Chapter 12: Risk, Return, and Capital Budgeting
12.1 Cost of equity
RS = 5 + 0.95 (9) = 13.55% NPV of the project 5 $340,000 = -$1.2 million + t t1 1.1355 = -$20,016.52 Do not undertake the project.
12.2 a. R D = (-0.05 + 0.05 + 0.08 + 0.15 + 0.10) / 5 = 0.066
R M = (-0.12 + 0.01 + 0.06 + 0.10 + 0.05) / 5 = 0.02 b. 2 RD - R D RM - R M (RM - R M ) (RD - R D )(RM - R M ) -0.116 -0.14 0.0196 0.01624 -0.016 -0.01 0.0001 0.00016 0.014 0.04 0.0016 0.00056 0.084 0.08 0.0064 0.00672 0.034 0.03 0.0009 0.00102 0.0286 0.02470
Beta of Douglas = 0.02470 / 0.0286 = 0.864
12.3 RS = 6% + 1.15 10% = 17.5%
RB = 6% + 0.3 10% = 9% a. Cost of equity = RS = 17.5% b. B / S = 0.25 B / (B + S) = 0.2 S / (B + S) = 0.8 WACC = 0.8 17.5% + 0.2 9% (1 - 0.35) = 15.17%
1/2 12.4 C = (0.004225) = 0.065 1/2 M = (0.001467) = 0.0383 Beta of ceramics craftsman 2 = CM C M / M = CM C / M = (0.675) (0.065) / 0.0383 = 1.146
12.5 a. To compute the beta of Mercantile Manufacturing’s stock, you need the product of the deviations of Mercantile’s returns from their mean and the deviations of the market’s returns from their mean. You also need the squares of the deviations of the market’s returns from their mean.
The mechanics of computing the means and the deviations were presented in an earlier chapter.
1 R T = 0.196 / 12 = 0.016333
R M = 0.236 / 12 = 0.019667
E(RT - R T ) (RM - R M ) = 0.003226 2 E(RM - R M ) = 0.003086 = 0.003226 / 0.003086 = 1.0453 b. The beta of the average stock is 1. Mercantile’s beta is higher. This indicates that Mercantile’s stock is riskier than the average stock.
12.6 a. RM can have three values, 0.16, 0.18 or 0.20. The probability that RM takes one of these values is the sum of the joint probabilities of the return pair that include
the particular value of RM. For example, if RM is 0.16, RJ will be 0.16, 0.18 or 0.22. The probability that RM is 0.16 and RJ is 0.16 is 0.10. The probability that RM is 0.16 and RJ is 0.18 is 0.06. The probability that RM is 0.16 and RJ is 0.22 is 0.04. The probability that RM is 0.16 is, therefore, 0.10 + 0.06 + 0.04 = 0.20. The same procedure is used to calculate the probability that RM is 0.18 and the probability that RM is 0.20. Remember, the sum of the probability must be one.
RM Probability 0.16 0.20 0.18 0.60 0.20 0.20
b. i. R M = 0.16 (0.20) + 0.18 (0.60) + 0.20 (0.20) = 0.18 2 2 2 ii. M = (0.16 - 0.18) (0.20) + (0.18 - 0.18) (0.60) + (0.20 - 0.18) 2 (0.20) = 0.00016 1/2 iii. M = (0.00016) = 0.01265
c. RJ Probability .16 .10 .18 .20 .20 .40 .22 .20 .24 .10
d. i. E j = .16 (.10) + .18 (.20) + .20 (.40) + .22 (.20) + .24(.10) = .20 2 2 2 2 ii. j = (.16 - .20) (.10) + (.18 - .20) (.20) + (.20 - .20) (.40) + (.22 - .20)2 (.20) + (.24 - .20)2 (.10) = .00048 1/2 iii. j = (.00048) = .02191
e. Covmj = (.16 - .18) (.16 - .20) (.10) + (.16 - .18) (.18 - .20) (.06) + (.16 - .18) (.22 - .20) (.04) + (.20 - .18) (.18 - .20) (.02) + (.20 - .18) (.22 - .20) (.04) + (.20 - .18) (.24 - .20) (.10) = .000176
2 Corrmj = (0.000176) / (0.01265) (0.02191) = 0.635
f. j = (.635) (.02191) / (.01265) = 1.10
12.7 i. The risk of the new project is the same as the risk of the firm without the project. ii. The firm is financed entirely with equity.
12.8 a. Pacific Cosmetics should use its stock beta in the evaluation of the project only if the risk of the perfume project is the same as the risk of Pacific Cosmetics. b. If the risk of the project is the same as the risk of the firm, use the firm’s stock beta. If the risk differs, then use the beta of an all-equity firm with similar risk as the perfume project. A good way to estimate the beta of the project would be to average the betas of many perfume producing firms.
12.9 E(RS) = 0.1 3 + 0.3 8 + 0.4 20 + 0.2 15 = 13.7%
E(RB) = 0.1 8 + 0.3 8 + 0.4 10 + 0.2 10 = 9.2%
E(RM) = 0.1 5 + 0.3 10 + 0.4 15 + 0.2 20 = 13.5%
State {RS - E(RS)}{RM - E(RM)}Pr {RB - E(RB)}{RM - E(RM)}Pr 1 (0.03-0.137)(0.05-0.135)0.1 (0.08-0.092)(0.05-0.135)0.1 2 (0.08-0.137)(0.10-0.135)0.3 (0.08-0.092)(0.10-0.135)0.3 3 (0.20-0.137)(0.15-0.135)0.4 (0.10-0.092)(0.15-0.135)0.4 4 (0.15-0.137)(0.20-0.135)0.2 (0.10-0.092)(0.20-0.135)0.2 Sum 0.002056 0.00038
= Cov(RS, RM) = Cov(RB, RM)
2 2 2 M = 0.1 (0.05 - 0.135) + 0.3 (0.10-0.135) + 0.4 (0.15-0.135)2 + 0.2 (0.20-0.135)2 = 0.002025 2 a. Beta of debt = Cov(RB, RM) / M = 0.00038 / 0.002025 = 0.188 2 b. Beta of stock = Cov(RS, RM) / M = 0.002055 / 0.002025 = 1.015 c. B / S = 0.5 Thus, B / (S + B) = 1 / 3 = 0.3333 S / (S + B) = 2 / 3 = 0.6667 Beta of asset = 0.188 0.3333 + 1.015 0.6667 = 0.739
12.10 The discount rate for the project should be lower than the rate implied by the use of the Security Market Line. The appropriate discount rate for such projects is the weighted average of the interest rate on debt and the cost of equity. Since the interest rate on the debt of a given firm is generally less than the firm’s cost of equity, using only the stock’s beta yields a discount rate that is too high. The concept and practical uses of a weighted average discount rate will be in a later chapter.
3 12.11 i. Revenues The gross income of the firm is an important factor in determining beta. Firms whose revenues are cyclical (fluctuate with the business cycle) generally have high betas. Firms whose revenues are not cyclical tend to have lower betas. ii. Operating leverage Operating leverage is the percentage change in earnings before interest and taxes (EBIT) for a percentage change in sales, [(Change in EBIT / EBIT) (Sales / Change in sales)]. Operating leverage indicates the ability of the firm to service its debt and pay stockholders. iii. Financial leverage Financial leverage arises from the use of debt. Financial leverage indicates the ability of the firm to pay stockholders. Since debt holders must be paid before stockholders, the higher the financial leverage of the firm, the riskier its stock.
The beta of common stock is a function of all three of these factors. Ultimately, the riskiness of the stock, of which beta captures a portion, is determined by the fluctuations in the income available to the stockholders. (As was discussed in the chapter, whether income is paid to the stockholders in the form of dividends or it is retained to finance projects are irrelevant as long as the projects are of similar risk as the firm.) The income available to common stock, the net income of the firm, depends initially on the revenues or sales of the firm. The operating leverage indicates how much of each dollar of revenue will become EBIT. Financial leverage indicates how much of each dollar of EBIT will become net income.
12.12 a. Cost of equity for National Napkin = 7 + 1.29 (13 - 7) = 14.74% b. B / (S + B) = S / (S + B) = 0.5 WACC = 0.5 7 0.65 + 0.5 14.74 = 9.645%
12.13 B = $60 million 1.2 = $72 million S = $20 5 million = $100 million B / (S + B) = 72 / 172 = 0.4186 S / (S + B) = 100 / 172 = 0.5814 WACC = 0.4186 12% 0.75 + 0.5814 18% = 14.23%
12.14 S = $25 20 million = $500 million B = 0.95 $180 million = $171 million B / (S + B) = 0.2548 S / (S + B) = 0.7452 WACC = 0.7452 20% + 0.2548 10% 0.60 = 16.43%
12.15 B / S = 0.75
4 B / (S + B) = 3 / 7 S / (S + B) = 4 / 7 WACC = (4 / 7) 15% + (3 / 7) 9% (1 - 0.35) = 11.08% 5 $7million NPV = -$25 million + t t1 (1 0.1108) = $819,299.04 Undertake the project.
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