Chapter 13 - The Weighted-Average Cost of Capital and Company Valuation
Solutions to Chapter 13
The Weighted-Average Cost of Capital and Company Valuation
1.
The yield to maturity for the bonds (since maturity is now 19 years) is the interest
rate (r) that is the solution to the following equation:
[$80  annuity factor (r, 19 years)] + [$1,000/(1 + r)19] = $1,050
1
 $1,000
1
$80   

 $1,050  r = 7.50%
19 
19
 r r  (1  r )  (1  r )
Using a financial calculator, enter n = 19, FV = 1,000, PV = (–)1,050, PMT = 80;
then compute i = 7.50%.
Therefore, the after-tax cost of debt is 7.50% (1 – 0.35) = 4.88%.
Est time: 01–05
2.
r
DIV
$4

 0.100  10.0%
P0
$40
Est time: 01–05
3.
D
 P
 E

WACC    rdebt  (1  TC )    rpreferred     requity 
V
 V
 V

= [0.3  7.50%  (1 – 0.35)] + [0.2  10%] + [0.5  12.0%] = 9.46%
Est time: 01–05
4.
r
DIV1
DIV0 (1  g )
$5  1.05
g
g
 0.05  0.1375  13.75%
P0
P0
$60
Est time: 01–05
5.
The total value of the firm is $80 million. The weights for each security class
are as follows:
Debt:
Preferred:
Common:
D/V = 20/80 = 0.250
P/V = 10/80 = 0.125
E/V = 50/80 = 0.625
13-1
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 13 - The Weighted-Average Cost of Capital and Company Valuation
D
 P
 E

WACC    rdebt  (1  TC )    rpreferred     requity 
V
 V
 V

= [0.250  6%  (1 – 0.35)] + [0.125  8%] + [0.625  12.0%] = 9.475%
Est time: 01–05
6.
Executive Fruit should use the WACC of Geothermal, not its own WACC,
when evaluating an investment in geothermal power production. The risk of the
project determines the discount rate, and in this case Geothermal’s WACC is
more reflective of the risk of the project in question. The proper discount rate,
therefore, is not 12.3%. It is more likely to be 11.4%.
Est time: 01–05
7.
a.
The weighted-average cost of capital, with a tax rate of 40%, is:
D
 E

WACC    rdebt  (1  TC )    requity
V
 V

= [0.30  6%  (1 – 0.40)] + [0.70  11%] = 8.78%
Free cash flow next year is $68 million – $30 million = $38 million.
Since the cash flows are in the form of a growing perpetuity, with a growth
rate of 4%, the total value of Icarus is:
PV 
b.
$38 million
$38 million

 $795 million
rg
0.0878  0.04
Since management will maintain the company’s debt at 30% of the
present value of the company, the company’s equity is:
0.70 × $795 million = $556.5 million
Est time: 06–10
8.
The rate on Buildwell’s debt is 5%. The cost of equity capital is the required rate of
return on equity, which can be calculated from the CAPM as follows:
4% + (0.90  8%) = 11.2%
The weighted-average cost of capital, with a tax rate of 40%, is:
D
 E

WACC    rdebt  (1  TC )    requity
V
V

 

= [0.30  5%  (1 – 0.40)] + [0.70  11.2%] = 8.74%
13-2
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 13 - The Weighted-Average Cost of Capital and Company Valuation
Est time: 01–05
9.
The internal rate of return, which is 12%, exceeds the cost of capital. Therefore,
BCCI should accept the project.
The present value of the project cash flows is:
$100,000  annuity factor (8.74%, 8 years) =
 1

1
$100,000  

 $558,870.94
8
 0.0874 0.0874  (1.0874) 
This is the most BCCI should pay for the project.
Est time: 01–05
10.
Line numbers in the table below are from the text:
Year
3. EBITDA
4. Depreciation
5. Profit before tax = 3 – 4
6. Tax at 40%
7. Profit after tax = 5 – 6
8. Operating cash flow = 4 + 7
9. Investment
10. Free cash flow = 8 – 9
1
80
20
60
24
36
56
12
44
2
100
30
70
28
42
72
15
57
3
115
35
80
32
48
83
18
65
4
120
40
80
32
48
88
20
68
Since free cash flow from year 5 onward will remain unchanged at year-4 levels, the
horizon value at year 4 is:
$68 million
$68 million

 $778 million
r
0.0874
The company’s total value is:
PV 
$44
$57
$65
$68
$778




 $744.3 million
2
3
4
1.0874 (1.0874)
(1.0874)
(1.0874)
(1.0874) 4
Since the capital structure is 30% debt, the value of the firm’s debt is:
0.30 × $744.3 million = $223.3 million
The value of the equity is:
0.70 × $744.3 million = $521.0 million
Est time: 06–10
13-3
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 13 - The Weighted-Average Cost of Capital and Company Valuation
11.
Security
Debt
Equity
Total
Market Value
$ 5.5 million
$15.0 million
$20.5 million
*Number of shares =
Explanation
1.10  par value of $5 million
$30 per share  500,000 shares*
$10 million book value
 500,000
$20 book value per share
D
 E

WACC    rdebt  (1  TC )    requity
V
 V

 5.5
  15


 9%  (1  0.40)  
 15%  12.42%
 20.5
  20.5

Est time: 01–05
12. Since the firm is all-equity financed, asset beta = equity beta = 0.8.
The WACC is the same as the cost of equity, which can be calculated using the CAPM:
requity = rf + (rm – rf) = 4% + (0.80  10%) = 12%
Est time: 01–05
13. The 12.5% value calculated by the analyst is the current yield of the firm’s outstanding
debt: interest payments/bond value. This calculation ignores the fact that bonds selling
at discounts from, or premiums over, par value provide expected returns determined in
part by expected price appreciation or depreciation. The analyst should be using yield
to maturity instead of current yield to calculate cost of debt. (This answer assumes the
value of the debt provided is the market value. If it is the book value, then 12.5%
would be the average coupon rate of outstanding debt, which would also be a poor
estimate of the required rate of return on the firm’s debt.)
Est time: 01–05
14. a.
Using the recent growth rate of 30% and the dividend yield of 2%, one
estimate would be:
DIV1/P0 + g = 0.02 + 0.30 = 0.32 = 32%
Another estimate, based on the CAPM, would be:
r = rf + (rm – rf) = 4% + (1.2  8%) = 13.6%
13-4
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 13 - The Weighted-Average Cost of Capital and Company Valuation
b.
The estimate of 32% seems far less reasonable. It is based on a historic growth rate
that is impossible to sustain. The DIV1/P0 + g rule requires that the growth rate of
dividends per share must be viewed as highly stable over the foreseeable future. In
other words, it requires the use of the sustainable growth rate.
Est time: 01–05
15. a.
The 9% coupon bond has a yield to maturity of 10% and sells for 93.86% of face
value, as shown below:
 1
 $1,000
1
PV  $90  


 $938.55
10 
10
 0.10 0.10(1.10)  1.10
Using a financial calculator, enter n = 10, i = 10%, PMT = 90, FV = 1,000;
compute PV = $938.55.
Therefore, the market value of the issue is:
0.9386  $20 million = $18.77 million
The 10% coupon bond sells for 94% of par value and has a yield to maturity of
10.83%, as shown below:
1
 $1,000
1
 r = 10.83%
$940  $100   

15 
15
 r r  (1  r )  (1  r )
Using a financial calculator, enter n = 15, PV = ()940, PMT = 100, FV = 1,000;
compute i = 10.83%.
The market value of the issue is 0.94  $25 million = $23.50 million.
Therefore, the weighted-average before-tax cost of debt is:
18.77
23.50

 

18.77  23.50  10%  18.77  23.50  10.83%  10.46%
b.
The after-tax cost of debt is (1 – 0.35)  10.46% = 6.80%.
Est time: 06–10
16.
The bonds are selling below par value because the yield to maturity is greater than
the coupon rate.
The price per $1,000 par value is:
PV = [$80  annuity factor (9%, 10 years)] + ($1,000/1.0910)
13-5
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 13 - The Weighted-Average Cost of Capital and Company Valuation
 1
 $1,000
1
 $80  


 $935.82
10 
10
 0.09 0.09(1.09)  1.09
The total market value of the bonds is:
$10 million par value 
$935.82 market value
 $9.36 million
$1,000 par value
There are $2 million/$20 = 100,000 shares of preferred stock.
The market price of the preferred stock is $15 per share, so the total market value of
the preferred stock is $1.5 million.
There are $0.1 million/$0.10 = 1 million shares of common stock.
The market price of the common stock is $20 per share, so the total market value of
the common stock is $20 million.
Therefore, the capital structure is:
Security
Bonds
Preferred Stock
Common Stock
Total
Market Value
$9.36 million
$1.50 million
$20.00 million
$30.86 million
Percent
30.3%
4.9%
64.8%
100.0%
Est time: 06–10
17. The yield to maturity for the firm’s debt is rdebt = 9%.
The rate for the preferred stock is rpreferred = $2/$15 = 0.1333 = 13.33%.
The rate for the common stock is:
requity = rf + (rm – rf) = 6% + (0.8  10%) = 14%
Using the capital structure derived in the previous problem, we can calculate WACC as:
D
 P
 E

WACC    rdebt  (1  TC )    rpreferred     requity 
V
 V
 V

= [0.303  9%  (1 – 0.40)] + [0.049  13.33%] + [0.648  14%] = 11.36%
Est time: 01–05
13-6
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 13 - The Weighted-Average Cost of Capital and Company Valuation
18. The IRR on the computer project is less than the WACC of firms in the computer
industry. Therefore, the project should be rejected. However, the WACC of the firm
(based on its existing mix of projects) is only 11.36%. If the firm uses this figure as the
hurdle rate, it will incorrectly go ahead with the venture in home computers.
Est time: 01–05
19. a.
r = rf + (rm – rf) r = 4% + (1.2  10%) = 16%
b.
Weighted-average beta = (0.4  0) + (0.6  1.2) = 0.72
c.
D
 E

WACC    rdebt  (1  TC )    requity
V
 V

= [0.4  4%  (1 – 0.4)] + [0.6  16%] = 10.56%
d.
If the company plans to expand its present business, then the WACC is a
reasonable estimate of the discount rate since the risk of the proposed project
is similar to the risk of the existing projects. Use a discount rate of 10.56%.
e.
The WACC of optical projects should be based on the risk of those projects.
Using a beta of 1.4, the discount rate for the new venture is:
r = 4% + 1.4  10% = 18%
Est time: 06–10
20. If Big Oil does not pay taxes, then the after-tax and before-tax costs of debt are
identical. WACC would then become:
D
 E

WACC    rdebt  (1  TC )    requity
V
 V

= [0.243  9%  (1 – 0)] + [0.757  12.0%] = 11.27%
If Big Oil issues new equity and uses the proceeds to pay off all of its debt, the cost
of equity will decrease. There is no longer any leverage, so the equity becomes safer
and therefore commands a lower risk premium. In fact, with all-equity financing, the
cost of equity would be the same as the firm’s WACC, which is 11.27%. This is less
than the previous value of 12.0%. (We use the WACC derived in the absence of
interest tax shields since, for the all-equity firm, there is no interest tax shield.)
Est time: 06–10
13-7
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 13 - The Weighted-Average Cost of Capital and Company Valuation
21.
The net effect of Big Oil’s transaction is to leave the firm with $200 million more
debt (because of the borrowing) and $200 million less equity (because of the
dividend payout). Total assets and business risk are unaffected. The WACC will
remain unchanged because business risk is unchanged. However, the cost of
equity will increase. With the now-higher leverage, the business risk is spread
over a smaller equity base, so each share is now riskier.
The new financing mix for the firm is E = $1,000 and D = $585.7.
Therefore:
D $585.7
E
$1,000

 0.369 and 
 0.631
V $1,585.7
V $1,585.7
If the cost of debt is still 9%, then we solve for the new cost of equity as follows.
Use the fact that, even at the new financing mix, WACC must still be 11.27%.
D
 E

WACC    rdebt  (1  TC )    requity
V
V

 

= [0.369  9%  (1 – 0)] + [0.631  requity] = 11.27%
We solve to find that requity = 12.60%.
Est time: 06–10
22. Even if the WACC were lower when the firm’s tax rate is higher, this does not imply that
the firm would be worth more. The after-tax cash flows the firm would generate for its
owners would also be lower. This would reduce the value of the firm even if the cash
flows were discounted at a lower rate. If the tax authority is collecting more income from
the firm, the value of the firm will fall.
Est time: 06–10
23. This reasoning is faulty in that it implicitly treats the discount rate for the project as the
cost of debt if the project is debt-financed and as the cost of equity if the project is equityfinanced. In fact, if the project poses risk comparable to the risk of the firm’s other
projects, the proper discount rate is the firm’s cost of capital, which is a weighted average
of the costs of both debt and equity.
Est time: 06–10
13-8
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 13 - The Weighted-Average Cost of Capital and Company Valuation
Solution to Minicase for Chapter 13
Bernice needs to explain to her boss, Mr. Brinestone, that appropriate rates of return for
cost of capital calculations are the rates of return that investors can earn on comparable
risk investments in the capital market. Mr. Brinestone’s estimate of the cost of equity is
his target value for the book return on equity; it is not the expected rate of return that
investors demand on shares of stock with the same risk as Sea Shore Salt.
Bernice’s CAPM calculation indicates that the correct value for the equity rate is 10.5%.
This value is broadly consistent with the rate one would infer from the constant-growth
dividend discount model (which seems appropriate for a mature firm like this one with
stable growth prospects). The dividend discount model implies a cost of equity of a bit
more than 11%:
requity 
DIV1
$2
g
 0.067  0.117  11.7%
P0
$40
Thus, it appears that Bernice’s calculation is correct.
Similarly, Mr. Brinestone’s returns for other securities should be modified to reflect the
expected returns these securities currently offer to investors. The bank loan and bond
issue offer pretax rates of 8% and 7.75%, respectively, as in Mr. Brinestone’s memo. The
preferred stock, however, is not selling at par, so Mr. Brinestone’s assertion that the rate
of return on preferred is 6% is incorrect. In fact, with the preferred selling at $70 per
share, the rate of return is:
rpreferred 
DIV
P0

$6
 0.086  8.6%
$70
This makes sense: The pretax return on preferred should exceed that on the firm’s debt.
Finally, the weights used to calculate the WACC should reflect market, not book, values.
These are the prices that investors would pay to acquire the securities. The market value
weights are computed as follows:
Amount
(millions)
$120
80
70
400
$670
Comment
Bank loan
Bond issue
Preferred stock
Common stock
Valued at face amount
Valued at par
$70  1 million shares
$40  10 million shares
Percent of
Total
17.91
11.94
10.45
59.70
100.00
Rate of
Return (%)
8.00
7.75
8.60
10.50
Therefore, the WACC, which serves as the corporate hurdle rate, should be 8.70%:
WACC = [0.1791  8%  (1 – 0.35)] + [0.1194  7.75%  (1 – 0.35)]
+ (0.1045  8.6%) + (0.5970  10.5%) = 8.70%
13-9
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.