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To what future value will money invested at a given interest rate grow after a given period of time?

An investment of \$1 earning an interest rate of r will increase in value each period by the factor (1 + r). After t periods its value will grow to \$(1 + r)t. This is the future value of the \$1 investment with compound interest.

What is the present value of a cash flow to be received in the future?

The present value of a future cash payment is the amount that you would need to invest today to match that future payment. To calculate present value we divide the cash payment by (1 + r)t or, equivalently, multiply by the discount factor 1/(1 + r)t. The discount factor measures the value today of \$1 received in period t.

How can we calculate present and future values of streams of cash payments?

A level stream of cash payments that continues indefinitely is known as a perpetuity; one that continues for a limited number of years is called an annuity. The present value of a stream of cash flows is simply the sum of the present value of each individual cash flow. Similarly, the future value of an annuity is the sum of the future value of each individual cash flow. Shortcut formulas make the calculations for perpetuities and annuities easy.

What is the difference between real and nominal cash flows and between real and nominal interest rates?

A dollar is a dollar but the amount of goods that a dollar can buy is eroded by inflation. If prices double, the real value of a dollar halves. Financial managers and economists often find it helpful to reexpress future cash flows in terms of real dollars—that is, dollars of constant purchasing power.

Be careful to distinguish the nominal interest rate and the real interest rate—that is, the rate at which the real value of the investment grows. Discount nominal cash flows (that is, cash flows measured in current dollars) at nominal interest rates. Discount real cash flows (cash flows measured in constant dollars) at real interest rates. Never mix and match nominal and real.

How should we compare interest rates quoted over different time intervals—for example, monthly versus annual rates?

Interest rates for short time periods are often quoted as annual rates by multiplying the per-period rate by the number of periods in a year. These annual percentage rates (APRs) do not recognize the effect of compound interest, that is, they annualize assuming simple interest. The effective annual rate annualizes using compound interest. It equals the rate of interest per period compounded for the number of periods in a year.

http://invest-faq.com/articles/analy-fut-prs-val.html Understanding the concepts of present and future value

www.bankrate.com/brm/default.asp Different interest rates for a variety of purposes, and some calculators

www.financenter.com/ Calculators for evaluating financial decisions of all kinds http://www.financialplayerscenter.com/Overview.html An introduction to time value of money with several calculators http://ourworld.compuserve.com/homepages More calculators, concepts, and formulas

### Key Terms

Quiz future value compound interest simple interest present value (PV) discount rate discount factor annuity perpetuity annuity factor annuity due inflation real value of \$1

nominal interest rate real interest rate effective annual interest rate annual percentage rate (APR)

1. Present Values. Compute the present value of a \$100 cash flow for the following combinations of discount rates and times:

a. r = 10 percent. t = 10 years b. r = 10 percent. t = 20 years c. r = 5 percent. t = 10 years d. r = 5 percent. t = 20 years

2. Future Values. Compute the future value of a \$100 cash flow for the same combinations of rates and times as in problem 1.

3. 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. How much was each entitled to if the interest rate was 5 percent? What if it was 10 percent?

4. Future Values. You deposit \$1,000 in your bank account. If the bank pays 4 percent simple interest, how much will you accumulate in your account after 10 years? What if the bank pays compound interest? How much of your earnings will be interest on interest?

5. Present Values. You will require \$700 in 5 years. If you earn 6 percent interest on your funds, how much will you need to invest today in order to reach your savings goal?

6. Calculating Interest Rate. Find the interest rate implied by the following combinations of present and future values:

Present Value Years Future Value

7. Present Values. Would you rather receive \$1,000 a year for 10 years or \$800 a year for 15 years if a. the interest rate is 5 percent?

b. the interest rate is 20 percent?

8. Calculating Interest Rate. Find the annual interest rate.

Present Value Future Value Time Period

100 115.76 3 years 200 262.16 4 years 100_110.41_5 years

9. Present Values. What is the present value of the following cash-flow stream if the interest rate is 5 percent?

Year Cash Flow

10. Number of Periods. How long will it take for \$400 to grow to \$1,000 at the interest rate specified?

a. 4 percent b. 8 percent c. 16 percent

11. Calculating Interest Rate. Find the effective annual interest rate for each case: APR Compounding Period

12. Calculating Interest Rate. Find the APR (the stated interest rate) for each case:

Effective Annual Compounding

Interest Rate Period

13. Growth of Funds. If you earn 8 percent per year on your bank account, how long will it take an account with \$100 to double to \$200?

14. Comparing Interest Rates. Suppose you can borrow money at 8.6 percent per year (APR) compounded semiannually or 8.4 percent per year (APR) compounded monthly. Which is the better deal?

15. Calculating Interest Rate. Lenny Loanshark charges "one point" per week (that is, 1 percent per week) on his loans. What APR must he report to consumers? Assume exactly 52 weeks in a year. What is the effective annual rate?

16. Compound Interest. Investments in the stock market have increased at an average compound rate of about 10 percent since 1926.

a. If you invested \$1,000 in the stock market in 1926, how much would that investment be worth today?

b. If your investment in 1926 has grown to \$1 million, how much did you invest in 1926?

17. Compound Interest. Old Time Savings Bank pays 5 percent interest on its savings accounts. If you deposit \$1,000 in the bank and leave it there, how much interest will you earn in the first year? The second year? The tenth year?

18. Compound Interest. New Savings Bank pays 4 percent 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.

19. Calculating Interest Rate. A zero-coupon bond which will pay \$1,000 in 10 years is selling today for \$422.41. What interest rate does the bond offer?

20. Present Values. A famous quarterback just signed a \$15 million contract providing \$3 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. Who is better paid? The interest rate is 12 percent.

### Practice Problems

21. Loan Payments. If you take out an \$8,000 car loan that calls for 48 monthly payments at an APR of 10 percent, what is your monthly payment? What is the effective annual interest rate on the loan?

22. Annuity Values.

a. What is the present value of a 3-year annuity of \$100 if the discount rate is 8 percent?

b. What is the present value of the annuity in (a) if you have to wait 2 years instead of 1 year for the payment stream to start?

23. Annuities and Interest Rates. Professor's Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of \$80,000 at age 65, the firm will pay the retiring professor \$600 a month until death.

a. If the professor's remaining life expectancy is 20 years, what is the monthly rate on this annuity? What is the effective annual rate?

b. If the monthly interest rate is .5 percent, what monthly annuity payment can the firm offer to the retiring professor?

24. Annuity Values. You want to buy a new car, but you can make an initial payment of only \$2,000 and can afford monthly payments of at most \$400.

a. If the APR on auto loans is 12 percent and you finance the purchase over 48 months, what is the maximum price you can pay for the car?

b. How much can you afford if you finance the purchase over 60 months?

25. Calculating 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 percent, the borrower "pays" .10 X \$10,000 = \$1,000 immediately, thereby receiving net funds of \$9,000 and repaying \$10,000 in a year.

a. What is the effective interest rate on this loan?

b. If you call the discount d (for example, d = 10% using our numbers), express the effective annual rate on the loan as a function of d.

c. Why is the effective annual rate always greater than the stated rate d?

26. Annuity Due. Recall that an annuity due is like an ordinary annuity except that the first payment is made immediately instead of at the end of the first period.

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 ordinary annuity?

27. Rate on a Loan. If you take out an \$8,000 car loan that calls for 48 monthly payments of \$225 each, what is the APR of the loan? What is the effective annual interest rate on the loan?

28. Loan Payments. Reconsider the car loan in the previous question. What if the payments are made in four annual year-end installments? What annual payment would have the same present value as the monthly payment you calculated? Use the same effective annual interest rate as in the previous question. Why is your answer not simply 12 times the monthly payment?

29. Annuity Value. 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. If the interest rate your company can earn on its funds is 7 percent, is it cheaper to buy or lease?

30. Annuity Due Value. Reconsider the previous problem. What if the lease payments are an annuity due, so that the first payment comes immediately? Is it cheaper to buy or lease?

31. Annuity Due. A store offers two payment plans. Under the installment plan, you pay 25 percent down and 25 percent of the purchase price in each of the next 3 years. If you pay the entire bill immediately, you can take a 10 percent discount from the purchase price. Which is a better deal if you can borrow or lend funds at a 6 percent interest rate?

32. Annuity Value. Reconsider the previous question. How will your answer change if the payments on the 4-year installment plan do not start for a full year?

33. Annuity and Annuity Due Payments.

a. If you borrow \$1,000 and agree to repay the loan in five equal annual payments at an interest rate of 12 percent, what will your payment be?

b. What if you make the first payment on the loan immediately instead of at the end of the first year?

34. Valuing 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 6 percent, what is the present value of this stream of payments?

35. Mortgage with Points. Home loans typically involve "points," which are fees charged by the lender. Each point charged means that the borrower must pay 1 percent of the loan amount as a fee. For example, if the loan is for \$100,000, and two points are charged, the loan repayment schedule is calculated on a \$100,000 loan, but the net amount the borrower receives is only \$98,000. What is the effective annual interest rate charged on such a loan assuming loan repayment occurs over 360 months? Assume the interest rate is 1 percent per month.

36. Amortizing Loan. You take out a 30-year \$100,000 mortgage loan with an APR of 8 percent 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?

37. Amortizing Loan. Consider a 4-year amortizing loan. You borrow \$1,000 initially, and repay it in four equal annual year-end payments.

a. If the interest rate is 10 percent, show that the annual payment is \$315.47.

b. Fill in the following table, which shows how much of each payment is comprised of interest versus principal repayment (that is, amortization), and the outstanding balance on the loan at each date.

Loan Year-End Interest Year-End Amortization

Time Balance Due on Balance Payment of Loan

c. Show that the loan balance after 1 year is equal to the year-end payment of \$315.47 times the 3-year annuity factor.

38. Annuity Value. You've borrowed \$4,248.68 and agreed to pay back the loan with monthly payments of \$200. If the interest rate is 12 percent stated as an APR, how long will it take you to pay back the loan? What is the effective annual rate on the loan?

39. Annuity Value. The \$40 million lottery payment that you just won actually pays \$2 million per year for 20 years. If the discount rate is 10 percent, and the first payment comes in 1 year, what is the present value of the winnings? What if the first payment comes immediately?

40. 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?

EAR versus APR. You invest \$1,000 at a 6 percent annual interest rate, stated as an APR. Interest is compounded monthly. How much will you have in 1 year? In 1.5 years? Annuity Value. You 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. What monthly interest rate are you paying on the loan? What is the effective annual rate on that loan? What rate is the lender more likely to quote on the loan?

EAR. If a bank pays 10 percent interest with continuous compounding, what is the effective annual rate?

Annuity Values. You can buy a car that is advertised for \$12,000 on the following terms: (a) pay \$12,000 and receive a \$1,000 rebate from the manufacturer; (b) pay \$250 a month for 4 years for total payments of \$12,000, implying zero percent financing. Which is the better deal if the interest rate is 1 percent per month?

Continuous Compounding. How much will \$100 grow to if invested at a continuously compounded interest rate of 10 percent for 6 years? What if it is invested for 10 years at 6 percent?

Future Values. I now have \$20,000 in the bank earning interest of .5 percent 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?

Perpetuities. A local bank advertises the following deal: "Pay us \$100 a year for 10 years and then we will pay you (or your beneficiaries) \$100 a year forever." Is this a good deal if the interest rate available on other deposits is 8 percent?

Perpetuities. 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? Perpetuities. A property will provide \$10,000 a year forever. If its value is \$125,000, what must be the discount rate?

Applying Time Value. 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.)

a. If the interest rate is 8 percent, what is the present value of the sales price?

b. Is the property investment attractive to you? Why or why not?

c. Would your answer to (b) change if you also could earn \$200,000 per year rent on the property?

Applying Time Value. A factory costs \$400,000. You forecast that it will produce cash inflows of \$120,000 in Year 1, \$180,000 in Year 2, and \$300,000 in Year 3. The discount rate is 12 percent. Is the factory a good investment? Explain.

Applying Time Value. You invest \$1,000 today and expect to sell your investment for \$2,000 in 10 years.

a. Is this a good deal if the discount rate is 5 percent?

b. What if the discount rate is 10 percent?

Calculating Interest Rate. A store will give you a 3 percent 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. What is the implicit borrowing rate being paid by customers who choose to defer payment for the month?

Quoting Rates. Banks sometimes quote interest rates in the form of "add-on interest." In this case, if a 1-year loan is quoted with a 20 percent interest rate and you borrow \$1,000, then you pay back \$1,200. But you make these payments in monthly installments of \$100 each. What are the true APR and effective annual rate on this loan? Why should you have known that the true rates must be greater than 20 percent even before doing any calculations?

Compound Interest. Suppose you take out a \$1,000, 3-year loan using add-on interest (see previous problem) with a quoted interest rate of 20 percent per year. What will your monthly payments be? (Total payments are \$1,000 + \$1,000 X .20 X 3 = \$1,600.) What are the true APR and effective annual rate on this loan? Are they the same as in the previous problem?

56. Calculating Interest Rate. What is the effective annual rate on a one-year loan with an interest rate quoted on a discount basis (see problem 25) of 20 percent?

57. Effective Rates. First National Bank pays 6.2 percent interest compounded semiannually. Second National Bank pays 6 percent interest, compounded monthly. Which bank offers the higher effective annual rate?

58. Calculating 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 initiation fee?

59. Retirement Savings. You believe you will need to have saved \$500,000 by the time you retire in 40 years in order to live comfortably. If the interest rate is 5 percent per year, how much must you save each year to meet your retirement goal?

60. Retirement Savings. How much would you need in the previous problem if you believe that you will inherit \$100,000 in 10 years?

61. Retirement Savings. You believe you will spend \$40,000 a year for 20 years once you retire in 40 years. If the interest rate is 5 percent per year, how much must you save each year until retirement to meet your retirement goal?

62. Retirement Planning. A couple thinking about retirement decide to put aside \$3,000 each year in a savings plan that earns 8 percent interest. In 5 years they will receive a gift of \$10,000 that also can be invested.

a. How much money will they have accumulated 30 years from now?

b. If their goal is to retire with \$800,000 of savings, how much extra do they need to save every year?

63. Retirement Planning. A couple will retire in 50 years; they plan to spend about \$30,000 a year in retirement, which should last about 25 years. They believe that they can earn 10 percent interest on retirement savings.

a. If they make annual payments into a savings plan, how much will they need to save each year? Assume the first payment comes in 1 year.

b. How would the answer to part (a) change if the couple also realize that in 20 years, they will need to spend \$60,000 on their child's college education?

### Challenge Problems

64. Real versus Nominal Dollars. An engineer in 1950 was earning \$6,000 a year. Today she earns \$60,000 a year. However, on average, goods today cost 6 times what they did in 1950. What is her real income today in terms of constant 1950 dollars?

65. Real versus Nominal Rates. If investors are to earn a 4 percent real interest rate, what nominal interest rate must they earn if the inflation rate is:

66. Real Rates. If investors receive an 8 percent interest rate on their bank deposits, what real interest rate will they earn if the inflation rate over the year is:

67. Real versus Nominal Rates. You will receive \$100 from a savings bond in 3 years. The nominal interest rate is 8 percent.

a. What is the present value of the proceeds from the bond?

b. If the inflation rate over the next few years is expected to be 3 percent, what will the real value of the \$100 payoff be in terms of today's dollars?

c. What is the real interest rate?

d. Show that the real payoff from the bond (from part b) discounted at the real interest rate (from part c) gives the same present value for the bond as you found in part a.

68. Real versus Nominal Dollars. Your consulting firm will produce cash flows of \$100,000 this year, and you expect cash flow to keep pace with any increase in the general level of prices. The interest rate currently is 8 percent, and you anticipate inflation of about 2 percent.

a. What is the present value of your firm's cash flows for Years 1 through 5?

b. How would your answer to (a) change if you anticipated no growth in cash flow?

69. Real versus Nominal Annuities. Good news: you will almost certainly be a millionaire by the time you retire in 50 years. Bad news: the inflation rate over your lifetime will average about 3 percent.

a. What will be the real value of \$1 million by the time you retire in terms of today's dollars?

b. What real annuity (in today's dollars) will \$1 million support if the real interest rate at retirement is 2 percent and the annuity must last for 20 years?

70. Rule of 72. Using the Rule of 72, if the interest rate is 8 percent per year, how long will it take for your money to quadruple in value?

71. Inflation. Inflation in Brazil in 1992 averaged about 23 percent per month. What was the annual inflation rate?

72. Perpetuities. British government 4 percent perpetuities pay £4 interest each year forever. Another bond, 2V2 percent perpetuities, pays £2.50 a year forever. What is the value of 4 percent perpetuities, if the long-term interest rate is 6 percent? What is the value of 21/2 percent perpetuities?

### 73. Real versus Nominal Annuities.

a. You plan to retire in 30 years and want to accumulate enough by then to provide yourself with \$30,000 a year for 15 years. If the interest rate is 10 percent, how much must you accumulate by the time you retire?

b. How much must you save each year until retirement in order to finance your retirement consumption?

c. Now you remember that the annual inflation rate is 4 percent. If a loaf of bread costs \$1.00 today, what will it cost by the time you retire?

d. You really want to consume \$30,000 a year in real dollars during retirement and wish to save an equal real amount each year until then. What is the real amount of savings that you need to accumulate by the time you retire?

e. Calculate the required preretirement real annual savings necessary to meet your consumption goals. Compare to your answer to (b). Why is there a difference?

f. What is the nominal value of the amount you need to save during the first year? (Assume the savings are put aside at the end of each year.) The thirtieth year?

74. Retirement and Inflation. Redo part (a) of problem 63, but now assume that the inflation rate over the next 50 years will average 4 percent.

a. What is the real annual savings the couple must set aside?

b. How much do they need to save in nominal terms in the first year?

c. How much do they need to save in nominal terms in the last year?

d. What will be their nominal expenditures in the first year of retirement? The last?

75. Annuity Value. What is the value of a perpetuity that pays \$100 every 3 months forever? The discount rate quoted on an APR basis is 12 percent.

76. Changing Interest Rates. If the interest rate this year is 8 percent and the interest rate next year will be 10 percent, what is the future value of \$1 after 2 years? What is the present value of a payment of \$1 to be received in 2 years?

77. Changing Interest Rates. 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 8 percent. Since then, rates have been only 6 percent. Now you are 21 years old and ready to cash in. How much is in your account?

Solutions to

Self-Test

Questions

1 Value after 5 years would have been 24 X (1.05)5 = \$30.63; after 50 years, 24 X (1.05)50 = \$275.22.

2 Sales double each year. After 4 years, sales will increase by a factor of 2 X 2 X 2 X 2 = 24 = 16 to a value of \$.5 X 16 = \$8 million.

3 Multiply the \$1,000 payment by the 10-year discount factor:

\$441.06

4 If the doubling time is 12 years, then (1 + r)12 = 2, which implies that 1 + r = 21/12 = 1.0595, or r = 5.95 percent. The Rule of 72 would imply that a doubling time of 12 years is consistent with an interest rate of 6 percent: 72/6 = 12. Thus the Rule of 72 works quite well in this case. If the doubling period is only 2 years, then the interest rate is determined by (1 + r)2 = 2, which implies that 1 + r = 21/2 = 1.414, or r = 41.4 percent. The Rule of 72 would imply that a doubling time of 2 years is consistent with an interest rate of 36 percent: 72/36 = 2. Thus the Rule of 72 is quite inaccurate when the interest rate is high.

Present Value

Present Value

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Do you have annuity you dont want? Discover When is it Time to Sell Your Annuity? What can I do? Where can I get the money I need? I have an annuity, but I dont know that I can sell it. Is there a good time to sell my annuity? I already have a home improvement loan, but it was used before the roof needed replacing.

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### Responses

• anssi
Can he safely spend all the interest from his investment?
8 years ago
• Celedor Button
What is the real annual saving the couple must set aside?
8 years ago
• HIEWAN
How long will it take me to accumulate the \$30,000 future values?
8 years ago
• FRE-QALSI
How much is the bank interest rate on \$315.47 for 21 years?
8 years ago
• roberta
What annual interest rate, compounded monthy, is needed for eva to accumulate 200000?
8 years ago
• marina
What is the nominal value of the amount you need to save during the first year?
8 years ago
• bungo
What is the value of a perpetuity that pays \$100 every 3 months forever?
8 years ago
How long will it take for \$400 to grow to \$1000?
8 years ago
• quintina
What is the implicit borrowing rate being paid by customers who choose to defer?
8 years ago
• BILLIE DICKSON
How much interest is 5percent per year on 450000?
8 years ago
• peter
What is the value of 4% perpetuities if the long term interest rate is 6%?
8 years ago
• Antti
What is the value of % perpetuities if the longterm interest rate is %?
8 years ago
• eliano
What is the compounded annual rate implied by this 16 percent rate charged for only four weeks?
8 years ago
• tegan
What annual compound rate for an investment to double in value during a 19year period?
8 years ago
• michael
What is the monthly payment due on a 48 month, \$12,250 installment note at 7.2596%?
7 years ago