance of BP with the market portfolio (σBP, Market) is the mean of the eight respective covariances between BP and each of the eight stocks in the portfolio. (The covariance of BP with itself is the variance of BP.) Therefore, σBP, Market is equal to the average of the eight covariances in the second row or, equivalently, the average of the eight covariances in the second column. Beta for BP is equal to the covariance divided by the market variance, which we calculated at 0.03178 in problem 17. The covariances and betas are displayed in the table below: Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter 07 - Introduction to Risk and Return BHP BP Fiat Heineken Korea Nestlè Sony Tata Covariance 0.0218391 0.0353842 0.0548225 0.0161372 0.0227976 0.0036314 0.0562007 0.0434278 Market Variance 0.0317801 0.0317801 0.0317801 0.0317801 0.0317801 0.0317801 0.0317801 0.0317801 Beta 0.6871943 1.1134082 1.7250607 0.5077763 0.7173556 0.1142674 1.7684269 1.3665106 Est. Time: 21 – 25 Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.