Chapter 02 - How to Calculate Present Values Solution

Chapter 02 - How to Calculate Present Values 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. Ct = PV × (1 + r)t C8 = $100 × 1.158 C8 = $305.90 Est time: 01-05 2. PV = Ct / DFt DFt = $125 / $139 DFt = .8993 Est time: 01-05 3. PV = Ct / (1 + r)t PV = $374 / 1.099 PV = $172.20 Est time: 01-05 4. a. PV = C1 / (1 + r)1 + C2 / (1 + r)2 + C3 / (1 + r)3 PV = $432 / 1.15 + $137 / 1.152 + $797 / 1.153 PV = $1,003.28 b. NPV = PV – investment NPV = $1,003.28 – 1,200 NPV = –$196.72 Est time: 01-05 5. 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 6. NPV = C / r – investment Chapter 02 - How to Calculate Present Values NPV = $138 / .09 − $1,548 NPV = −$14.67 Est time: 01-05 7. PV = C / (r – g) PV = $4 / (.14 − .04) PV = $40 Est time: 01-05 8. 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 9. The basic present value formula is: PV = C / (1 + r)t a. PV = $100 / 1.0110 PV = $90.53 b. PV = $100 / 1.1310 PV = $29.46 c. PV = $100 / 1.2515 PV = $3.52 d. PV = C1 / (1 + r) + C2 / (1 + r)2 + C3 / (1 + r)3 PV = $100 / 1.12 + $100 / 1.122 + $100 / 1.123 PV = $240.18 Est time: 01-05 10. a. FV = C × ert FV = $1,000 × e.12 x 5 FV = $1,822.12 b. PV = C / ert Chapter 02 - How to Calculate Present Values 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 11. 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 b. Ct = PV × ert Ct = $10,000,000 × e.06 × 4 Ct = $12,712,492 Est time: 01-05 12. a. PV = Ct / (1 + r)t PV = $10,000 / 1.055 PV = $7,835.26 b. PV = C((1 / r) – {1 / [r(1 + r)t]}) PV = $12,000((1 / .08) – {1 / [.08(1.08)6]}) PV = $55,474.56 c. Ct = PV × (1 + r)t Ct = ($60,476 − 55,474.56) × 1.086 Ct = $7,936.66 Est time: 06-10 13. 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 Chapter 02 - How to Calculate Present Values e. PVAF3 = PVAF2 + DF3 DF3 = 2.465 – 1.7240 DF3 = .7410 Est time: 06-10 14. 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 five remaining year’s of cash flows. PV = $170,000 × ((1 / .14) – {1 / [.14(1.14)(10 – 5)]}) PV = $583,623.76 Est time: 01-05 15. 10 NPV = ∑ t=0 Ct (1.12) t NPV = –$380,000 + $50,000 / 1.12 + $57,000 / 1.122 + $75,000 / 1.123 + $80,000 / 1.124 + $85,000 / 1.125 + $92,000 / 1.126 + $92,000 / 1.127 + $80,000 / 1.128 + $68,000 / 1.129 + $50,000 / 1.1210 NPV = $23,696.15 Est time: 01-05 16. The opportunity cost of capital refers to the rate of return a firm’s shareholders could earn on their own by investing at the same level of risk. Thus, when a firm considers a new project, it is the risk level of the project that determines opportunity cost of capital for that project. Est time: 06-10 17. NPV = C0 + C1 / (1 + r) + C2 / (1 + r)2 + C3 / (1 + r)3 NPV = –$400,000 + $100,000 / 1.12 + $200,000 / 1.12 2 + $300,000 / 1.123 NPV = $62,258.56 Period 0 1 2 3 $100,000 / 1.12 = $200,000 / 1.122 = $300,000 / 1.123 Present Value –$400,000.00 89,285.71 159,438.78 213,534.07 Chapter 02 - How to Calculate Present Values $ 62,258.56 Est time: 01-05 18. 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 19. a. PV = C0 PV = $100,000 b. PV = Ct / (1 + r)t PV = $180,000 / 1.125 PV = $102,136.83 c. PV = C / r PV = $11,400 / .12 PV = $95,000 d. PV = C × ((1 / r) – {1 / – [r(1 + r)t]}) PV = $19,000 × ((1 / .12) – {1 / – [.12(1.12)10]}) PV = $107,354.24 e. PV = C / (r – g) PV = $6,500 / (.12 − .05) PV = $92,857.14 Prize (d) is the most valuable because it has the highest present value. Est time: 01-05 20. 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 Chapter 02 - How to Calculate Present Values 21. 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 22. The fact that Kangaroo Autos is offering “free credit” tells us what the cash payments are; it does not change the fact that money has time value. Present value of payments to Kangaroo Auto: PV = Down payment + C × ((1 / r) – {1 / [r(1 + r)t]}) PV = $1,000 + $300 × ((1 + .0083) – {1 / [.0083(1 + .0083)30]}) PV= $8,938.02 Present value of car at Turtle Motors: PV = price of car – discount PV = $10,000 – 1,000 PV = $9,000 Kangaroo Autos offers the best deal because it has the lower present value of costs. Est time: 01-05 23. 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 Chapter 02 - How to Calculate Present Values NPV13.65% = –$36.83 c Est time: 06-10 24. a. PVOA = C / r PVOA = $100 / .07 PVOA = $1,428.57 b. PVAD = C / r × (1 + r) PVAD = $100 / .07 × (1 + .07) PVAD = $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 For informational purposes, 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 Chapter 02 - How to Calculate Present Values 25. a. PV = C / r PV = $1 billion / .08 PV = $12.5 billion b. PV = C / (r – g) PV = $1 billion / (.08 – .04) PV = $25.0 billion c. PV = C × ((1 / r) – {1 / [r (1 + r)t]}) PV = $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)]} PV = $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. Annual compounding: Ct = PV × (1 + r)t C20 = $100 × 1.1520 C20 = $1,636.65 Continuous compounding: Ct = PV × ert C20 = $100 × e.15 × 20 C20 = $2,008.55 Est time: 01-05 27. One way to approach this problem is to solve for the present value of: Chapter 02 - How to Calculate Present Values (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, these present values must be equal, and thus we can 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, we find that r = 7.18%. Est time: 06-10 28. 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 Chapter 02 - How to Calculate Present Values 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 29. First, find the semiannual rate that is equivalent to the annual rate: 1 + r = (1 + r / 2)2 1.08 = (1 + r / 2)2 r / 2 = 1.08.5 – 1 r = .039230, or 3.9230% PV = C 0 + C × ((1 / r) – {1 / [r(1 + r)t]}) PV = $100,000 + $100,000 × ((1 / .039230) – {1 / [.039230(1 + .039230)9]}) PV = $846,147.28 Est time: 06-10 30. 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 31. a. PV = C × ((1 / r) – {1 / [r(1 + r)t]}) PV = $70,000 × ((1 / .08) – {1 / [.08(1 + .08)8]}) PV = $402,264.73 b. Year 1 2 3 4 5 6 7 8 Beginning Balance $402,264.73 364,445.90 323,601.58 279,489.70 231,848.88 180,396.79 124,828.53 64,814.81 Interest Payment $32,181.18 29,155.67 25,888.13 22,359.18 18,547.91 14,431.74 9,986.28 5,185.19 Principal Payment $37,818.82 40,844.33 44,111.87 47,640.82 51,452.09 55,568.26 60,013.72 64,814.81 Ending Balance $364,445.90 323,601.58 279,489.70 231,848.88 180,396.79 124,828.53 64,814.81 0.00 Chapter 02 - How to Calculate Present Values Est time: 06-10 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 r = 1.08 / 1.04 – 1 r = .0385, or 3.85% 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. Est time: 06-10 33. 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 34. 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) Annual retirement shortfall = 12 × ($7,500 + 1,500 – 15,000) Annual retirement shortfall = –$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]} 13.693302 = 1 / [.035(1 + .035)t] .073028 / .035 = 1.035t t = ln2.086514 / ln1.035 Chapter 02 - How to Calculate Present Values t = 21.38 years Est time: 06-10 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. a. PV = C × ((1 / r) – {1 / [r(1 + r)t]}) C = PV / ((1 / r) – {1 / [r (1 + r)t]}) C = $200,000 / ((1 / .06) – {1 / [.06(1 + .06)20]}) C = $17,436.91 b. Year Beginningof-Year Balance Year-end Interest on Balance 1 $200,000.00 $12,000.00 $17,436.91 $5,436.91 194,563.09 2 194,563.09 11,673.79 17,436.91 5,763.13 188,799.96 3 188,799.96 11,328.00 17,436.91 6,108.91 182,691.05 4 182,691.05 10,961.46 5 176,215.60 10,572.94 17,436.91 6,863.98 6 169,351.63 10,161.10 17,436.91 7,275.81 162,075.81 7 162,075.81 9,724.55 17,436.91 7,712.36 154,363.45 8 154,363.45 9,261.81 17,436.91 8,175.10 146,188.34 9 146,188.34 8,771.30 17,436.91 Total Year-end Payment 17,436.91 Amortization of Loan 6,475.45 End-of-Year Balance 176,215.60 169,351.63 137,522.73 Chapter 02 - How to Calculate Present Values 8,665.61 c. 10 137,522.73 8,251.36 17,436.91 9,185.55 128,337.19 11 128,337.19 7,700.23 17,436.91 9,736.68 118,600.51 12 118,600.51 7,116.03 17,436.91 10,320.88 108,279.62 13 108,279.62 6,496.78 17,436.91 10,940.13 97,339.49 14 97,339.49 5,840.37 17,436.91 11,596.54 85,742.95 15 85,742.95 5,144.58 17,436.91 12,292.33 73,450.61 16 73,450.61 4,407.04 17,436.91 13,029.87 60,420.74 17 60,420.74 3,625.24 17,436.91 13,811.67 46,609.07 18 46,609.07 2,796.54 17,436.91 14,640.37 31,968.71 19 31,968.71 1,918.12 17,436.91 15,518.79 16,449.92 20 16,449.92 986.99 17,436.91 16,449.92 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 Interest percent of last 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]}) PV10 = $17,436.91 × {1.06 – [1 / (.06 × 1.0610)]} PV10 = $128,337.19 0.00 Chapter 02 - How to Calculate Present Values Fraction of loan paid off = (Loan amount – PV10) / Loan amount Fraction of loan paid off = ($200,000 – 128,337.19) / $200,000 Fraction of loan paid off = .3583, or 35.83% Est time: 16-20 37. 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 38. Spreadsheet exercise Est time: 11-15 39. 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 Chapter 02 - How to Calculate Present Values 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