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29. Annuities. You can buy a car that is advertised for $24,000 on the following terms: (a) pay $24,000 and receive a $2,000 rebate from the manufacturer; (b) pay $500 a month for 4 years for total payments of $24,000, implying zero percent financing. Which is the better deal if the inter est rate is 1% per month? (LO5-3) 30. Annuities. You have just borrowed $100,000 to buy a condo. You will repay the loan in equal monthly payments of $804.62 over the next 30 years. (LO5-4) a. What monthly interest rate are you paying on the loan? b. What is the APR? c. What is the effective annual rate on that loan? d. What rate is the lender more likely to quote on the loan? 31. Future Value of Annuities. I now have $20,000 in the bank earning interest of .5% per month I need $30,000 to make a down payment on a house. I can save an additional $100 per month. How long will it take me to accumulate the $30,000? (LO5-3) 32. Real Annuities. A retiree wants level consumption in real terms over a 30-year retirement. If the inflation rate equals the interest rate she earns on her $450,000 of savings, how much can she spend in real terms each year over the rest of her life? (LO5-3) 33. Delayed Annuities. Suppose that you will receive annual payments of $10,000 for a period of 10 years. The first payment will be made 4 years from now. If the interest rate is 5%, what is the present value of this stream of payments? (LO5-3) 34. Annuity Due. Your landscaping company can lease a truck for $8,000 a year (paid at year-end) for 6 years. It can instead buy the truck for $40,000. The truck will be valueless after 6 years. The interest rate your company can earn on its funds is 7%. (LO5-3) a. What is the present value of the cost of leasing? b. Is it cheaper to buy or lease? c. What is the present value of the cost of leasing if the lease payments are an annuity due, so the first payment comes immediately? d. Is it now cheaper to buy or lease? 35. Annuity Due. Recall that an annuity due is like an ordinary annuity except that the first pay- ment is made immediately instead of at the end of the first period. (LO5-3) a. Why is the present value of an annuity due equal to (1 + r) times the present value of an ordinary annuity? b. Why is the future value of an annuity due equal to (1 + r) times the future value of an ordi- nary annuity? 36. Annuity Due. A store offers two payment plans. Under the installment plan, you pay 25% down and 25% of the purchase price in each of the next 3 years. If you pay the entire bill immediately, you can take a 10% discount from the purchase price. (LO5-3) a. Which is a better deal if you can borrow or lend funds at a 5% interest rate? b. How will your answer change if the payments on the 4-year installment plan do not start for a full year? 37. Annuity Due. (LO5-3) a. If you borrow $1,000 and agree to repay the loan in five equal annual payments at an interest rate of 12%, what will your payment be? b. What will your payment be if you make the first payment on the loan immediately instead of at the end of the first year? 38. Annuity Due. The $40 million lottery payment that you have just won actually pays $2 million per year for 20 years. The interest rate is 8%. (LO5-3) a. If the first payment comes in 1 year, what is the present value of the winnings? b. What is the present value if the first payment comes immediately? 39. Amortizing Loan. You take out a 30-year $100,000 mortgage loan with an APR of 6% and monthly payments. In 12 years you decide to sell your house and pay off the mortgage. What is the principal balance on the loan? (LO5-3) 40. Amortizing Loan. Consider a 4-year amortizing loan. You borrow $1,000 initially and repay it in four equal annual year-end payments. (LO5-3) a. If the interest rate is 8%, what is the annual payment? b. Fill in the following table, which shows how much of each payment is interest versus princi- pal repayment (that is, amortization) and the outstanding balance on the loan at each date.Chapter 5 The 'I'Irne Value of Money 157 41. 42. 43. 45. 46. 47. 48. Loan Year-End Interest on. on rent YIItvEl'ld Amot'tluol'l of Loan Bel-nee l$l Payment [5} l0." l3) lietlrement Savings. A couple will retire in 50 years; they Plan 10 SW9d "'0'" "um I 3"!" (tn current dollars} in retirement. which should last about 2.5 years. 1116:! Mic": "'5' "'3' can earn a real interest rate of 8% on retirement savings. (L053) :1. if they nuke annuai payments into a savings plan. how much will they "05d to 53"" "Ch ital"? Assume the rst payment comes in 1 year. I:- How would the answer to pan (a) change it 01.: couple also realize that in 20 years they will need '0 Spend $60,000 in current dollars on their child's college education? iletirement Savings. You believe you will need to have saved 5500.000 by the time 3'0" mire In 40 years to order to live comfortably. If the interest rate is 6% per year. 110'" much mm you save each year to meet your retirement goal? (L054) Batirement Savings. You believe you will need to have saved $500,000 by the Lime you retire in 40 years in order to live comfortably. You also believe that you will inherit 5100.000 in '0 years. If the interest rate is 6% per year, how much must you save each year to meet your retirement goal? (L05-3) Retirement Savings. You believe you will spend 540.000 a year for 20 years one: you retire in 40 years. If the interest rate is 6% per year. how much must you save each year until retirement to meet your retirement goal? (L053) Retirement Savings. A couple thinking about retirement decide to put aside $3.000 each year in a savings plan that earns 8% interest. in 5 years they will receive a gift of 510.000 that 5130 can be invested. (MS-3) a. How much money will they have accumulated 30 years from now? b. If their goal is to retire with $300.000 of savings. how much extra do they need to save every year? Perpenrllius and Effective Interest Rate. What is the value of a perpetuity that pays $100 every 3 months forever? The interest rate quoted on an APR basis is 6%. (L053) Amortizing Loans and Ination. Suppose you take out a 3100.000. 20-year mortgage loan to buy a condo. The interest rate on the loan is 6%. To keep things simple. we will assume you make payments on the loan annually at the end of each year. (005-3) a. What is your annual payment on the loan? it. Construct a mortgage amortization table in Excel similar to Table 5.5 in which you compute the interest payment each year. the amortization of the loan. and the loan balance each year. (Allow the interest rate to he an input that the user of the Spreadsheet can enter and change.) What fraction of your initial loan payment is interest? What fraction of your initial loan payment is amortization? e. What fraction of the loan has been paid off after 10 years (halfWay through the life of the loan)? Why is the answer not 50%? f. If the inflation rate is 2%. what is the real value of the rst (year~end) payment? If the ination rate is 2%. what is the real value of the last (yeanend) payment? Now assume the inflation rate is 8% and the real interest rate on the loan is unchanged. What must be the new nominal interest rate? i. Recompute the amortization table. What is the real value of the rst {year-end) payment in this high-ination scenario? j. What is the real value of the last payment in this high-ination scenario? so we Mortgage with Points. Home loans often involve "points." which are fees charged by the lender. Each. point charged means that the borrower must pay 1% of the loan amount as :1 ice. For example. if the loan is for $100,000 and 2 points are charged. the loan repayment schedule is calculated on a $100,000 loan but the net amount the borrower receives is only $93,000 Assume the interest rate is 1% per month. What is the effective annual interest rate charged on such a loan. assuming loan repayment occurs over 360 months? (L054) QUESTIONS AND PROBLEMS connect 1. Compound Interest. Old Time Savings Bank pays 4% interest on its savings accounts. If you deposit $1,000 in the bank and leave it there: (LO5-1) a. How much interest will you earn in the first year? b. How much interest will you earn in the second year? c. How much interest will you earn in the 10th year? 2. Compound Interest. New Savings Bank pays 4% interest on its deposits. If you deposit $1,000 in the bank and leave it there, will it take more or less than 25 years for your money to double? You should be able to answer this without a calculator or interest rate tables. (LO5-1) 3. Compound Interest. Suppose that the value of an investment in the stock market has increased at an average compound rate of about 5% since 1900. It is now 2019. (LO5-1) a. If your great-grandfather invested $1,000 in 1900, how much would that investment be worth today? b. If an investment in 1900 has grown to $1 million, how much was invested in 1900?4. Future Values. Compute the future value of a $100 cash flow for the following combinations of rates and times. (LO5-1) a. r= 8%, r = 10 years b. r = 8%, t = 20 years c. r = 4%, 1 = 10 years d. r = 4%, t = 20 years 5. Future Values. You deposit $1,000 in your bank account. (LOS-1) a. If the bank pays 4% simple interest, how much will you accumulate in your account after 10 years? b. How much will you accumulate if the bank pays compound interest? 6. Future Values. If you earn 6% per year on your bank account, how long will it take an account with $100 to double to $200? (LOS-1) 7. Future Values. In 1880 five aboriginal trackers were each promised the equivalent of 100 Australian dollars for helping to capture the notorious outlaw Ned Kelley. In 1993 the granddaughters of two of the trackers claimed that this reward had not been paid. The Victorian prime minister stated that if this was true, the government would be happy to pay the $100. However, the granddaughters also claimed that they were entitled to compound interest. (LO5-1) a. How much was each granddaughter entitled to if the interest rate was 4%? b. How much was each entitled to if the interest rate was 8%? 8. Future Values. How long will it take for $400 to grow to $1,000 at the following interest rates? (LO5-1) a. 4% b. 8% c. 16% 9. Future Values. You invest $1,000 today and expect to sell your investment for $2,000 in 10 years. (LO5-1) a. Is this a good deal if the interest rate is 6%? b. What if the interest rate is 10%? 10. Future Values. Your wealthy uncle established a $1,000 bank account for you when you were born. For the first 8 years of your life, the interest rate earned on the account was 6%. Since then, rates have been only 4%. Now you are 21 years old and ready to cash in. How much is in your account? (LO5-1) 11. Present Values. You can buy property today for $3 million and sell it in 5 years for $4 million. (You earn no rental income on the property.) (LO5-2) a. If the interest rate is 8%, what is the present value of the sales price? b. Is the property investment attractive to you? c. Would your answer to part (b) change if you also could earn $200,000 per-year rent on the property? The rent is paid at the end of each year. 12. Present Values. Compute the present value of a $100 cash flow for the following combinations of discount rates and times. (LO5-2) a. r= 8%, 1 = 10 years b. r = 8%, 1 = 20 years c. r = 4%, 1 = 10 years d. r= 4%, t = 20 years 13. Present Values. You will require $700 in 5 years. If you earn 5% interest on your funds, how much will you need to invest today in order to reach your savings goal? (LO5-2) 14. Present Values. What is the present value of the following cash-flow stream if the interest rate is 6%? (LO5-2) Year Cash Flow $200 W N 400 300CHIN-er 5 The Time Value of Money 15. 16. 1?. 18. 19. 21. 22. 23. 25. 26. 27. 28. 155 Calculating the Interest Rate. Find the interest rate implied by [ht following mmmm of present and future values. (L054) Calculating the Interest Rate. Find the annual interest rate. (1-05-21! Present Value Future Value "Time Period 5! 15.76 262.16 110.41 Calculating the Interest Rate. A vs. Treasury strip that will pay 51.000 in 10 years is selling today for $422.41. What interest rate does the strip offer? (11.05-2} Present Values. A factory costs mill}. You forecast that it will produce cash inflows of 51mm in year I. $18011!) in yearZ. and 5300.000 in year3. The discount rate is 12%. (1.05-2} a. What is the value of the factory? b. Is the factory a good investment? Present Values. The 2-year discount factor is .92. What is the present value of $1 to be received in year '2'? What is the present value of $2,000? (L05-2]: Annuities. A famous quarterback just signed a. $15 million contract providing 33 million a year for 5 years. A less famous receiver signed a $14 million 5-year contract providing $4 million now and $2 million a year for 5 years. The interest rate is 10%. Who is better paid? (L05-3] Annuities. Would you rather receive 31.000 :1 year for 10 years or $800 a year for 15 years if the interest rate is 5%? What if the interest rate is 20%? (L053) Perpetuitles. A local bank advertises the following deal: "Pay us $100 a year for 10 years and then we will pay you (or your beneciaries} $100 a year forever." 13 this a good deal ifthe inter- est rate is 6%? (L05-3) Perpetuitiets. A local bank will pay you $100 a year for your lifetime if you deposit $2,500 in the bank today. if you plan to live forever. what interest rate is the bank. paying? (LBS-3) Perpetuities. A property will provide $10,000 a year forever. lfits value is $125,000, what must be the discount rate? (IDS-3] Perpetuities. British government 4% perpetuities pay 4 interest each year forever. Another bond. 2.5% perpetuities. pays 2.50 a year forever. {1105-3) 3. What is the value of 4% perpetuities if the long-term interest rate is 6%? b. What is the value of 2.5% perpeuiilies? Annuities. (L054) a. What is the present value of a 3-year annuity of $100 if the discount rate is 6%? b. What is the present value of the annuity in part (a) if you have to wait 2 years instead of 1 year for the rst payment? Annuities. Professor's Annuity Corp. offers a lifetime annuity to retiring professors. For a paymeu' of $30,030 at age 65. the furn will pay the tetiring professor $600 a month until death. (L0?!) it. If the professor's remaining life expectancy is 20 years. what is the monthly interest rate on this annuity? b. What is the effective annual interest rate? c. If the monthly interest rate is 5%. what monthly annuity payment can the rm offer to the retiring professor? Annuities. You want to buy a new car. but you can make an initial payment of only 32.000 and can afford monthly payments of at most $400. (1205-3) a. If the APR on auto loans is 12% and you nance the purchase over 48 months. what is the maximum price you can pay for the car? b. How much can you afford if you nance the purchase over 60 months? 2..." Jan-1'1 ,,-.,_._. - >IJ Part Two Value 49. Effective Interest Rate. A store will give you a 3% discount on the cost of your purchase if you pay cash today. Otherwise, you will be billed the full price with payment due in 1 month. Why month? (LO5-4) is the implicit borrowing rate being paid by customers who choose to defer payment for the 50. Effective Interest Rate. You've borrowed $4,248.68 and agreed to pay back the loan with monthly payments of $200. If the interest rate is 12% stated as an APR, how long will it take you to pay back the loan? What is the effective annual rate on the loan? (LO5-4) 51. Effective Interest Rate. You invest $1,000 at a 6% annual interest rate, stated as an APR. Inter. est is compounded monthly. How much will you have in 1 year? In 1.5 years? (105-4) 52. Effective Interest Rate. If a bank pays 6% interest with continuous compounding, what is the effective annual rate? (LOS-4) 53. Effective Interest Rate. In a discount interest loan, you pay the interest payment up front. For example, if a 1-year loan is stated as $10,000 and the interest rate is 10%%, the borrower "pays" .10 x $10,000 = $1,000 immediately, thereby receiving net funds of $9.000 and repaying $10,000 in a year. (LO5-4) a. What is the effective interest rate on this loan? b. What is the effective annual rate on a 1-year loan with an interest rate quoted on a discount basis of 20%? 54. Effective Interest Rate. Banks sometimes quote interest rates in the form of "add-on inter- est." In this case, if a 1-year loan is quoted with a 20% interest rate and you borrow $1,000, then you pay back $1,200. But you make these payments in monthly installments of $100 each. (LO5-4) a. What is the true APR on this loan? b. What is the effective annual rate on the loan? 55. Effective Interest Rate. You borrow $1,000 from the bank and agree to repay the loan over the next year in 12 equal monthly payments of $90. However, the bank also charges you a loan initiation fee of $20, which is taken out of the initial proceeds of the loan. What is the effective annual interest rate on the loan, taking account of the impact of the initia- tion fee? (LO5-4) 56. Effective Interest Rate. First National Bank pays 6.2% interest compounded semiannually. Second National Bank pays 6% interest compounded monthly. Which bank offers the higher effective annual interest rate? (LOS-4) 57. Loan Payments. You take out an $8,000 car loan that calls for 48 monthly payments starting after 1 month at an APR of 10%. (LO5-4) a. What is your monthly payment? b. What is the effective annual interest rate on the loan? c. Now assume the payments are made in four annual year-end installments. What annual pay- ment would have the same present value as the monthly payment you calculated? 58. Continuous Compounding. How much will $100 grow to if invested at a continuously com- pounded interest rate of 10% for 8 years? What if it is invested for 10 years at 8%? (LOS-4) 59. Effective Interest Rate. Find the effective annual interest rate for each case. (LO5-4) APR Compounding Period 12% 1 month 8 10 a W 60. Effective Interest Rate. Find the APR (the stated interest rate) for each case. (LO5-4) Effective Annual Compounding Interest Rate Period 10.00% 1 month 6.09 8.24 W O
