Solutions to Chapter 11 Risk, Return, and Capital Budgeting a False Investors require higher expected rates of return on investments with high market risk, not high total risk Variability of returns is a measure of total risk Stocks with high total risk (highly variable returns) can have low market risk That is, their returns have low correlation with the market b False If beta = 0, the asset’s expected return should equal the risk-free rate, not zero c False The portfolio is one-third invested in Treasury bills and two-thirds in the market Its beta will be 1/3 × + 2/3 × 1.0 = 2/3 d True High exposure to macroeconomic changes cannot be diversified away in a portfolio Thus stocks with higher sensitivity to macroeconomic risks have higher market risk and higher expected returns when compared to stocks with lower sensitivity to macroeconomic changes e True For similar reasons as in (d) Sensitivity to fluctuations in the stock market cannot be diversified away Such stocks have higher systematic risk and higher expected rates of return The risks of deaths of individual policyholders are largely independent, and therefore are diversifiable Therefore, the insurance company is satisfied to charge a premium that reflects actuarial probabilities of death, without an additional risk premium In contrast, flood damage is not independent across policyholders If my coastal home floods in a storm, there is a greater chance that my neighbor's will too Because flood risk is not diversifiable, the insurance company may not be satisfied to charge a premium that reflects only the expected value of payouts The actual returns on the Snake Oil fund exhibit considerable variation around the regression line This indicates that the fund is subject to diversifiable risk: it is not well diversified The variation in the fund's returns is influenced by more than just market-wide events Investors would buy shares of firms with high degrees of diversifiable risk, and earn high risk premiums But by holding these shares in diversified portfolios, they would not necessarily bear a high degree of portfolio risk This would represent a profit opportunity, however As investors seek these shares, we would expect their 11-1 Copyright © 2006 McGraw-Hill Ryerson Limited prices to rise, and the expected rate of return to investors buying at these higher prices to fall This process would continue until the reward for bearing diversifiable risk dissipated a Required return = rf + β(rm – rf) = 4% + (11% – 4%) = 8.2% With an IRR of 14%, the project is attractive b If beta = 1.6, required return increases to: 4% + 1.6 (11% – 4%) = 15.2% which is greater than the project IRR You should now reject the project c Given its IRR, the project is attractive when its risk and therefore its required return are low At a higher risk level, the IRR is no longer higher than the expected return on comparable risk assets available elsewhere in the capital market a The expected cash flows from the firm are in the form of a perpetuity The discount rate is: rf + β(rm – rf) = 5% + 4(11% – 5%) = 7.4% Therefore, the value of the firm would be: P0 = = = $135,135 b If the true beta is actually 6, the discount rate should be: rf + β(rm – rf) = + 6(11 – 5) = 8.6% Therefore, the value of the firm is: P0 = = = $116,279 By underestimating beta, you would overvalue the firm by $135,135 – $116,279 = $18,856 11-2 Copyright © 2006 McGraw-Hill Ryerson Limited Required return = rf + β(rm – rf) = 4% + 1.25(11% – 4%) = 12.75% Expected return = 11% The stock’s expected return is less than the required return given its risk Thus the stock is overpriced Why? Given the stock’s future cash flows and its current price, investors can expect to earn only 11% Comparable risk investments earn 12.75% At the current price, investors are better off investing in these other investments This lack of demand will cause the stock price to fall until its expected return increases to the required return of 12.75% Required return = riskfree rate + beta × [ expected return on market – riskfree rate] = rf + β(rm – rf) For the stock, we know that 12% = rf + ( 14% - rf ) Using the CAPM, solve for the riskfree rate of interest: rf = (Required return - β rm) / ( - β) = (12% - × 14%) / (1 - 8) = 4% We assume that the riskfree rate is not changed Therefore, if the market return turns out to be 10%, we expect that the stock’s return will be 4% + 8(10% - 4%) = 8.8% a A diversified investor will find the highest-beta stock most risky This is Nike, which has a beta of 1.20 b Nike has the highest total volatility; the standard deviation of its returns is 31% c β = (.61 + 53 + 1.20)/3 = 78 d The portfolio will have the same beta as Exxon, 61 The total risk of the portfolio will be 61 times the total risk of the market portfolio because the effect of firm-specific risk will be diversified away The standard deviation of the portfolio is therefore 61 × 20% = 12.2% e Using the CAPM, we compute the expected rate of return on each stock from the equation r = rf + β × (rm – rf) In this case, rf = 4% and (rm – rf) = 7% Exxon: r = 4% + 61(7%) = 8.27% Polaroid: r = 4% + 53(7%) = 7.71% Nike: r = 4% + 1.20(7%) = 12.4% 11-3 Copyright © 2006 McGraw-Hill Ryerson Limited 10 The following table shows the average return on Tumblehome for various values of the market return It is clear from the table that, when the market return increases by 1%, Tumblehome’s return increases on average by 1.5% Therefore, β = 1.5 If you prepare a plot of the return on Tumblehome as a function of the market return, you will find that the slope of the line through the points is 1.5 Market return(%) −2 −1 Average return on Tumblehome(%) −3.0 −1.5 0.0 1.5 3.0 Note: If your calculator supports statistics then you can estimate this Enter points as X,Y values In stats linear mode you see that b = 1.5 which is the slope of the line Using the SLOPE function in Excel will also calculate the slope of 1.5 11 a Beta is the responsiveness of each stock's return to changes in the market return Then: βA = = = = 1.2 βD = = = = 75 D is considered to be a more defensive stock than A because its return is less sensitive to the return of the overall market In a recession, D will usually outperform both stock A and the market portfolio b We take an average of returns in each scenario to obtain the expected return rm = (32% – 8%)/2 = 12% rA = (38%– 10%)/2 = 14% rD = (24% – 6%)/2 = 9% 11-4 Copyright © 2006 McGraw-Hill Ryerson Limited c According to the CAPM, the expected returns that investors will demand of each stock, given the stock betas and given the expected return on the market, are: r = rf + β(rm – rf) rA = 4% + 1.2(12% – 4%) = 13.6% rD = 4% + 75(12% – 4%) = 10.0% d 12 The return you actually expect for stock A, 14%, is above the fair return, 13.6% The return you expect for stock D, 9%, is below the fair return, 10% Therefore stock A is the better buy Figure follows below Cost of capital = risk-free rate + beta × market risk premium Since the risk-free rate is 4% and the market risk premium is 7%, we can write the cost of capital as: Cost of capital = 4% + beta × 7% Cost of capital (from CAPM) = 10% + beta × 8% 4% + 75 × 7% = 9.25% 4% + 1.75 × 7% = 16.25% Beta 75 1.75 r SML 11% 7% = market risk premium 4% beta 1.0 The cost of capital of each project is calculated using the above CAPM formula Thus, for Project P, its cost of capital is: 4% + 1.0 ì 7% = 11% 11-5 Copyright â 2006 McGraw-Hill Ryerson Limited If the cost of capital is greater than IRR, then the NPV is negative If the cost of capital equals the IRR, then the NPV is zero Otherwise, if the cost of capital is less than the IRR, the NPV is positive Project P Q R S T 13 Beta 1.0 0.0 2.0 0.4 1.6 Cost of capital 11.0% 4.0 18.0 6.8 15.2 IRR 11% 17 16 NPV + − + + The appropriate discount rate for the project is: r = rf + β(rm – rf) = 4% + 1.4(11% – 4%) = 13.8% Therefore: NPV = –100 + 15 × annuity factor(13.8%, 10 years) = –100 + 78.8563 = -$21.14 You should reject the project 14 We need to find the discount rate for which: 15 × annuity factor(r, 10 years) = 100 Solving this equation on the calculator, we find that the project IRR is 8.14% The IRR is less than the opportunity cost of capital, 13.8% Therefore you should reject the project, just as you found from the NPV rule 15 From the CAPM, the appropriate discount rate is: r = rf + β(rm – rf) = 4% +.75(7%) = 9.25% r = 0925 = = P1 = $52.625 11-6 Copyright © 2006 McGraw-Hill Ryerson Limited 16 If investors believe the year-end stock price will be $54, then the expected return on the stock is: = 12 = 12%, which is greater than the opportunity cost of capital Alternatively, the “fair” price of the stock (that is, the present value of the investor's expected cash flows) is (2 + 54)/1.0925 = $51.26, which is greater than the current price Investors will want to buy the stock, in the process bidding up its price until it reaches $51.26 At that point, the expected return is a “fair” 9.25%: = 0925 = 9.25% 17 a The expected return of the portfolio is the weighted average of the returns on the TSX and T-bills Similarly, the beta of the portfolio is a weighted average of the beta of the TSX (which is 1.0) and the beta of T-bills (which is zero) (i) (ii) (iii) (iv) (v) 18 E(r) = × 13% + 1.0 × 5% = 5% E(r) = 25 × 13% + 75 × 5% = 7% E(r) = 50 × 13% + 50 × 5% = 9% E(r) = 75 × 13% + 25 × 5% = 11% E(r) = 1.00 × 13% + × 5% = 13% β= 0×1 + 1×0 = β = 25 × + 75 × = 25 β = 50 × + 50 × = 50 β = 75 × + 25 × = 75 β = 1.0 × + × = 1.0 b For every increase of 25 in the β of the portfolio, the expected return increases by 2% The slope of the relationship (additional return per unit of additional risk) is therefore 2%/.25 = 8% c The slope of the return per unit of risk relationship is the market risk premium: rM – rf = 13% – 5% = 8%, which is exactly what the SML predicts The SML says that the risk premium equals beta times the market risk premium a Call the weight in the TSX w and the weight in T-bills (1 – w) Then w must satisfy the equation: w × 10% + (1 – w) × 5% = 8% which implies that w = The portfolio would be 60% in the TSX and 40% in T-bills The beta of the portfolio would be the weighted average of the betas of the TSX and T-Bills Since T-Bills are risk-free, their beta is zero The beta of the portfolio is: 6×1 + 4×0 = 11-7 Copyright © 2006 McGraw-Hill Ryerson Limited b To form a portfolio with a beta of 4, use a weight of 40 in the TSX and a weight of 60 in T-bills Then, the portfolio beta would be: β = 40 × + 60 × = 40 The expected return on this portfolio is × 10% + × 5% = 7% c Both portfolios have the same ratio of risk premium to beta: = = 5% Notice that the ratio of risk premium to risk (i.e., beta) equals the market risk premium (5%) for both stocks 19 If the systematic risk were comparable to that of the market, the discount rate would be 12.5% The property would be worth $50,000/.125 = $400,000 20 The CAPM states that r = rf + β(rm – rf) If β < 0, then r < rf Investors would invest in a security with an expected return below the risk-free rate because of the hedging value such a security provides for the rest of the portfolio Investors get their “reward” in terms of risk reduction rather than in the form of high expected return 21 The historical risk premium on the market portfolio has been about 7% Therefore, using this value and the assumed risk-free rate of 4%, we can use the CAPM to derive the cost of capital for these firms as 4% + β × 7% ATI Technologies Nova Chemicals Quebecor World Shaw Communications 22 Beta 2.48 95 41 Cost of capital 21.36% 10.65 6.87 10.3 r = rf + β(rm – rf) = rf + 5(rm – rf) (stock A) 13 = rf + 1.5(rm – rf) (stock B) Solve these simultaneous equations to find that r f = 1% and rm = 9% Thus the market risk premium is 9% - 1%, or 8% 11-8 Copyright © 2006 McGraw-Hill Ryerson Limited 23 r = rf + β(rm – rf) 10 = + β(14 – 6) β = 24 Internet: Applying the CAPM Tips: If your school's library subscribes to Financial Post Advisor, students have access to betas for Canadian stocks They are found in FP Analyzer, in the Spot Data section Betas can also be collected at a Bloomberg terminal, if your students have access to one An interesting extension is to ask students to estimate their own betas However, getting stock returns and market index returns is more work If your school has access to the CFMRC database, it is an easy task to get rates of return To put together rates of return using internet resources is more challenging Historical stock prices and index levels can be downloaded from ca.finance.yahoo.ca Ideally, dividends and the ex-dividend dates would be matched to the stock prices before rates of return are calculated (Most beta services, including Bloomberg, calculate betas without including dividends in the stock returns because it is much more work to match up the dividends with stock prices) If you want students to estimate their own betas, the Slope function in Excel produces an estimate Expected results: This exercise is self-explanatory 25 Shaw Communications should use the beta of ATI Technologies (which is 2.48) to find that the required rate of return is 21.36% The project is an investment in graphics hardware and the beta of ATI reflects the risk of a firm in the graphics hardware business The beta of Shaw Communications reflects the risk of a cable and satellite communications company 26 a False The stock’s risk premium, not its expected rate of return, is twice as high as the market’s b True The stock’s unique risk does not affect its contribution to portfolio risk but its market risk does c False A stock plotting below the SML offers too low an expected return relative to the expected return indicated by the CAPM The stock is overpriced Investors will not want to pay that price to receive the stock’s cash flows The price must fall to increase the stock’s rate of return 11-9 Copyright © 2006 McGraw-Hill Ryerson Limited 27 d True If the portfolio is diversified to such an extent that it has negligible unique risk, then the only source of volatility is its market exposure A beta of then implies twice the volatility of the market portfolio e False An undiversified portfolio has more than twice the volatility of the market In addition to the fact that it has double the sensitivity to market risk, it also has volatility due to unique risk The CAPM implies that the expected rate of return that investors will demand of the portfolio is: r = rf + β(rm – rf) = 4% + 8(11% – 4%) = 9.6% If the portfolio is expected to provide only a 9% rate of return, it’s an unattractive investment The portfolio does not provide an expected return that is sufficiently high relative to its risk 28 A portfolio invested 80% in a stock market index fund (with a beta of 1.0) and 20% in a money market fund (with a beta of zero) would have the same beta as this manager's portfolio: β = 80 × 1.0 + 20 × = 80 However, it would provide an expected return of 80 × 11% + 20 × 4% = 9.6% which is better than the portfolio manager's expected return 29 The security market line provides a benchmark expected return that an investor can earn by mixing index funds with money market funds Before you place your funds with a professional manager, you will need to be convinced that he or she can earn an expected return (net of fees) in excess of the expected return available on an equally risky index fund strategy 30 a r = rf + β(rm − rf) = 5% + [–.2 × (12% – 5%)] = 3.6% b.Portfolio beta = 90 × βmarket + 10 × βlaw firm = 90 × 1.0 + 10 × (−.2) = 88 31 Expected income on stock fund: $2 million × 12 Interest paid on borrowed funds: $1 million × 04 Net expected earnings: 11-10 Copyright © 2006 McGraw-Hill Ryerson Limited = 24 million = 04 million $0.20 million Expected rate of return on the $1 million you invest is: = 20 = 20% Risk premium = 20% – 4% = 16% This is double the risk premium of the market index fund (which is 8%, = 12% - 4%) The risk is also double that of holding a market index fund You have $2 million at risk, but the net value of your portfolio is only $1 million A 1% change in the rate of return on the market index will change your profits by 01 × $2 million = $20,000 But this changes the rate of return on your portfolio by $20,000/$1,000,000 = 2%, double that of the market So your risk is in fact double that of the market index 32 a Expected rate of return = rf + β(rm − rf) = 04 + × (.11 - 04) = 103 = 10.3% b The appropriate discount rate to evaluate ChemCo is one that reflects the riskiness of ChemCo’s cash flows Since we know that ChemCo's current beta is 1.4, it is reasonable to use this in the calculation of the appropriate discount rate Note that the discount rate of BigCo is irrelevant because BigCo has three different divisions, of which only one is in the same business of ChemCo Discount rate = rf + β(rm − rf) = 04 + 1.4 × (.11 - 04) = 138 = 13.8% c Assuming that these are after-tax cash flows and using the constant dividend growth model, the value of ChemCo is Value of ChemCo = = $91.837 million d Think of BigCo as a portfolio consisting of the original three divisions plus the new ChemCo division Thus, the new beta of BigCo will equal the weighted average of its old beta and the beta ChemCo, with the weights based on the market values of three original divisions, ChemCo and the new combined BigCo The value of BigCo is its original $1,000 million plus the value of ChemCo, for a total of $1,091.837 million Weight for ChemCo division = = = 084 = 8.4% Weight for original BigCo = = = 916 = 91.6% New beta of BigCo = 084 × 1.4 + 916 × = 942 As expected, adding ChemCo, with its higher beta, causes the beta of BigCo 11-11 Copyright © 2006 McGraw-Hill Ryerson Limited to increase 33 a We take advantage of the formula for the present value of a growing [ () ] annuity, found in Chapter 4: × - T for valuing the Year to cash flows and recognize that starting in Year 6, each stock has a constant perpetual growth rate and can be valued using the constant dividend growth model, DIV/(r – g) Food Express (FE) Required rate of return = rf + β(rm − rf) = 04 + 85 × (.10 - 04) = 091 Value of FE today: [ () ] + × = +× × 1- = $114.754 million Computer Power (CP) Required rate of return = rf + β(rm − rf) = 04 + 95 × (.10 - 04) = 097 Value of CP today: [ () ] =+×× 1- +× = $40.09 million Bridge Steel (BS) Required rate of return = rf + β(rm − rf) = 04 + 1.3 × (.10 - 04) = 118 Value of BS today: [ () ] =+× × 1- +× = $110.07 million b Total portfolio value = $114.754 + $40.09 + $110.07 = $264.914 million β Weight × β Company FE Weight in Portfolio = 4332 85 3682 CP = 1513 95 1437 11-12 Copyright © 2006 McGraw-Hill Ryerson Limited BS = 4155 1.3 5402 Total 1.0521 The portfolio beta is the weighted average of the individual stocks’ betas and is about 1.1 34 a This question is most easily handled with a spreadsheet but can be done using the formulas State of the Economy Recession Normal Boom Probability Division Division Division Division A B C D 0.2 8% -10% -1% -4% 0.6 8% 15% 7% 15% 0.2 9% 30% 10% 20% Expected return Standard deviation Firm Market -1.75% 11.25% 17.25% -3% 11% 22% 8.2% 13.0% 6.0% 12.2% 9.9% 10.4% 0.40% 12.88% 3.69% 8.33% 6.25% 7.94% b Using the formula βj = , where corrjm is the correlation between stock j’s return and the market return, σj is the standard deviation of stock j’s return and σ m is the standard deviation of the market return: βmarket = = = βA = = = 0368 βB = = = 1.61 βC = = = 451 βD = = = 991 Since each of the four divisions is worth about ¼ of the firm’s market value, the beta of the firm is the equal-weighted average of the four divisions’ betas: βFirm = = (.0368 + 1.61 + 451 + 991)/4 = 77 c According to the CAPM, the required rate of return is rj = rf + βj (rm − rf) Assuming the riskfree rate is percent, and using the expected return on the market calculated in (a), the required rate of return to each division is: rA = 04 + 0368 × (.104 - 04) = 0424 = 4.24% rB = 04 + 1.61 × (.104 - 04) = 143 = 14.3% 11-13 Copyright © 2006 McGraw-Hill Ryerson Limited rC = 04 + 451 × (.104 - 04) = 0689 = 6.89% rD = 04 + 991 × (.104 - 04) = 103 = 10.3% d Compare each division’s expected return, calculated in (a), with its required return, calculated according to the CAPM and reported in (c) Divisions with expected return at least equal to or greater than the required return are generating positive NPV Those whose expected return is less than the required return are underperforming and provide negative NPV Conglomerate should a buyer who can improve the performance of these negative NPV divisions Hopefully, Conglomerate will sell these poorly performing divisions for more than they are worth under its control, capturing some of gains from the improved performance (See Chapter 23 for more on how companies share gains from mergers) Both Divisions A and D have expected returns greater than the CAPM required rate of return However, Division B, with a required rate of return of 14.3%, has an expected return of 13% Likewise, Division C has a required rate of return of 6.89% but its expected rate of return is 6% This means Divisions B and C are likely candidates for sale However, Conglomerate may want to consider whether improvements in performance can be made to increase the expected rates of return, without resorting to selling the divisions 35 Standard & Poor's Expected results: This will give students hands-on experience with beta estimations 36 Standard & Poor's Expected results: Use Market Insight to provide beta estimates Student hopefully will be able to see patterns in the betas that relate business activities of the companies Encourage the students to learn more about each company's main businesses and think about how market-wide factors affect their revenues and costs 11-14 Copyright © 2006 McGraw-Hill Ryerson Limited ... the weighted average of the betas of the TSX and T-Bills Since T-Bills are risk-free, their beta is zero The beta of the portfolio is: 6×1 + 4×0 = 1 1-7 Copyright © 2006 McGraw-Hill Ryerson Limited... return of the portfolio is the weighted average of the returns on the TSX and T-bills Similarly, the beta of the portfolio is a weighted average of the beta of the TSX (which is 1.0) and the beta of. .. (.10 - 04) = 097 Value of CP today: [ () ] =+×× 1- +× = $40.09 million Bridge Steel (BS) Required rate of return = rf + β(rm − rf) = 04 + 1.3 × (.10 - 04) = 118 Value of BS today: [ () ] =+× × 1-