VIEWS: 7 PAGES: 9 CATEGORY: Business POSTED ON: 2/24/2011 Public Domain
CHAPTER 18 CAPITAL ASSET PRICING THEORY ANSWERS TO END OF CHAPTER QUESTIONS & PROBLEMS Mini-case 1 HP TOOT TOY 1. ERs = .05 + [.086] = .173 .133 .152 2. Graph 3. PR = {Pt+1 - Pt + DIV]/Pt = .187 .141 .086 4. If PR > ERs it is undervalued undervalued undervalued overvalued If PR < ERs it is overvalued 5. Pt = [Pt+1 + DIV]/(1+ERs) = [$77 + $3.55]/(1+.133) = $71.09 6. Pt = [Pt+1 + DIV]/(1+ERs) = [$38 + 0.0]/(1+.152) = $32.99 Mini-case 2 1. Run the regression using LINEST from Excel. 2. Intepret the R2 as the percentage of the variance in the stock return explained by the market return. The square root of R2 is the correlation coefficient between the stock return and the market return, or CORR(s,M). 3. The total risk for each stock is the standard deviation risk. 4. The systematic risk can be calculated as R2 X SDs2 and the diversifiable risk is calculated as (1-R2) X SDs2. To the extent that R2 is closer to 1.0 (the highest value possible), it has higher systematic risk. That makes sense because the greater correlation between the stock and the market returns indicates the risk contribution of the stock is greater to the market risk. Discuss Questions and Problems 1. Assumptions 1, 2, and 3 are necessary in order to have the same opportunities (assumption 1), assumption 2 enable investors to create their optimal portfolios , and assumption 3’s risk aversion allows one to generalize decisions on investment choices. Assumptions 4 and 5 simplify the model by assuming it is an one-period model (assumption 4) and that all investors can borrow and lend at the riskfree rate (assumption 5). 2. a. ERp = 10.6% b. SDp = .80(SDp3) = 12% c. graph 3. The Separation Theorem results from the introduction of the riskfree asset. the theorem is important because it states that investors can separate their individual preference from the choice of a risky portfolio. All investors will choose portfolio M and then using their individual preference functions will decide on the proportional investment in M and the riskfree asset. b. The Capital Market Line (CML) results from the introduction of the riskfree asset and the resulting Separation Theorem. c. Portfolio M is preferred over all other risky portfolio by all risk averse investors. 4. a. The beta or systematic risk and the diversifiable or firm specific risk are the two types of risk. b. Only the beta risk relevant since all investors hold a well-diversified portfolio M and the diversifiable risk is eliminated through diversification. 5. UAL predicted return is PR = 16.2% and the CAPM required return for UAL is 16.75%. UAL is priced approximately correctly, so it is neither overvalued nor undervalued. An investor would not earn extraordinary returns, but just what is required (about 16%). 6. ERs = .176 PR = .073 Since PR < ERs the stock is expected to earn a return less than required. So the stock is not a good investment and is overvalued. 7. See graph below. AMR’s required return with beta equaled to 1.35 and ERs= .146 should also lie on the line. SML for American Airlines 0.2 0.15 ERs 0.1 SML 0.05 0 0 0.5 1 1.5 Beta 8. ERs = .185 PR = .125 Since Delta’s PR < ERs, it is not a good investment. It is overvalued. 9. Delta has the same SML as American Airlines because RF = .06 and the market risk premium [ERM - RF] = .086. 10. a. Beta = .90 for GM and PEP b. The two betas are equal even if GM’s total risk is almost half of PEP’s total risk, SD. The reason is that GM’s correlation with the market is much greater than that of PEP’s and so the two variables create an offsetting effect. c. The example shows that both the correlation and the total risk (SD) of the security are important factors to determine betas. Both should be considered when betas are evaluated for securities. 11. a. Beta = CORR(s, M)[SDs/SDM] BetaGE = (.65)[.20/.50] = .26 BetaAT&T = (.45)[.60/.50] = .54 b. Disagree. Though the correlation is an important component for calculating beta, it also depends on the Sds. In this example, GE has a higher correlation, but lower beta because the SD for GE is only 1/3 of AT&T’s SD. 12. a. ERA = 11.46% or rounded to 11.5% ERB = 8.88% or rounded to 9.0% b. graph beta on x axis and ER on y axis with beta of 1.10 corresponding to ER of 11.5% and beta of .80 corresponding to Er of 9%. Fund A falls above the SML and Fund B falls below the SML. c. PRA = 16% and PRB = 8% so Fund A is undervalued and Fund B is overvalued. d. Price today =( Pt+1 + DIVt+1)/(1+ERs) = $55.61 for A and $76.15 for B. 13. Stock X Stock Y a. ERs = .114 .184 b. graph c. PR = .346 .256 Both stocks are undervalued because PR > ERs and they are good investments. d. Price = $48.92 $66.55 14. Stock W Stock Z a. Ers = .118 .130 b. graph c. PR = .156 .314 Both stocks are undervalued because PR > ERs and they are good investments. d. Price = $23.26 $40.71 15. Beta for C = .67 and ERc = .12 and Price = $93.75. 16. Beta = (CORR(s,M)){SDs/SDM] = (-.25)[.30/.40] = -.1875 ERs = .03 +- .1875[.13 - .03] = .01125 PR = [$80 - $65 + 4.00]/$65 = 29.2% Invest because PR > ERs. 17. a. Vs0 = Pt+1 + DIV/(1 + Ers) = ($80 + $4)/(1+.1125) = $75.51 b. The stock is unvrvalued as its intrinsic value = $75.51 and its current stock price = $65. 18. a. BetaIBM = 1.70 b. BetaIBM = 0.25 c. If your investment portfolio consist oflarger firms like the Dow, then the Dow would better represent IBM’s systematic risk for your investment purpose; however, if you hold a more diversified, larger portfolio, the beta calculated using the Wilshire Index may better represent your beta risk. Another point, the beta estimated using the Dow as the market proxy seems “low”. 19. The slope of the SML is steeper if the Wilshire Index is used as the market proxy. IBM falls below the SML if the Wilshire Index is used. On the other hand, if the Dow is used as the market proxy, IBM falls above the SML. Suppose the predicted return for IBM is 28%. Using the beta estimated by the Dow and its corresponding SML(DOW), IBM is overvalued; however, using the beta estimated by the Wilshire and its corresponding SML (WIL), shows that IBM is undervalued. Of course, the drawback from using the wrong index is making the wrong investment decision. 20. a. DJIA Beta = CORR(s,M)[SDs/SDM] = (.55)[.60/.20] = 1.65 b. WIL Beta = (.5)[.60/.30] = 1.30 c. If the investor’s portfolio holding is diverse including small company stocks, the the WIL beta is a more appropriate measure of security risk. If the investor’s portfolio holding consist of large company stocks that represented by the Dow 30 stocks, then the investor’s security risk is better reflected by the DJIA beta. 21. a. ERs = .03 + 1.65[.18 - .03] = .2775 b. Ers = .03 + 1.30[.20 - .03] = .251 c. PR = [$169 - $134]/$134 = .26 d. Using DJIA, Microsoft is overvalued because PR < ERs. Using WIL, Microsoft is undervalued because PR > ERs. e. The problems are obvious. If the wrong index is used, an investor can make the wrong investment decision about Microsoft as well as other stocks. 22. Graph 23. a. [R2]1/2 = correlation coefficient = [.35]1/2 = .59 b. beta = 1.32 c. SD2 = (R2)SD2 + (1-R2)SD2 or .04 = .014 + .026 Thirty five percent of .04 total risk is systematic and 65% is diversifiable risk or .014 is systematic and .026 is diversifiable. 24. a. SDs2 = R2(SDs2) + (1 - R2)(SDs2) = .16 = (.25)2(.40)2 + (1 - (.25)2)(.40)2 (.0625)(.16) + (.9375)(.16) = .01 + .15 = .16 b. SDs2 = .25 = (.852)(.25) + (1 - .852)(.25) = (.7225)(.25) + (.2775)(.25) = .181 + .069 = .25 c. If an investor holds a diversified mutual fund, systematic risk is the only relevant risk to consider. So Stock J is less risky, but NOT because the SD risk is less, but because the systematic risk is less. However, to determine whether it is a good investment, it depends on the PR compared to the ERs and one should not compare just the riskiness of the stocks. 25. A negative beta means that the security return moves opposite to the market portfolio return. For example, -.54 beta means that the XYZ’s stock return may godown by .54% for every increase in the market portfolio return. The benefits of a negative beta stock is that it adds diversification benefits to the market portfolio and actually reduces the risk of the market portfolio when combined together. Therefore, even if a negative beta stock has an expected return lower than the riskfree rate, it is still desireable. 26. a. Beased on portfolio theory, a pair of securities with the lowest correlation coefficient has the greatest diversification potential. This would be stocks MAC and WM with a -.60 correlation. b. Holding a well-diversified portfolio, the systematic risk is the only relevant risk measure because the diversifiable risk is eliminated. This means that WM with a negative beta, should be recommended to your client. c. Though it has the highest total risk, the systematic risk is the only relevant risk to consider since your client holds a well-diversified portfolio. It also has the highest ER (20%) as compared to MAC and ABT (10% and 15% respectively). Further, calculating the CV, it also shows that WM has the lowest risk perreturn (1.0 for WM, 1.22 for MAC, and 1.6 for ABT). 27. Problem 27 was deleted. The problem states: The following data for IBM (IBM), PepsiCO (PEP), and Duke Power (DUK) were compiled for your information: Expected Standard Systematic Diversifiable Stock Return Deviation Risk Risk IBM 0.15 0.20 0.10 0.05 PEP 0.20 0.25 0.15 0.10 DUK 0.35 0.05 0.05 0.00 Correlation coefficients: CORR(IBM,PEP) = 0.50 CORR(IBM,DUK) = 0.10 CORR(PEP,DUK) = 0.30 a. If a client wants to invest in only one stock, which one would you recommend? Can you unequivocally recommend one? b. Suppose your client already holds a well-diversified portfolio such as the S&P 500 index. Which one stock would you recommend? Why? c. Your client says that PEP is far too risky with a standard deviation of 0.25, especially compared to the other two firms’ standard deviations of 0.20 and 0.05. How would you address his concern? Carefully explain, assuming that your client holds a well-diversified portfolio. ANSWER: a. A risk averse investor will unequivocally choose DUK over IBM because it has a lower risk for the same return. However, the choice between PEP and DUK is based on the individual risk preference. b. If your client holds a well-diversified mutual fund, then the only relevant risk is the systematic risk. So the client should invest based on the return expected via CAPM. In this case, it is impossible to judge which is better because the returns and systematic risks are relatively very close. However, DUK has the highest expected return (35%) and the lowest systematic risk (5%), so it is the most likely candidate to be a favorable investment. However, others could also be undervalued. c. This time the client is right. (In marketing, isn’t the client always right?). PEP has a relatively lower expected return (20%) compared to the relevant systematic risk (15%). Though we must calculate the required return (via CAPM) and compare to a predicted return to really determine the answer, the likelihood that it’s a good investment is small. 28. a. Graph b. According to portfolio theory, the lower correlation provides a greater potential to diversify. Stocks DIS and JNJ have a negative correlation and the lowest, so it should have the greatest diversification potential. c. If the client holds a well diversified portfolio then the only relevant risk is the systematic risk. Since JNJ’s systematic risk is negative, it would reduce the risk of the well-diversified portfolio. d. Your client should only focus on the systematic risk since he/she is holding a well-diversified portfolio. Therefore, JNJ’s systematic risk of -.02 is much lower than the risk of the other two stocks, .20 and .15. Further, JNJ has the highest expected return, making it a very attractive investment. 29. The differences between CML and SML are: (1) CML uses total risk (SD) while the SML uses beta or systematic risk on the x axis. (2) CML defines only efficient portfolios while the SML defines both efficient and noneffcient portfolios (which includes securities). (3) Every point on the CML is a proportional combination between RF and M. The SML graphs all portfolios and securities which lies on and off the CML. 30. The differences between the SML and the security characteristic line are: (1) SML graphs beta versus expected return while the security characteristic line graphs time series of security returns versus the market index returns. SML is a cross-sectional graph and the security characteristic line is a time series graph. (2) The security characteristic line is used to estimate beta and also to determine how a security return correlates to a market index return. The SML is used for estimating the expected return for a security relative to its beta risk. The beta estimate for the SML comes from the slope estimate of the security characteristic line. Extra Problems 1. You have chosen a risky portfolio, P13 with an expected return of 0.18 and standard deviation of 0.20. The risk-free rate, RF, equals 0.06. a. What is the CML for P13? b. Suppose you prefer to reduce your risk by investing 30 percent in RF and 70 percent in portfolio P13. What is your expected return now? c. What is your standard deviation risk of the 30-70 mix between RF and P13? d. Show graphically where it would be on the CML defined in part a. 2. Southwest Airlines is selling at $28 and will pay dividends of .10 this year. It expects to hit $35 in one year and its beta is estimated at 1.95. The market risk premium is .086 and the T-bill is yielding .06. Is Southwest Airlines a good investment? Explain. 3. Display Southwest Airlines required and predicted returns on a SML graph. Include the riskfree asset and the market portfolio return on the SML graph. 4. The correlation coefficient of Kmart (s) with the market portfolio is +0.70, SDs = .35 and SDM = .20. a. What is Kmart’s beta? b. If the market risk premium is .086 and the Tbill yields .06, calculate Kmart’s required return using the CAPM. c. If the predicted return equals 15 percent, is Kmart a good investment? Explain. 5. You are given the following information about two mutual funds, J and K. Current Expected Expected Estimated Price Price Dividends Beta Fund J $54 $60 $2.50 1.25 Fund K $83 $100 0.0 1.50 (ERM - RF) = .09 RF = .05 a. Estimate the required return using the CAPM. b. Graphically represent the CAPM returns, the market portfolio return, the risk-free asset. c. Calculate the predicted returns for the two funds and determine whether they are good investments or not. d. At what current price would the two funds be at equilibrium. 6. Stock W is expected to sell for $150 in one year and to pay $1.00 in dividends. The correlation coefficient of Stock W and the market portfolio is +.45, SDW = .40, SDM = .25, RF = .05, and ERM = .17. a. At what price will the stock sell for today? b. If the stock sellf for $125, what rate of return will you earn on this stock? 7. You are interested in estimating Hewlett Packard (HP)’s beta. HP’s correlation with the Dow Jones Industrial Average (DJ) is +.75 and SD HP equals .40 while SDD is 20 percent. You also decide to look into the Wilshire 5000 stock index as the market proxy for the market portfolio. The SDW is 30 percent and its correlation with HP is +.90. a. Calculate HP’s beta using DJ as the market portfolio. b. Calculate HP’s beta using the Wilshire index as the market portfolio. c. Discuss the differences if any exist. 8. suppose the Dow (DJ)’s expected return is 20 percent and the Tbill yields .05. Suppose the Wilshire Index (W) expects a 30 percent return. a. Using the beta estimate from problem 7, estimate HP’s required return using the Dow as the market portfolio. b. Using the beta estimate from problem 7, estimate HP’s required return using the Wilshire as the market portfolio. c. Suppose HP’s expected price is $78 a year from now and also expects dividends equal to $1.25. HP’s price today is $60. Calculate HP’s predicted return. d. Conduct security analysis on HP using the Dow as the market portfolio and then redo the analysis using the Wilshire as the market portfolio. 9. Graph the two SMLs for HP using the Dow and the Wilshire as market portfolios. Also incude the predicted return for HP on the graph. Discuss the problems of using the wrong market index. 10. The following information has been compiled on Merck, Advanced Micro Devices (AMD), and American Home Products (AHP). Stock Predicted Standard Systematic Diversifiable Return Deviation Risk Risk Merck .20 .10 .08 .02 AMD .25 .20 .06 .14 AHP .15 .15 .03 .12 CORR(Merck, AMD) = -.20 Correlation coefficient CORR(Merck, AHP) = +.80 CORR(AMD,AHP) = +.06 a. Using the concept from portfolio theory and without performing any calculations, which combination of two stocks would you recommend to your client? Explain. b. Suppose the market standard deviation risk equals 0.20 and beta can be calculated as: = systematic risk / SDM What are the beta risks for each security? c. Calculate the CAPM required return for each stock if the market expected return equals .18 and the Tbill yields .07. d. Given the required returns from part c which stock would you invest in? Explain. e. Your client complains that AMD is the riskiest with a 20 percent total risk level. Your client, however, holds a well-diversified portfolio. How would you address his complaint. (Otherwise, he’s a good client and you don’t want to lose him). Answers 1. a. ERp = .06 + [(.18-.06)/.20]SDp or ERp = .06 + .60SDp b. ERp = .30(.06) + .70(.18) = .144 c. SDp = (.70)(.20) = .14 d. The graph should have .06 return at 0 SD risk, .18 return at .20 SD risk, and .144 return at .14 risk all on one line. 2. ERs = .06 + 1.95(.086) = .23 PR = ($35 - $28 + $.10)/$28 = .25 Since the PRs > ERs, it is a good investment. 3. The graph should show .06return at 0 beta, .146 return at 1.0 beta, and .23 return at 1.95 beta. Above the 1.95 beta, an X at .25 return will mark the predicted return, PR. 4. a. = 1.225 b. ERs = .165 c. If predicted return equals 15 percent, Kmart is not a good investment because investors require a 16.5 percent return. 5. a. ERJ = .1625 ERK = .185 b. Graph should have a .05 return at 0 beta, .14 return at 1.0 beta, .1625 return at 1.25 beta and .185 return at 1.5 beta. c. PRJ = .157 PRK = .205 Since PRs > ERs for Fund K only, Fund K is the only good investment available between the two funds. d. PJ = $53.76 PK = $84.39 6. a. = .75 ERs = .14 Ps = ($150+$1.00)/(1+.14) = $132.46 b. If it sells for $125, you earn = Ret = ($150 + $1.00 - $125)/$125 = .208 7. a. DJ = 1.50 b. W = 1.20 c. The difference is due to the correlation of HP with the two market indexes. Even though the HP is much more highly correlated with the Wilshire Index, the Wilshire is riskier than the Dow, so its effect is a lower beta. HP is less correlated with the Dow, but the lower risk level of the Dow (.20) results in a higher beta. 8. a. ERs = .275 using the Dow as the market portfolio. b. ERs = .35 using the Wilshire as the market portfolio. c. PR for HP is .32 d. Using the Dow as the market, HP’s PR > ERs (.275); however, using the Wilshire as the market, PR < ERs (.35). So using the Dow as the market we would invest in HP, but using the Wilshire as the market, we would not invest in HP. Of course, it is important to recognize which market portfolio best represents your investment before making this decision. 9. The two SML graphs will be as follows: The Dow SML has .05 return at zero beta and .20 return at 1.0 beta. The Wilshire SML has .05 return at zero beta and .30 return at 1.0 beta. Draw a straight horizontal line at 32 percent for HP’s predicted return. The graph should show that PR lies below Wilshire’s SML line at the 1.2 beta level and Dow’s SML line should show that PR lies above Dow’s SML at the 1.5 beta level. 10. a. Using portfolio theory, the two stocks that are negatively correlated would be the best combination of stocks to hold. b. m = 1.414 amd = 1.22 ahp = .866 c. ERm = .226 ERamd = .204 ERahp = .165 PRm = .20 PRamd = .25 PRahp = .15 as given in the problem. d. Based on the expected (or predicted) returns given in the table, it seems only AMD’s predicted return exceeds the required return, so it is the best one to invest. It is important to use only the beta (or systematic) risk in order to make investment decisions if your client holds a diversified portfolio. e. Even though AMD is riskiest with 20 percent total risk (standard risk), if your client holds a diversified portfolio, the only relevant risk is the systematic risk as compared to its expected return and required return. Since AMD has a predicted return greater than the required return (and is the only one out of the three stocks with that condition), it should be selected. Your client should recognze that systematic risk must be incorporated to calculate the required return and only if the predicted rate exceed that required, it is an appropriate investment.