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Fundamentals of Coporate Finance Ch4~5 Ross

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Fundamentals of Coporate Finance Ch4~5 Ross

CHAPTER 4

    DISCOUNTED CASH FLOW VALUATION

Answers to Concepts Review and Critical Thinking Questions

    1. Assuming positive cash flows and interest rates, the future value increases and the present value

    decreases.

    2. Assuming positive cash flows and interest rates, the present value will fall and the future value will

    rise.

3. The better deal is the one with equal installments.

    4. Yes, they should. APRs generally don’t provide the relevant rate. The only advantage is that they are

    easier to compute, but, with modern computing equipment, that advantage is not very important.

    5. A freshman does. The reason is that the freshman gets to use the money for much longer before

    interest starts to accrue.

6. It’s a reflection of the time value of money. GMAC gets to use the $500 immediately. If GMAC uses

    it wisely, it will be worth more than $10,000 in thirty years.

    7. Oddly enough, it actually makes it more desirable since GMAC only has the right to pay the full

    $10,000 before it is due. This is an example of a “call” feature. Such features are discussed at length

    in a later chapter.

    8. The key considerations would be: (1) Is the rate of return implicit in the offer attractive relative to

    other, similar risk investments? and (2) How risky is the investment; i.e., how certain are we that we

    will actually get the ?10,000? Thus, our answer does depend on who is making the promise to repay.

    9. The Treasury security would have a somewhat higher price because the Treasury is the strongest of

    all borrowers.

    10. The price would be higher because, as time passes, the price of the security will tend to rise toward

    $10,000. This rise is just a reflection of the time value of money. As time passes, the time until

    receipt of the $10,000 grows shorter, and the present value rises. In 2010, the price will probably be

    higher for the same reason. We cannot be sure, however, because interest rates could be much higher,

    or GMAC’s financial position could deteriorate. Either event would tend to depress the security’s

    price.

     CHAPTER 4 B- 2

    Solutions to Questions and Problems

NOTE: All-end-of chapter problems were solved using a spreadsheet. Many problems require multiple

    steps. Due to space and readability constraints, when these intermediate steps are included in this

    solutions manual, rounding may appear to have occurred. However, the final answer for each problem is

    found without rounding during any step in the problem.

     Basic

1. The simple interest per year is:

     $5,000 × .07 = $350

     So, after 5 years, you will have:

     $350 × 5 = $1,750 in interest.

     The total balance will be $5,000 + 1,750 = $6,750

     With compound interest, we use the future value formula:

     t FV = PV(1 +r) 5 FV = $5,000(1.07) = $7,012.76

     The difference is:

     $7,012.76 6,750 = $262.76

. To find the FV of a lump sum, we use: 2

     t FV = PV(1 + r)

     10 a. FV = ?1,000(1.06) = ?1,790.85 10 b. FV = ?1,000(1.07) = ?1,967.15 20 c. FV = ?1,000(1.06) = ?3,207.14

     d. Because interest compounds on the interest already earned, the future value in part c is more

    than twice the future value in part a. With compound interest, future values grow exponentially.

3. To find the PV of a lump sum, we use:

     t PV = FV / (1 + r)

     6 PV = ?15,451 / (1.05) = ?11,529.77 9 PV = ?51,557 / (1.11) = ?20,154.91 18 PV = ?886,073 / (1.16) = ?61,266.87 23 PV = ?550,164 / (1.19)= ?10,067.28

     CHAPTER 4 B- 3

    4. To answer this question, we can use either the FV or the PV formula. Both will give the same answer

    since they are the inverse of each other. We will use the FV formula, that is:

     t FV = PV(1 + r)

     Solving for r, we get:

     1 / t r = (FV / PV) 1

     21/2 FV = $307 = $265(1 + r); r = ($307 / $265) 1 = 7.63% 91/9 FV = $896 = $360(1 + r); r = ($896 / $360) 1 = 10.66% 151/15 FV = $162,181 = $39,000(1 + r); r = ($162,181 / $39,000) 1 = 9.97% 301/30 FV = $483,500 = $46,523(1 + r); r = ($483,500 / $46,523) 1 = 8.12%

    5. To answer this question, we can use either the FV or the PV formula. Both will give the same answer

    since they are the inverse of each other. We will use the FV formula, that is:

     t FV = PV(1 + r)

     Solving for t, we get:

     t = ln(FV / PV) / ln(1 + r)

     t FV = ?1,284,000 = ?625,000 (1.09); t = ln(?1,284,000/ ?625,000) / ln 1.09 = 8.35 yrs t FV = ?4,341,000 = ?810,000 (1.07); t = ln(?4,341,000/ ?810,000) / ln 1.07 = 24.81 yrs t FV = ?402,662,000 = ?18,400,000 (1.21); t = ln(?402,662,000 / ?18,400,000) / ln 1.21 = 16.19 yrs t FV = ?173,439,000 = ?21,500,000 (1.29); t = ln(?173,439,000 / ?21,500,000) / ln 1.29 = 8.20 yrs

    6. To find the length of time for money to double, triple, etc., the present value and future value are

    irrelevant as long as the future value is twice the present value for doubling, three times as large for

    tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the

    same answer since they are the inverse of each other. We will use the FV formula, that is:

     t FV = PV(1 + r)

     Solving for t, we get:

     t = ln(FV / PV) / ln(1 + r)

     The length of time to double your money is:

     t FV = $2 = $1(1.06)

     t = ln 2 / ln 1.06 = 11.90 years

     The length of time to quadruple your money is:

     t FV = $4 = $1(1.06)

     t = ln 4 / ln 1.06 = 23.79 years

     CHAPTER 4 B- 4

     Notice that the length of time to quadruple your money is twice as long as the time needed to double

    your money (the difference in these answers is due to rounding). This is an important concept of time

    value of money.

7. To find the PV of a lump sum, we use:

     t PV = FV / (1 + r)20 PV = $800,000,000 / (1.095) = $130,258,959.12

    8. To answer this question, we can use either the FV or the PV formula. Both will give the same answer

    since they are the inverse of each other. We will use the FV formula, that is:

     t FV = PV(1 + r)

     Solving for r, we get:

     1 / t r = (FV / PV) 1 1/4 r = ($10,311,500 / $12,377,500) 1 = 4.46%

     Notice that the interest rate is negative. This occurs when the FV is less than the PV.

9. A consol is a perpetuity. To find the PV of a perpetuity, we use the equation:

     PV = C / r

     PV = ?120 / .15

     PV = ?800.00

    10. To find the future value with continuous compounding, we use the equation:

     Rt FV = PVe

     .12(5) a. FV = $1,000e = $1,822.12 .10(3) b. FV = $1,000e = $1,349.86 .05(10) c. FV = $1,000e = $1,648.72 .07(8) d. FV = $1,000e = $1,750.67

    11. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a

    lump sum, we use:

     t PV = FV / (1 + r)

     234 PV@10% = ?1,200 / 1.10 + ?600 / 1.10 + ?855 / 1.10 + ?1,480 / 1.10 = ?3,240.01

     234 PV@18% = ?1,200 / 1.18 + ?600 / 1.18 + ?855 / 1.18 + ?1,480 / 1.18 = ?2,731.61

     234 PV@24% = ?1,200 / 1.24 + ?600 / 1.24 + ?855 / 1.24 + ?1,480 / 1.24 = ?2,432.40

     CHAPTER 4 B- 5

    12. To find the PVA, we use the equation:

     t PVA = C({1 [1/(1 + r)] } / r )

     At a 5 percent interest rate:

    9 X@5%: PVA = 4,000{[1 (1/1.05) ] / .05 } = 28,431.29

    5 Y@5%: PVA = 6,000{[1 (1/1.05) ] / .05 } = 25,976.86

     And at a 22 percent interest rate:

    9 X@22%: PVA = 4,000{[1 (1/1.22) ] / .22 } = 15,145.14

    5 Y@22%: PVA = 6,000{[1 (1/1.22) ] / .22 } = 17,181.84

    Notice that the PV of Cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a

    22 percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the

    total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a

    higher interest rate, Y is more valuable since it has larger cash flows. At a higher interest rate, these

    bigger cash flows early are more important since the cost of waiting (the interest rate) is so much

    greater.

13. To find the PVA, we use the equation:

     t PVA = C({1 [1/(1 + r)] } / r )

     15 PVA@15 yrs: PVA = Rs.3,600{[1 (1/1.10) ] / .10} = Rs.27,381.89

     40 PVA@40 yrs: PVA = Rs.3,600{[1 (1/1.10) ] / .10} = Rs.35,204.58

     75 PVA@75 yrs: PVA = Rs.3,600{[1 (1/1.10) ] / .10} = Rs.35,971.70

     To find the PV of a perpetuity, we use the equation:

     PV = C / r

     PV = Rs.3,600 / .10

     PV = Rs.36,000.00

     Notice that as the length of the annuity payments increases, the present value of the annuity

    approaches the present value of the perpetuity. The present value of the 75-year annuity and the

    present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years

    is only Rs.28.30.

    14. This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:

     PV = C / r

     PV = $15,000 / .08 = $187,500.00

     CHAPTER 4 B- 6

     To find the interest rate that equates the perpetuity cash flows with the PV of the cash flows. Using

    the PV of a perpetuity equation:

     PV = C / r

     $195,000 = $15,000 / r

     We can now solve for the interest rate as follows:

     r = $15,000 / $195,000 = 7.69%

15. For discrete compounding, to find the EAR, we use the equation:

     m EAR = [1 + (APR / m)] 1

     4 EAR = [1 + (.11 / 4)] 1 = 11.46%

     12 EAR = [1 + (.07 / 12)] 1 = 7.23%

     365 EAR = [1 + (.09 / 365)] 1 = 9.42%

     To find the EAR with continuous compounding, we use the equation:

     q EAR = e 1 .17 EAR = e 1 = 18.53%

16. Here, we are given the EAR and need to find the APR. Using the equation for discrete

    compounding:

     m EAR = [1 + (APR / m)] 1

     We can now solve for the APR. Doing so, we get:

     1/m APR = m[(1 + EAR) 1]

     21/2 EAR = .081 = [1 + (APR / 2)] 1 APR = 2[(1.081) 1] = 7.94%

     121/12 EAR = .076 = [1 + (APR / 12)] 1 APR = 12[(1.076) 1] = 7.35%

     521/52 EAR = .168 = [1 + (APR / 52)] 1 APR = 52[(1.168) 1] = 15.55%

     Solving the continuous compounding EAR equation:

     q EAR = e 1

     We get:

     APR = ln(1 + EAR)

     APR = ln(1 + .262)

     APR = 23.27%

     CHAPTER 4 B- 7

    17. For discrete compounding, to find the EAR, we use the equation:

     m EAR = [1 + (APR / m)] 1

     So, for each bank, the EAR is:

     12 First National: EAR = [1 + (.122 / 12)] 1 = 12.91%

     2 First United: EAR = [1 + (.125 / 2)] 1 = 12.89%

     Notice that the higher APR does not necessarily mean the higher EAR. The number of compounding

    periods within a year will also affect the EAR.

18. The cost of a case of wine is 10 percent less than the cost of 12 individual bottles, so the cost of a

    case will be:

     Cost of case = (12)(?10)(1 – .10)

     Cost of case = ?108

     Now, we need to find the interest rate. The cash flows are an annuity due, so:

     t PVA = (1 + r) C({1 [1/(1 + r)] } / r) 12 ?108 = (1 + r) ?10({1 [1 / (1 + r)] / r )

     Solving for the interest rate, we get:

     r = .0198 or 1.98% per week

     So, the APR of this investment is:

     APR = .0198(52)

     APR = 1.0277 or 102.77%

     And the EAR is:

     52 EAR = (1 + .0198) 1

     EAR = 1.7668 or 176.68%

     The analysis appears to be correct. He really can earn about 177 percent buying wine by the case.

    The only question left is this: Can you really find a fine bottle of Bordeaux for ?10?

19. Here, we need to find the length of an annuity. We know the interest rate, the PV, and the payments.

    Using the PVA equation:

     t PVA = C({1 [1/(1 + r)] } / r) t ?16,500 = ?500{ [1 (1/1.009) ] / .009}

     CHAPTER 4 B- 8

     Now, we solve for t:

     t 1/1.009 = 1 [(?16,500)(.009) / (?500)] t 1.009= 1/(0.703) = 1.422

     t = ln 1.422 / ln 1.009 = 39.33 months

    20. Here, we are trying to find the interest rate when we know the PV and FV. Using the FV equation:

     FV = PV(1 + r)

     $4 = $3(1 + r)

     r = 4/3 1 = 33.33% per week

     The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks

    in a year, so:

     APR = (52)33.33% = 1,733.33%

     And using the equation to find the EAR:

     m EAR = [1 + (APR / m)] 1 52 EAR = [1 + .3333] 1 = 313,916,515.69%

     Intermediate

21. To find the FV of a lump sum with discrete compounding, we use:

     t FV = PV(1 + r)

     3 a. FV = $1,000(1.08) = $1,259.71 6 b. FV = $1,000(1 + .08/2) = $1,265.32 36 c. FV = $1,000(1 + .08/12) = $1,270.24

     To find the future value with continuous compounding, we use the equation:

     Rt FV = PVe

     .08(3) d. FV = $1,000e = $1,271.25

     e. The future value increases when the compounding period is shorter because interest is earned

    on previously accrued interest. The shorter the compounding period, the more frequently

    interest is earned, and the greater the future value, assuming the same stated interest rate.

    22. The total interest paid by First Simple Bank is the interest rate per period times the number of

    periods. In other words, the interest by First Simple Bank paid over 10 years will be:

     .08(10) = .8

     CHAPTER 4 B- 9

     First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor

    of $1, or:

     10 (1 + r)

     Setting the two equal, we get:

     10 (.08)(10) = (1 + r) 1

     1/10 r = 1.8 1 = 6.05%

    23. We need to find the annuity payment in retirement. Our retirement savings ends at the same time the

    retirement withdrawals begin, so the PV of the retirement withdrawals will be the FV of the

    retirement savings. So, we find the FV of the stock account and the FV of the bond account and add

    the two FVs.

     360 Stock account: FVA = Rs.700[{[1 + (.11/12) ] 1} / (.11/12)] = Rs.1,963,163.82

     360 Bond account: FVA = Rs.300[{[1 + (.07/12) ] 1} / (.07/12)] = Rs.365,991.30

     So, the total amount saved at retirement is:

     Rs.1,963,163.82 + 365,991.30 = Rs.2,329,155.11

     Solving for the withdrawal amount in retirement using the PVA equation gives us:

     300 PVA = Rs.2,329,155.11 = C[1 {1 / [1 + (.09/12)]} / (.09/12)]

     C = Rs.2,329,155.11 / 119.1616 = Rs.19,546.19 withdrawal per month

    24. Since we are looking to quardruple our money, the PV and FV are irrelevant as long as the FV is

    four times as large as the PV. The number of periods is four, the number of quarters per year. So:

     (12/3) FV = $4 = $1(1 + r)

     r = 41.42%

    25. Here, we need to find the interest rate for two possible investments. Each investment is a lump sum,

    so:

     5 G: PV = ?50,000 = ?85,000 / (1 + r) 5 (1 + r) = ?85,000 / ?50,000 1/5 r = (1.70) 1 = 11.20%

     10 H: PV = ?50,000 = ?175,000 / (1 + r) 10 (1 + r) = ?175,000 / ?50,000 1/10 r = (3.50) 1 = 13.35%

     CHAPTER 4 B- 10

    26. This is a growing perpetuity. The present value of a growing perpetuity is:

     PV = C / (r g)

     PV = Rs.200,000 / (.10 .05)

     PV = Rs.4,000,000

     It is important to recognize that when dealing with annuities or perpetuities, the present value

    equation calculates the present value one period before the first payment. In this case, since the first

    payment is in two years, we have calculated the present value one year from now. To find the value

    today, we simply discount this value as a lump sum. Doing so, we find the value of the cash flow

    stream today is:

     t PV = FV / (1 + r) 1 PV = Rs.4,000,000 / (1 + .10)

     PV = Rs.3,636,363.64

    27. The dividend payments are made quarterly, so we must use the quarterly interest rate. The quarterly

    interest rate is:

     Quarterly rate = Stated rate / 4

     Quarterly rate = .12 / 4

     Quarterly rate = .03

     Using the present value equation for a perpetuity, we find the value today of the dividends paid must

    be:

     PV = C / r

     PV = ?1,000 / .03

     PV = ?33,333.33

    28. We can use the PVA annuity equation to answer this question. The annuity has 20 payments, not 19

    payments. Since there is a payment made in Year 3, the annuity actually begins in Year 2. So, the

    value of the annuity in Year 2 is:

     t PVA = C({1 [1/(1 + r)] } / r ) 20 PVA = $2,000({1 [1/(1 + .08)] } / .08)

     PVA = $19,636.29

     This is the value of the annuity one period before the first payment, or Year 2. So, the value of the

    cash flows today is:

     t PV = FV/(1 + r) 2 PV = $19,636.29/(1 + .08)

     PV = $16,834.96

    29. We need to find the present value of an annuity. Using the PVA equation, and the 15 percent interest

    rate, we get:

     t PVA = C({1 [1/(1 + r)] } / r ) 15 PVA = $500({1 [1/(1 + .15)] } / .15)

     PVA = $2,923.69

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