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Unformatted text preview: present value of the 75year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years is only $74.51. 14. This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation: PV = C / r PV = $20,000 / .065 = $307,692.31 54 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 $340,000 = $20,000 / r We can now solve for the interest rate as follows: r = $20,000 / $340,000 = .0588 or 5.88% 15. For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m)]m – 1 EAR = [1 + (.08 / 4)]4 – 1 EAR = [1 + (.18 / 12)]12 – 1 = .0824 or 8.24% = .1956 or 19.56% EAR = [1 + (.12 / 365)]365 – 1 = .1275 or 12.75% To find the EAR with continuous compounding, we use the equation: EAR = eq – 1 EAR = e.14 – 1 = .1503 or 15.03% 16. Here, we are given the EAR and need to find the APR. Using the equation for discrete compounding: EAR = [1 + (APR / m)]m – 1 We can now solve for the APR. Doing so, we get: APR = m[(1 + EAR)1/m – 1] EAR = .1030 = [1 + (APR / 2)]2 – 1 EAR = .0940 = [1 + (APR / 12)]12 – 1 EAR = .0720 = [1 + (APR / 52)]52 – 1 APR = 2[(1.1030)1/2 – 1] APR = 12[(1.0940)1/12 – 1] APR = 52[(1.0720)1/52 – 1] = .1005 or 10.05% = .0902 or 9.02% = .0696 or 6.96% Solving the continuous compounding EAR equation: EAR = eq – 1 We get: APR = ln(1 + EAR) APR = ln(1 + .1590) APR = .1476 or 14.76% 55 17. For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m)]m – 1 So, for each bank, the EAR is: First National: EAR = [1 + (.1010 / 12)]12 – 1 = .1058 or 10.58% First United: EAR = [1 + (.1040 / 2)]2 – 1 = .1067 or 10.67% A 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: PVA = (1 + r) C({1 – [1/(1 + r)]t } / r) $108 = (1 + r) $10({1 – [1 / (1 + r)12] / 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: EAR = (1 + .0198)52 – 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: PVA = C({1 – [1/(1 + r)]t } / r) $18,400 = $600{ [1 – (1/1.009)t ] / .009} 56 Now, we solve for t: 1/1.009t = 1 – [($18,400)(.009) / ($600)] 1.009t = 1/(0.724) = 1.381 t = ln 1.381 / ln 1.009 = 36.05 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: EAR = [1 + (APR / m)]m – 1 EAR = [1 + .3333]52 – 1 = 313,916,515.69% Intermediate 21. To find the FV of a lump sum with discrete compounding, we use: FV = PV(1 + r)t a. b. c. d. FV = $1,000(1.08)7 = $1,713.82 FV = $1,000(1 + .08/2)14 = $1,731.68 FV = $1,000(1 + .08/12)84 = $1,747.42 To find the future value with continuous compounding, we use the equation: FV = PVeRt FV = $1,000e.08(7) = $1,750.67 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: .06(10) = .6 57 First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor of $1, or: (1 + r)10 Setting the two equal, we get: (.06)(10) = (1 + r)10 – 1 r = 1.61/10 – 1 = .0481 or 4.81% 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. Stock account: FVA = $700[{[1 + (.10/12) ]360 – 1} / (.10/12)] = $1,582,341.55 Bond account: FVA = $300[{[1 + (.06/12) ]360 – 1} / (.06/12)] = $301,354.51 So, the total amount saved at retirement is: $1,582,341.55 + 301,354.51 = $1,883,696.06 Solving for the withdrawal amount in retirement using the PVA equation gi...
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 Spring '10
 eshmalwi
 Finance, Corporate Finance

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