Practice

APR of 10 percent, what is your monthly payment? What is the effective annual interest rate

Problems 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 retir-

ing 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 — $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 effec-

tive 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 pay-

ment 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 or-

dinary 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 pres-

ent 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?

The Time Value of Money 73

31. Annuity Due. A store offers two payment plans. Under the installment plan, you pay 25 per-

cent 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 pay-

ments 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 in-

terest 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 per-

cent and monthly payments. In 12 years you decide to sell your house and pay off the mort-

gage. 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 in-

terest 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

0 $1,000 $100 $315.47 $215.47

1 ””” ””” 315.47 ”””

2 ””” ””” 315.47 ”””

3 ””” ””” 315.47 ”””

4 0 0 ” ”

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?

74 SECTION ONE

41. 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?

42. 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?

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

annual rate?

44. 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?

45. 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?

46. 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?

47. 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?

48. 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?

49. Perpetuities. A property will provide $10,000 a year forever. If its value is $125,000, what

must be the discount rate?

50. 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?

51. Applying Time Value. A factory costs $400,000. You forecast that it will produce cash in-

flows 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.

52. 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?

53. 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?

54. 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?

55. Compound Interest. Suppose you take out a $1,000, 3-year loan using add-on interest (see

The Time Value of Money 75

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 — .20 — 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 in-

terest 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 re-

tire 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 re-

tire 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 per-

cent 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?

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.

Challenge What is her real income today in terms of constant 1950 dollars?

Problems 65. Real versus Nominal Rates. If investors are to earn a 4 percent real interest rate, what nom-

inal interest rate must they earn if the inflation rate is:

a. zero

b. 4 percent

c. 6 percent

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:

a. zero

b. 3 percent

c. 6 percent

76 SECTION ONE

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 re-

tirement 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, 21„2 percent perpetuities, pays £2.50 a year forever. What is the value of 4 per-

cent 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 con-