CH-11-12 Factor Models and the Arbitrage Pricing Theory – Corporate Finance : Book Guide

This Document Contains Chapters 11 to 12 Chapter 11 Factor Models and the Arbitrage Pricing Theory 1. An expected return is an ex-ante measure or prediction of what the return on an asset will be in the future. An observed return is an ex-post or historical measure of the return on an asset in the past. It is impossible to predict the surprise component in a return because it is a random innovation in the price change of an asset. 2. a. Real GNP was higher than anticipated. Since returns are positively related to the level of GNP, returns should rise based on this factor. b. Inflation was exactly the amount anticipated. Since there was no surprise in this announcement, it will not affect Lewis-Striden returns. c. Interest rates are lower than anticipated. Since returns are negatively related to interest rates, the lower than expected rate is good news. Returns should rise due to interest rates. d. The chairman’s death is bad news. Although the chairman was expected to retire, his retirement would not be effective for six months. During that period he would still contribute to the firm. His untimely death means that those contributions will not be made. Since he was generally considered an asset to the firm, his death will cause returns to fall. However, since his departure was expected soon, the drop might not be very large. e. The poor research results are also bad news. Since Lewis-Striden must continue to test the drug, it will not go into production as early as expected. The delay will affect expected future earnings, and thus it will dampen returns now. f. The research breakthrough is positive news for Lewis Striden. Since it was unexpected, it will cause returns to rise. g. The competitor’s announcement is also unexpected, but it is not a welcome surprise. This announcement will lower the returns on Lewis-Striden. The systematic factors in the list are real GNP, inflation, and interest rates. The unsystematic risk factors are the president’s ability to contribute to the firm, the research results, and the competitor. 3. Using a benchmark composed of Greek equities is wrong because the equities included are not of the same style as those in an Irish growth fund. 4. See Section 11.4 for a full discussion of this issue. 5. The market portfolio has to lie on the security market line by definition. If a security plots below the line it means that the security is overpriced. In this situation nobody would want to hold it, and its price would fall until the expected return was high enough to plot back on the Security Market Line. 6. Assuming the market portfolio is properly scaled, it can be shown that the one-factor model is identical to the CAPM. 7. Any return can be explained with a large enough number of systematic risk factors. However, for a factor model to be useful as a practical matter, the number of factors that explain the returns on an asset must be relatively limited. 8. For an automobile firm, like Renault, you should consider deviations from macro economic factors since the company’s performance is linked to the performance of the large countries it exports automobiles to. However, you could also have gone for a standard factor model that uses HML, SMB, and MOM factors as well as the market return. This Document Contains Chapters 11 to 12 9. The main difference is that the market model assumes that only one factor, usually a stock market aggregate, is enough to explain stock returns, while the Carhart model assumes a 4- factor model relying on 4 factors (SML, HML, Momentum, market risk premium) to explain returns. 10. The fact that APT does not give any guidance about the factors that influence security returns is a commonly-cited criticism. However, in choosing factors, we should choose factors that have an economically valid reason for potentially affecting security returns. For example, a smaller company has more risk than a large company. Therefore, the size of a company can affect the returns of the company shares. 11. It is the weighted average of expected returns plus the weighted average of each security's beta times a factor F plus the weighted average of the unsystematic risks of the individual securities. 12. We have a three factor model and so first the expected return must be calculated using our original forecasts. This means that  = 9.1%. The revised expected return is: 13. Since we have the expected return of the stock, the revised expected return can be determined using the innovation, or surprise, in the risk factors. So, the revised expected return is: R = 11% + 1.2(4.2% – 3%) – 0.8(4.6% – 4.5%) R = 12.36% 14. a. If m is the systematic risk portion of return, then: m = rm-rfΔ(Rm – rf) + HMLΔHML + SMBΔSMB m = 1.5(8.1% – 8.4%) – 1.40(3.80% – 3.10%) – .67(10.30% – 9.50%) m = -1.966% b. The unsystematic return is the return that occurs because of a firm specific factor such as the bad news about the company. So, the unsystematic return of the security is –2.6 percent. The total return is the expected return, plus the two components of unexpected return: the systematic risk portion of return and the unsystematic portion. So, the total return of the security is: R = + m +  R = 9.50% -1.966% – 2.6% R = 4.94% 15. a. If m is the systematic risk portion of return, then: m = rm-rfΔ(Rm – rf) + HMLΔHML + SMBΔSMB+ MomΔMom m = 1.04(4.8% – 3.5%) – 1.90(7.80% – 7.10%)+0.60(6.4% - 2.4%) + 0.44(-3.2% - 0.23%) E(R) =a+.8(6%)+.3(2%)-.1(-5%) =a+5.9% =15% E(R) =9.1%+.8(-8%)+.3(-.3%)-.1(9%) =1.71% R m = 0.91% b. The unsystematic is the return that occurs because of a firm specific factor such as the increase in market share. If  is the unsystematic risk portion of the return, then:  = 0.36(27% – 23%)  = 1.44% c. The total return is the expected return, plus the two components of unexpected return: the systematic risk portion of return and the unsystematic portion. So, the total return of the equity is: R = + m +  R = 10.50% + 0.91% + 1.44% R = 12.85% 16. The beta for a particular risk factor in a portfolio is the weighted average of the betas of the assets. This is true whether the betas are from a single factor model or a multi-factor model. So, the betas of the portfolio are: F1 = .20(1.20) + .20(0.80) + .60(0.95) F1 = 0.97 F2 = .20(0.90) + .20(1.40) + .60(–0.05) F2 = 0.43 F3 = .20(0.20) + .20(–0.30) + .60(1.50) F3 = 0.88 So, the expression for the return of the portfolio is: Ri = 5% + 0.97F1 + 0.43F2 + 0.88F3 Which means the return of the portfolio is: Ri = 5% + 0.97(5.50%) + 0.43(4.20%) + 0.88(4.90%) Ri = 16.45% 17. We can express the multifactor model for each portfolio as: E(RP ) = RF +βF1 + γF2 + δ F3 where F1, F2 and F3 are the respective risk premiums for each factor. Expressing the return equation for each portfolio, we get: 18% = 6% + 0.75F1 + 1.2F2 +0.04 F3 14% = 6% + 1.60F1 – 0.2F2 + 0.07 F3 22% = 6% +.85 F1 + 1.2 F2 = 0.65 F3 We can now solve the system of three equations with three unknowns. R F1 = 5.546%, F2 = 6.345%, F3 = 5.648% 18. a. The market model is specified by: R = + (RM – ) +  so applying that to each equity: Equity A: RA = + A(RM – ) + A RA = 10.5% + 1.2(RM – 14.2%) + A Equity B: RB = + B(RM – ) + B RB = 13.0% + 0.98(RM – 14.2%) + B Equity C: RC = + C(RM – ) + C RC = 15.7% + 1.37(RM – 14.2%) + C b. Since we don't have the actual market return or unsystematic risk, we will get a formula with those values as unknowns: RP = .30RA + .45RB + .30RC RP = .30[10.5% + 1.2(RM – 14.2%) + A] + .45[13.0% + 0.98(RM – 14.2%) + B] + .25[15.7% + 1.37(RM – 14.2%) + C] RP = .30(10.5%) + .45(13%) + .25(15.7%) + [.30(1.2) + .45(.98) + .25(1.37)](RM – 14.2%) + .30A + .45B + .30C RP = 12.925% + 1.1435(RM – 14.2%) + .30A + .45B + .30C c. Using the market model, if the return on the market is 15 percent and the systematic risk is zero, the return for each individual equity is: RA = 10.5% + 1.20 (15% – 14.2%) RA = 11.46% RB = 13% + 0.98 (15% – 14.2%) RB = 13.78% RC = 15.70% + 1.37(15% – 14.2%) RC = 16.80% To calculate the return on the portfolio, we can use the equation from part b, so: R R M R A R M RB R M RC R M RP = 12.925% + 1.1435(15% – 14.2%) RP = 13.84% Alternatively, to find the portfolio return, we can use the return of each asset and its portfolio weight, or: RP = X1R1 + X2R2 + X3R3 RP = .30(11.46%) + .45(13.78%) + .25(16.80%) RP = 13.84% 19. a. Since five equities have the same expected returns and the same betas, the portfolio also has the same expected return and beta. However, the unsystematic risks might be different, so the expected return of the portfolio is: = 11% + 0.72F1 + 1.69F2 + (1/5)(1 + 2 + 3 + 4 + 5) b. Consider the expected return equation of a portfolio of five assets we calculated in part a. Since we now have a very large number of securities in the portfolio, as: N → , → 0 But, the js are infinite, so: (1/N)(1 + 2 + 3 + 4 +…..+ N) → 0 Thus: = 11% + 0.72F1 + 1.69F2 20. a) The returns will be correlated because they have exposures to the same underlying factors. For example, when factor 1 increases, Equities A and B will also increase but C will decrease. b) If there are no arbitrage opportunities and the returns for A and B are equal, this means that If there are no arbitrage opportunities and the returns on A and C are equal, this means that Solve for F1 and F2. Subtract one equation from the other. R P N 1 R P 2.4% +0.5F1 =4.6% +1.2F1 -0.5F2 -.7F1 =2.2% -0.5F2 F1 =3.143% +.714F2 2.4% +0.5F1 =4.1% -0.4F1 +0.6F2 0.9F1 =1.7% +0.6F2 F1 =1.889% +0.667F2 0 = 1.254% + 0.048F2 F2 = -26.33% If F2 = -26.33%, F1 = 2.4% + 0.5(-26.33%) = -10.77% If there are no arbitrage opportunities and the returns for A and D are equal, 21. We have a two factor model and so the expected return must be calculated using our original forecasts. This means that  = 12.2%. The revised expected return is: 22. To determine which investment an investor would prefer, you must compute the variance of portfolios created by many equities from either market. Because you know that diversification is good, it is reasonable to assume that once an investor has chosen the market in which she will invest, she will buy many equities in that market. Known: EF = 0 and  = 0.10 E = 0 and Si = 0.20 for all i If we assume the equities in the portfolio are equally-weighted, the weight of each equity is , that is: Xi = for all i If a portfolio is composed of N equities each forming 1/N proportion of the portfolio, the return on the portfolio is 1/N times the sum of the returns on the N securities. To find the variance of the respective portfolios in the 2 markets, we need to use the definition of variance from Statistics: Var(x) = E[x – E(x)]2 In our case: 2.4% +0.5F1 =a% +F1 +F2 -0.5F1 =(a-2.4%) +F2 -0.5(-10.77%) =(a-2.4%) -26.33% 5.3825% =a-28.73% a =34.11% E(R) =12% =a+1.4(0.5%)-0.3(3%) =a-0.2% E(R) =12.2%+.1.4(3%)+.3(6%) =18.2% N 1 N 1 Var(RP) = E[RP – E(RP)]2 Note however, to use this, first we must find RP and E(RP). So, using the assumption about equal weights and then substituting in the known equation for Ri: RP = RP = (0.10 + F + i) RP = 0.10 + F + Also, recall from Statistics a property of expected value, that is: If: where a is a constant, and , , and are random variables, then: and E(a) = a Now use the above to find E(RP): E(RP) = E E(RP) = 0.10 + E(F) + E(RP) = 0.10 + (0) + E(RP) = 0.10 Next, substitute both of these results into the original equation for variance: Var(RP) = E[RP – E(RP)]2 Var(RP) = E Var(RP) = E N1 Ri N1  N1 i Z~ = aX~ + Y~ Z~ X~ Y~ E(Z~) = E(a)E(X~) + E(Y~)       + +  F i N 1 0.10 β N1 E(i ) N1 0 2 - 0.10 N 1 0.10 + βF + εi  2 N 1 βF + ε Var(RP) = E Var(RP) = Finally, since we can have as many stocks in each market as we want, in the limit, as N → , → 0, so we get: Var(RP) = 22 + Cov(i,j) and, since: Cov(i,j) = ij(i,j) and the problem states that 1 = 2 = 0.10, so: Var(RP) = 22 + 12(i,j) Var(RP) = 2(0.01) + 0.04(i,j) So now, summarize what we have so far: R1i = 0.10 + 1.5F + 1i R2i = 0.10 + 0.5F + 2i E(R1P) = E(R2P) = 0.10 Var(R1P) = 0.0225 + 0.04(1i,1j) Var(R2P) = 0.0025 + 0.04(2i,2j) Finally we can begin answering the questions a, b, & c for various values of the correlations: a. Substitute (1i,1j) = (2i,2j) = 0 into the respective variance formulas: Var(R1P) = 0.0225 Var(R2P) = 0.0025 Since Var(R1P) > Var(R2P), and expected returns are equal, a risk averse investor will prefer to invest in the second market. b. If we assume (1i,1j) = 0.9, and (2i,2j) = 0, the variance of each portfolio is: Var(R1P) = 0.0225 + 0.04(1i,1j) Var(R1P) = 0.0225 + 0.04(0.9) Var(R1P) = 0.0585 2 2 N12 ( )2 2 N 1 β F + 2βF ε +   2 2 2 2 Cov( , ) N 1 σ 1 - N 1 β σ            + +   i  j N 1 Var(R2P) = 0.0025 + 0.04(2i,2j) Var(R2P) = 0.0025 + 0.04(0) Var(R2P) = 0.0025 Since Var(R1P) > Var(R2P), and expected returns are equal, a risk averse investor will prefer to invest in the second market. c. If we assume (1i,1j) = 0, and (2i,2j) = 0, the variance of each portfolio is: Var(R1P) = 0.0225 + 0.04(1i,1j) Var(R1P) = 0.0225 + 0.04(0) Var(R1P) = 0.0225 Var(R2P) = 0.0025 + 0.04(2i,2j) Var(R2P) = 0.0025 + 0.04(0.5) Var(R2P) = 0.0225 Since Var(R1P) = Var(R2P), and expected returns are equal, a risk averse investor will be indifferent between the two markets. d. Since the expected returns are equal, indifference implies that the variances of the portfolios in the two markets are also equal. So, set the variance equations equal, and solve for the correlation of one market in terms of the other: Var(R1P) = Var(R2P) 0.0225 + 0.04(1i,1j) = 0.0025 + 0.04(2i,2j) (2i,2j) = (1i,1j) + 0.5 Therefore, for any set of correlations that have this relationship (as found in part c), a risk adverse investor will be indifferent between the two markets. 23. a. In order to find standard deviation, , you must first find the Variance, since  = . Recall from Statistics a property of Variance: If: where a is a constant, and , , and are random variables, then: and: Var(a) = 0 The problem states that return-generation can be described by: Ri,t = i + i(RM) + i,t Realize that Ri,t, RM, and i,t are random variables, and i and i are constants. Then, applying the above properties to this model, we get: Var Z~ = aX~ + Y~ Z~ X~ Y~ Var(Z~) = a2Var(X~) + Var(Y~) Var(Rj) = Var(RM) + Var(i) and now we can find the standard deviation for each asset: = 0.72(0.0121) + 0.01 = 0.015929 = = .1262 or 12.62% = 1.22(0.0121) + 0.0144 = 0.031824 = = .1784 or 17.84% = 1.52(0.0121) + 0.0225 = 0.049725 = = .2230 or 22.30% b. From above formula for variance, note that as N → , → 0, so you get: Var(Ri) = Var(RM) So, the variances for the assets are: = 0.72(.0121) = 0.005929 = 1.22(.0121) = 0.017424 = 1.52(.0121) = 0.027225 c. We can use the model: = RF + i( – RF) which is the CAPM (or APT Model when there is one factor and that factor is the Market). So, the expected return of each asset is: = 3.5% + 0.7(10.6% – 3.5%) = 8.47% = 3.5% + 1.2(10.6% – 3.5%) = 12.02% = 3.5% + 1.5(10.6% – 3.5%) = 14.15% We can compare these results for expected asset returns as per CAPM or APT with the expected returns given in the table. This shows that assets A & B are accurately priced, but asset C is overpriced (the model shows the return should be higher). Thus, rational investors will not hold asset C. d. If short selling is allowed, rational investors will sell short asset C, causing the price of asset C to decrease until no arbitrage opportunity exists. In other words, the price of asset C should decrease until the return becomes 14.15 percent. βi2 σA2 σA 0.015929 σB2 σB 0.031824 σC2 σC 0.049725 N Var(εi ) βi2 σA2 σB2 σC2 Ri R M R A RB RC 24. a. Let: X1 = the proportion of Security 1 in the portfolio and X2 = the proportion of Security 2 in the portfolio and note that since the weights must sum to 1.0, X1 = 1 – X2 Recall from Chapter 10 that the beta for a portfolio (or in this case the beta for a factor) is the weighted average of the security betas, so P1 = X111 + X221 P1 = X111 + (1 – X1)21 Now, apply the condition given in the hint that the return of the portfolio does not depend on F1. This means that the portfolio beta for that factor will be 0, so: P1 = 0 = X111 + (1 – X1)21 P1 = 0 = X1(1.0) + (1 – X1)(0.5) and solving for X1 and X2: X1 = – 1 X2 = 2 Thus, sell short Security 1 and buy Security 2. To find the expected return on that portfolio, use RP = X1R1 + X2R2 so applying the above: E(RP) = –1(20%) + 2(20%) E(RP) = 20% P1 = –1(1) + 2(0.5) P1 = 0 b. Following the same logic as in part a, we have P2 = 0 = X331 + (1 – X3)41 P2 = 0 = X3(1) + (1 – X3)(1.5) and X3 = 3 X4 = –2 Thus, sell short Security 4 and buy Security 3. Then, E(RP2) = 3(10%) + (–2)(10%) E(RP2) = 10% P2 = 3(0.5) – 2(0.75) P2 = 0 Note that since both P1 and P2 are 0, this is a risk free portfolio! c. The portfolio in part b provides a risk free return of 10%, which is higher than the 4.9% return provided by the risk free security. To take advantage of this opportunity, borrow at the risk free rate of 4.9% and invest the funds in a portfolio built by selling short security four and buying security three with weights (3,–2) as in part b. d. First assume that the risk free security will not change. The price of security four (that everyone is trying to sell short) will decrease, and the price of security three (that everyone is trying to buy) will increase. Hence the return of security four will increase and the return of security three will decrease. The alternative is that the prices of securities three and four will remain the same, and the price of the risk-free security drops until its return is 10%. Finally, a combined movement of all security prices is also possible. The prices of security four and the risk-free security will decrease and the price of security three will increase until the opportunity disappears. 25. The returns on an equally weighted portfolio of the three equities is simply the sum of the weighted average of the loadings on each factor. This means that: Rpt = 5% + 0.533RHML + 0.25RSMB 26. We must do this question in a number of steps: 1) Step one And (2) step two (3)step three ( ) ( ) N ( N N N N NK K N ) K K K K x r F F F x r F F F x r F F F                ~ ~ ~ ... ~ ~ . . . ~ ~ ~ ... ~ ~ ~ ~ ~ ... ~ ~ 1 1 2 2 2 2 2 21 1 22 2 2 2 1 1 1 11 1 12 2 1 1 = + + + + + = + + + + + = + + + + + p N N pK K K N NK p N N p N N p N N p N N p p p pK K p x x x x x x x x x x x x x x x R x r x r x r R F F F                          ~ ~ ~ ~ . . . ~ ~ ~ ~ ~ ~ ~ ... ~ ~ 1 1 2 2 1 1 2 2 2 1 12 2 22 2 1 1 11 2 21 1 1 1 2 2 1 1 2 2 1 1 2 2 = + + + = + + + = + + + = + + + = + + + = + + + =  + + + + + KK KK KK KK KK KK ( ) ( ) ( K N NK ) K ( N N ) N N N N N N x x F x x x r x r x x x x F         ~ ~ ~ ~ ~ ~ 1 1 1 1 1 1 1 1 1 11 1 1 + + + + + + + + = + + + + + + K K K K K K ( ) K p K N K N K K N N N N N N F F r F x F r F x F F r F x F r F x r x r x x x    ~ ~ var( ~ ) cov(~ , ~ ) var( ~ ) cov(~, ~ ) ~ var( ~ ) cov(~ , ~ ) var( ~ ) ~ ~ cov(~, ~ ) 1 1 1 1 1 1 1 1 1 1 1 1 1  +      + + + +       + + = + + + + + K K K K K (4) Step four The analogous equation holds for all factors Fj, where j = 1,…,K. Therefore, we can write: 27. Four factors generate four risk premia plus one implicit risk-free rate that can exactly explain any five expected returns. 28. ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( 1 1 ) ( 1 1 ) 1 1 1 1 1 1 1 1 1 1 2 2 1 , 1 ~ cov ~, ~ cov ~ , ~ cov ~, ~ cov ~ , ~ cov ~ ~ , ~ ~ cov ~ , ~ cov ~, ~ cov ~ , ~ cov x r F x r F x r F x r F x r x r F R F x r F x r F x r F N N N N N N N N = + + = + + = + +  = + + + K K K K ( ) ( ) ( ) ( ) ( ) ( ) ( ) var (~ ) . ~ cov ~ , . . . ~ . var ~ cov ~ , ~ ~ ~ var ~ ~ cov ~ , ~ var ~ ~ cov ~ , 1 1 2 2 1 1 1 1 11 2 21 1 1 1 1 K p K pK K K N NK p p N N K p K p p K p p F R F x x x F R F x x x F F R F F F R F R  = + + + =  = + + + = = + + + +           K K K ( ) ( ) 0.53 36 0.01 16 0.01 0.01 2 62 var( 1 ) 42 var( 2 ) var( ) = = + +  A = F + F +  A ( ) ( ) ( ) ( ) ( ) 0.12 4 0.01 4 0.01 0.04 2 22 var 1 22 var 2 var = = + + B = F + F + B ( ) ( ) ( ) ( ) ( ) 0.28 25 0.01 1 0.01 0.02 2 52 var 1 ( 1)2 var 2 var = = + + C = F + − F + c ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0.20 12 0.01 8 0.01 12 var 8 var cov 6 ,2 cov 4 ,2 cov 6 4 ,2 2 1 2 1 1 2 2 1 2 1 2 = = + = + = + = + +  + +  F F F F F F  AB F F  F F  29. a. b. 30. ( ) ( ) ( ) ( ) ( ) ( ) 0.08 10 0.01 2 0.01 10var 2 var cov 2 ,5 cov 2 , 1 1 2 1 1 2 2 = = − = − = + − F F BC F F F F ( ) ( ) ( ) ( ) 0.26 30 0.01 4 0.01 cov 6 1 ,5 1 cov 4 2 , 1 2 = = −  AC = F F + F − F 0.675 (0.53)(0.28) 0.26 0.436 (0.28)(0.12) 0.08 0.793 (0.53)(0.12) 0.20 = = = = = = = = =     A C AC AC B C BC BC             ( A ) ( B ) C p A B C F F F F F F R r r r = + + + − + + + + + − + = − + 2 0.13 6 1 4 2 2 0.15 2 1 2 2 .07 5 1 2 2 2 1 ( ) .03 3.0% .03 13 1 3 2  = = = + + + p p p R R F F  Rp rA rB rC 7 rC 3 7 10 7 20 7 = 20 − + − ( ) ( ) ( ) p p A B C A B C F F F F F F F F F F F F F F r r r      = + + + = + + − − − + + − + = + + + − + + + + + − + = − + 1 2 1 2 1 2 1 2 1 2 1 2 1 2 7 33 7 115 .0129 0.07 5 7 40 7 40 0.429 7 80 7 120 0.371 0.15 2 2 .07 5 7 20 0.13 6 4 7 20 7 20 7 20 (Rp )= .0129 =12.9% ( ) ( ) 3 1 , 6 7 , 2 1 Portfolio 1: 6 3 5 3 1 3 4 4 2 11 0 6 2 5 1 1 = − = − = − = − + = − = − + − − − = + + − − = A B C A A B A B A B A B A B A B x x x x x x x x x x x x x x x x ( ) ( ) , 0 2 3 , 2 1 Portfolio 2 : 6 3 5 3 2 3 5 4 2 11 1 6 2 5 1 0 = − = = = − + = − = − + − − − = + + − − = A B C A A B A B A B A B A B A B x x x x x x x x x x x x x x x x ( ) ( ) ( ) ( ) (0.15) 0.16 16.0% 2 3 0.13 2 1 0.07 .1333 13.33% 3 1 0.15 6 7 0.13 2 1 2 1 = − + = = = − + + = = p p R R 6 4 5 3 1 3 5 4 2 1 0 6 2 5 5 5 0 = − + = − = − + − + + = + + − − = A A B A B A B A B A B A B x x x x x x x x x x x x x 9 2 9 13 3 2 = = = − C B A x x x ( ) ( ) ( ) .16 .1456 .0144 1.44% 0.1333 0.1456 0.0123 1.23% 0.07 0.1456 14.56% 9 2 0.15 9 13 0.13 3 2 2 1 = − = = = − = − = − = − + + = =   Rp Chapter 12 Risk, Cost of Capital, and Capital Budgeting 1. The cost of equity capital is the return (from a manager’s perspective) that must be promised to investors for buying shares in your firm. The cost of equity capital (return that is promised) must be comparable to that given by other financial securities of similar risk. In an all equity firm, the cost of equity capital is equal to the risk of the assets. For an all- equity firm, the cost of equity capital will be the discount rate in the capital budgeting analysis. 2. The beta of a company is its systematic risk and, through the capital asset pricing model, is used to estimate the cost of equity capital. Beta is normally calculated from historical data. However, given that the past does not determine the future, the historical beta may not be reflective of the future beta. In addition, the estimation methodology (ordinary least squares regression) is a statistical technique that estimates beta with error. You can increase the precision of your beta estimate by increasing the sample size but this may then make the beta estimate inaccurate if the company’s beta has changed very recently because of an increase in financial leverage. 3. The three factors that influence beta are a) Cyclicity of revenues, b) Financial leverage, and c) Operating leverage. Firms have cyclic revenues if their revenues go up and down with the seasons or in cyclic manner. Firms with a high financial leverage have high levels of debt in their capital structure and firms with high operating leverage have high levels of fixed costs to variable costs. Beta measures the responsiveness of a security's returns to movements in the market. Beta is determined by the cyclicality of a firm's revenues. This cyclicality is magnified by the firm's operating and financial leverage. The following three factors will impact the firm’s beta. (1) Revenues. The cyclicality of a firm's sales is an important factor in determining beta. In general, stock prices will rise when the economy expands and will fall when the economy contracts. As we said above, beta measures the responsiveness of a security's returns to movements in the market. Therefore, firms whose revenues are more responsive to movements in the economy will generally have higher betas than firms with less-cyclical revenues. (2) Operating leverage. Operating leverage is the percentage change in earnings before interest and taxes (EBIT) for a percentage change in sales. A firm with high operating leverage will have greater fluctuations in EBIT for a change in sales than a firm with low operating leverage. In this way, operating leverage magnifies the cyclicality of a firm's revenues, leading to a higher beta. (3) Financial leverage. Financial leverage arises from the use of debt in the firm's capital structure. A levered firm must make fixed interest payments regardless of its revenues. The effect of financial leverage on beta is analogous to the effect of operating leverage on beta. Fixed interest payments cause the percentage change in net income to be greater than the percentage change in EBIT, magnifying the cyclicality of a firm's revenues. Thus, returns on highly-levered stocks should be more responsive to movements in the market than the returns on stocks with little or no debt in their capital structure. 4. If the investment has a different risk from the company, you should use the cost of equity capital for a firm that undertakes projects of comparable type and not the cost of equity capital of your firm. If you have debt in your capital structure, you should use the weighted average cost of capital instead of the cost of equity capital. 5. The main problem is finding comparison firms to estimate the cost of equity. Every firm is unique and their mix of operations can lead to different cost of equities. 6. The company can list on a liquid stock exchange, ensure its share price is within an acceptable range, make sure that it is being followed by analysts and that information is being released regularly to the market. 7. There is no consensus because estimating cost of equity is not an exact science. There is no deterministic way to calculate it. In an emerging market, the task of calculating cost of equity is even more difficult. The best way may even to survey financial experts in the country and ask them. Alternatively, you could use simple methods such as the Gordon growth model. 8. EVA or Economic Value Added is an ex-post measure of performance that takes the return earned from investments over the weighted average cost of capital and multiplies it by the amount that is invested. The formula is: EVA = [ROA − Weighted average cost of capital]  Total capital 9. No. The cost of capital depends on the risk of the project, not the source of the money. 10. Although Ericsson is a technology firm and you would expect its beta to be high, because of its size, it the beta is low in relation to other companies in Sweden. Students should be encourage to consider the issues of cyclicity, financial leverage and operating leverage with respect to Ericsson by looking at the company’s annual report and financial accounts. 11. Interest expense is tax-deductible. There is no difference between pre-tax and after-tax equity costs. 12. You are assuming that the new project’s risk is the same as the risk of the firm as a whole, and that the firm is financed entirely with equity. Two primary advantages of the SML approach are that the model explicitly incorporates the relevant risk of the stock and the method is more widely applicable than is the DCF model, since the SML doesn’t make any assumptions about the firm’s dividends. The primary disadvantages of the SML method are (1) three parameters (the risk-free rate, the expected return on the market, and beta) must be estimated, and (2) the method essentially uses historical information to estimate these parameters. The risk-free rate is usually estimated to be the yield on very short maturity T- bills and is, hence, observable; the market risk premium is usually estimated from historical risk premiums and, hence, is not observable. The stock beta, which is unobservable, is usually estimated either by determining some average historical beta from the firm and the market’s return data, or by using beta estimates provided by analysts and investment firms. 13. An industry beta may be more appropriate than a company beta if the financial leverage or operating leverage of the firm is unusual or temporarily out of synch with other companies in the industry. An industry beta cancels out random firm specific variations in beta and therefore may be more appropriate. 14. The appropriate after-tax cost of debt to the company is the interest rate it would have to pay if it were to issue new debt today. Hence, if the YTM on outstanding bonds of the company is observed, the company has an accurate estimate of its cost of debt. If the debt is privately- placed, the firm could still estimate its cost of debt by (1) looking at the cost of debt for similar firms in similar risk classes, (2) looking at the average debt cost for firms with the same credit rating (assuming the firm’s private debt is rated), or (3) consulting analysts and investment bankers. Even if the debt is publicly traded, an additional complication arises when the firm has more than one issue outstanding; these issues rarely have the same yield because no two issues are ever completely homogeneous. 15. The manager can attempt several things but the company essentially wishes to increase the liquidity of its shares. For example, the company can improve the level of disclosure and transparency in business operations. This will reduce adverse selection risk and encourage investment in the firm. The company would attempt to reduce the cyclicity of its revenues, the degree of operating leverage and the level of debt in the firm. This is covered in great detail in Section 12.6 of the textbook. 16. a. This only considers the dividend yield component of the required return on equity. b. This is the current yield only, not the promised yield to maturity. In addition, it is based on the book value of the liability, and it ignores taxes. c. Equity is inherently riskier than debt (except, perhaps, in the unusual case where a firm’s assets have a negative beta). For this reason, the cost of equity exceeds the cost of debt. If taxes are considered in this case, it can be seen that at reasonable tax rates, the cost of equity does exceed the cost of debt. 17. RShire = .03 + .75(.08) = .09 or 9.00% Both should proceed. The appropriate discount rate does not depend on which company is investing; it depends on the risk of the project. Since Superior is in the business, it is closer to a pure play. Therefore, its cost of capital should be used. With a 9% cost of capital, the project has an NPV of £10 million regardless of who takes it. 18. If the different operating divisions were in much different risk classes, then separate cost of capital figures should be used for the different divisions; the use of a single, overall cost of capital would be inappropriate. If the single hurdle rate were used, riskier divisions would tend to receive more funds for investment projects, since their return would exceed the hurdle rate despite the fact that they may actually plot below the SML and, hence, be unprofitable projects on a risk-adjusted basis. The typical problem encountered in estimating the cost of capital for a division is that it rarely has its own securities traded on the market, so it is difficult to observe the market’s valuation of the risk of the division. Two typical ways around this are to use a pure play proxy for the division, or to use subjective adjustments of the overall firm hurdle rate based on the perceived risk of the division. 19. The main issue that faces the company is whether the risk of the operations in each sector in the new country is of greater or lower risk. The risk of the investment is key and this will depend on country factors. 20. The discount rate for the projects should be lower that the rate implied by the security market line. The security market line is used to calculate the cost of equity. The appropriate discount rate for projects is the firm’s weighted average cost of capital. Since the firm’s cost of debt is generally less that the firm’s cost of equity, the rate implied by the security market line will be too high. 21. With the information given, we can find the cost of equity using the CAPM. The cost of equity is: RE = .045 + 1.30 (.12 – .045) = .1425 or 14.25% 22. The pre-tax cost of debt is the YTM of the company’s bonds, so: P0 = £920 = £20(PVIFAR%,14) + £1,000(PVIFR%,14) R = 2.693% YTM = 2 × 2.693% = 5.387% And the after-tax cost of debt is: RD = .05387(1 – .24) = .04094 or 4.094% 23. a. The pre-tax cost of debt is the YTM of the company’s bonds, so: P0 = €1,080 = €50(PVIFAR%,46) + €1,000(PVIFR%,46) R = 4.58% YTM = 2 × 4.58% = 9.16% b. The after-tax cost of debt is: RD = .0916(1 – .35) = .0595 or 5.95% c. The after-tax rate is more relevant because that is the actual cost to the company. 24. The book value of debt is the total par value of all outstanding debt, so: BVD = €20M + 80M = €100M To find the market value of debt, we find the price of the bonds and multiply by the number of bonds. Alternatively, we can multiply the price quote of the bond times the par value of the bonds. Doing so, we find: MVD = 1.08(€20M) + .58(€80M) = €68M The YTM of the zero coupon bonds is: PZ = €580 = €1,000(PVIFR%,7) R = 8.09% So, the after-tax cost of the zero coupon bonds is: RZ = .0809(1 – .35) = .0526 or 5.26% The after-tax cost of debt for the company is the weighted average of the after-tax cost of debt for all outstanding bond issues. We need to use the market value weights of the bonds. The total after-tax cost of debt for the company is: RD = .0595(€21.6/€68) + .0526(€46.4/€68) = .0548 or 5.48% 25. Using the equation to calculate the WACC, we find: WACC = .55 (.16) + .45(.09)(1 – .35) = .1143 or 11.43% 26. The formula to calculate beta is: b =sSiracha rSiracha,m sm So: b =.33(.62) .15 =1.364 27. Here we need to use the debt-equity ratio to calculate the WACC. Doing so, we find: WACC = .18(1/1.60) + .10(.6/1.60)(1 – .35) = .1369 or 13.69% 28. Here we need to find the pre-tax cost of debt. If the debt to assets ratio is .23, then the equity to assets ratio is .77. We can now calculate the pre-tax cost of debt. WACC = .209 = .23rD(1-.27) + .77(.25) Therefore rD is 9.83% 29. Here we have the WACC and need to find the debt-equity ratio of the company. Setting up the WACC equation, we find: WACC = .1150 = .16(E/V) + .085(D/V)(1 – .35) Rearranging the equation, we find: .115(V/E) = .16 + .085(.65)(D/E) Now we must realize that the V/E is just the equity multiplier, which is equal to: V/E = 1 + D/E .115(D/E + 1) = .16 + .05525(D/E) Now we can solve for D/E as: .05975(D/E) = .0450 D/E = .7531 30. a. The book value of equity is the book value per share times the number of shares, and the book value of debt is the face value of the company’s debt, so: BVE = 9.5M(£5) = £47.5M BVD = £75M + 60M = £135M So, the total value of the company is: V = £47.5M + 135M = £182.5M And the book value weights of equity and debt are: E/V = £47.5/£182.5 = .2603 D/V = 1 – E/V = .7397 b. The market value of equity is the share price times the number of shares, so: MVE = 9.5M(£53) = £503.5M Using the relationship that the total market value of debt is the price quote times the par value of the bond, we find the market value of debt is: MVD = .93(£75M) + .965(£60M) = £127.65M This makes the total market value of the company: V = £503.5M + 127.65M = £631.15M And the market value weights of equity and debt are: E/V = £503.5/£631.15 = .7978 D/V = 1 – E/V = .2022 c. The market value weights are more relevant. 31. First, we will find the cost of equity for the company. The information provided allows us to solve for the cost of equity using the CAPM, so: RE = .052 + 1.2(.09) = .16 or 16.00% Next, we need to find the YTM on both bond issues. Doing so, we find: P1 = £930 = £40(PVIFAR%,20) + £1,000(PVIFR%,20) R = 4.54% YTM = 4.54% × 2 = 9.08% P2 = £965 = £37.5(PVIFAR%,12) + £1,000(PVIFR%,12) R = 4.13% YTM = 4.13% × 2 = 8.25% To find the weighted average after-tax cost of debt, we need the weight of each bond as a percentage of the total debt. We find: wD1 = .93(£75M)/£127.65M = .546 wD2 = .965(£60M)/£127.65M = .454 Now we can multiply the weighted average cost of debt times one minus the tax rate to find the weighted average after-tax cost of debt. This gives us: RD = (1 – .35)[(.546)(.0908) + (.454)(.0825)] = .0566 or 5.66% Using these costs and the weight of debt we calculated earlier, the WACC is: WACC = .7978(.1600) + .2022(.0566) = .1391 or 13.91% 32. a. Using the equation to calculate WACC, we find: WACC = .105 = (1/1.8)(.15) + (.8/1.8)(1 – .35)RD RD = .0750 or 7.50% b. Using the equation to calculate WACC, we find: WACC = .105 = (1/1.8)RE + (.8/1.8)(.064) RE = .1378 or 13.78% 33. We will begin by finding the market value of each type of financing. We find: MVD = 40,000(£100)(1.03) = £4,120,000 MVE = 90,000(£57) = £5,130,000 And the total market value of the firm is: V = £4,120,000 + 5,130,000 = £9,250,000 Now, we can find the cost of equity using the CAPM. The cost of equity is: RE = .06 + 1.10(.08) = .1480 or 14.80% The cost of debt is the YTM of the bonds, so: P0 = £103 = £3.50(PVIFAR%,40) + £100(PVIFR%,40) R = 3.36% YTM = 3.36% × 2 = 6.72% And the after-tax cost of debt is: RD = (1 – .28)(.0672) = .0484 or 4.84% Now we have all of the components to calculate the WACC. The WACC is: WACC = .0484(4.12/9.25) + .1480(5.13/9.25) = .1036 or 10.36% Notice that we didn’t include the (1 – tC) term in the WACC equation. We simply used the after-tax cost of debt in the equation, so the term is not needed here. 34. Total Asset Value is (15.77 + 1.21) = £16.98 billion. This means that the weight of Debt is 0.07126 and the weight of equity is 0.92874. WACC = 0.07126(0.05)(1-.24) + 0.92874(.1996) = .1880 or 18.8% 35. a. We will begin by finding the market value of each type of financing. We find: MVD = 1,200,000(£100)(0.93) = £111,600,000 MVE = 9,000,000(£34) = £306,000,000 And the total market value of the firm is: V = £111,600,000 + 306,000,000 = £417,600,000 So, the market value weights of the company’s financing is: D/V = £111,600,000/£417,600,000 = .2672 E/V = £306,000,000/£417,600,000 = .7328 b. For projects equally as risky as the firm itself, the WACC should be used as the discount rate. First we can find the cost of equity using the CAPM. The cost of equity is: RE = .05 + 1.20(.10) = .1700 or 17.00% The cost of debt is the YTM of the bonds, so: P0 = £93 = £4.25(PVIFAR%,30) + £100(PVIFR%,30) R = 4.69% YTM = 4.69% × 2 = 9.38% And the after-tax cost of debt is: RD = (1 – .28)(.0938) = .0675 or 6.75% Now we can calculate the WACC as: WACC = .1700(.7328) + .0675 (.2672) = .1426 or 14.26% 36. a. Project Z. b. Using the CAPM to consider the projects, we need to calculate the expected return of each project given its level of risk. This expected return should then be compared to the expected return of the project. If the return calculated using the CAPM is lower than the project expected return, we should accept the project; if not, we reject the project. After considering risk via the CAPM: E[W] = .03 + .80(.075 – .03) = .066 > .06, so reject W E[X] = .03 + .70(.075 – .03) = .0615 > .05, so reject X E[Y] = .03 + 1.15(.075 – .03) = .0818 < .09, so accept Y E[Z] = .03 + 1.70(.075 – .03) = .1065 < .13, so accept Z c. Project Y would be incorrectly rejected. 37. Using the debt-equity ratio to calculate the WACC, we find: WACC = (.65/1.65)(.055) + (1/1.65)(.15) = .1126 or 11.26% Since the project is riskier than the company, we need to adjust the project discount rate for the additional risk. Using the subjective risk factor given, we find: Project discount rate = 11.26% + 2.00% = 13.26% We would accept the project if the NPV is positive. The NPV is the PV of the cash outflows plus the PV of the cash inflows. Since we have the costs, we just need to find the PV of inflows. The cash inflows are a growing perpetuity. If you remember, the equation for the PV of a growing perpetuity is the same as the dividend growth equation, so: PV of future CF = €3,500,000/(.1326 – .05) = €42,385,321 The project should only be undertaken if its cost is less than €42,385,321 since costs less than this amount will result in a positive NPV. 38. We will begin by finding the market value of each type of financing. We will use D1 to represent the coupon bond, and D2 to represent the zero coupon bond. So, the market value of the firm’s financing is: MVD1 = 50,000(€1,000)(1.1980) = €59,900,000 MVD2 = 150,000(€1,000)(.1385) = €20,775,000 MVP = 120,000(€112) = €13,440,000 MVE = 2,000,000(€65) = €130,000,000 And the total market value of the firm is: V = €59,900,000 + 20,775,000 + 13,440,000 + 130,000,000 = €224,115,000 Now, we can find the cost of equity using the CAPM. The cost of equity is: RE = .04 + 1.10(.09) = .1390 or 13.90% The cost of debt is the YTM of the bonds, so: P0 = €1,198 = €40(PVIFAR%,50) + €1,000(PVIFR%,50) R = 3.20% YTM = 3.20% × 2 = 6.40% And the after-tax cost of debt is: RD1 = (1 – .40)(.0640) = .0384 or 3.84% And the after-tax cost of the zero coupon bonds is: P0 = €138.50 = €1,000(PVIFR%,60) R = 3.35% YTM = 3.35% × 2 = 6.70% RD2 = (1 – .40)(.0670) = .0402 or 4.02% Even though the zero coupon bonds make no payments, the calculation for the YTM (or price) still assumes semi-annual compounding, consistent with a coupon bond. Also remember that, even though the company does not make interest payments, the accrued interest is still tax deductible for the company. To find the required return on preference shares, we can use the preference share pricing equation, which is the level perpetuity equation, so the required return on the company’s preference shares is: RP = D1 / P0 RP = €6.50 / €112 RP = .0580 or 5.80% Now we have all of the components to calculate the WACC. The WACC is: WACC = .0384(59.9/224.115) + .0402(20.775/224.115) + .1390(130/224.115) + .0580(13.44/224.115) WACC = .0981 or 9.81% 39. We can use the debt-equity ratio to calculate the weights of equity and debt. The debt of the company has a weight for long-term debt and a weight for trade payables. We can use the weight given for trade payables to calculate the weight of trade payables and the weight of long-term debt. The weight of each will be: trade payables weight = .20/1.20 = .17 Long-term debt weight = 1/1.20 = .83 Since the trade payables have the same cost as the overall WACC, we can write the equation for the WACC as: WACC = (1/2.3)(.17) + (1.3/2.3)[(.20/1.2)WACC + (1/1.2)(.09)(1 – .28)] Solving for WACC, we find: WACC = .0739 + .5652[(.20/1.2)WACC + .054] WACC = .0739 + (.0942)WACC + .0305 (.9058)WACC = .1044 WACC = .1153 or 11.53% Since the cash flows go to perpetuity, we can calculate the future cash inflows using the equation for the PV of a perpetuity. The NPV is: NPV = –£45,000,000 + (£5,700,000/.1153) NPV = –£45,000,000 + 49,436,253 = £4,436,253 40. The £7 million cost of the land 3 years ago is a sunk cost and irrelevant; the £6.5 million appraised value of the land is an opportunity cost and is relevant. The relevant market value capitalization weights are: MVD = 150,000(£100)(0.92) = £13,800,000 MVE = 300,000(£75) = £22,500,000 MVP = 20,000(£72) = £1,440,000 The total market value of the company is: V = £13,800,000 + 22,500,000 + 1,440,000 = £37,740,000 Next we need to find the cost of funds. We have the information available to calculate the cost of equity using the CAPM, so: RE = .05 + 1.3(.08) = .1540 or 15.40% The cost of debt is the YTM of the company’s outstanding bonds, so: P0 = £92 = £3.50(PVIFAR%,30) + £100(PVIFR%,30) R = 3.96% YTM = 3.96% × 2 = 7.92% And the after-tax cost of debt is: RD = (1 – .28)(.0792) = .0570 or 5.70% The cost of the preference shares is: RP = £5/£72 = .0694 or 6.94% a. The initial cost to the company will be the opportunity cost of the land, the cost of the plant, and the net working capital cash flow, so: CF0 = –£6,500,000 – 15,000,000 – 900,000 = –£22,400,000 b. To find the required return on this project, we first need to calculate the WACC for the company. The company’s WACC is: WACC = [(£22.5/£37.74)(.1540) + (£1.44/£37.74)(.0694) + (£13.8/£37.74)(.0570)] = .1153 The company wants to use the subjective approach to this project because it is located overseas. The adjustment factor is 2 percent, so the required return on this project is: Project required return = .1153 + .02 = .1353 c. The depreciation schedule for the equipment will be: Year 1 2 3 4 5 (a) Starting Value £15,000,000 £12,000,000 £9,600,000 £7,680,000 £6,144,000 (b) Depreciation 20% £3,000,000 £2,400,000 £1,920,000 £1,536,000 £1,144,000 (c) Accumulated Depreciation £3,000,000 £5,400,000 £7,320,000 £8,856,000 £10,000,000 (d) Residual Value £12,000,000 £9,600,000 £7,680,000 £6,144,000 £5,000,000 d. The operating cash flow for this project is: 1 2 3 4 5 Sales 12,000 12,000 12,000 12,000 12,000 Revenues £120,000,000 £120,000,000 £120,000,000 £120,000,000 £120,000,000 Variable Costs £108,000,000 £108,000,000 £108,000,000 £108,000,000 £108,000,000 Fixed Costs £400,000 £400,000 £400,000 £400,000 £400,000 Depreciation 20% £3,000,000 £2,400,000 £1,920,000 £1,536,000 £1,144,000 EBT £8,600,000 £9,200,000 £9,680,000 £10,064,000 £10,456,000 Tax £2,408,000 £2,576,000 £2,710,400 £2,817,920 £2,927,680 Net Income £6,192,000 £6,624,000 £6,969,600 £7,246,080 £7,528,320 Operating Cash Flow £9,192,000 £9,024,000 £8,889,600 £8,782,080 £8,672,320 e. Using Solver, the accounting breakeven sales figure for this project is 2,400 units. 1 2 3 4 5 Sales 2,400 2,400 2,400 2,400 2,400 Revenues £24,000,000 £24,000,000 £24,000,000 £24,000,000 £24,000,000 Variable Costs £21,600,000 £21,600,000 £21,600,000 £21,600,000 £21,600,000 Fixed Costs £400,000 £400,000 £400,000 £400,000 £400,000 Depreciation 20% £3,000,000 £2,400,000 £1,920,000 £1,536,000 £1,144,000 EBT -£1,000,000 -£400,000 £80,000 £464,000 £856,000 Tax -£280,000 -£112,000 £22,400 £129,920 £239,680 Net Income -£720,000 -£288,000 £57,600 £334,080 £616,320 Operating Cash Flow £2,280,000 £2,112,000 £1,977,600 £1,870,080 £1,760,320 The sum of the Net Income over five years is zero. f. We have calculated all cash flows of the project. The capital budgeting analysis is set out as follows: 0 1 2 3 4 5 Operating Cash Flow £9,192,000 £9,024,000 £8,889,600 £8,782,080 £8,672,320 Net Working Capital -£900,000 £900,000 Land -£6,500,000 £4,500,000 Investment -£15,000,000 £5,000,000 Net Cash Flow -£22,400,000 £9,192,000 £9,024,000 £8,889,600 £8,782,080 £19,072,320 PV Cash Flows @ 13.53% -£22,400,000 £8,096,538 £7,001,286 £6,075,056 £5,286,337 £10,112,309 NPV £14,171,527 IRR 34.71% Solution Manual for Corporate Finance David Hillier, Stephen Ross, Randolph Westerfield, Jeffrey Jaffe, Bradford Jordan 9780077139148