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Problems Problem 4-12 Problem 4-13 Problem 4-14 Problem 4-15 Problem 4-23 Problem 4-24 Problem 4-27 Problem 4-28 Problem 4-29 Problem 4-30 Problem 4-32 Problem 4-34 Problem 4-35 Problem 4-36 Problem 4-44 Problem 4-45 Problem 4-46 Problem 4-12 You have just received a windfall from an investment you made in a friend’s business. He will be paying you $10,000 at the end of this year, $20,000 at the end of the following year, and $30,000 at the end of the year after that (three years from today). The interest rate is 3.5% per year. Interest Rate: 3.50% Year 0 1 2 3 Cash flow: $10,000.00 $20,000.00 $30,000.00 a. What is the present value of your windfall? Present Value Using formulas 55,390.33 Using Excel functions 55,390.33 b. What is the future value of your windfall in three years (on the date of the last payment)? Future Value Using formulas 61,412.25 Using Excel functions 61,412.25 Problem 4-13 You have a loan outstanding. It requires making three annual payments at the end of the next three years of $1000 each. Your bank has offered to allow you to skip making the next two payments in lieu of making one large payment at the end of the loan’s term in three years. If the interest rate on the loan is 5%, what final payment will the bank require you to make so that it is indifferent between the two forms of payment? Interest Rate: 5.00% Year 0 1 2 Cash flow: $1,000.00 $1,000.00 Future value ments at the end of the you to skip making the nd of the loan’s term in nt will the bank require ment? 3 $1,000.00 3,152.50 Problem 4-14 You have been offered a unique investment opportunity. If you invest $10,000 today, you will receive $500 one year from now, $1500 two years from now, and $10,000 ten years from now. Interest rate 6.00% NPV (2,609.36) Year 0 1 2 3 4 5 6 7 8 9 10 Cash flow: ($10,000.00) $500.00 $1,500.00 $0.00 $0.00 $0.00 $0.00 $0.00 $0.00 $0.00 $10,000.00 Discounted value (10,000.00) 471.70 1,334.99 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5,583.95 NPV (2,609.36) a. What is the NPV of the opportunity if the interest rate is 6% per year? Should you take the opportunity? At an interest rate of 6.00% the NPV is (2,609.36) so reject the project. b. What is the NPV of the opportunity if the interest rate is 2% per year? Should you take it now? At an interest rate of 2.00% the NPV is 135.43 so accept the project. The relationship is illustrated by the chart below: Data for NPV profile: NPV 0% 2,000.00 1% 1,018.36 2% 135.43 3% -659.73 4% -1,376.75 3,000.00 5% -2,024.13 2,000.00 6% -2,609.36 7% -3,139.06 1,000.00 NPV 0.00 0% 1% 2% 3% 4% 5% 6% 7% NPV -1,000.00 -2,000.00 -3,000.00 -4,000.00 Discount rate Problem 4-15 Marian Plunket owns her own business and is considering an investment. If she undertakes the investment, it will pay $4000 at the end of each of the next three years. The opportunity requires an initial investment of $1000 plus an additional investment at the end of the second year of $5000. What is the NPV of this opportunity if the interest rate is 2% per year? Should Marian take it? Interest rate 2.00% NPV using Excel function 5,729.69 Year 0 1 2 Cash flow: ($1,000.00) $4,000.00 ($1,000.00) Discounted cash flow (1,000.00) 3,921.57 (961.17) NPV the "long" way 5,729.69 Marion should take the investment. an investment. If she each of the next three 000 plus an additional at is the NPV of this take it? 3 $4,000.00 3,769.29 Problem 4-23 Your grandmother has been putting $1000 into a savings account on every birthday since your first (that is, when you turned 1). The account pays an interest rate of 3%. How much money will be in the account on your 18th birthday immediately after your grandmother makes the deposit on that birthday? PMT $1,000.00 Rate 3.00% Years 18 FV $23,414.44 Problem 4-24 A rich relative has bequeathed you a growing perpetuity. The first payment will occur in a year and will be $1000. Each year after that, you will receive a payment on the anniversary of the last payment that is 8% larger than the last payment. This pattern of payments will go on forever. If the interest rate is 12% per year, a. What is today’s value of the bequest? First Payment $1,000.00 Growth Rate 8.00% Interest Rate 12.00% Value $25,000.00 b. What is the value of the bequest immediately after the first payment is made? 2 ways: Value This Year Times 1 + Growth Rate $27,000.00 - OR - Second Payment $1,080.00 Growth Rate 8.00% Interest Rate 12.00% Value $27,000.00 The first payment will will receive a payment n the last payment. This 12% per year, st payment is made? Problem 4-27 Your oldest daughter is about to start kindergarten at a private school. Tuition is $10,000 per year, payable at the beginning of the school year. You expect to keep your daughter in private school through high school. You expect tuition to increase at a rate of 5% per year over the 13 years of her schooling. What is the present value of the tuition payments if the interest rate is 5% per year? Look at 4-28 for an explanation. In this problem, the difference is that the first payment occurs today. (If you want to look at this like 4-28, change that sheet just a little and you can see the balance go to zero.) Payment $10,000.00 Years 13 Value $130,000.00 This is true only because the growth rate equals the interest rate. Problem 4-28 A rich aunt has promised you $5000 one year from today. In addition, each year after that, she has promised you a payment (on the anniversary of the last payment) that is 5% larger than the last payment. She will continue to show this generosity for 20 years, giving a total of 20 payments. If the interest rate is 5%, what is her promise worth today? Discount rate 5.00% Number of payments 20 Payment amount $5,000.00 PV = 95,238.10 which is equal to the payment amount times the number Growth Rate= 5.00% of periods discounted for one period. Here's why: In order to have enough to make all her payments to you, auntie would have to deposit that much now. Here's how that would be depleted: Amount of - amount principal used Beginning paid out end for that balance +interest of year ending balance payment 1 5000.00 95,238.10 4,761.90 5,000.00 95,000.00 2 5250.00 95,000.00 4,750.00 5,250.00 94,500.00 (500.00) 3 5512.50 94,500.00 4,725.00 5,512.50 93,712.50 (787.50) 4 5788.13 93,712.50 4,685.63 5,788.13 92,610.00 (1,102.50) 5 6077.53 92,610.00 4,630.50 6,077.53 91,162.97 (1,447.03) 6 6381.41 91,162.97 4,558.15 6,381.41 89,339.71 (1,823.26) 7 6700.48 89,339.71 4,466.99 6,700.48 87,106.22 (2,233.49) 8 7035.50 87,106.22 4,355.31 7,035.50 84,426.03 (2,680.19) 9 7387.28 84,426.03 4,221.30 7,387.28 81,260.05 (3,165.98) 10 7756.64 81,260.05 4,063.00 7,756.64 77,566.41 (3,693.64) 11 8144.47 77,566.41 3,878.32 8,144.47 73,300.26 (4,266.15) 12 8551.70 73,300.26 3,665.01 8,551.70 68,413.57 (4,886.68) 13 8979.28 68,413.57 3,420.68 8,979.28 62,854.97 (5,558.60) 14 9428.25 62,854.97 3,142.75 9,428.25 56,569.47 (6,285.50) 15 9899.66 56,569.47 2,828.47 9,899.66 49,498.29 (7,071.18) 16 10394.64 49,498.29 2,474.91 10,394.64 41,578.56 (7,919.73) 17 10914.37 41,578.56 2,078.93 10,914.37 32,743.12 (8,835.44) 18 11460.09 32,743.12 1,637.16 11,460.09 22,920.18 (9,822.94) 19 12033.10 22,920.18 1,146.01 12,033.10 12,033.10 (10,887.09) 20 12634.75 12,033.10 601.65 12,634.75 (0.00) (12,033.10) Problem 4-29 You are running a hot Internet company. Analysts predict that its earnings will grow at 30% per year for the next five years. After that, as competition increases, earnings growth is expected to slow to 2% per year and continue at that level forever. Your company has just announced earnings of $1,000,000. What is the present value of all future earnings if the interest rate is 8%? (Assume all cash flows occur at the end of the year.) Starting growth rate 30.00% Number of years of high growth 5 Later growth rate 2.00% Discount rate 8.00% Year 0 1 2 3 4 5 6 Cash flow $1,000,000.00 $1,300,000.00 $1,690,000.00 $2,197,000.00 $2,856,100.00 $3,712,930.00 $3,787,188.60 PV(cash flow) $1,203,703.70 $1,448,902.61 $1,744,049.43 $2,099,318.76 $2,526,957.77 PV(infinite cash flows) 42,958,282.09 Value of company 51,981,214.36 Problem 4-30 Your brother has offered to give you $100, starting next year, and after that growing at 3% for the next 20 years. You would like to calculate the value of this offer by calculating how much money you would need to deposit in the local bank so that the account will generate the same cash flows as he is offering you. Your local bank will guarantee a 6% annual interest rate so long as you have money in the account. Initial payment $100 Annual growth rate 3% Number of periods 20 Discount rate 6.00% a. How much money will you need to deposit into the account today? PV of growing annuity $ 1,456.15 b. Using an Excel spreadsheet, show explicitly that you can deposit this amount of money into the account, and every year withdraw what your brother has promised, leaving the account with nothing after the last withdrawal. Amount of - amount paid out principal used for Beginning balance +interest end of year ending balance that payment 1 100.00 1,456.15 87.37 100.00 1,443.52 2 103.00 1,443.52 86.61 103.00 1,427.13 (16.39) 3 106.09 1,427.13 85.63 106.09 1,406.67 (20.46) 4 109.27 1,406.67 84.40 109.27 1,381.80 (24.87) 5 112.55 1,381.80 82.91 112.55 1,352.16 (29.64) 6 115.93 1,352.16 81.13 115.93 1,317.36 (34.80) 7 119.41 1,317.36 79.04 119.41 1,276.99 (40.36) 8 122.99 1,276.99 76.62 122.99 1,230.63 (46.37) 9 126.68 1,230.63 73.84 126.68 1,177.79 (52.84) 10 130.48 1,177.79 70.67 130.48 1,117.98 (59.81) 11 134.39 1,117.98 67.08 134.39 1,050.66 (67.31) 12 138.42 1,050.66 63.04 138.42 975.28 (75.38) 13 142.58 975.28 58.52 142.58 891.22 (84.06) 14 146.85 891.22 53.47 146.85 797.84 (93.38) 15 151.26 797.84 47.87 151.26 694.45 (103.39) 16 155.80 694.45 41.67 155.80 580.32 (114.13) 17 160.47 580.32 34.82 160.47 454.67 (125.65) 18 165.28 454.67 27.28 165.28 316.67 (138.00) 19 170.24 316.67 19.00 170.24 165.43 (151.24) 20 175.35 165.43 9.93 175.35 (0.00) (165.43) Problem 4-32 You are thinking of purchasing a house. The house costs $350,000. You have $50,000 in cash that you can use as a down payment on the house, but you need to borrow the rest of the purchase price. The bank is offering a 30-year mortgage that requires annual payments and has an interest rate of 7% per year. What will your annual payment be if you sign up for this mortgage? Cost of House $350,000.00 Down Payment $50,000.00 Number of Years 30 Interest Rate 7.00% Annual Payment Amount (24,175.92) sts $350,000. You have the house, but you need is offering a 30-year est rate of 7% per year. mortgage? Problem 4-34 You would like to buy the house and take the mortgage described in Problem 32. You can afford to pay only $23,500 per year. The bank agrees to allow you to pay this amount each year, yet still borrow $300,000. At the end of the mortgage (in 30 years), you must make a balloon payment; that is, you must repay the remaining balance on the mortgage. How much will this balloon payment be? Payment $23,500.00 Cost $300,000.00 Term 30 Rate 7.00% Balloon 63,848.03 - OR - PV of Payments (291,612.47) PV of difference 8,387.53 FV of difference 63,848.03 escribed in Problem 32. ees to allow you to pay d of the mortgage (in 30 ust repay the remaining ent be? Problem 4-35 You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Today is your 30th birthday, and you decide, starting today and continuing on every birthday up to and including your 65th birthday, that you will put the same amount into a savings account. If the interest rate is 5%, how much must you set aside each year to make sure that you will have $2 million in the account on your 65th birthday? Cumulative value of account Your age Payment (end of year) 30 (20,868.91) (21,912.36) Number of payments 36 31 (20,868.91) (44,920.34) Interest rate 5.00% 32 (20,868.91) (69,078.71) Future value $2,000,000.00 33 (20,868.91) (94,445.01) 34 (20,868.91) (121,079.62) Payment amount (20,868.91) 35 (20,868.91) (149,045.96) 36 (20,868.91) (178,410.62) 37 (20,868.91) (209,243.51) You'll get the same answer if you calculate 38 (20,868.91) (241,618.05) the payment that will accumulate to 2 39 (20,868.91) (275,611.31) million and then multiply it by 1.05. 40 (20,868.91) (311,304.23) 41 (20,868.91) (348,781.81) 42 (20,868.91) (388,133.26) 43 (20,868.91) (429,452.28) 44 (20,868.91) (472,837.25) 45 (20,868.91) (518,391.48) 46 (20,868.91) (566,223.41) 47 (20,868.91) (616,446.94) 48 (20,868.91) (669,181.65) 49 (20,868.91) (724,553.09) 50 (20,868.91) (782,693.10) 51 (20,868.91) (843,740.12) 52 (20,868.91) (907,839.48) 53 (20,868.91) (975,143.82) 54 (20,868.91) (1,045,813.37) 55 (20,868.91) (1,120,016.40) 56 (20,868.91) (1,197,929.58) 57 (20,868.91) (1,279,738.42) 58 (20,868.91) (1,365,637.70) 59 (20,868.91) (1,455,831.94) 60 (20,868.91) (1,550,535.90) 61 (20,868.91) (1,649,975.05) 62 (20,868.91) (1,754,386.17) 63 (20,868.91) (1,864,017.83) 64 (20,868.91) (1,979,131.09) 65 (20,868.91) (2,000,000.00) Problem 4-36 You realize that the plan in Problem 35 has a flaw. Because your income will increase ove be more realistic to save less now and more later. Instead of putting the same amount asid to let the amount that you set aside grow by 3% per year. Under this plan, how much will today? (Recall that you are planning to make the first contribution to the account today.) Cumulative value of account Your age Payment (end of year) 30 (13,823.91) (14,515.10) 31 (14,791.58) (30,772.02) 32 (15,826.99) (48,928.96) 33 (16,934.88) (69,157.04) 34 (18,120.32) (91,641.23) 35 (19,388.75) (116,581.47) 36 (20,745.96) (144,193.80) 37 (22,198.18) (174,711.57) 38 (23,752.05) (208,386.80) 39 (25,414.69) (245,491.57) 40 (27,193.72) (286,319.55) 41 (29,097.28) (331,187.67) 42 (31,134.09) (380,437.85) 43 (33,313.48) (434,438.89) 44 (35,645.42) (493,588.53) 45 (38,140.60) (558,315.58) 46 (40,810.44) (629,082.32) 47 (43,667.17) (706,386.96) 48 (46,723.87) (790,766.38) 49 (49,994.54) (882,798.97) 50 (53,494.16) (983,107.79) 51 (57,238.75) (1,092,363.87) 52 (61,245.47) (1,211,289.80) 53 (65,532.65) (1,340,663.57) 54 (70,119.93) (1,481,322.68) 55 (75,028.33) (1,634,168.55) 56 (80,280.31) (1,800,171.31) 57 (85,899.93) (1,980,374.81) 58 (91,912.93) (2,175,902.12) 59 (98,346.83) (2,387,961.40) 60 ######### (2,617,852.14) 61 ######### (2,866,971.90) 62 ######### (3,136,823.55) 63 ######### (3,429,023.00) 64 ######### (3,745,307.50) 65 ######### (3,892,899.58) aw. Because your income will increase over your lifetime, it would er. Instead of putting the same amount aside each year, you decide per year. Under this plan, how much will you put into the account e first contribution to the account today.) Number of payments 36 Interest rate 5.00% Present value (345,314.83) Growth rate 3.00% PV interest factor (formula) 24.979538 Payment (13,823.91) Problem 4-44 You are thinking of making an investment in a new plant. The plant will generate revenues of $1 million per year for as long as you maintain it. You expect that the maintenance cost will start at $50,000 per year and will increase 5% per year thereafter. Assume that all revenue and maintenance costs occur at the end of the year. You intend to run the plant as long as it continues to make a positive cash flow (as long as the cash generated by the plant exceeds the maintenance costs). The plant can be built and become operational immediately. If the plant costs $10 million to build, and the interest rate is 6% per year, should you invest in the plant? The question is, how long will it take for $50,000.00 to equal $1,000,000.00 if it grows at 5.00% per year? You can solve this using logarithms, by solving the following equation for n: 1000000 = 50000 * 1.05 ^(n-1) divide both sides by $ 50,000.00 1.05 ^(n-1) = $20.00 (n-1)* ln(1.05) = ln(20) (n-1)* 0.048790164 = 2.995732274 Divide both sides by 0.048790164 and add 1 to get 62.40 Or you can use the Excel function: 62.40 Problem 4-45 You have just turned 30 years old, have just received your MBA, and have accepted your f must decide how much money to put into your retirement plan. The plan works as follows: plan earns 7% per year. You cannot make withdrawals until you retire on your sixty-fifth b point, you can make withdrawals as you see fit. You decide that you will plan to live to 10 you turn 65. You estimate that to live comfortably in retirement, you will need $100,000 p the end of the first year of retirement and ending on your one hundredth birthday. You wil same amount to the plan at the end of every year that you work. How much do you need to year to fund your retirement? Interest rate assumption 7.00% You will need $100,000.00 PV as of age 65 = (1,294,767.23) You can accumulate this amount in 35 Note: The above uses future value of an annuity. You can also calculate this using PV of an annuity, by figuring the PV of the FV a annuity implied by that. PV as of age 30 = 121,271.70 PMT 9,366.29 your MBA, and have accepted your first job. Now you ment plan. The plan works as follows: Every dollar in the ls until you retire on your sixty-fifth birthday. After that decide that you will plan to live to 100 and work until retirement, you will need $100,000 per year starting at your one hundredth birthday. You will contribute the you work. How much do you need to contribute each per year for 35 years years by paying 9,366.29 per year. nuity. nnuity, by figuring the PV of the FV and computing the Problem 4-46 Problem 45 is not very realistic because most retirement plans do not allow you to spec contribute every year. Instead, you are required to specify a fixed percentage of your sa contribute. Assume that your starting salary is $75,000 per year and it will grow 2% per Assuming everything else stays the same as in Problem 45, what percentage of your inc contribute to the plan every year to fund the same retirement income? Interest rate assumption 7.00% You will need $100,000.00 PV as of age 65= (1,294,767.23) PV as of age 30= (121,271.70) Your contributions will grow at an annual rate of 2.00% Starting salary $75,000.00 PV interest factor 16.253689 You can accumulate this amount in 35 Age Contribution 31 $7,461.18 32 $7,610.40 33 $7,762.61 34 $7,917.86 35 $8,076.22 36 $8,237.75 37 $8,402.50 38 $8,570.55 39 $8,741.96 40 $8,916.80 41 $9,095.14 42 $9,277.04 43 $9,462.58 44 $9,651.83 45 $9,844.87 46 $10,041.77 47 $10,242.60 48 $10,447.45 49 $10,656.40 50 $10,869.53 51 $11,086.92 52 $11,308.66 53 $11,534.83 54 $11,765.53 55 $12,000.84 56 $12,240.86 57 $12,485.67 58 $12,735.39 59 $12,990.10 60 $13,249.90 61 $13,514.90 62 $13,785.19 63 $14,060.90 64 $14,342.11 65 $14,628.96 rement plans do not allow you to specify a fixed amount to specify a fixed percentage of your salary that you want to ,000 per year and it will grow 2% per year until you retire. oblem 45, what percentage of your income do you need to etirement income? per year for 35 years years by paying 7,461.18 per year. or 9.95% of your annual income. Balance $7,461.18 $15,593.87 $24,448.05 $34,077.28 $44,538.91 $55,894.38 $68,209.48 $81,554.70 $96,005.49 $111,642.67 $128,552.80 $146,828.53 $166,569.11 $187,880.78 $210,877.30 $235,680.48 $262,420.72 $291,237.62 $322,280.66 $355,709.83 $391,696.44 $430,423.85 $472,088.36 $516,900.07 $565,083.92 $616,880.65 $672,547.97 $732,361.71 $796,617.13 $865,630.22 $939,739.23 $1,019,306.17 $1,104,718.50 $1,196,390.91 $1,294,767.23