Unformatted text preview: Chapter 4: Net Present Value 4.1 a. b. c. d. $1,000 1.0510 = $1,628.89 $1,000 1.0710 = $1,967.15 $1,000 1.0520 = $2,653.30 Interest compounds on the I nterest already earned. Therefore, the interest earned in part c, $1,653.30, is more than double the amount earned in part a, $1,000 / 1.17 = $513.16 $2,000 / 1.1 = $1,818.18 $500 / 1.18 = $233.25 $628.89. 4.2 a. b. c. 4.3 You can make your decision by computing either the present value of the $2,000 that you can receive in ten years, or the future value of the $1,000 that you can receive now. Present value: $2,000 / 1.0810 = $926.39 Future value: $1,000 1.0810 = $2,158.93 Either calculation indicates you should take the $1,000 now. Since this bond has no interim coupon payments, its present value is simply the present value of the $1,000 that will be received in 25 years. Note: As will be discussed in the next chapter, the present value of the payments associated with a bond is the price of that bond. PV = $1,000 /1.125 = $92.30 PV = $1,500,000 / 1.0827 = $187,780.23 At a discount rate of zero, the future value and present value are always the same. Remember, FV = PV (1 + r) t. If r = 0, then the formula reduces to FV = PV. Therefore, the values of the options are $10,000 and $20,000, respectively. You should choose the second option. b. Option one: $10,000 / 1.1 = $9,090.91 Option two: $20,000 / 1.15 = $12,418.43 Choose the second option. c. Option one: $10,000 / 1.2 = $8,333.33 Option two: $20,000 / 1.25 = $8,037.55 Choose the first option. d. You are indifferent at the rate that equates the PVs of the two alternatives. You know that rate must fall between 10% and 20% because the option you would choose differs at these rates. Let r be the discount rate that makes you indifferent between the options. $10,000 / (1 + r) = $20,000 / (1 + r)5 (1 + r)4 = $20,000 / $10,000 = 2 1 + r = 1.18921 r = 0.18921 = 18.921% PV of Joneses' offer = $150,000 / (1.1)3 = $112,697.22 Since the PV of Joneses' offer is less than Smiths' offer, $115,000, you should choose Smiths' offer. a. P0 = $1,000 / 1.0820 = $214.55 a. 4.4 4.5 4.6 4.7 4.8 Answers to EndofChapter Problems B49 4.9 b. P10 = P0 (1.08)10 = $463.20 c. P15 = P0 (1.08)15 = $680.59 The $1,000 that you place in the account at the end of the first year will earn interest for six years. The $1,000 that you place in the account at the end of the second year will earn interest for five years, etc. Thus, the account will have a balance of $1,000 (1.12)6 + $1,000 (1.12)5 + $1,000 (1.12)4 + $1,000 (1.12)3 = $6,714.61 4.10 PV = $5,000,000 / 1.1210 = $1,609,866.18 4.11 a. The cost of investment is $900,000. PV of cash inflows = $120,000 / 1.12 + $250,000 / 1.122 + $800,000 / 1.123 = $875,865.52 Since the PV of cash inflows is less than the cost of investment, you should not make the investment. NPV = $900,000 + $875,865.52 = $24,134.48 NPV = $900,000 + $120,000 / 1.11 + $250,000 / 1.112 + $800,000 / 1.113 = $4,033.18 Since the NPV is still negative, you should not make the investment. b. c. 4.12 NPV = ($340,000 + $10,000) + ($100,000  $10,000) / 1.1 + $90,000 / 1.12 + $90,000 / 1.13 + $90,000 / 1.14 + $100,000 / 1.15 = $2,619.98 Since the NPV is negative, you should not buy it. If the relevant cost of capital is 9 percent, NPV = $350,000 + $90,000 / 1.09 + $90,000 / 1.092 + $90,000 / 1.093 + $90,000 / 1.094 + $100,000 / 1.095 = $6,567.93 Since the NPV is positive, you should buy it. 4.13 a. b. Profit = PV of revenue  Cost = NPV NPV = $90,000 / 1.15  $60,000 = $4,117.08 No, the firm will not make a profit. Find r that makes zero NPV. $90,000 / (1+r)5  $60,000 = $0 (1+r)5 = 1.5 r = 0.08447 = 8.447% 4.14 The future value of the decision to own your car for one year is the sum of the tradein value and the benefit from owning the car. Therefore, the PV of the decision to own the car for one year is $3,000 / 1.12 + $1,000 / 1.12 = $3,571.43 Since the PV of the roommate's offer, $3,500, is lower than the aunt's offer, you should accept aunt's offer. 4.15 a. b. c. $1.000 (1.08)3 = $1,259.71 $1,000 [1 + (0.08 / 2)]2 3 = $1,000 (1.04)6 = $1,265.32 $1,000 [1 + (0.08 / 12)]12 3 = $1,000 (1.00667)36 = $1,270.24 B50 Answers to EndofChapter Problems sg* h d. e. $1,000 e0.08 3 = $1,271.25 The future value increases because of the compounding. The account is earning interest on interest. Essentially, the interest is added to the account balance at the end of every compounding period. During the next period, the account earns interest on the new balance. When the compounding period shortens, the balance that earns interest is rising faster. $1,000 e0.12 5 = $1,822.12 $1,000 e0.1 3 = $1,349.86 $1,000 e0.05 10 = $1,648.72 $1,000 e0.07 8 = $1,750.67 12 4.16 a. b. c. d. 4.17 PV = $5,000 / [1+ (0.1 / 4)]4 = $1,528.36 4.18 Effective annual interest rate of Bank America = [1 + (0.041 / 4)]4  1 = 0.0416 = 4.16% Effective annual interest rate of Bank USA = [1 + (0.0405 / 12)]12  1 = 0.0413 = 4.13% You should deposit your money in Bank America. 4.19 The price of the consol bond is the present value of the coupon payments. Apply the perpetuity formula to find the present value. PV = $120 / 0.15 = $800 4.20 Quarterly interest rate = 12% / 4 = 3% = 0.03 Therefore, the price of the security = $10 / 0.03 = $333.33 4.21 The price at the end of 19 quarters (or 4.75 years) from today = $1 / (0.15 4) = $26.67 The current price = $26.67 / [1+ (.15 / 4)]19 = $13.25 4.22 a. b. c. $1,000 / 0.1 = $10,000 $500 / 0.1 = $5,000 is the value one year from now of the perpetual stream. Thus, the value of the perpetuity is $5,000 / 1.1 = $4,545.45. $2,420 / 0.1 = $24,200 is the value two years from now of the perpetual stream. Thus, the value of the perpetuity is $24,200 / 1.12 = $20,000. 4.23 The value at t = 8 is $120 / 0.1 = $1,200. Thus, the value at t = 5 is $1,200 / 1.13 = $901.58. 4.24 P = $3 (1.05) / (0.12  0.05) = $45.00 4.25 P = $1 / (0.1  0.04) = $16.67 4.26 The first cash flow will be generated 2 years from today. The value at the end of 1 year from today = $200,000 / (0.1  0.05) = $4,000,000. Thus, PV = $4,000,000 / 1.1 = $3,636,363.64. 4.27 A zero NPV  $100,000 + $50,000 / r = 0  r = 0.5 Answers to EndofChapter Problems B51 4.28 Apply the NPV technique. Since the inflows are an annuity you can use the present value of an annuity factor. 8 NPV = $6,200 + $1,200 0.1 = $6,200 + $1,200 (5.3349) = $201.88 Yes, you should buy the asset. 4.29 Use an annuity factor to compute the value two years from today of the twenty payments. Remember, the annuity formula gives you the value of the stream one year before the first payment. Hence, the annuity factor will give you the value at the end of year two of the 20 stream of payments. Value at the end of year two = $2,000 0.08 = $2,000 (9.8181) = $19,636.20 The present value is simply that amount discounted back two years. PV = $19,636.20 / 1.082 = $16,834.88 4.30 The value of annuity at the end of year five 15 = $500 0.15 = $500 (5.84737) = $2,923.69 The present value = $2,923.69 / 1.125 = $1,658.98 4.31 The easiest way to do this problem is to use the annuity factor. The annuity factor must be equal to $12,800 / $2,000 = 6.4; remember PV =C Atr. The annuity factors are in the appendix to the text. To use the factor table to solve this problem, scan across the row labeled 10 years until you find 6.4. It is close to the factor for 9%, 6.4177. Thus, the rate you will receive on this note is slightly more than 9%. You can find a more precise answer by interpolating between nine and ten percent. 10% a r 9% b 6.1446 c 6.4 d 6.4177 By interpolating, you are presuming that the ratio of a to b is equal to the ratio of c to d. (9  r ) / (9  10) = (6.4177  6.4 ) / (6.4177  6.1446) r = 9.0648% The exact value could be obtained by solving the annuity formula for the interest rate. Sophisticated calculators can compute the rate directly as 9.0626%. B52 Answers to EndofChapter Problems 4.32 a. 1.075]. b. The annuity amount can be computed by first calculating the PV of the $25,000 which you need in five years. That amount is $17,824.65 [= $25,000 / Next compute the annuity which has the same present value. 5 $17,824.65 = C 0.07 $17,824.65 = C (4.1002) C = $4,347.26 Thus, putting $4,347.26 into the 7% account each year will provide $25,000 five years from today. The lump sum payment must be the present value of the $25,000, i.e., $25,000 / 1.075 = $17,824.65 The formula for future value of any annuity can be used to solve the problem (see footnote 14 of the text). 4.33 The amount of loan is $120,000 0.85 = $102,000. 20 C 0.10 = $102,000 The amount of equal installments is 20 C = $102,000 / 0.10 = $102,000 / 8.513564 = $11,980.88
36 4.34 The present value of salary is $5,000 0.01 = $150,537.53 The present value of bonus is $10,000 0.1268 = $23,740.42 (EAR = 12.68% is used since bonuses are paid annually.) The present value of the contract = $150,537.53 + $23,740.42 = $174,277.94
3 4.35 The amount of loan is $15,000 0.8 = $12,000. 48 C 0.0067 = $12,000 The amount of monthly installments is 48 C = $12,000 / 0.0067 = $12,000 / 40.96191 = $292.96 4.36 Option one: This cash flow is an annuity due. To value it, you must use the aftertax amounts. The aftertax payment is $160,000 (1  0.28) = $115,200. Value all except the first payment using the standard annuity formula, then add back the first payment of $115,200 to obtain the value of this option. 30 Value = $115,200 + $115,200 0.10 = $115,200 + $115,200 (9.4269) = $1,201,178.88 Option two: This option is valued similarly. You are able to have $446,000 now; this is already on an aftertax basis. You will receive an annuity of $101,055 for each of the next thirty years. Those payments are taxable when you receive them, so your aftertax payment is $72,759.60 [= $101,055 (1  0.28)]. 30 Value = $446,000 + $72,759.60 0.10 = $446,000 + $72,759.60 (9.4269) = $1,131,897.47 Since option one has a higher PV, you should choose it. Answers to EndofChapter Problems B53 4.37 The amount of loan is $9,000. The monthly payment C is given by solving the equation: 60 C 0.008 = $9,000 C = $9,000 / 47.5042 = $189.46 In October 2000, Susan Chao has 35 (= 12 5  25) monthly payments left, including the one due in October 2000. Therefore, the balance of the loan on November 1, 2000 34 = $189.46 + $189.46 0.008 = $189.46 + $189.46 (29.6651) = $5,809.81 Thus, the total amount of payoff = 1.01 ($5,809.81) = $5,867.91 4.38 Let r be the rate of interest you must earn. $10,000(1 + r)12 = $80,000 (1 + r)12 = 8 r = 0.18921 = 18.921% 4.39 First compute the present value of all the payments you must make for your children's education. The value as of one year before matriculation of one child's education is 4 $21,000 0.15 = $21,000 (2.8550) = $59,955. This is the value of the elder child's education fourteen years from now. It is the value of the younger child's education sixteen years from today. The present value of these is PV = $59,955 / 1.1514 + $59,955 / 1.1516 = $14,880.44 You want to make fifteen equal payments into an account that yields 15% so that the present value of the equal payments is $14,880.44. 15 Payment = $14,880.44 / 0.15 = $14,880.44 / 5.8474 = $2,544.80 4.40 The NPV of the policy is 3 3 NPV = $750 0.06  $800 0.06 / 1.063 + $250,000 / [(1.066) (1.0759)] = $2,004.76  $1,795.45 + $3,254.33 = $545.88 Therefore, you should not buy the policy. 4.41 The NPV of the lease offer is 9 NPV = $120,000  $15,000  $15,000 0.08  $25,000 / 1.0810 = $105,000  $93,703.32  $11,579.84 = $283.16 Therefore, you should not accept the offer. 4.42 This problem applies the growing annuity formula. The first payment is $50,000(1.04)2(0.02) = $1,081.60. PV = $1,081.60 [1 / (0.08  0.04)  {1 / (0.08  0.04)}{1.04 / 1.08}40] = $21,064.28 This is the present value of the payments, so the value forty years from today is $21,064.28 (1.0840) = $457,611.46 B54 Answers to EndofChapter Problems 4.43 Use the discount factors to discount the individual cash flows. Then compute the NPV of the project. Notice that the four $1,000 cash flows form an annuity. You can still use the factor tables to compute their PV. Essentially, they form cash flows that are a six year annuity less a two year annuity. Thus, the appropriate annuity factor to use with them is 2.6198 (= 4.3553  1.7355). Year Cash Flow Factor PV 1 $700 0.9091 $636.37 2 900 0.8264 743.76 3 1,000 4 1,000 2.6198 2,619.80 5 1,000 6 1,000 7 1,250 0.5132 641.50 8 1,375 0.4665 641.44 Total $5,282.87 NPV = $5,000 + $5,282.87 = $282.87 Purchase the machine. 4.44 Weekly inflation rate = 0.039 / 52 = 0.00075 Weekly interest rate = 0.104 / 52 = 0.002 PV = $5 [1 / (0.002  0.00075)] {1 [(1 + 0.00075) / (1 + 0.002)]52 30} = $3,429.38 4.45 Engineer: 4 NPV = $12,000 0.05 + $20,000 / 1.055 + $25,000 / 1.056  $15,000 / 1.057
25  $15,000 / 1.058 + $40,000 0.05 / 1.058 = $352,533.35 Accountant: NPV = $13,000 4 0.05 + $31,000 3005 / 1.054 0. = $345,958.81 Become an engineer. After your brother announces that the appropriate discount rate is 6%, you can recalculate the NPVs. Calculate them the same way as above except using the 6% discount rate. Engineer NPV = $292,419.47 Accountant NPV = $292,947.04 Your brother made a poor decision. At a 6% rate, he should study accounting. 4.46 Since Goose receives his first payment on July 1 and all payments in one year intervals from July 1, the easiest approach to this problem is to discount the cash flows to July 1 then use the six month discount rate (0.044) to discount them the additional six months. PV = $875,000 / (1.044) + $650,000 / (1.044)(1.09) + $800,000 / (1.044)(1.092) + $1,000,000 / (1.044)(1.093) + $1,000,000/(1.044)(1.094) + $300,000 / (1.044)(1.095) Answers to EndofChapter Problems B55 17 10 + $240,000 0.09 / (1.044)(1.095) + $125,000 0.09 / (1.044)(1.0922) = $5,051,150 Remember that the use of annuity factors to discount the deferred payments yields the value of the annuity stream one period prior to the first payment. Thus, the annuity factor applied to the first set of deferred payments gives the value of those payments on July 1 of 1989. Discounting by 9% for five years brings the value to July 1, 1984. The use of the six month discount rate (4.4%) brings the value of the payments to January 1, 1984. Similarly, the annuity factor applied to the second set of deferred payments yields the value of those payments in 2006. Discounting for 22 years at 9% and for six months at 4.4% provides the value at January 1, 1984. The equivalent fiveyear, annual salary is the annuity that solves: $5,051,150 = C 0.09 C = $5,051,150/3.8897 C = $1,298,596
5 The student must be aware of possible rounding errors in this problem. The difference between 4.4% semiannual and 9.0% and for six months at 4.4% provides the value at January 1, 1984. 4.47 PV 4.48 NPV = $10,000 + ($35,000 + $3,500) [1 / (0.12  0.04)] [1  (1.04 / 1.12) 25 ] = $415,783.60 = $40,000 + $10,000 [1 / (0.10  0.07)] [1  (1.07 / 1.10)5 ] = $3,041.91 Revise the textbook. 4.49 The amount of the loan is $400,000 (0.8) = $320,000 360 The monthly payment is C = $320,000 / 0.0067 = $ 2,348.10 0. Thirty years of payments $ 2,348.10 (360) = $ 845,316.00 Eight years of payments $2,348.10 (96) = $225,417.60 The difference is the balloon payment of $619,898.40 4.50 The lease payment is an annuity in advance 23 C + C 0.01 = $4,000 C (1 + 20.4558) = $4,000 C = $186.42 4.51 The effective annual interest rate is [ 1 + (0.08 / 4) ] 4 1 = 0.0824 The present value of the tenyear annuity is 10 PV = 900 0.0824 = $5,974.24 Four remaining discount periods PV = $5,974.24 / (1.0824) 4 = $4,352.43 B56 Answers to EndofChapter Problems 4.52 The present value of Ernie's retirement income 20 PV = $300,000 0.07 / (1.07) 30 = $417,511.54 The present value of the cabin PV = $350,000 / (1.07) 10 = $177,922.25 The present value of his savings 10 PV = $40,000 0.07 = $280,943.26 In present value terms he must save an additional $313,490.53 In future value terms FV = $313,490.53 (1.07) 10 = $616,683.32 He must save 20 C = $616.683.32 / 0.07 = $58,210.54 Answers to EndofChapter Problems B57 ...
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 Finance, Time Value Of Money, Interest, Net Present Value

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