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Required Returns and the Cost
of Capital
To guess is cheap. To guess wrong is expensive.
CHINESE PROVERB.
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ANSWERS TO QUESTIONS
1. If the weights used in the calculations do not correspond to the proportions of financing the
2. The principal qualification to its use is that existing as well as new investment proposals are
alike with respect to risk. In other words, the proposal being judged should not alter the risk
3. Yes, these funds have a cost. In most cases, however, the cost is ignored because these
4. The tax shield associated with the use of debt funds would be lost, at least until profits were
restored. As a result, we would no longer multiply the before-tax cost of debt by one minus
5. Dividends per share are estimated out into the future, preferably out to infinity. The
discount rate necessary to equate the present value of the expected future stream of
6. The critical assumption is that capital markets are perfect and that only the systematic risk
of the firm is important. With market imperfections, such as bankruptcy costs, the total risk
7. The firm’s before-tax cost of debt is used as a base to which a risk premium is added. The
risk premium is the difference in required return between stocks and bonds. For companies
8. Proxy companies are used in place of the project or group of projects under consideration.
The idea is to find a group of proxy companies that closely parallel the business represented
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9. A project-specific required return refers to the hurdle rate for a specific project as derived
10. Management determines the acceptability of the project on the basis of the project’s
expected return in relation to the probability distribution of possible returns. The usual
11. The RADR approach to project selection calls for “adjusting” the required return, or
discount rate, upward (downward) from the firm’s overall cost of capital for projects or
12. For a group of projects, the correlation between returns of the various projects must be
13. Empirically, companies in the same industry tend to have similar betas and required rates of
return. However, there are many exceptions. It depends on how similar the industry
14. No. Eventually the equity base will need to be rebuilt and this will require retained earnings
15. An increase in bankruptcy costs would increase the required rate of return for companies.
The change should make companies more conscious of avoiding bankruptcy and analyzing
16. If the divisions have significantly different risks, a company should use different costs of
capital for them. One approach is the capital-asset pricing model context using outside
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17. Value is created when projects are accepted, whose expected returns exceed the required
18. Through investment in assets, value is created by industry attractiveness and competitive
SOLUTIONS TO PROBLEMS
1. (1) (2) (1) × (2)
Cost Proportion of Total Financing Weighted Cost
Bonds ki B/(B+S) ki[B/(B+S)]
Thus, ko = ki[B/(B+S)] + ke[S/(B+S)]
or alternatively written
ie
o
k(B)+k(S)
k(B + S)
=
$7,000,000 += +=
ko = ki[B/(B+S)] + ke[S/(B+S)]
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3. (January 20X1)
1
0
D$3
P$300
(January 20X2)
1
0
D$3.45
P = = = $57.50
4.
End of Year
Dividend
Per Share
Present Value
at 12 Percent
Present Value
at 13 Percent
1 $ 2.240 $ 2.000 $ 1.982
0.12 $25.511
X $0.511
k
e $25
0.01
0.13 $23.411
$2.10
x $0.511 (0.01)($0.511)
= Therefore, x = = 0.0024
ke = 0.12 + X = 0.12 + 0.0024 = 12.24 percent
* Implied Value of Stock at End of Year 5 = D$3.347
6= = $27.89
(0.12 – g) (0.12 – g)
**Implied Value of Stock at End of Year 5 = $3.347
D6
(0.13–g) (0.13 – 0)
= = $25.75
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5.
(1)
After-tax Cost
(2)
Proportion of Total Financing
(1) × (2)
Weighted Cost
$90,000
.145 −−=−
The proposal should be rejected. Without flotation costs, however, the net present value
would have been positive and the proposal acceptable.
7. a. Cost of equity = 0.12 + (0.18 – 0.12) 1.28 = 19.68 percent
b. The approach assumes that unsystematic risk is not a factor of importance, which may
8. a. Required Return = 0.10 + (0.15 – 0.10)(1.10) = 15.5 percent
c. Required Return
Required Return
Expected value of required rate of return
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9. Using the RADR approach we would calculate the project’s net present value at the
management-determined risk-adjusted discount rate:
Year
————
(1)
——————————
Expected Cash Flow
——————————
(2)
————————————
Discount Factor at 15%
————————————
(1) × (2)
———————
Present Value
———————
0 $–400,000 1.0000 $–400,000
1 50,000 0.8695 43,480
Since the NPV is negative ($–19,960), we would reject the project. Alternatively, we could
10. The selection will depend on the risk preferences of the individual. Graphs of the plots are
shown below. For the reasonable risk averter, the selection will probably be combination
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Solution to Appendix A Problem:
11. a.
c)]
β
Peerless unlevered beta = [1+(B/S) (1 T
j
–
1.15 1.15 1.00
===
An adjusted beta of 1.45 is appropriate for the new venture if the assumptions of the
capital-asset pricing model hold, except for corporate taxes.
b. ki = kd(1 – Tc) = 0.15(1 – 0.4) = 0.09
ke = Rf + ( m
R– Rf)ß = 0.13 + (0.17 – 0.13)1.45 = 0.188
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Solution to Appendix B Problem:
12. a.
Schedule for determining the present value of the interest tax-shield benefits related to
the new snow plow truck
End of
Year
(1)
Debt Owed
At Year End
(1)t–1 – $3,000
(2)
Annual Interest
(1)t–1 × 0.12
(3)
Tax-Shield
Benefits
(2) × 0.30
(4)
PV of Benefits
at 12%
0 $18,000 – – – – – –
1 15,000 $2,160 $648 $ 579
To an all-equity financed firm, the net present value of the project’s after-tax operating cash
flows would be,
t
while the adjusted present value would be,
The project is acceptable.
b. If the cash flows are $8,000 per year instead of $10,000, the net present value of the
project’s after-tax operating cash flows becomes –$520, while the adjusted present
value becomes,
The project is still acceptable – but, barely.
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SOLUTIONS TO SELF-CORRECTION PROBLEMS
1. a. ke = D1/P0 + g D1 = D0(1.12) = $1(1.12) = $1.12
b. Through the trial-and-error approach illustrated in Chapters 3 and 4, one ends up
determining that the discount rate necessary to discount the cash dividend stream to $20
must fall somewhere between 18 and 19 percent as follows:
End of Year Dividend Per Share Present Value At 18% Present Value At 19%
1 $1.20 $1.02 $1.01
Year 6 dividend = $2.49 (1.10) = $2.74
Market prices at the end of year 5 using a constant growth dividend
valuation model: P5 = D6/(ke – g)
Present value at time 0 for amounts received at end of year 5:
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18% 19%
Present value of years 1 – 5 $ 5.26 $ 5.13
Therefore, the discount rate is closer to 18 percent than it is to 19 percent. Interpolating,
we get
and ke = 0.18 + X = 0.18 + 0.0010 = 18.10 percent, which is the estimated return on equity
that the market requires.
2.
Situation Equation: Rf + ( m
R– Rf)ß Return Required
1 10% + (15% – 10%) 1.00 15.0%
The greater the risk-free rate, the greater the expected return on the market portfolio, and the
greater the beta, the greater will be the required return on equity, all other things being the
same. In addition, the greater the market risk premium mf
(R R )–, the greater the required
return, all other things being the same.
3. Cost of debt = 15%(1 – 0.4) = 9%
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As mentioned in the text, a conceptual case can be made for adjusting the nonequity costs of
financing the two divisions for differences in systematic risks. However, we have not
done so.
4. a. The coefficients of variation (standard deviation/
N
PV ) for the alternatives are as follows:
Existing projects (E) 0.50
Graphs of risk versus return are shown below. A moderately risk-averse decision maker will
probably prefer the existing projects plus both new projects to any of the other three
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b. If the CAPM approach leads to a different decision, the key to deciding would be the
importance of market imperfections. As indicated earlier, if a company’s stock is traded