Chapter 4 Solutions (5, 8, 9, 14, 19, and 23)
Problem 5
Heavy Tech Company has a current beta of 1.2 with a stock return variance (σ2) of 50%. The stock is selling for $50, down $4 from last year. The company paid a $2 dividend last year and will (expected) pay a $2.50 dividend this coming year. The tax rate is 40% for the company.
The variance of the market for this same period is 20%. The NYSE composite (broad market index) is down 8% from last year and the dividend yield for this composite is 3%.
Current Treasury Bill is 3% (risk-free rate)
a. What is the expected return for Heavy Tech for the coming year? We will use the CAPM for this but need to find the market risk premium.
Unfortunately the author did not provide enough information to find the market premium. We could try the Discounted Dividend Model approach but we do not know the growth rate on dividends…so from his answer book he states we will use the market return premium over the risk-free rate and gives 8.76% as the market risk premium and now with the CAPM we have:
Expected return = 3% + 1.2 (8.76%) = 13.512%
This should be…Expected return = 3% + 1.2 (5.5%) = 9.60%,
b. What would you expect the price to be one year from now?
If it does return 9.60% one year from now we can put this into the return formula
E(r) = (Ending Price + Dividends – Beginning Price) / Beginning Price
Substituting the known variables we have
9.60% = (Ending Price + $2.50 - $50.00) / $50.00
And with the magic of algebra we eventually have
Ending Price = $50 + 0.096 x $50 - $2.50 = $52.30
c. What would you expect the return over the past year if, the risk-free rate was 5% and we used the same market risk premium of 5.5%?
E(r) = 5% + 1.2 (5.5%) = 11.60%
d. What was the actual return of Heavy Tech the past year?
Return = (Ending Price + Dividends – Beginning Price) / Beginning Price
Return = ($50.00 + $2.00 - $54.00) / $54.00 = -$2.00 / $54.00 = -0.037 or -3.7%
e. Estimate the new beta if Heavy Tech goes 100% equity (an unlevered firm).
Here the author meant to say that the $50 million in debt would be retired, not the $5 million in debt from the problem…so
Unlevered Beta = Beta / (1 + (1 –Tax Rate) (D/E))
Unlevered Beta = 1.2 / ((1 + (1 -0.40) (50/100)) = 1.2 / (1 + 0.6 x 0.5)
= 1.2 /1.3 = 0.92
Because the new firm is 100% equity, its beta should be 0.92 the unlevered beta
Problem 8
Hewlett-Packard has four divisions and four division betas. The nice thing about beta is you can do a weighted combination of the betas (weights based on market value of divisions) to find the beta of the firm.
Division Market Value Beta
Mainframes 2.0 Billion 1.10
PCs 2.0 Billion 1.50
Software 1.0 Billion 2.00
Printers 3.0 Billion 1.00
Total 8.0 Billion
a. What is the beta of the company?
Unlevered Beta = 2.0/8.0 x 1.1 + 2.0/8.0 x 1.5 + 1.0/8.0 x 1.50 + 3.0/8.0 x 1.00
= 0.275 + 0.375 + 0.25 + 0.375 = 1.275
Now with 1.0 billion of debt the total value of the firm is 9.0 billion. Because the company borrows for all divisions under the HP name we can allocate the debt evenly across the four divisions. The levered beta of HP with a tax rate of 36% and a 1/8 D/E ratio is:
Levered Beta = Unlevered x (1 + (1 –Tax Rate) (D/E))
= 1.275 x (1+ (1 – 0.36) (1/8)) = 1.275 x 1.08 = 1.377
Is this the same beta you would get by regressing the HP returns on the market? Probably not as the size of the divisions have changed over time and so it would not have the same percentage invested in each division each year. If the percentage invested each year was the same and the D/E ratio was same over time theoretically you could get the same beta.
b. If the Treasury bond rate is 7.5% (and you are looking at the long horizon) estimate the cost of equity for HP. Again we need the market risk premium which the author states at the start of the problems…so using 5.5% (expected return of the market at 13% and premium of 13% - 7.5% = 5.5%)
Using our beta of 1.377 the required return to equity holders should be:
Re = 7.5% + 1.377 (5.5%) = 15.0735%
What about each division should you calculate the required return for equity for each division? First we have to find each division’s levered beta if the ones given are unlevered (which we assumed for part a)
Levered Beta = Unlevered x (1 + (1 –Tax Rate) (D/E))
Mainframes = 1.10 x (1+ (1 – 0.36) (1/8)) = 1.10 x 1.08 = 1.188
PCs = 1.50 x (1+ (1 – 0.36) (1/8)) = 1.50 x 1.08 = 1.62
Software = 2.00 x (1+ (1 – 0.36) (1/8)) = 2.00 x 1.08 = 2.16
Printers = 1.00 x (1+ (1 – 0.36) (1/8)) = 1.00 x 1.08 = 1.08
And now the required returns for each division
Mainframes Re = 7.5% + 1.188 (5.5%) = 14.034%
PCs Re = 7.5% + 1.62 (5.5%) = 16.41%
Software Re = 7.5% + 2.16 (5.5%) = 19.38%
Printers Re = 7.5% + 1.08 (5.5%) = 13.44%
c. If the Mainframe division is sold we will assume it is sold at market value (equity market value plus its proportion of the debt or 25% of the 1 billion in debt) for $225 billion. The author says that the $225 was paid out as a dividend but let’s assume the portion of the debt is also retired (because how can you reduce the equity value of the other divisions?) The current value of the firm is now $6.75 billion and its debt/equity ratio is 0.75/6.0 (as 25% of the debt was retired).
Unlevered beta of HP = (2.0/6.0) (1.50) + (1.0/6.0)(2.00) + (3.0/6.0)(1.0) = 1.33
Levered Beta = Unlevered x (1 + (1 –Tax Rate) (D/E))
= 1.3333 x (1+ (1 – 0.36) (0.75/6.0)) = 1.3333 x 1.083478 = 1.445
Problem 9
Four Firms with the following data
Company % Change in Revenue % Change in Op Income Beta
PharmaCorp 27% 25% 1.00
SynerCorp 25% 32% 1.15
BioMed 23% 36% 1.30
SafeMed 21% 40% 1.40
a. Calculate the degree of operating leverage for each firm.
Operating Leverage = %Change in EBIT / %Change in Sales
Here EBIT is the same as operating income and Sales is the same as Revenue
PharmaCorp Operating Leverage = 0.25 / 0.27 = 0.9259
SynerCorp Operating Leverage = 0.32 / 0.25 = 1.28
BioMed Operating Leverage = 0.36 / 0.23 = 1.5652
SafeMed Operating Leverage = 0.40 / 0.21 = 1.9048
b. Can you explain the betas by operating leverage? First note that the greater the operating leverage the greater the beta. High operating leverage means that the income is more sensitive to changes in the sales. High operating income means higher fixed costs (relative to variable costs) and thus a riskier firm and a higher beta.
Problem 14
This problem wants to re-establish the relationships in the statistics from a regression with the variances and covariances of the random variables (here the returns of AMR and the S&P 500).
a. You know that the R-squared of the regression is 0.36 and the stock has a variance of 0.67. The market variance is 0.12 so…what is the beta of the stock?
Recall R2 = (beta)2 (variance of market) / (variance of the stock)
Substituting we have, 0.36 = (beta)2 (0.12) / (0.67)
And rearranging via algebra we have beta = [(0.36) (0.67) / (0.12)]1/2 = 1.4177
b. You know that AMR did worse than expected by 0.39 percent (0.0039) during the regression and the risk-free rate during the regression period was 4.84%. What is the intercept of the regression? This question is not correct statistically speaking. A company can not under perform in a regression. What the author should have done was to give you the average return of the two assets over this period. Let’s assume that the market’s average return was 13.5% and the average return for AMR was 18.75%. Now find the intercept of the regression given the beta of 1.4177 in part a…
Intercept = (mean of the AMR) – (beta) (mean of market)
Intercept = (0.1875) – (1.4177) (0.135) = - 0.0039 or - 0.39%
c. Another firm has an R-squared of 0.48 (I think he meant 0.36 otherwise this question does not make sense)…would the two firms have the same beta?
The answer is no…the beta is a function of covariance of each company and the market while the R-squared is a function of the variance of each company and the market. Think of this as a proof…
R2i = (β1) (σ2i) / (σ2M)
So (β 1) (σ21) / (σ2M) = (β 2) (σ22) / (σ2M)
And (β 1) (σ21) = (β 2) (σ22)
So if β1 = β2 then (σ21) = (σ22) …and we do not know this…
Problem 19
Trying to estimate the beta of a private company…this is called pure play…
Other home appliance manufacturing companies that are public have the following beta, debt, and equity…
Company Beta Debt Equity
Black and Decker 1.40 $2,500 $3,000
Fedders 1.20 $5 $200
Maytag 1.20 $540 $2,250
National Presto 0.70 $8 $300
Whirlpool 1.50 $2,900 $4,000
a. Estimate the beta of a private company with a debt-equity ratio of 25% and a tax rate of 40%. These firms also have a tax rate of 40%.
One way is to compute the unlevered betas of each of the five firms and then average these unlevered betas and substitute it as the unlevered beta of the private company…
Unlevered Beta = Beta / (1 + (1 –Tax Rate) (D/E))
Black & Decker = 1.4 / (1+ (1-0.40) (2,500/3,000) = 0.9333
Fedders = 1.2 / (1+ (1-0.40) (5/200) = 1.1823
Maytag = 1.2 / (1+ (1-0.40) (540/2,250) = 1.0490
National Presto = 0.7 / (1+ (1-0.40) (8/300) = 0.6890
Whirlpool = 1.5 / (1+ (1-0.40) (2,900/4,000) = 1.0453
Average Unlevered Beta = (0.9333 + 1.1823 + 1.0490 + 0.6890 + 1.0453) / 5
Average Unlevered Beta = 4.8989 / 5 = 0.9798
Levered beta of private firm = 0.9798 (1+ (1 – 0.40) (1/4)) = 1.1268
b. What are your concerns on using this approach?
First, the wide range of betas in the industry makes it hard to justify a point estimate of the beta of the private firm. Second, the debt/equity ratios and size of these vary and it would probably be better to find a firm similar in size and debt/equity funding for the private firm and just use that beta.
Problem 23
Look at Tiffany’s information and estimate its beta based on the industry…
You know that regression estimate of the beta is 0.75 and the standard error for the beta is 0.50. You also note that the average unlevered beta of comparable specialty retailing firms is 1.15.
a. If Tiffany’s has a d/e ratio of 20%, estimate the beta for the company based on comparable firms using a 40% tax rate…
Beta = Unlevered Beta (1 + (1 –Tax Rate) (D/E))
Beta of Tiffany’s = 1.15 (1+ (1-0.4)(1/5)) = 1.288
b. Estimate a range for the beta from the regression…
Here we must assume that we have a desired range, which is within one standard deviation of the mean estimate…or two standard deviations of the mean…
One standard deviation range: 0.75 – (0.50) to 0.75 + (0.50) = 0.25 to 1.25
Two standard deviation range: 0.75 – 2(0.50) to 0.75 + 2(0.50) = -0.25 to 1.75
The one standard deviation range has a confidence interval of 67% and the two standard deviation mean has a confidence interval of 95%.
c. Assume Tiffany’s has a triple B rating and the spread is 1 percent above the T-bond rate of 6.5%. Estimate the cost of capital for Tiffany.
Need the market risk premium (use 5.5% as specified at the start of the problems)
Cost of Equity using the 0.75 beta = 6.5% + 0.75 (5.5%) = 10.625%
Cost of Debt = 6.5% + 1.0% = 7.5%
Cost of Capital (WACC) = 5/6 (10.625%) + 1/6 (7.5%) (1 – 0.40)
= 8.8542 % + 0.75% = 9.6042%
Where does the 5/6 and 1/6 come from in the WACC?
Recall the debt-equity ratio is 20% or 1/5. If we set E = 5 and D = 1 then in the WACC we have E/V = 5 / (5 +1) and D/V = 1 / (5+1) or 5/6 and 1/6…