SOLUTIONS
Exercises (9)
1. SOLUTION
rA 
ExpectedOperatingIncome
MarketValueofAllSecurities
= (proportion in debt X expected return on debt) + (proportion in equity X
expected return on equity)
= (weight of debt X cost of debt) + (weight of equity X cost of equity)
= W DrD + WErE
= [D/(D + E) X rD] + [E/(D + E) X rE]
= (0.25 X 0.12) + (0.75 X 0.16)
= 0.03 + 0.12
= 0.15 = l5 percent
2. SOLUTION
The return on assets is give by:
rA = expected operating income/market value of all securities
= $9,000/$50,000
= 0.18 = 18 percent
In an all-equity-financed firm, the return on equity rE is equal to the return on assets;
so
rE = rA = 18 percent
While borrowing changes the return on equity, it will not change the return on assets.
This will remain the same because the value of the firm does not change and the
expected operating income does not change. Therefore, the return on equity is:
rE = rA + (D/E)(rA - rD)
= 0.18 + ($15,000/$35,000)(0.18 - 0.12)
= 0.18 + 0.4286(0.06)
= 0.18 + 0.0257
= 0.2057 = 20.57 percent
3. SOLUTION
The beta of the equity without any debt financing is the same as the beta of the firm
because no other securities are outstanding. After the financing, the beta of the
equity changes as follows:
I
E = A + (D/E)(A - D)
= 1.2 + 0.4286(1.2 - 0.5)
= 1.2 + 0.3 = 1.5
For the other debt proportions, E is given in the table below.
D/E (%)
30
50
60
70
E
1.41
1.55
1.62
1.69
4. SOLUTION
The values for E for different debt/equity ratios and different D are given in the table
below.
D/E (%)
0
5
10
20
40
60
80
100
E for different value of D
D = 0
D = 1.2
D = 0.5
1.20
1.20
1.20
1.26
1.20
1.24
1.32
1.20
1.27
1.44
1.20
1.34
1.68
1.20
1.48
1.92
1.20
1.62
2.16
1.20
1.76
2.40
1.20
1.90
While the data above are somewhat contrived, several interesting results emerge.
You can clearly see that the increase in the risk or beta of the equity with increased
debt levels is a function of the beta or risk of the debt itself. First, when the beta of
the debt is equal to the beta of the firm, when it is all-equity-financed, for all practical
purposes, the company has issued another dose of equity and not debt.
Consequently, the beta of the equity does not change. If the company were able to
issue debt at the zero-beta level, the risk-free rate, the betas of the equity would tend
to increase substantially with additional debt. In practice and in most cases, the beta
of corporate debt is greater than zero but less than the beta of the all-equity-financed
firm.
5. SOLUTION
First, estimate the value of the firm before the debt financing. The operating earnings
are capitalized at 20 percent, so the value of the firm when it is all-equity-financed is
$125,000 ($25,000/0.20). The value per share is $12.50; earnings per share are
$2.50 ($25,000/10,000 shares); with the price-earnings ratio being 5.0 times
($12.50/$2.50).
Next, determine the earnings per share after the debt financing. The company must
sell $62,500 of debt at the going market rate of 10 percent in order to repurchase an
equivalent amount of equity ($125,000/2). Remember, because the operating
income remains unchanged, the firm's total value remains unchanged at $125,000.
The equity earnings change, however, as follows:
Operating income $25,000
less interest
6,250
Equity earnings
$18,750
With 5,000 shares now outstanding, earnings per share increase to $3.75
($18,750/5000 shares). Ms. Black assumes that she can still get the same priceearnings ratio of 5. Then, the shares have a market price of $18.75 (5 x $3.75). But
going into debt entails additional risk to the shareholders. The beta will increase
substantially. Using the formula:
E = A + D/E(A - D)
Before borrowing:
 = 1.2 + 0.0(1.2 - 0) = 1.2
After borrowing:
 = 1.2 + 1.0(1.2 - 0.4) = 1.2 + 0.8 = 2.0
Risk increases by two-thirds. With increased risk, the required rate of return on
equity increases too. Assuming that the return on asset stays same, we can
calculate the required return as follows:
rE = rA + (D/E)(rA - rD)
= 0.20 + 1.0(0.20 - 0.10)
= 0.20 + 0.10
= 0.30 or 30 percent
The value of the shares will then be: $3.75/0.30 = $12.50
Thus, there is no change in the value of the stock and leverage does not increase the
value of the stock. Any increase in earnings is fully neutralized by the corresponding
increase in risk and required rate of return. The price-earnings ratio will go down and
the market price of the shares will remain the same.