VIEWS: 130 PAGES: 19 POSTED ON: 5/5/2011
Time Value of Money Problems 1. What is the PV of $100 received in: a. Year 10 at a discount rate of 1 percent. b. Year 10 at a discount rate of 13 percent. c. Year 15 at a discount rate of 25 percent. d. Each of years 1 through 3 at a discount rate of 12 percent. a. PV = $100/1.0110 = $90.53 b. PV = $100/1.1310 = $29.46 c. PV = $100/1.2515 = $ 3.52 d. PV = $100/1.12 + $100/1.122 + $100/1.123 = $240.18 a b c d N = 10 10 15 3 I= 1 13 25 12 Cpt. PV = 90.53 29.46 3.52 240.18 Pmt = 0 0 0 100 FV = 100 100 100 0 2. For each of the following, compute the future value: Present Value Years Interest Rate Future Value $1,000 4 10% $2,500 6 12.25% N= 4 6 I = 10% 12.25% PV = 1,000 2,500 Pmt = 0 0 Cpt. FV = 1,464.10 5,001.01 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 1 3. For each of the following, compute the interest rate: Present Value Years Interest Rate Future Value $5,500 8 $12,000 $7,500 15 $60,000 N= 8 15 Cpt. I = 10.2433% 14.8698% PV = -5,500 -7,500 Pmt = 0 0 FV = $12,000 $60,000 4. For each of the following, compute the number of years: Present Value Years Interest Rate Future Value $300 5% 450 $27,500 10.125% $60,000 Cpt. N = 8.3104 8.0891 I= 5 10.125 PV = -300 -27,500 Pmt = 0 0 FV = 450 60,000 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 2 5. A factory costs $800,000. You believe that it will produce a cash flow of $170,000 a year for 10 years. If the opportunity cost of capital is 14 percent, what is the NPV of the factory? What will the factory be worth at the end of five years? The present value of the 10-year stream of cash inflows is: 1 1 PV $170,000 10 $886,739.66 0.14 0.14 (1.14) Thus: NPV = –$800,000 + $886,739.66 = +$86,739.66 CF0 = -800,000 I = 14 CF1 = 170,000 F1= 10 Cpt NPV = 86,739.66 At the end of five years, the factory’s value will be the present value of the five remaining $170,000 cash flows: 1 1 PV $170,000 5 $583,623.76 0.14 0.14 (1.14) N= 5 I = 14 Cpt. PV = 583,623.76 Pmt = 170,000 FV = 0 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 3 6. A machine costs $380,000 and it is expected to produce the following cash flows. Year 1 2 3 4 5 6 7 8 9 10 CF ($000S) 50 57 75 80 85 92 92 80 68 50 If the cost of capital is 12 percent, what is the machine’s NPV? 10 Ct $50,000 $57,000 $75,000 $80,000 $85,000 NPV t $380,000 t 0 (1.12) 1.12 1.122 1.123 1.124 1.125 $92,000 $92,000 $80,000 $68,000 $50,000 $23,696.15 1.126 1.127 1.128 1.129 1.1210 Or CF0 -380 I= 12% CF1 50 Cpt NPV = 23.69615 CF2 57 CF3 75 CF4 80 CF5 85 CF6 92 CF7 92 CF8 80 CF9 68 CF10 50 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 4 7. Mike Polanski is 30 years of age and his salary next year will be $40,000. Mike forecasts that his salary will increase at a steady rate of 5 percent per year until his retirement at age 60. a. If the discount rate is 8 percent, what is the PV of these future salary payments? b. If Mike saves 5 percent of his salary each year and invests these savings at an interest rate of 8 percent, how much will he have saved by age 60? c. If Mike plans to spend these savings in even amounts over the subsequent 20 years, how much can he spend each year? a. Let St = salary in year t 30 St 30 40,000 (1.05)t 1 30 (40,000/1. 05) 30 38,095.24 PV t 1 (1.08)t t 1 (1.08)t t 1 (1.08 / 1.05) t t 1 (1.0286) t 1 1 38,095.24 30 $760,379.21 0.0286 0.0286 (1.0286) PV(salary) x 0.05 = $38,018.96 Future value = $38,018.96 x (1.08)30 = $382,571.75 1 1 PV C t r r (1 r) 1 1 $382,571.75 C 20 0.08 0.08 (1.08) 1 1 C $382,571.75 $38,965.78 0.08 0.08 (1.08)20 Or a b c N = 30 30 20 I = 2.86 8 8 Cpt. PV = 760,379.21 (0.05)(760,379.21) = 38,018.96 382,571.75 Pmt = 38,095.24 0 Cpt Pmt = 38,965.78 FV = 0 Cpt FV = 382,571.75 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 5 8. A factory costs $400,000. It will produce a cash inflow of $100,000 in year 1, $200,000 in year 2, and $300,000 in year 3. The opportunity cost of capital is 12 percent. Calculate the NPV Period Present Value 0 400,000.00 1 +100,000/1.12 = + 89,285.71 2 2 +200,000/1.12 = +159,438.78 3 3 +300,000/1.12 = +213,534.07 Total = NPV = $62,258.56 CF0 -400,000 I = 12% CF1 100,000 Cpt NPV = $62,258.56 CF2 200,000 CF3 300,000 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 6 9. Halcyon Lines is considering the purchase of new bulk carrier for $8 million. The forecasted revenues are $5 million a year and operating costs are $4 million. A major refit costing $2 million will be required after both the fifth and tenth years. After 15 years, the ship is expected to be sold for scrap at $1.5 million. If the discount rate is 8 percent, what is the ship’s NPV? We can break this down into several different cash flows, such that the sum of these separate cash flows is the total cash flow. Then, the sum of the present values of the separate cash flows is the present value of the entire project. (All dollar figures are in millions.) Cost of the ship is $8 million PV = $8 million Revenue is $5 million per year, operating expenses are $4 million. Thus, operating cash flow is $1 million per year for 15 years. 1 1 PV $1million 15 $8.559 million 0.08 0.08 (1.08) Major refits cost $2 million each, and will occur at times t = 5 and t = 10. PV = ($2 million)/1.085 + ($2 million)/1.0810 = $2.288 million Sale for scrap brings in revenue of $1.5 million at t = 15. PV = $1.5 million/1.0815 = $0.473 million Adding these present values gives the present value of the entire project: NPV = $8 million + $8.559 million $2.288 million + $0.473 million NPV = $1.256 million CF0 = -8 I=8 CF1 = 1 F1= 4 Cpt NPV =-1.2552 CF2 = 1 – 2 = -1 F2 = 1 CF3 = 1 F3= 4 CF4 = 1 – 2 = -1 F4 = 1 CF5 = 1 F5= 4 CF6 = 1 + 1.5 = 2.5 F6 = 1 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 7 10. As winner of a breakfast cereal competition, you can choose one of the following prizes: a. $100,000 now. b. $180,000 at the end of 5 years. c. $11,400 a year forever. d. $19,000 for each of 10 years. e. $6,500 next year and increasing thereafter by 5 percent a year forever. If the interest rate is 12 percent, which is the most valuable prize? a. PV = $100,000 b. PV = $180,000/1.125 = $102,137 c. PV = $11,400/0.12 = $95,000 1 1 d. PV $19,000 10 $107,354 0.12 0.12 (1.12) e. PV = $6,500/(0.12 0.05) = $92,857 Prize (d) is the most valuable because it has the highest present value. a b c d N= 0 5 10 I = 12 12 12 Cpt. PV = 100,000 102,137 11,400/0.12 = 95,000 107,354 Pmt = 0 0 19,000 FV = 100,000 180,000 0 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 8 11. Siegfried Basset is 65 years of age and has a life expectancy of 12 more years. He wishes to invest $20,000 in an annuity that will make a level payment at the end of each year until his death. If the interest rate is 8 percent, what income can Mr. Basset expect to receive each year? Mr. Basset is buying a security worth $20,000 now. That is its present value. The unknown is the annual payment. Using the present value of an annuity formula, we have: 1 1 PV C t r r (1 r) 1 1 $20,000 C 12 0.08 0.08 (1.08) 1 1 C $20,000 $2,653.90 0.08 0.08 (1.08)12 N = 12 I= 8 PV = 20,000 Cpt. Pmt = 2,653.90 FV = 0 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 9 12. David and Helen Zhang are saving to buy a boat at the end of five years. If the boat costs $20,000 and they can earn 10 percent a year on their savings, how much do they need to put aside at the end of years 1 through 5? Assume the Zhangs will put aside the same amount each year. One approach to solving this problem is to find the present value of the cost of the boat and then equate that to the present value of the money saved. From this equation, we can solve for the amount to be put aside each year. PV(boat) = $20,000/(1.10)5 = $12,418 1 1 PV(savings) = Annual savings 5 0.10 0.10 (1.10) Because PV(savings) must equal PV(boat): 1 1 Annual savings 5 $12,418 0.10 0.10 (1.10) 1 1 Annual savings $12,418 5 $3,276 0.10 0.10 (1.10) Another approach is to find the value of the savings at the time the boat is purchased. Because the amount in the savings account at the end of five years must be the price of the boat, or $20,000, we can solve for the amount to be put aside each year. If x is the amount to be put aside each year, then: x(1.10)4 + x(1.10)3 + x(1.10)2 + x(1.10)1 + x = $20,000 x(1.464 + 1.331 + 1.210 + 1.10 + 1) = $20,000 x(6.105) = $20,000 x = $ 3,276 Or N= 5 I = 10 PV = 0 Cpt. Pmt = 3,275.95 FV = 20,000 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 10 13. Kangaroo Autos is offering free credit on a new $10,000 car. You pay $1,000 down and then $300 a month for the next 30 months. Turtle Motors next door does not offer free credit but will give you $1,000 off the list price. If the rate of interest is 10 percent a year, which company is offering the better deal? 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. A 10 percent annual rate of interest is equivalent to a monthly rate of 0.83 percent: rmonthly = rannual /12 = 0.10/12 = 0.0083 = 0.83% The present value of the payments to Kangaroo Autos is: 1 1 $1,000 $300 30 $8,938 0.0083 0.0083 (1.0083) A car from Turtle Motors costs $9,000 cash. Therefore, Kangaroo Autos offers the better deal, i.e., the lower present value of cost. Or N = 30 I = 10/12 = 0.8333 Cpt. PV = 7,934.11 7,934.11 + 1,000 = 8,934.11 Pmt = 300 FV = 0 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 11 14. You are building an office building and construction will require two years. The contractor requires a $120,000 down payment now and commitment of the land with a market value of $50,000. The contractor will be paid $100,000 in 1 year and a final payment of $100,000 at the completion of construction in 2 years. Your real estate advisor estimates the office building will be worth $420,000 when completed. What is the NPV if the cost of capital is 5 percent, 10 percent, and 15 percent? Draw a NPV profile. At what rate would the NPV be zero? Check your answer. Time T=0 T=1 T=2 Land -50,000 Construction -120,000 -100,000 -100,000 Payoff 420,000 Total CFs -170,000 -100,000 320,000 The NPVs are: $100,000 $320,000 at 5 percent NPV $170,000 $25,011 1.05 (1.05) 2 $100,000 320,000 at 10 percent NPV $170,000 $3,554 1.10 (1.10) 2 $100,000 320,000 at 15 percent NPV $170,000 $14,991 1.15 (1.15) 2 The figure below shows that the project has zero NPV at about 11 percent. As a check, NPV at 11 percent is: $100,000 320,000 NPV $170,000 $371 1.11 (1.11) 2 CF0 -170,000 I= 5% 10% 15% CF1 -100,000 Cpt NPV = $25,011.34 $3,553.72 -$14,990.55 CF2 320,000 30 NPV 20 10 0 -10 -20 0.05 0.10 0.15 Rate of Interest b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 12 15. You have just read an advertisement stating, “Pay us $100 a year for ten years and we will pay you $100 a year thereafter in perpetuity.” If this is a fair deal, what is the rate of interest? 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, 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: 1 1 PV $100 10 r (r) (1 r) The present value, as of year 10, of $100 per year forever, with the first payment in year 11, is: PV10 = $100/r At t = 0, the present value of PV10 is: 1 $100 PV 10 (1 r) r Equating these two expressions for present value, we have: 1 1 1 $100 $100 10 10 r (r) (1 r) (1 r) r Using trial and error or algebraic solution, we find that r = 7.18%. b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 13 16. Which would you prefer? a. An investment paying interest of 12 percent compounded annually. b. An investment paying interest of 11.7 percent compounded semiannually. c. An investment paying 11.5 percent compounded continuously. Assume the amount invested is one dollar. Let A represent the investment at 12 percent, compounded annually. Let B represent the investment at 11.7 percent, compounded semiannually. Let C represent the investment at 11.5 percent, compounded continuously. After one year: FVA = $1 (1 + 0.12)1 = $1.1200 FVB = $1 (1 + 0.0585) 2 = $1.1204 (0.115 1) FVC = $1 e = $1.1219 After five years: FVA = $1 (1 + 0.12)5 = $1.7623 FVB = $1 (1 + 0.0585) 10 = $1.7657 (0.115 5) FVC = $1 e = $1.7771 After twenty years: FVA = $1 (1 + 0.12)20 = $9.6463 FVB = $1 (1 + 0.0585) 40 = $9.7193 FVC = $1 e(0.115 20) = $9.9742 The preferred investment is C. b c c Nom 11.7 Nom 11.5 Nom 11.5 Cpt. Eff. 12.0422% Cpt. Eff. 12.185% Cpt. Eff. 12.186% C/Y 2 C/Y 365 C/Y 730 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 14 17. Fill in the blanks in the following table: Nominal Inflation Rate Real Interest Rate Interest Rate 6.00% 1.00% 10.00% 12.00% 9.00% 3.00% 1 + rnominal = (1 + rreal) (1 + inflation rate) Nominal Inflation Rate Real Interest Rate Interest Rate 6.00% 1.00% 4.95% 23.20% 10.00% 12.00% 9.00% 5.83% 3.00% 1.06 = (1 + rreal)(1.01) rreal = .0495 1 + rnominal = (1.1)(1.12) rnominal = 0.2320 1.09 = (1.03) (1 + inflation rate) inflation rate = 0.0583 18. A leasing contract calls for an immediate payment of $100,000 and nine subsequent $100,000 semiannual payments at six-month intervals. What is the PV of these payments if the discount rate is 8 percent? Because the cash flows occur every six months, we use a six-month discount rate, here 8%/2, or 4%. Thus: 1 1 PV $100,000 $100,000 9 $843,533 0.04 0.04 (1.04) N= 9 I = 8/2 = 4 Cpt. PV = 743,533 743,533 + 100,000 =843,533 Pmt = 100,000 FV = 0 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 15 19. You estimate that by the time you retire in 35 years, you will have accumulated savings of $2 million. If the interest rate is 8 percent and you live 15 years after retirement, what annual level of expenditure will these savings support? Unfortunately, inflation will eat into the value of your retirement income. Assume a 4 percent inflation rate and work out a spending program for your retirement that will allow you to maintain a level real expenditure during retirement. This is an annuity problem with the present value of the annuity equal to $2 million (as of your retirement date), and the interest rate equal to 8 percent, with 15 time periods. Thus, your annual level of expenditure (C) is determined as follows: 1 1 PV C t N = 15 r r (1 r) I= 8 PV = 2,000,000 1 1 $2,000,000 C 15 or Cpt. Pmt = 233,659 0.08 0.08 (1.08) FV = 0 1 1 C $2,000,000 $233,659 0.08 0.08 (1.08)15 With an inflation rate of 4 percent per year, we will still accumulate $2 million as of our retirement date. However, because we want to spend a constant amount per year in real terms (R, constant for all t), the nominal amount (C t ) must increase each year. For each year t: R = C t /(1 + inflation rate)t Therefore: PV [all C t ] = PV [all R (1 + inflation rate)t] = $2,000,000 (1 0.04)1 (1 0.04)2 (1 0.04)15 R . . . $2,000,000 (1 0.08) (1 0.08) (1 0.08)15 1 2 R [0.9630 + 0.9273 + . . . + 0.5677] = $2,000,000 R 11.2390 = $2,000,000 R = $177,952 Thus C1 = ($177,952 1.04) = $185,070, C2 = $192,473, etc. N = 15 Remember: I = 3.8462 1 + rnominal = (1 + rreal) (1 + inflation rate) PV = 2,000,000 1.08 = (1 + rreal)(1.04) rreal = .038462 Cpt. Pmt = 177,952 FV = 0 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 16 20. You are considering the purchase of an apartment complex that currently generates a net cash flow of $400,000 per year. You normally demand a 10 percent rate of return on such investments. Future cash flows are expected to grow with inflation at 4 percent year from today’s level. How much would you be willing to pay for the complex if it: a. Will produce cash flows forever? b. Will have to be torn down in 20 years? Assume that the site will be worth $5 million at the time of demolition costs. (The $5 million includes 20 years’ inflation.) Now calculate the real discount rate corresponding to the 10 percent nominal rate. Redo the calculations for part (a) and (b) using real cash flows. (Your answers should not change.) First, with nominal cash flows: a. The nominal cash flows form a growing perpetuity at the rate of inflation, 4%. Thus, the cash flow in one year will be $416,000 and: PV = $416,000/(0.10 – 0.04) = $6,933,333 b. The nominal cash flows form a growing annuity for 20 years, with an additional payment of $5 million at year 20: 1 1 ( 1 g)t $5,000,000 PV C1 t (r-g) (r g) ( 1 r) ( 1 r)t 1 1 ( 1.04 )20 $5,000,000 $ 416,000 20 ( 0.10-0.04 ) ( 0.10-0.04 ) ( 1.10 ) ( 1.10 )20 $5,418,389 Second, with real cash flows: a. Here, the real cash flows are $400,000 per year in perpetuity, and we can find the real rate (r) by solving the following equation: (1 + 0.10) = (1 + r)(1 + 0.04) r = 0.05769 = 5.769% PV = $400,000/0.057692 = $6,933,333 b. Now, the real cash flows are $400,000 per year for 20 years and $5 million (nominal) in 20 years. In real terms, the $5 million dollar payment is: $5,000,000/(1.04)20 = $2,281,935 Thus, the present value of the project is: 1 1 $ 2,281,935 PV $ 400,000 20 20 $5,418,510 ( 0.05769 ) ( 0.05769 )(1.05769 ) ( 1.05769 ) [As noted in the statement of the problem, the answers agree, to within rounding errors.] N = 20 I = 5.7692 Cpt. PV = 5,418,389 Pmt = 400,000 FV = 2,281,935 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 17 21. Vernal Pool, a self-employed herpetologist, wants to put aside a fixed fraction of her annual income as savings for retirement. Ms. Pool is now 40 years old and makes $40,000 a year. She expects her income to increase by 2 percentage points over inflation (e. g., 4 percent inflation means a 6 percent increase in income). She wants to accumulate $500,000 in real terms to retire at age 70. What fraction of her income does she need to set aside? Assume her retirement funds are conservatively invested at an expected real rate of return of 5 percent a year. Ignore taxes. Let x be the fraction of Ms. Pool’s salary to be set aside each year. At any point in the future, t, her real income will be: ($40,000)(1 + 0.02) t The real amount saved each year will be: (x)($40,000)(1 + 0.02) t The present value of this amount is: x $40,0001 0.02t 1 0.05t Ms. Pool wants to have $500,000, in real terms, 30 years from now. The present value of this amount (at a real rate of 5 percent) is: $500,000/(1 + 0.05)30 Thus: $500,000 30 x $40,0001.02 t 1.0530 t 1 1.05t $500,000 x 30 $40,0001.02 t 1.0530 t 1 1.05t $115,688.72 = (x)($790,012.82) x = 0.146 b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 18 22. You own a pipeline which will generate a $2 million cash flow over the coming year. The pipeline’s operating costs are negligible, and it is expected to last for a very long time. Unfortunately, the volume of oil shipped is declining, and cash flows are expected to decline by 4 percent per year. The discount rate is 10 percent. a. What is the PV of the pipeline’s cash flows if its cash flows are assumed to last forever? b. What is the PV of the cash flows if the pipeline is scrapped after 20 years? a. This calls for the growing perpetuity formula with a negative growth rate (g = –0.04): $ 2 million $ 2 million PV $14 .29 million 0.10 ( 0.04 ) 0.14 b. The pipeline’s value at year 20 (i.e., at t = 20), assuming its cash flows last forever, is: C21 C ( 1 g)20 PV20 1 rg rg With C1 = $2 million, g = –0.04, and r = 0.10: ($2 million) ( 1 0.04 )20 $0.884 million PV20 $6.314 million 0.14 0.14 Next, we convert this amount to PV today, and subtract it from the answer to Part (a): $ 6.314 million PV $14 .29 million $13.35 million ( 1.10 )20 Most of these problems and part of the solutions are from Chapter 3 in Principles of Corporate Finance by Brealey, Myers, and Allen 8th edition. Part of the solutions were generated by Danny Ervin b0deb235-ce52-462e-bbba-a3a1066034ce.doc Page 19