Solution Manual for Principles of Corporate Finance 14th Edition by Richard Brealey, Stewart Myers, Verified Chapters 1 - 34, Complete Newest Version.pdf
Solution Manual for Principles of Corporate Finance 14th Edition by Richard Brealey, Stewart Myers, Verified Chapters 1 - 34, Complete Newest Version.pdf
Solution Manual for Principles of Corporate Finance 14th Edition by Richard Brealey, Stewart Myers, Verified Chapters 1 - 34, Complete Newest Version.pdf
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Solution Manual for Principles of Corporate Finance 14th Edition by Richard Brealey, Stewart Myers, Verified Chapters 1 - 34, Complete Newest Version.pdf
1. ALL Chapters (1 - 34)
14th Edition By Richard Brealey, Stewart Myers,
Principles Of Corporate Finance
SOLUTION MANUAL FOR
2. Chapter 1: Introduction to Corporate Finance
Chapter 2: How to Calculate Present Values
Chapter 3: Valuing Bonds
Chapter 4: Valuing Stocks
Chapter 5: Net Present Value and Other Investment Criteria
Chapter 6: Making Investment Decisions with the Net Present Value Rule
Chapter 7: Introduction to Risk, Diversification, and Portfolio Selection Chapter 8: The
Capital Asset Pricing Model
Chapter 9: Risk and the Cost of Capital
Chapter 10: Project Analysis
Chapter 11: How to Ensure That Projects Truly Have PositiveNPVs
Chapter 12: Efficient Markets and Behavioral Finance Chapter
13: An Overview of Corporate Financing Chapter 14: How
Corporations Issue Securities
Chapter 15: Payout Policy
Chapter 16: Does Debt Policy Matter?
Chapter 17: How Much Should a Corporation Borrow?
Chapter 18: Financing and Valuation
Chapter 19: Agency Problems and Corporate Governance Chapter 20:
Stakeholder Capitalism and Responsible Business
Chapter 21: Understanding Options
Chapter 22: Valuing Options Chapter 23:
Real Options
Chapter 24: Credit Risk and the Value of Corporate Debt Chapter 25:
The Many Different Kinds of Debt
Chapter 26: Leasing
Chapter 27: Managing Risk
Chapter 28: International Financial Management
Chapter 29: Financial Analysis Chapter
30: Financial Planning
Chapter 31: Working Capital Management
Chapter 32: Mergers
Chapter 33: Corporate Restructuring
TABLE OF CONTENTS
3. Chapter 34: Conclusion: What We Do and Do Not Know about Finance
CHAPTER 1
Introduction to Corporate Finance
The values shown in the solutions may be rounded for display purposes. However, the answers were derived using a spreadsheet without any
intermediate rounding.
Answers to Problem Sets
1. a. real
b. executive airplanes
c. brand names
d. financial
e. bonds
*f. investment or capital expenditure
*g. capital budgeting or investment h.
financing
*Note that f and g are interchangeable in the question.
Est time: 01-05
2. A trademark, a factory, undeveloped land, and your work force (c, d, e, and g) are all real assets. Real assets are
identifiable as items with intrinsic value. The others in the list are financial assets, that is, these assets derive
value because of a contractual claim.
Est time: 01-05
3. a. Financial assets, such as stocks or bank loans, are claims held by investors. Corporations sell
financial assets to raise the cash to invest in real assets such as plant and equipment. Some real
assets are intangible.
b. Capital expenditure means investment in real assets. Financing means raising the cash for this
investment.
4. c. The shares of public corporations are traded on stock exchanges and can be purchased by a wide
range of investors. The shares of closely held corporations are not publicly traded and are held by a
small group of private investors.
d. Unlimited liability: Investors are responsible for all the firm‘s debts. A sole proprietor has unlimited
liability. Investors in corporations have limited liability. They can lose their investment, but no more.
Est time: 01-05
5. 4. Items c and d apply to corporations. Because corporations have perpetual life, ownership can be transferred
without affecting operations, and managers can be fired with no effect on ownership. Other forms of business
may have unlimited liability and limited life.
Est time: 01-05
5. Separation of ownership facilitates the key attributes of a corporation, including limited liability for investors,
transferability of ownership, a separate legal personality of the corporation, and delegated centralized
management. These four attributes provide substantial benefit for investors, including the ability to diversify
their investment among many uncorrelated returns—a very valuable tool explored in later chapters. Also, these
attributes allow investors to quickly exit, enter, or short sell an investment, thereby generating an active liquid
market for corporations.
However, these positive aspects also introduce substantial negative externalities as well. The separation of
ownership from management typically leads to agency problems, where managers prefer to consume private
perks or make other decisions for their private benefit—rather than maximize shareholder wealth. Shareholders
tend to exercise less oversight of each individual investment as their diversification increases. Finally, the
corporation‘s separate legal personality makes it difficult to enforce accountability if they externalize costs onto
society.
Est time: 01-05
6. Shareholders will only vote to maximize shareholder wealth. Shareholders can modify their pattern of
consumption through borrowing and lending, match risk preferences, and hopefully balance their own
checkbooks (or hire a qualified professional to help them with these tasks).
Est time: 01-05
7. If the investment increases the firm‘s wealth, it increases the firm‘s share value. Ms. Espinoza could then
sell some or all these more valuable shares to provide for her retirement income.
Est time: 01-05
8. a. Assuming that the encabulator market is risky, an 8% expected return on the F&H
encabulator investments may be inferior to a 4% return on U.S.
government securities, depending on the relative risk between the two assets.
b. Unless the financial assets are as safe as U.S. government securities, their cost of capital would be
higher. The CFO could consider expected returns on assets with similar risk.
Est time: 06-10
9. Managers would act in shareholders‘ interests because they have a legal duty to act in their interests. Managers
may also receive compensation— bonuses, stock, and option payouts with value tied (roughly) to firm
performance. Managers may fear personal reputational damage from not acting in shareholders‘ interests. And
managers can be fired by the board of directors (elected by shareholders). If managers still fail to act in
shareholders‘ interests, shareholders may sell their shares, lowering the stock price and potentially creating the
possibility of a takeover, which can again lead to changes in the board of directors and senior
management.
Est time: 01-05
6. 10. Managers that are insulated from takeovers may be more prone to agency problems and therefore more likely
to act in their own interests rather than in shareholders‘. If a firm instituted a new takeover defense, we
might expect to see the value of its shares decline as agency problems increase and less shareholder value
maximization occurs. The counterargument is that defensive measures allow managers to negotiate for a higher
purchase price in the face of a takeover bid—to the benefit of shareholder value.
Est time: 01-05
AppendixQuestions:
1. Both would still invest in their friend‘s business. A invests and receives $121,000 for his investment at the
end of the year—which is greater than the $120,000 that would be received from lending at 20% ($100,000
× 1.20 = $120,000). G also invests, but borrows against the
$121,000 payment, and thus receives $100,833 ($121,000 / 1.20) today.
Est time: 01-05
2. a. He could consume up to $200,000 now (forgoing all future consumption) or up to $216,000 next year
($200,000 × 1.08, forgoing all consumption this year). He should invest all of his wealth to earn $216,000 next
year. To choose the same consumption (C) in both years, C = ($200,000
– C) × 1.08 = $103,846.
Dollars Next Year
220,000
216,000
203,704
200,000 Dollars Now
b. He should invest all of his wealth to earn $220,000 ($200,000 × 1.10) next year. If he consumes all this year,
he can now have a total of $203,703.70 ($200,000 × 1.10/1.08) this year or $220,000 next year. If he
consumes C this year, the amount available for next year‘s consumption is ($203,703.70 – C) × 1.08. To get
equal consumption in both years, set the amount consumed today equal to the amount next year:
C = ($203,703.70 – C) × 1.08
C = $105,769.20
Est time: 06-10
7. CHAPTER 2
How to Calculate Present Values
The values shown in the solutions may be rounded for display purposes. However, the answers were derived using a spreadsheet without
any intermediate rounding.
Answers to Problem Sets
1. a. False. The opportunity cost of capital varies with the risks associated with each individual project or
investment. The cost of borrowing is unrelated to these risks.
b. True. The opportunity cost of capital depends on the risks associated with each project and its cash
flows.
c. True. The opportunity cost of capital is dependent on the rates of returns shareholders can earn on the
own by investing in projects with similar risks
d. False. Bank accounts, within FDIC limits, are considered to be risk-free. Unless an investment is also risk-
free, its opportunity cost of capital must be adjusted upward to account for the associated risks.
Est time: 01-05
2. a. In the first year, you will earn $1,000 × 0.04 = $40.00
b. In the second year, you will earn $1,040 × 0.04 = $41.60
c. By the end of the ninth year, you will accrue a principle of $1,040 × (1.049) = $1,423.31. Therefore,
in the Tenth year, you will earn $1,423.31 × 0.04 = $56.93
Est time: 01-05
3.
Transistors Transistors (1 r)t
2019 1972
32,000,000,000 2,250 (1 r)48
r 40.94% 59.00% rPredicted
Est time: 01-05
4. The ―Rule of 72‖ is a rule of thumb that says with discrete compounding the time it takes for an investment
to double in value is roughly 72/interest rate (in percent).
Therefore, without a calculator, the Rule of 72 estimate is: Time to
double = 72 / r
Time to double = 72 / 4
Time to double = 18 years, so less than 25 years.
8. If you did have a calculator handy, this estimate is verified as followed:
Ct = PV × (1 + r)t
t = ln2 / ln1.04
t = 17.67 years
Est time: 01-05
5. a. Using the inflation adjusted 1958 price of $1,060, the real return per annum is:
$450,300,000 = $1,060 × (1 + r)(2017-1958)
r = [$450,300,000/$1,060](1/59 ) – 1 = 0.2456 or 24.56% per annum
b. Using the inflation adjusted 1519 price of $575,000, the real return per annum is:
$450,300,000 = $575,000 × (1 + r)(2017-1519)
Est time: 01-05
r = [$450,300,000/$575,000](1/498 ) – 1 = 0.0135 or 1.35% per annum
6. Ct = PV × (1 + r)t
C8 = $100 × 1.158
C8 = $305.90
Est time: 01-05
7. a. Ct = PV × (1 + r)t
C10 = $100 × 1.0610
C10 = $179.08
b. Ct = PV × (1 + r)t C20
= $100 × 1.0620 C20 =
$320.71
c. Ct = PV × (1 + r)t C10
= $100 × 1.0410 C10 =
$148.02
d. Ct = PV × (1 + r)t C20
= $100 × 1.0420 C20 =
$219.11
Est time: 01-05
8. a. PV = Ct × DFt
DFt = $125 / $139
DFt = .8993
10. 13. a. NPV = – Investment + C × ((1 / r) – {1 / [r(1 + r)t]})
NPV = –$800,000 + $170,000 × ((1 / .14) – {1 / [.14(1.14)10]}) NPV =
$86,739.66
b. After five years, the factory‘s value will be the present value of the remaining cash flows: PV =
$170,000 × ((1 / .14) – {1 / [.14(1.14)(10 – 5)]})
PV = $583,623.76
Est time: 01-05
14. Use the formula: NPV = –C0 + C1 / (1 + r) + C2 / (1 + r)2
NPV5% = –$700,000 + $30,000 / 1.05 + $870,000 / 1.052 NPV5%
= $117,687.07
NPV10% = –$700,000 + $30,000 / 1.10 + $870,000 / 1.102
NPV10% = $46,280.99
NPV15% = –$700,000 + $30,000 / 1.15 + $870,000 / 1.152
NPV15% = –$16,068.05
The figure below shows that the project has a zero NPV at about 13.65%.
NPV13.65% = –$700,000 + $30,000 / 1.1365 + $870,000 / 1.13652 NPV13.65% = –
$36.83
Est time: 11-15
11. 15. a. NPV = –Investment + PVAoperating cash flows – PVrefits + PVscrap value
NPV = –$8,000,000 + ($5,000,000 – 4,000,000) × ((1 / .08) – {1 / [.08(1.08)15]}) – ($2,000,000
/ 1.085 + $2,000,000 / 1.0810) + $1,500,000 / 1.0815
NPV = –$8,000,000 + 8,559,479 – 2,287,553 + 472,863 NPV =
–$1,255,212
b. The cost of borrowing does not affect the NPV because the opportunity cost of capital depends
on the use of the funds, not the source.
Est time: 06-10
16. NPV = C / r – investment
NPV = $138 / .09 − $1,548
NPV = −$14.67
Est time: 01-05
17. One way to approach this problem is to solve for the present value of:
(1) $100 per year for 10 years, and
(2) $100 per year in perpetuity, with the first cash flow at year 11.
If this is a fair deal, the present values must be equal, thus solve for the interest rate (r).
The present value of $100 per year for 10 years is: PV = C
× ((1 / r) – {1 / [r × (1 + r)t]})
PV = $100 × ((1 / r) – {1 / [r × (1 + r)10]})
The present value, as of year 0, of $100 per year forever, with the first payment in year 11, is: PV = (C / r) /
(1 + r)t
PV = ($100 / r) / (1 + r)10
Equating these two present values, we have:
$100 × ((1 / r) – {1 / [r × (1 + r)10]}) = ($100 / r) / (1 + r)10
Using trial and error or algebraic solution, r = 7.18%.
Est time: 06-10
12. 18. a. PV = C / r
PV = $1 / .10
PV = $10
b. PV7 = (C8 / r)
PV0 approx = (C8 / r) / 2
PV0 approx = ($1 / .10) / 2
PV0 approx = $5
c. A perpetuity paying $1 starting now would be worth $10 (part a), whereas a perpetuity starting in
year 8 would be worth roughly $5 (part b). Thus, a payment of $1 for the next seven years would
also be worth approximately $5 (= $10 – 5).
d. PV = C / ( r − g)
PV = $10,000 / (.10 − .05) PV
= $200,000
Est time: 06-10
19. a. DF1 = 1 / (1 + r)
r = (1 – .905) / .905
r = .1050, or 10.50%
b. DF2 = 1 / (1 + r)2
DF2 = 1 / 1.1052
DF2 = .8190
c. PVAF2 = DF1 + DF2
PVAF2 = .905 + .819
PVAF2 = 1.7240
d. PVA = C PVAF3
PVAF3 =$24.65 / $10
PVAF3 = 2.4650
e. PVAF3 = PVAF2 + DF3
DF3 = 2.465 – 1.7240
DF3 = .7410
Est time: 06-10
20. PV = Ct / (1 + r)t
PV = $20,000 / 1.105
PV = $12,418.43
C = PVA / ((1 / r) – {1 / [r(1 + r)t]})
C = $12,418.43 / ((1 / .10) – {1 / [.10 (1 + .10)5]})
C = $3,275.95
Est time: 06-10
13. 21. C = PVA / ((1 / r) – {1 / [r(1 + r)t]})
C = $20,000 / ((1 / .08) – {1 / [.08(1 + .08)12]})
C = $2,653.90
Est time: 01-05
22. a. PV = C × ((1 / r) – {1 / [r(1 + r)t]})
PV = ($9,420,713 / 19) × ((1 / .08) – {1 / [.08(1 + .08)19]}) PV =
$4,761,724
b. PV = C × ((1 / r) – {1 / [r(1 + r)t]})
$4,200,000 = ($9,420,713 / 19) × ((1 / r) – {1 / [r(1 + r)t]})
Using Excel or a financial calculator, we find that r = 9.81%.
Est time: 06-10
23. a. PV = C × ((1 / r) – {1 / [r(1 + r)t]})
PV = $50,000 × ((1 / .055) – {1 / [.055(1 + .055)12]}) PV
= $430,925.89
b. Since the payments now arrive six months earlier than previously:
PV = $430,925.89 × {1 + [(1 + .055).5 – 1]}
PV = $442,617.74
Est time: 06-10
24. Ct = PV × (1 + r)t
Ct = $1,000,000 × (1.035)3
Ct = $1,108,718
Annual retirement shortfall = 12 × (monthly aftertax pension + monthly aftertax Social Security –
monthly living expenses)
= 12 × ($7,500 + 1,500 – 15,000)
= –$72,000
The withdrawals are an annuity due, so:
PV = C × ((1 / r) – {1 / [r(1 + r)t]}) × (1 + r)
$1,108,718 = $72,000 × ((1 / .035) – {1 / [.035(1 + .035)t]}) × (1 + .035)
14.878127 = (1 / .035) – {1 / [.035(1 + .035)t]}
14. Est time:06-10
13.693302 = 1 / [.035(1 + .035)t]
.073028 / .035 = 1.035t
t = ln2.086514 / ln1.035
t = 21.38 years
25. a. PV = C / r = $1 billion / .08
PV = $12.5 billion
b. PV = C / (r – g) = $1 billion / (.08 – .04)
PV = $25.0 billion
c. PV = C × ((1 / r) – {1 / [r (1 + r)t]}) = $1 billion × ((1 / .08) – {1 / [.08(1 + .08)20]})
PV = $9.818 billion
d. The continuously compounded equivalent to an annually compounded rate of 8% is
approximately 7.7%, which is computed as:
Ln(1.08) = .077, or 7.7%
PV = C × {(1 / r) – [1 / (r × ert)]} = $1 billion × {(1 / .077) – [1 / (.077 – e.077 × 20)]}
PV = $10.206 billion
This result is greater than the answer in Part (c) because the endowment is now earning interest
during the entire year.
Est time:06-10
26. a. PV = C × ((1 / r) – {1 / [r(1 + r)t]})
PV = $2.0 million × ((1 / .08) – {1 / [.08(1.08)20]}) PV
= $19.64 million
b. If each cashflow arrives one year earlier, then you can simply compound the PV calculated
in part a by (1+r) $19.64 million × (1.08) = $21.21 million
Est time: 01-05
27. a. Start by calculating the present value of an annuity due assuming a price of $1: PV = 0.25
+ 0.25 × ((1 / .05) – {1 / [.05(1.05)3]})
PV = 0.93, therefore it is better to pay instantly at a lower cost of 0.90 [= 1 × 0.9]
b. Recalculate, except this time using an ordinary annuity: PV =
0.25 × ((1 / .05) – {1 / [.05(1.05)4]})
PV = 0.89, therefore it is better to take the financing deal as it costs less than 0.90.
Est time: 06-10
16. c. Interest percent of first payment = Interest1 / Payment
Interest percent of first payment = (.06 × $200,000) / $17,436.91 Interest
percent of first payment = .6882, or 68.82%
Interest percent of last payment = Interest20 / Payment = $986.99 / $17,436.91 Interest
percent of last payment = .0566, or 5.66%
Without creating an amortization schedule, the interest percent of the last payment can be
computed as:
Interest percent of last payment = 1 – {[Payment / (1 + r)] / Payment}
Interest percent of last payment = 1 – [($17,436.91 / 1.06) / $17,436.91] Interest
percent of last payment = .0566, or 5.66%
After 10 years, the balance is:
PV10 = C × ((1 + r) – {1 / [r × (1 + r)t]}) = $17,436.91 × {1.06 – [1 / (.06 × 1.0610)]} PV10
= $128,337.19
Fraction of loan paid off = (Loan amount – PV10) / Loan amount
= ($200,000 – 128,337.19) / $200,000
Fraction of loan paid off = .3583, or 35.83%
Though 50% of time has passed, only 35.83% of the loan has been paid off; this is because
interest comprises a higher portion of the monthly payments at the beginning of the loan (e.g.,
Interest percent of first payment > interest percent of last payment).
Est time:16-20
30. a. PV = Ct / (1 + r)t = $10,000 / 1.055
PV = $7,835.26
b. PV = C((1 / r) – {1 / [r(1 + r)t]}) = $12,000((1 / .08) – {1 / [.08(1.08)6]})
PV = $55,474.56
c. Ct = PV × (1 + r)t = ($60,476 − 55,474.56) × 1.086
Ct = $7,936.66
Est time: 06-10
17. $4.00
31. PV $40.00
stock
.14 .04
Est time: 01-05
32. a. PV = C × ((1 / r) – {1 / [r(1 + r)t]})
C = $2,000,000 / ((1 / .08) – {1 / [.08(1 + .08)15]})
C = $233,659.09
b. r = (1 + R) / (1 + h) – 1 = 1.08 / 1.04 – 1
r = .0385, or 3.85%
Est time: 06-10
PV = C × ((1 / r) – {1 / [r(1 + r)t]})
C = $2,000,000 / ((1 / .0385) – {1 / [.0385(1 +.0385)15]})
C = $177,952.49
The retirement expenditure amount will increase by 4% annually.
33. Calculate the present value of a growing annuity for option 1, then compare this amount with the option to
pay instantly $12,750:
PV = C × ([1 / (r – g)] – {(1 + g)t / [(r – g) × (1 + r)t]})
PV = $5,000 × ([1 / (.10 – .06)] – {(1 + .06)3 / [(.10 – .06) × (1 + .10)3]}) PV =
$13,146.51
Since the $13,147 present value of the three year growing annual membership dues exceeds the single $12,750
payment for three years, it is better to pay the lower upfront 3-year dues.
Est time: 06-10
34. a. PV = C0
PV = $100,000
b. PV = Ct / (1 + r)t = $180,000 / 1.125
PV = $102,136.83
c. PV = C / r = $11,400 / .12
PV = $95,000
d. PV = C × ((1 / r) – {1 / – [r(1 + r)t]}) = $19,000 × ((1 / .12) – {1 / – [.12(1.12)10]})
PV = $107,354.24
e. PV = C / (r – g) = $6,500 / (.12 .05)
PV = $92,857.14
Prize (d) is the most valuable because it has the highest present value.
Est time: 06-10
18. 35. a. PV = C / r
PV = $2,000,000 / .12 PV
= $16,666,667
b. PV = C × ((1 / r) – {1 / [r(1 + r)t]})
PV = $2,000,000 × ((1 / .12) – {1 / [.12(1 + .12)20]}) PV =
$14,938,887
c. PV = C / (r – g)
PV = $2,000,000 / (.12 – .03) PV
= $22,222,222
d. PV = C × ([1 / (r – g)] – {(1 + g)t / [(r – g) × (1 + r)t]})
PV = $2,000,000 × ([1 / (.12 – .03)] – {(1 + .03)20 / [(.12 – .03) × (1 + .12)20]}) PV
= $18,061,473
Est time: 06-10
36. First, find the semiannual rate that is equivalent to the annual rate: 1 + r =
(1 + rsemi )2
1.08 = (1 + rsemi)2
rsemi = 1.08.5 – 1
rsemi = .039230, or 3.9230%
PV = C 0 + C × ((1 / rsemi) – {1 / [rsemi × (1 + rsemi)t]})
PV = $100,000 + $100,000 × ((1 / .039230) – {1 / [.039230(1 + .039230)9]}) PV =
$846,147.28
Est time:06-10
37. a. Ct = PV × (1 + r)t
C1 = $1 × 1.121 = $1.1200 C5
= $1 × 1.125 = $1.7623 C10 =
$1 × 1.1210 = $9.6463
b. Ct = PV × (1 + r / m)mt
C1 = $1 × [1 + (.117 / 2)2 × 1 = $1.1204 C5
= $1 × [1 + (.117 / 2)2 × 5 = $1.7657 C10 =
$1 × [1 + (.117 / 2)2 × 20 = $9.7193
c. Ct = PV × emt
C1 = $1 × e(.115 × 1) = $1.1219 C5
= $1 × e(.115 × 5) = $1.7771 C10 =
$1 × e(.115 × 20) = $9.9742
The preferred investment is (c) because it compounds interest faster and produces the highest future value at
any point in time.
Est time: 06-10
19. 38.
a. Ct = PV × (1 + r)t
Ct = $10,000,000 x (1.06)4
Ct = $12,624,770
b. Ct = PV × [1+ (r / m)mt
Ct = $10,000,000 × [1 + (.06 / 12)]12 × 4
Ct = $12,704,892
c. Ct = PV × ert
Ct = $10,000,000 × e.06 × 4 Ct
= $12,712,492
Est time: 01-05
39. a. PVend of year = C / r
PVend of year = $100 / .07 PVend of
year = $1,428.57
b. PVbeginning of year = (C / r) × (1 +r) PVbeginning of
year = ($100 / .07) × (1 + .07) PVbeginning of year
= $1,528.57
c. To find the present value with payments spread evenly over the year, use the continuously
compounded rate that equates to 7% compounded annually. This rate is found using natural
logarithms.
PVCC = C / rCC
PVCC = $100 / ln(1 + .07)
PVCC = $1,478.01
[Note: the continuously compounded rate is :Ln(1 + .07) = .0677, or 6.77%] The sooner
payments are received, the more valuable they are.
Est time: 06-10
40. Annualcompounding:
Ct = PV × (1 + r)t C20
= $100 1.1520 C20
= $1,636.65
20. Continuouscompounding:
Ct = PV × ert
C20 = $100 e.15 × 20
C20 = $2,008.55
Est time: 01-05
41. a. FV = C × ert
FV = $1,000 × e.12 x 5
FV = $1,822.12
b. PV = C / ert
PV =$5,000,000 / e.12 ×8 PV
= $1,914,464
c. PV = C (1 / r – 1 / rert)
PV = $2,000 (1 / .12 – 1 / .12e .12 x 15) PV
= $13,911.69
Est time: 01-05
42. Spreadsheet exercise, answers will vary
Est time: 11-15
43. a. PV = C / (r – g)
PV = $2,000,000 / [.10 – (–.04)]
PV = $14,285,714
b. PV20 = C21 / (r – g)
PV20 = {$2,000,000 × [1 + (–.04)]20} / [.10 – (–.04)] PV20
= $6,314,320
PV cash flows 1-20 = PV – PV20 / (1 + r)20
PV cash flows 1-20 = $14,285,714 – ($6,314,320 / 1.1020) PV cash
flows 1-20 = $13,347,131
Est time:06-10
21. 44. a. Rule of 72 estimate:
Time to double = 72 / r Time
to double = 72 / 12 Time to
double = 6 years
Exact time to double:
Ct = PV × (1 + r)t
t = ln2 / ln1.12
t = 6.12 years
b. With continuous compounding for interest rate r and time period t: e rt = 2
rt = ln2
Solving for t when r is expressed as a decimal:
rt = .693
t = .693 / r
Est time:06-10
CHAPTER 3
Valuing Bonds
The values shown in the solutions may be rounded for display purposes. However, the answers were derived using a spreadsheet without any
intermediate rounding.
Answers to Problem Sets
1. a. Does not change. The coupon rate is set at time of issuance.
b. Price falls. The yield to maturity and the price are inversely related.
c. Yield to maturity rises. Since the price falls, the bond‘s yield to maturity will rise.
Est. Time: 01-05
2. a. If the coupon rate is higher than the yield to maturity, then investors must be expecting a
decline in the capital value of the bond over its remaining life. Thus, the bond’s price must
be greater than its face value.
b. Conversely, if the yield to maturity is greater than the coupon, the price will be below
face value. The price will rise and equal face value at maturity.
22. Est. Time: 01-05
3. Answers will vary.
a. Fall. Assume a one-year, 10 percent bond. If the interest rate is 10
percent, the bond is worth $110 / 1.1 = $100. If the interest rate rises to 15 percent, the bond is worth
$110 / 1.15 = $95.65.
b. Less. Using the example in part a, if the bond yield to maturity is 15 percent but the coupon
rate is lower at 10 percent, the price of the bond is less than $100.
c. Less. If r = 5 percent, then a 1-year 10 percent bond is worth $110 / 1.05 = $104.76.
d. Higher. If r = 10 percent, a 1-year 10 percent bond is worth $110 / 1.1 = $100, while a 1- year 8
percent bond is worth $108 / 1.1 = $98.18.
e. No. Low-coupon bonds have longer durations (unless there is only one
period to maturity) and are therefore more volatile. For example. if r falls from 10 percent to 5
percent, the value of a 2-year 10 percent annual coupon bond rises from $100 to
$109.30, which is an increase of 9.3 percent. The value of a 2-year 5 percent annual coupon bond
rises from $91.32 to $100, which is an increase of 9.5 percent.
Est. Time: 06-10
4. PV = (.05 × €100) × ((1 / .06) – {1 / [.06 × (1 + .06)10]}) + €100 / (1 + .06)10 PV
= €92.64
Est. Time: 01-05
5. Semiannual discount rate =.0132 / 2 = 0066, or 0.66%
Number of time periods = (2040 – 2020) × 2 = 40
PV = [(.0425 × $1,000) / 2] × ((1 / .0066) – {1 / [.0066 × (1 + .0066)40]}) + $1,000 / (1 + .0066)40 PV =
$1,513.55
Est. Time: 01-05
6. a. PV = (.0275 × $1,000) × ((1 / .026) – {1 / [.026(1 + .026)10×2 ]}) + $1,000 / (1 + .026)10×2 PV =
$1,023.16
b.
Yield to
Maturity
PV of Bond
1% $1,427.22
2% 1,315.80
3% 1,214.60
4% 1,122.64
5% 1,038.97
23. 6% 962.81
7% 893.41
8% 830.12
9% 772.36
10% 719.60
11% 671.36
12% 627.23
13% 586.81
14% 549.75
15% 515.76
Est. Time: 06-10
7. Answers will differ. Generally, we would expect yield changes to have the greatest impact on long-maturity
and low-coupon bonds.
Est. Time: 06-10
8. Answers will vary by the interest rates chosen.
a. Suppose the YTM on a four-year 3 percent coupon bond is 2 percent:
PV = (.03 × $1,000) × ((1 / .02) – {1 / [.02(1 + .02)4]}) + $1,000 / (1 + .02)4 PV
= $1,038.08
If the YTM stays the same, one year later the bond will sell for:
PV = (.03 × $1,000) × ((1 / .02) – {1 / [.02(1 + .02)3]}) + $1,000 / (1 + .02)3 PV =
$1,028.84
r = ($30 + 1,028.84 – 1,038.08) / $1,038.08
r = .02, or 2%, which is equal to the yield to maturity
b. Suppose the YTM on a four-year 3 percent coupon bond is 4 percent:
PV = (.03 × $1,000) × ((1 / .04) – {1 / .04 × (1 + .04)4]}) + $1,000 / (1 + .04)4 PV =
$963.70
If the YTM stays the same, one year later the bond will sell for:
PV = (.03 × $1,000) × ((1 / .04) – {1 / .04 × (1 + .04)3]}) + $1,000 / (1 + .04)3 PV =
$972.25
24. r = ($30 + 972.25 – 963.70) / $963.70
r = .04, or 4%, which is equal to the yield to maturity
Est. Time: 11-15
9. a. PV5-year = (.08 × $1,000) × ((1 / .06) – {1 / .06 × (1 + .06)5]}) + $1,000 / (1 + .06)5 PV5-
year = $1,084.25
PV4-year = (.08 × $1,000) × ((1 / .06) – {1 / .06 × (1 + .06)4]}) + $1,000 / (1 + .06)4 PV4-
year = $1,069.30
b. Rate of return = ($1,069.30 + 80.00) / $1,084.25 – 1 = 0.06 or 6%, the YTM.
c. An investor will earn the yield-to-maturity on a bond if the YTM is unchanged.
Est. Time: 06-10
10. One-year rate of 3 percent:
P0 = (.05 × $1,000) × ((1 / .03) – {1 / [.03(1 + .03)6]}) + $1,000 / (1 + .03)6 P0 =
$1,108.34
P1 = (.05 × $1,000) × ((1 / .03) – {1 / .03(1 + .03)5]}) + $1,000 / (1 + .03)5 P1 =
$1,091.59
r = [(.05 × $1,000) + $1,091.59 – 1,108.34] / $1,108.34
r = .0300, or 3.00%
One year rate of 2 percent:
P0 = (.05 × $1,000) × ((1 / .03) – {1 / [.03(1 + .03)6]}) + $1,000 / (1 + .03)6 P0 =
$1,108.34
P1 = (.05 × $1,000) × ((1 / .02) – {1 / [.02(1 + .02)5]}) + $1,000 / (1 + .02)5 P1 =
$1,141.40
r = [(.05 × $1,000) + $1,141.40 – 1,108.34] / $1,108.34
r = .0749, or 7.49%
Est. Time: 06-10
11. a. False. Duration depends on the coupon as well as the maturity.
25. b. False. Given the yield to maturity, volatility is proportional to duration.
c. True. A lower coupon rate means longer duration and therefore higher volatility.
d. False. A higher interest rate reduces the relative present value of distant principal repayments.
Est. Time: 01-05
12. Using Exce (or a financial calculator)l:
Bond 1 YTM = 4.30% Highest Yield to Maturity Bond 2
YTM = 4.20%
Bond 3 YTM = 3.90% Lowest Yield to Maturity
Bond 1 Duration = 9.05 Longest Duration Bond 2
Duration = 8.42
Bond 3 Duration = 7.65 Shortest Duration
Est. Time: 01-05
13.
Year Ct PV(Ct)
Proportion of
Total Value
Proportion
× Time
Volatility =
(Duration / (1 +
r)
r = 8%
Security A 1 40 37.04 .3593 .3593
2 40 34.29 .3327 .6654
3 40 31.75 .3080 .9241
Total PV = 103.08 1.0000 Duration = 1.9487 1.80
Security B 1 20 18.52 .1414 .1414
2 20 17.15 .1310 .2619
3 120 95.26 .7276 2.1828
Total PV = 130.93 1.0000 Duration = 2.5861 2.39
Security C 1 10 9.26 .0881 .0881
2 10 8.57 .0815 .1631
3 110 87.32 .8304 2.4912
Total PV = 105.15 1.00 Duration = 2.7424 2.54
Est. Time: 06-10
26. 14. To calculate the duration, consider the following table like Table 3.4:
Year 1 2 3 4 5 6 7 Totals
Payment ($) 30 30 30 30 30 30 1,030
PV(Ct) at 4% ($) 28.846 27.737 26.670 25.644 24.658 23.709 782.715 939.979
Fraction of total value
[PV(Ct)/PV] .031 .030 .028 .027 .026 .025 .833 1.000
Year × fraction of total value .031 .059 .085 .109 .131 .151 5.829
Duration (Years) 6.395
The duration is the sum of the year × fraction of total value row, or 6.395 years. The
modified duration, or volatility, is 6.395 / (1 + .04) = 6.15.
The price of the 3 percent coupon bond at 3.5 percent, and 4.5 percent equals $969.43 and
$911.61, respectively. The price difference is $57.82, or 6.15 percent of the bond‘s value at the 4 percent
discount rate. The percentage difference is equal to the 1 percent change in the discount rate × modified
duration.
Est. Time: 06-10
15.
a. If the bond coupon payment changes from 9% as listed in Table 3.4 to 8%, then the following
calculation for duration can be made:
Year 1 2 3 4 5 6 7 Totals
Payment ($) 80 80 80 80 80 80 1,080
PV(Ct) at 4% ($) 76.923 73.964 71.120 68.384
65.75
4 63.225
820.71
1
1,240.08
2
Fraction of total value
[PV(Ct)/PV] .062 .060 .057 .055 .053 .051 .662 1.000
Year × fraction of total value .062 .119 .172 .221 .265 .306 4.633
Duration (years) 5.778
A decrease in the coupon payment will increase the duration of the bond, as the duration at an 8 percent
coupon payment is 5.778 years.
The volatility for the bond in Table 3.4 with an 8 percent coupon payment is: 5.778 / 1.04 = 5.556. The bond
therefore becomes less volatile if the coupon payment decreases.
b. For a 9 percent bond whose yield increases from 4 percent to 6 percent, the duration can be
calculated as follows:
Year 1 2 3 4 5 6 7 Totals
Payment ($) 90 90 90 90 90 90 1090
27. PV(Ct) at 6% ($) 84.906 80.100 75.566 71.288
67.25
3 63.446 724.912
1,167.47
1
Fraction of total value
[PV(Ct)/PV] .073 .069 .065 .061 .058 .054 .621 1.000
Year × fraction of total
value .073 .137 .194 .244 .288 .326 4.346
Duration (years) 5.609
There is an inverse relationship between the yield to maturity and the duration. When the yield goes up from 4
percent to 6 percent, the duration decreases slightly. The volatility can be calculated as follows: 5.609 / 1.06 =
5.291. This shows that the volatility decreases as well when the yield increases.
Est. Time: 11-15
16. The duration of a perpetual bond is: [(1 + yield) / yield]. The
duration of a perpetual bond with a yield of 5% is:
D5 = 1.05 / .05 = 21 years
The duration of a perpetual bond yielding 10 percent is:
D10 = 1.10 / .10 = 11 years
Because the duration of a zero-coupon bond is equal to its maturity, the 15-year zero-coupon bond has a
duration of 15 years.
Thus, comparing the 5 percent perpetual bond and the 15-year zero-coupon bond, the 5 percent perpetual
bond has the longer duration. Comparing the 10 percent perpetual bond and the 15- year zero, the zero has a
longer duration.
Est. Time: 06-10
17. a. Spot interest rates. Yield to maturity is a complicated average of the separate spot rates of interest.
b. Bond prices. The bond price is determined by the bond‘s cash flows and the spot rates of interest.
Once you know the bond price and the bond‘s cash flows, it is possible to calculate the yield to
maturity.
Est. Time: 01-05
18. a. 4%; each bond will have the same yield to maturity. b.
PV = $80 / (1.04) + $1,080 / (1.04)2
PV = $1,075.44
Est. Time: 01-05
19. The new calculations are shown in the table below:
1 2 3 4 Bond Price (PV) YTM (%)
Spot rates (%) 4.60 4.40 4.20 4.00
28. Discount factors .9560 .9175 .8839 .8548
Bond A (8% coupon):
Payment (Ct) $80 $1,080
PV(Ct) $76.48 $990.88 $1,067.37 4.407%
Bond B (8% coupon):
Payment (Ct) $80 $80 $1,080
PV(Ct) $76.48 $73.40 $954.60 $1,104.48 4.219%
Bond C (8% coupon):
Payment (Ct) $80 $80 $80 $1,080
PV(Ct) $76.48 $73.40 $70.71 $923.19 $1,143.78 4.036%
20. We will borrow $1,000 at a five-year loan rate of 2.5% and buy a four-year strip paying 4%. We may not
know what interest rates we will earn on the last year, but we can put it under our mattress earning 0 percent,
if necessary, to pay off the loan when it comes due.
Using the information from problem 19, the cost of the strip will be $1,000 × .8548 = $854.80. The proceeds
from the 2.5 percent loan = $1,000 / (1.025)5 = $883.85. We can pocket the difference of $29.05, smile, and
repeat.
The minimum sensible value would be to set the discount factor used in year 5 equal to that of year 4, which
would assume a 0 percent interest rate for year 5. We can solve for the interest rate where 1 / (1 + r)5 = .8548,
which is roughly 3.19%.
Est. Time: 06-10
21. a. PV = (.05 × $1,000) / (1 + r1) + [(.05 × $1,000) + $1,000] / (1 + r2)2
b.
PV = (.05 × $1,000) / (1 + y)+ [(.05 × $1,000) + $1,000] / (1 + y)2
c. Less; it is between the 1-year and the 2-year spot rates.
Est. Time: 01-05
22. The key here is to find a combination of these two bonds (i.e., a portfolio of bonds) that has a cash flow only at
t = 6. Then, knowing the price of the portfolio and the cash flow at t = 6, we can calculate the six-year spot
rate. We begin by specifying the cash flows of each bond and using these and their yields to calculate their
current prices:
Investment Yield C1 . . . C5 C6 Price
6% bond 12% 60 . . . 60 1,060 $753.32
10% bond 8% 100 . . . 100 1,100 $1,092.46
From the cash flows in years 1 through 5, we can see that buying two 6 percent bonds produces the same
annual payments as buying 1.2 of the 10 percent bonds. To see the value of a cash flow only in year 6,
consider the portfolio of two 6 percent bonds minus 1.2 10 percent bonds.
29. This portfolio costs:
($753.32 × 2) – (1.2 $1,092.46) = $195.68
The cash flow for this portfolio is equal to zero for years 1 through 5 and, for year 6, is equal to: ($1,060 × 2) –
(1.2 $1,100) = $800
Thus:
$195.68 (1 + r6)6 = $800
r6 = .2645, or 26.45%
Est. Time: 06-10
23. Downward sloping. This is because high-coupon bonds provide a greater proportion of their cash flows in the
early years. In essence, a high-coupon bond is a ―shorter‖ bond than a low-coupon bond of the same maturity.
Est. Time: 01-05
24. a.
Year Discount factor
1 1 / 1.05 = .952
2 1 / (1.054)2 = .900
3 1/ (1.057)3 = .847
4 1 / (1.059)4 = .795
5 1 / (1.060)5 = .747
b. i. 5 percent, two-year bond:
PV = $50 / 1.05 + $1,050 / 1.0542 PV
= $992.79
ii. 5 percent, five-year bond:
PV = $50 / 1.05 + $50 / 1.0542 + $50 / 1.0573 + $50 / 1.0594 + $1,050 / 1.0605 PV =
$959.34
iii. 10 percent, five-year bond:
PV = $100 / 1.05 + $100 / 1.0542 + $100 / 1.0573 + $100 / 1.0594 + $1,100 / 1.0605
30. Est. Time: 06-10
PV = $1,171.43
25. a. First, we calculate the yield for each of the two bonds. For the 5 percent bond, this means solving for r
in the following equation:
$959.34 = $50 / (1 + r) + $50 / (1 + r)2 + $50 / (1 + r)3 + $50 / (1 + r)4 + $1,050 / (1 +
r)5
r = .05964, or 5.964% For
the 10% bond:
$1,171.43 = $100 / (1 + r) + $100 / (1 + r)2 + $100 / (1 + r)3 + $100 / (1 + r)4 +
$1,100 / (1 + r)5
r = .05937, or 5.937%
The yield depends upon both the coupon payment and the spot rate at the time of the payment. The 10
percent bond has a slightly greater proportion of its total payments coming earlier, when interest rates are
low, than does the 5 percent bond. Thus, the yield of the 10 percent bond is slightly lower.
b. The yield to maturity on a five-year zero-coupon bond is the five-year spot rate, which is 6.0 percent.
c. First, we find the price of the five-year annuity, assuming that the annual payment is $1: PV = $1 /
1.05 + $1/ 1.0542 + $1 / 1.0573 + $1 / 1.0594 + $1 / 1.0605
PV = $4.2417
Now we find the yield to maturity for this annuity:
$4.2417= $1 / (1 + r) + $1 / (1 + r)2 + $1 / (1 + r)3 + $1 / (1 + r)4 + $1 / (1 + r)5
r = .0575 or 5.75%
d. The yield on the five-year note lies between the yield on a five-year zero-coupon bond and the yield
on a five-year annuity because the cash flows of the Treasury bond lie between the cash flows of
these other two financial instruments during a period of rising interest rates. That is, the annuity has
fixed, equal payments; the zero-coupon bond has one payment at the end; and the bond‘s payments
are a combination of these.
Est. Time: 06-10
26. a. The 2-year spot rate is r2 = (100 / 90.703).5 – 1 = 5%. The 3-
year spot rate is r3 = (100 / 85.892)1/3 – 1 = 5.2%. The 4-year
spot rate is r4 = (100 / 81.491).25 – 1 = 5.25%. The 5-year spot
rate is r5 = (100 / 77.243).2 – 1 = 5.3%.
b. Upward-sloping.
c. Higher; the yield on the coupon bond is a complicated average of the separate spot rates.
Est. Time: 01-05
31. 27. 1-year rate in 1 year = 1.062 / 1.05 – 1 1-
year rate in 1 year = .0701, or 7.01%
Est. Time: 01-05
28.
a. Based on a $100 investment:
$100 × (1 + .057)3 = $118.093
$100 × (1 + .059)4 = $125.772
1-year spot rate in three years:
($125.772 – 118.093) / $118.093 = .065, or 6.5%
b. If investing in long-term bonds carries additional risks, then the risk equivalent of a one- year spot
rate in three years would be less that the 6.5 percent, reflecting the fact that some risk premium must
be built into this 6.5 percent spot rate.
Est. Time: 06-10
29. a. r = 1.10 / 1 .05 – 1
r = .0476, or 4.76%
b-1. The real rate does not change. b-2.
The nominal rate increases to:
rNominal = 1.0476 × 1.07 – 1
rNominal = .1210, or 12.10%
Est. Time: 01-05
30.
a. Nominal 2-year return:
1.082 – 1 = .1664, or 16.64%
Real 2-year return:
(1.08 / 1.03) × (1.08 / 1.05) – 1 = .0785, or 7.85%
b. Nominal 2-year return:
1.082 – 1 = .1664, or 16.64%
Real 2-year return:
(1.08 × 1.03) × (1.08 × 1.05) – 1 = .2615, or 26.15%
Est. Time: 01-05
32. 31. PV = (.10 × $1,000) × ((1 / .0129) – {1 / [.0129 × (1 + .0129)5]}) + $1,000 / 1.01295 PV =
$1,419.14
PV = (.10 × $1,000) × ((1 / .0311) – {1 / [.0311 × (1 + .0311)5]}) + $1,000 / 1.03115 PV =
$1,314.55
Est. Time: 01-05
32. Spreadsheet problem; answers will vary.
Est. Time: 06-10
33. We begin with the definition of duration as applied to a bond with yield r and an annual payment of C in
perpetuity:
1C tC t
(1
33.
r) r)
DUR
1 r
C
C
(1
r)2
C
(1
r)3
C
1 r
(1 r)2 (1
r)
(1 r)
3 t
We first simplify by dividing both the numerator and the denominator by C:
1
(1 r)
2
(1 r)2
3
(1 r)3
t
(1 r)t
DUR 1 1 1 1
(1 r)2 (1 r)3
1 r (1 r)t
The denominator is the present value of a perpetuity of $1 per year, which is equal to (1/r). To simplify the
numerator, we first denote the numerator S and then divide S by (1 + r):
S 1 2 3 t
(1 r)t 1
(1 r)
(1
r)2
(1
r)3
(1 r)4
Note that this new quantity [S/(1 + r)] is equal to the square of denominator in the duration formula above,
that is:
2
S 1 1 1 1
Therefore:
(1 r) 1 r (1
r)2
2
(1
r)3
(1 r)t
S 1 1 r
(1 r) r S
r2
34.
Thus, for a perpetual bond paying C dollars per year:
DUR
1 r 1 1 r
r2 (1/ r) r
39. Chapter 01 – Introduction to Corporate Finance
Est. Time: 01-05
7. r = DIV1 / P0
r = $5 / $40
r = .125, or 12.5%
Est. Time: 01-05
8. P0 = DIV1 / (r – g)
P0 = $10 / (.08 − .05) P0
= $333.33
Est. Time: 01-05
9. DIV1 = $10
DIV2 = DIV1 × (1 + g) = $10 × 1.05 = $10.50 DIV3 =
DIV2 × (1 + g) = $10.50 × 1.05 = $11.03
P0 = DIV1 / (r – g) = $10 / (.08 – .05) = $333.33 P1
= P0 × (1 + g) = $333.33 × 1.05 = $350.00 P2 = P1
× (1 + g) = $350.00 × 1.05 = $367.50 P3 = P2 × (1
+ g) = $367.50 × 1.05 = $385.88
r1 = (DIV1 + P1 – P0) / P0 = ($10 + 350.00 – 333.33) / $333.33 = .08, or 8%
r2 = (DIV2 + P2 – P1) / P1 = ($10.50 + 367.50 – 350.00) / $350.00 = .08, or 8%
r3 = (DIV3 + P3 – P2) / P2 = ($11.03 + 385.88 – 367.50) / $367.50 = .08, or 8%
Since the rate of return each year is 8 percent, each investor should expect to earn 8%.
Est. Time: 06-10
10. a. P0 = [DIV0 × (1 + g)] / (r – g)
P0 = [($1.35 × (1 + .0275)] / (.095 – .0275)
P0 = $20.55
b. r = (1 + R) / (1 + h) – 1
r = (1 + .095) / (1 + .0275) – 1
r = .0657, or 6.57%
In real terms, g equals 0, so DIV1 equals DIV0.
P0 = $1.35 / .0657 P0
= $20.55
Est. Time: 06-10
11. Assuming steadygrowth of dividends for both stocks, the estimates are as follows:
40. Chapter 01 – Introduction to Corporate Finance
a. Stock A: Payout ratio = Dividends/Earnings = $1.00/$2.00 = 50% Stock
B: Payout ratio = Dividends/Earnings = $1.00/$1.50 = 67%
b. Stock A: Growth rate = plowback ratio × ROE = (1 – 50%) × 15% = 7.5% Stock
B: Growth rate = plowback ratio × ROE = (1 – 67%) × 10% = 3.3%
c. Stock A: PVA = DIV1/(r-g) = $1.00/(.15-.075) = $13.33
Stock B: PVB = DIV1/(r-g) = $1.00/(.15-.033) = $8.55
Est. Time: 01-05
12. Internet exercise; answers will vary.
Est. Time: 06-10
13. P4 = EPS5 / r
P4 = [EPS1 × (1 + g1)3
× (1 + g2)] / r P4
= [$15 × (1 + .05)3
× (1 + 0)] / .08 P4 =
$217.05
Note that $15 is the EPS for year 1. The 5 percent growth rate stops after year 4, so the exponent for the first
growth rate must be 3, (Year 4 – Year 1). There is no growth in year 5.
P0 = DIV1 / (1 + r) + [DIV1 × (1 + g)] / (1 + r)2
+ [DIV1 × (1 + g)2
] / (1 + r)3
+
[DIV1 × (1 + g)3
] / (1 + r)4
+ P4 / (1 + r)4
P0 = $10 / 1.08 + ($10 × 1.05) / 1.082
+ ($10 × 1.052
) / 1.083
+ ($10 × 1.053
) / 1.084
+
$217.05 / 1.084
P0 = $195.06
Est. Time: 06-10
14.
a. P0 Stock A = DIV1 / r = $10 / .10 = $100
b. P0 Stock B = DIV1 / (r – g) = $5 / (.10 – .04) = $83.33
c.
𝑃0 Stock C DIV1 DIV1 × (1 + 𝑔) DIV1 × (1 + 𝑔)2 DIV1 × (1 + 𝑔)3
= +
1 + 𝑟 (1 + 𝑟)2
+
(1 + 𝑟)3
+
(1 + 𝑟)4
DIV1 × (1 + 𝑔)4
+
(1 + 𝑟)5
+
DIV1 × (1 + 𝑔)5
(1 + 𝑟)6
+
[DIV1 × (1 + 𝑔)5 × (1 + 𝑔2)]
/ 𝑟
(1 + 𝑟)6
P0 Stock C = $5 / 1.1 + ($5 × 1.2) / 1.12 + ($5 × 1.22) / 1.13 + ($5 × 1.23) / 1.14 +
($5 × 1.24) / 1.15 + ($5 × 1.25) / 1.16 + {[$5 × 1.25 × (1 + 0)] / .1} / 1.16
P0 Stock C = $104.51
41. Chapter 01 – Introduction to Corporate Finance
At a 10% capitalization rate, Stock C has the largest present value.
Using the same formulas as above with a 7% capitalization rate, the values are:
P0 Stock A = $10 / .07 = $142.86
P0 Stock B = $5 / (.07 - .04) = $166.67
P0 Stock C = $5 / 1.07 + ($5 × 1.2) / 1.072 + ($5 × 1.22) / 1.073 + ($5 × 1.23) / 1.074 + ($5 ×
1.24) / 1.075 + ($5 × 1.25) / 1.076 + {[$5 × 1.25 × (1 + 0)] / .07} / 1.076
P0 Stock C = $156.50
At a 7% capitalization rate, Stock B has the largest present value.
Est. Time: 11-15
15. a. Plowback ratio = 1 – payout ratio = 1 – .5
Plowback ratio = .5
gYears 1-4 = plowback ratio × ROE = .5 × .14
gYears 1-4 = .07
EPS0 = ROE × book equity per share =.14 × $50 EPS0 =
$7.00
DIV0 = payout ratio × EPS0 = .5 × $7.00 DIV0
= $3.50
g Year 5 and later = plowback ratio × ROE = (1 – .8) × .115
g Year 5 and later = .023, or 2.3%
The annual EPS and DIV are as follows:
Year EPS DIV
0 $7.00
1 $7.00 × 1.07 = $7.49 $7.49 × .5 = $3.75
2 $7.00 × 1.072 = $8.01 $8.01 × .5 = $4.01
3 $7.00 × 1.073 = $8.58 $8.58 × .5 = $4.29
4 $7.00 × 1.074 = $9.18 $9.18 × .5 = $4.59
5 $7.00 × 1.074 × 1.023 = $9.39 $9.39 × .8 = $7.51
42. Chapter 01 – Introduction to Corporate Finance
b. PH = [DIV5 × (1 + g2)] / (r – g2)
PH = ($7.51 × 1.023) / (.115 - .023)
PH = $83.50
P0 = DIV1 / (1 + r) + DIV2 / (1 + r)2 + DIV3 / (1 + r)3 + DIV4 / (1 + r)4
+ DIV5 / (1 + r)5 + PH / (1 + r)5
P0 = $3.75 / 1.115 + $4.01 / 1.1152 + $4.29 / 1.1153 + $4.59 / 1.1154
+ $7.51 / 1.1155 + $83.50 / 1.1155
P0 = $65.45
The last term in the above calculation is dependent on the payout ratio and the growth rate after
year 4.
Est. Time: 11-15
16. a. r = DIV1 / P0 + g
r = $4 / $100 + .04
r = .08, or 8%
EPS1 = Div1 / (1 – reinvestment rate) EPS1 =
$4 / (1 – .40)
EPS1 = $6.67
P0 = EPS1 / r + PVGO
PVGO = P0 – EPS1 / r
PVGO = $100 – $6.67 / .08
PVGO = $16.67
b. DIV1 will decrease to: .20 $6.67 = $1.33.
By plowing back 80% of earnings, CSI will grow by 8% per year for five years before returning to its
long-run growth rate of 4%. The dividend will be 20% of earnings for years 1-5 and 60% of earnings
in year 6 and beyond.
Year 1 2 3 4 5 6
43. Chapter 01 – Introduction to Corporate Finance
EPSt $6.67 $7.20 $7.78 $8.40 $9.07 $9.80
DIVt 1.33 1.44 1.56 1.68 1.81 5.88
P5 = DIV6 / (r – g)
P5 = $5.88 / (.08 – .04) P5
= $146.93
P0 = DIV1 / (1 + r) + DIV2 / (1 + r)2 + DIV3 / (1 + r)3 + DIV4 / (1 + r)4 +
DIV5 / (1 + r)5 + P5 / (1 + r)5
P0 = $1.33 / 1.08 + $1.44 / 1.082 + $1.56 / 1.083 + $1.68 / 1.084 +
$1.81 / 1.085 + $146.93 / 1.085
P0 = $106.17
Est. Time: 11-15
17. A stock‘s capitalization rate equals EPS1 / P0 when PVGO = 0, that is when the firm pays out all its earnings
and is not growing.
Est. Time: 01-05
18. Answers will vary.
a. An Incorrect Application. Hotshot Semiconductor‘s earnings and dividends have grown by 30% per
year since the firm‘s founding 10 years ago. Current stock price is $100, and next year‘s dividend is
projected at $1.25. Thus:
r = DIV1 / P0 + g = $1.25 / $100 + .30 = .3125, or 31.25%
This is wrong because the formula assumes perpetual growth; it is not possible for Hotshot to grow at
30% per year forever.
A Correct Application. The formula might be correctly applied to the Old Faithful Railroad, which
has been growing at a steady 5% rate for decades. Its EPS1 = $10, DIV1
= $5, and P0 = $100. Thus:
r = Div1 / P0 + g = $5 / $100 + .05 = .10, or 10%
Even here, you should be careful not to blindly project past growth into the future. If Old Faithful
hauls coal, an energy crisis could turn it into a growth stock.
b. An Incorrect Application. Hotshot has current earnings of $5 per share. Thus:
r = EPS1 / P0 = $5 / $100 = .05, or 5%
This is too low to be realistic. The reason P0 is so high relative to earnings is not that r is low, but
rather that Hotshot is endowed with valuable growth opportunities. Suppose PVGO = $60:
44. NPVα
(rα .15)
21. Share price 1
NPV
r
Chapter 01 – Introduction to Corporate Finance
P0 = EPS1 / r + PVGO
$100 = $5 / r + $60
r = 12.5%
A Correct Application. Unfortunately, Old Faithful has run out of valuable growth
opportunities. Since PVGO = 0:
P0 = EPS1 / r + PVGO
$100 = $10 / r + $0
r = 10%
Est. Time: 11-15
19. P0 = DIV1 / (r – g)
P0 = $10 / (.08 – .05) P0
= $333.33
P0 = EPS1 / r + PVGO PVGO
= $333.33 – $15 / .08 PVGO =
$145.83
Est. Time: 01-05
20. Internet exercise; answers will vary.
Est. Time: 01-05
EPS
r r g
Therefore:
Ρα
EPSα1 NPV
(r
rα α .15)
Ρβ
EPSβ1
β
PVβ (rβ
.08)
The statement in the question implies the following:
EPSβ1 NPVβ EPSα1 NPVα
(r .08) r
(r .15)
r β
α
β
Rearranging, we have:
α
NPVα
rα NPVβ
rβ
EPSβ1
(rα .15) EPSα1
(rβ .08)
NPVβ
(rβ .08)
45. Chapter 01 – Introduction to Corporate Finance
Possibleexplanations:
a. NPV < NPV ; everything else equal
b. (r – .15) > (r – .08); everything else equal
c. NPVα
NPVβ ; everything else equal
(rα .15)
rα
(rβ .08)
rβ
d.
Est.Time: 06-10 EPSα1
EPSβ1
; everything else equal
22. a. P0 = Div1 / (1 + r) + Div2 / (1 + r)2 + Div3 / (1 + r)3 + (Div4 / (r – g)) / (1 + r)3 P0
= $.50 / 1.12 + $.60 / 1.122 + $1.15 / 1.123 + [$1.24 / (.12 – .08)] / 1.123 P0 =
$23.81
b. The horizon value P3 contributes:
[$1.24 / (.12 – .08)] / 1.123 = $22.07
c. Without PVGO, P3 would equal earnings for year 4 capitalized at 12%, so PVGO3 is valued as:
PVGO3 = [DIV4 / (r – g)] – EPS4 / r
PVGO3 = [$1.24 / (.12 – .08)] – $2.48 / .12
PVGO3 = $10.33
d. The PVGO of $10.33 is lost at Year 3. Therefore, the current stock price of $23.81 will decrease
by the present value of PVGO:
P0 No-growth = P0 – PVGO3 / (1 + r)3
P0
No-growth = $23.81 – $10.33 / 1.123 P0 No-
growth = $16.45
Est. Time: 11-15
23. Free cash flow is the amount of cash left over and available to pay out to investors after all investments
necessary for growth. In our simple examples, free cash flow equals operating cash flow minus capital
expenditures. Free cash flow can be negative if investments are large.
Est. Time: 01-05
24. Horizon value is the value of a firm at the end of a forecast period. Horizon value can be estimated using
the constant-growth DCF formula or by using price-earnings or market-book ratios for similar
companies.
Est. Time: 01-05
47. Chapter 01 – Introduction to Corporate Finance
412,437,817) / 1.093
PV2021 = $439,490,293
Price per share2021 = PV2021 / number of shares Price
per share2021 = $439,490,293 / 7,000,000 Price per
share2021 = $62.78
b. EPS2021 = net income2021 / number of shares
EPS2021 = $72,000,000 / 7,000,000
EPS2021 = $10.29
EPS/P = $10.29 / $62.78 EPS/P
= .164, or 16.4%
The EPS/P is greater than the cost of capital because production and earnings are declining.
Est. Time: 15-20
27. The free cash flow for years 1 through 8 is computed in the following table:
Year
($ in millions) 1 2 3 4 5 6 7 8
Asset value 10 11.2 12.54 14.05 15.31 16.69 18.19 19.29
Earnings 1.2 1.34 1.51 1.69 1.84 1.92 2.00 2.03
Investment 1.20 1.34 1.51 1.26 1.38 1.50 1.09 1.16
Free Cash Flow .00 .00 .00 .43 .46 .42 .91 .87
The present value of the business is:
PV = FCF1 / (1 + r) + FCF2 / (1 + r)2 + FCF3 / (1 + r)3 + FCF4 / (1 + r)4 +
FCF5 / (1 + r)5 + FCF6 / (1 + r)6 + FCF7 / (1 + r)7 + [FCF8 / (r – g)] / (1 + r)7
PV
= $0 / 1.1 + $0 / 1.12 + $0 / 1.13 + $.43 / 1.14 + $.46 / 1.15 + $.42 / 1.16 +
$.91 / 1.17 + [$.87 / (.1 – .08)] / 1.17 PV
= $23.47 million
Est. Time: 15-20
28. Currency amounts are in billions of pesos.
a. r = DIV1 / P0 + g
r = 8.5 / 200 + .075
48. Chapter 01 – Introduction to Corporate Finance
r = .1175, or 11.75%
b. g = ROE × (1 – reinvestment rate)
g = .12 × (1 – .50)
g = .06, or 6%
r = DIV1 / P0 + g
r = 8.5 / 200 + .06
r = .1025, or 10.25%
Est. Time: 15-20
29. a. PV2021 = DIV2022 / (1 + r) + DIV2023 / (1 + r)2 + DIV2024 / (1 + r)3 +
DIV2025 / (1 + r)4 + DIV2026 / (1 + r)5 + (DIV2025 / r) / (1 + r)5
PV2021 = $0 / 1.09 + $1 / 1.092 + $2 / 1.093 + $2.3 / 1.094 + $2.6 / 1.095
+ ($2.6 / .09) / 1.095
PV2021 = $24.48 million
b. Assuming no debt, the share price would be:
Price per share2021 = PV2021 / number of shares Price
pershare2021 =$24.48/12
Price per share2021 = $2.04
c. The PV of the cash flows at various points in time are as follows: PV2022 =
$1 / 1.09 + $2 / 1.092 + $2.3 / 1.093 + $2.6 / 1.094 +
($2.6 / .09) / 1.094. PV2022
= $26.68
PV2023 = $2 / 1.09 + $2.3 / 1.092 + $2.6 / 1.093 + ($2.6 / .09) / 1.093 PV2023 =
$28.09
PV2024 =$2.3 / 1.09+$2.6 /1.092 +($2.6/.09)/1.092 PV2024
= $28.61
PV2025 = $2.6 / .09
PV2025 = $28.89
PV2026 = $2.6 / .09
PV2026 = $28.89
Using the formula, r0 = (DIV1 + P1 – P0) / P0, the annual rates of return are:
Rate of return2023 = ($1 + 28.09 – 26.68 / $26.68 = .09, or 9% Rate of
return2024 = ($2 + 28.61 – 28.09 / $28.09 = .09, or 9% Rate of
return2025 = ($2.3 + 28.89 – 28.61) / $28.61 = .09, or 9%
Rate of return2026 = ($2.6 + 28.89 – 28.89) / $28.89 = .09, or 9%
Est. Time: 15-20
49. Chapter 01 – Introduction to Corporate Finance
30. P0 = [ROE × (1 – b) × BVPS] / (r – b × ROE)
P0 / BVPS = [ROE(1 – b)] / (r – b × ROE) P0
/ BVPS = (1 – b) / [(r / ROE) – b] Consider
three cases:
ROE < r (P0 / BVPS) < 1
ROE = r (P0 / BVPS) = 1
ROE > r (P0 / BVPS) > 1
Thus, as ROE increases, the price-to-book ratio also increases, and, when ROE = r, price-to- book equals
one.
Est. Time: 11-15
31. Value at a dividend yield of 5 percent:
r = dividend yield + g r
– g = dividend yield r –
g = .05, or 5%
Value = (annual fee × portfolio value) / (r – g)
Value = (.005 × $100 million) / .05 Value =
$10 million
Value at a dividend yield of 4 percent using the same formulas as above: Value = (.005 ×
$100 million) / .04
Value = $12.5 million
Est. Time: 6-10
32. a. Under the new faster growth assumption, Table 4.8 is reproduced here.
($ millions) 1 2 3 4
Year
5 6 7 8 9 10
Asset value 10.00 12.00 14.40 17.28 19.35 21.68 24.28 25.73 27.28 28.91
Earnings 1.20 1.44 1.73 2.07 2.32 2.60 2.91 3.09 3.27 3.47
Net investment 2.00 2.40 2.88 2.07 2.32 2.60 1.46 1.54 1.64 1.73
Free cash flow –.80 –.96 –1.15 .00 .00 .00 1.46 1.54 1.64 1.73
Return on equity .12 .12 .12 .12 .12 .12 .12 .12 .12 .12
Asset growth rate .20 .20 .20 .12 .12 .12 .06 .06 .06
Earnings growth rate .20 .20 .20 .12 .12 .12 .06 .06 .06
50. Chapter 01 – Introduction to Corporate Finance
The present value of the near-term cash flows for the first six years is computed using the FCF for
years 1 to 3 as the FCF is zero for years 4 to 6.
PV = FCF1 / (1 + r) + FCF2 / (1 + r)2 + FCF3 / (1 + r)3
PV = –$.80 / 1.1 + (–$.96) / 1.12 + (–$1.15) / 1.13 PV
= –$2.39 million
The present value at time 0 of the horizon value, computed as of year 6, is:
PVH = [$1.46 / (.1 – .06)] / 1.16 PVH
= $20.56 million
PV(business) = PV(free cash flow) + PV(horizon value) PV(business) = –
$2.39 + 20.56
PV(business) = $18.17 million
Under the original growth assumptions, the PV of free cash flow was $.9 million, the PV of the horizon value
was $15.4 million, and the PV of the business was $16.3 million. All three present values increased as a result of
the increased rate of growth.
b. Issuing new shares does not affect the overall value of the company as that value is dependent only on the
free cash flows. However, if new shares are issued to fund the negative present values of the new cash
flows, the value of the existing shares would be diluted.
Est. Time: 15-20
CHAPTER 5
Net Present Value and Other Investment Criteria
The values shown in the solutions may be rounded for display purposes. However, the answers were derived using a spreadsheet without any
intermediate rounding.
Answers to Problem Sets
1. a. A = 3 years; B = 2 years; C = 3 years
b. B
c. A, B, and C
51. Chapter 01 – Introduction to Corporate Finance
d. B and C (At 10%, NPVA = –$1,011; NPVB = $3,378; NPVC = $2,405)
e. True. The payback rule ignores all cash flows after the cutoff date, meaning that future years‘ cash
inflows are not considered. Thus, payback is biased towards short-term Projects.
Est. time: 06 – 10
2. a. NPVA = –$1,000 + $1,000 / (1 + .10) = –$90.91
NPVB = –$2,000 + $1,000 / (1 + .10) + $1,000 / (1 + .10)2 + $4,000 / (1 + .10)3 +
$1,000 / (1 + .10)4 + $1,000 / (1 + .10)5 NPVB =
$4,044.73
NPVC = –$3,000 + $1,000 / (1 + .10) + $1,000 / (1 + .10)2 + $1,000 / (1 + .10)4
+ $1,000 / (1 + .10)5 NPVC
= $39.47
Projects B and C have positive NPVs.
b. Payback A = 1 year
Payback B = 2 years
Payback C = 4 years
c. Accept projects A and B
Est. time: 11– 15
3.
NPV Rule Key Feature Payback Rule Accounting ROR
Clear benchmark with which to
compare a project‘s NPV (Accept
positive NPV, reject negative NPV
Yes – Reject projects that
have a payback period
longer than the cutoff
No – No nature benchmark for
judging if the project‘s returns are
attractive
NPV depends on all cash flows
No – Ignores Cash flows
beyond the payback period
No – does not forecast cash
flows but accounting profits,
which are subject to
accounting convention
52. IF YOU WANT THIS TEST BANK OR SOLUTION
MANUAL EMAIL ME kevinkariuki227@gmail.com TO
RECEIVE ALL CHAPTERS IN PDF FORMAT
IF YOU WANT THIS TEST BANK OR SOLUTION
MANUAL EMAIL ME kevinkariuki227@gmail.com TO
RECEIVE ALL CHAPTERS IN PDF FORMAT
