Time Value of Money

(Van & Horne)

VALUE OF MONEY

Future Value Present Value

Simple Interest Compound Interest

Unknown Discount Unknown Time

Rate Period

Annuity Due Ordinary Annuity

(Beginning of each (End of each Period)
Period)
 Time Value of Money
Introduction:

Time value of money is a critical consideration in financial and investment decisions.

For example, compound interest calculations are needed to determine future sums of
money resulting from an investment. Discounting, or the calculation of present value,
which is inversely related to compounding, is used to evaluate future cash flow
associated with capital budgeting projects. There are plenty of applications of time value
of money in finance.

Future Values—Compounding

A Rupee in hand today is worth more than a dollar to be received tomorrow because of
the interest it could earn from putting it in a savings account or placing it in an
investment account. Compounding interest means that interest earns interest. For the
discussion of the concepts of compounding and time value, let us define:

Fn = future value= the amount of money at the end of year n

P = principal
i= annual interest rate
n= number of years
Then
F1= the amount of money at the end of year 1
= principal and interest = P+I P= P (1+i)
F2= the amount of money at the end of year 2
= F1 (1+i) = P (1+i) (1+i) = P (1+i)2

The future value of an investment compounded annually at rate i for n year is

Fn = P (1+i)n = P. (FVIFi.n)

Where P. FVIFi.n is the future value interest factor for Rs.1 Table III at the end of this
book.

Example—1

Amjad placed Rs.1,000 in a saving account earning 8% interest compounded annually.

How much money will he have in the account at the end of 4 years?

Answer
 Fn = P (1+i)n

Fn = 1,000 (1+0.08)4
= 1,360

Example—2

Raheel Khan invested a large sum of money in the stock of AICL Corporation. The
company paid Rs.3 dividend per share. The dividend is expected to increase by 20
percent per year for the next 3 years. He wishes to project the dividends for years 1
through 3.

Answer

Fn = P (1+i)n

F1 = 3 (1+0.20)1 = 3.6
F2 = 3 (1+0.20)2 = 4.32
F3 = 3 (1+0.20)3 = 5.18

Example—3

In 1790 Jawaid Jabbar bought approximately an acre of land on the east side of Mehran
Valley for Rs.58. Jawaid, who was considered a shrewd (sharp) investor, made many
such purchases, How much would his descendants have in 2005, if instead of buying
the land, Jawaid had invested the Rs.58 at 5 percent compound annual interest?

Answer

Fn = P (1+i)n

F215 = 58 (1+0.05)215
= 2,084,883.09
Example—4

Determine the present value, if you invest something today and receive Rs.2000 after
10 years form now taking the interest rate 8 percent.

Answer

PVo = FV10 (1/ (1+i)n)

= 2000 (1/ (1.08)10)
= 926.38

Unknown Discount Rate and Time Period

Some times we are faced with a time-value-of-money situation in which we know both
future and present values, as well as the number of time periods involved. What is
unknown is the compound interest rate (i).

Example—5

If you invest Rs.1000 today, you will receive Rs.3000 in exactly 8 years. What is the
interest rate we earn/ offer to us.
Answer

FV8 = PVo (FVIFi.8 )

3000 = 1000 (FVIFi.8 )
FVIFi.8 = 3000/1000 = 3

(1+i)8 = 3
(1+i) = 31/8 = 30.125 = 1.1472
i= 0.1472 or 14.72%

Example—6

How much time the investment of Rs.1000 will reach to Rs.1900 if the interest rate is
10%

Answer

FVn = PVo (FVIF10%.n)

1900=1000 (FVIF10%.n)
FVIF10%.n = 1900/1000 = 1.9

(1+0.1)n =1.9
n(ln 1.1) = ln 1.9
n= (ln 1.9)/ (ln 1.1) = 6.73

Annuities

Ordinary Annuity, an annuity is a series of equal payments or receipts occurring over

specified number of periods. In an ordinary annuity, payments or receipts occur at the
end of each period and expressed algebraically, with FVA n, defined as the future
(compound) value of an annuity, R the periodic receipt (or payment), and n the length
of the annuity, the formula for FVAn, is

Future value of annuity

R ([(1 + i) n – 1] / i) or R x (1 + i) n -1 / i
Or equivalently FVA n = R (FAIFA n)

Present value of annuity

R ([(1 + i) n – 1] / i) or R x (1 + i) n -1/ i

Or equivalently FVA n = R (FAIFA n)

1. The following are exercises in future (terminal) values:
a) At the end of three years, how much is an initial deposit of $100 worth, assuming
a compound annual interest rate of (i) 100 percent? (ii) 10 percent? (iii) 0
percent?
b) At the end of five years, how much is an initial $500 deposit followed by five year-
end, annual $100 payments worth, assuming a compound annual interest rate of
(i) 10 percent? (ii) 5 percent? (iii) 0 percent?
c) At the end of six years, how much is an initial $500 deposit followed by five year-
end, annual $100 payments worth, assuming a compound annual interest rate of
(i) 10 percent? (ii) 5 percent? (iii) 0 percent?
d) At the end of three years, how much is an initial $100 deposit worth, assuming a
quarterly compounded annual interest rate of (i) 100 percent? (ii) 10 percent?
e) Why do your answers to Part (d) differ from those to Part (a)?
f) At the end of 10 years, how much is a $100 initial deposit worth, assuming an
annual interest rate of 10 percent compounded (i) annually? (ii) semiannually?
(iii) quarterly? (iv) continuously?
2. The following are exercises in present values:
a) $100 at the end of three years is worth how much today, assuming a discount
rate of (i) 100 percent? (ii) 10 percent? (iii) 0 percent?
b) What is the aggregate present value of $500 received at the end of each of the
next three years, assuming a discount rate of (i) 4 percent? (ii) 25 percent?
c) $100 is received at the end of one year, $500 at the end of two years, and
$1,000 at the end of three years. What is the aggregate present value of these
receipts, assuming a discount rate of (i) 4 percent? (ii) 25 percent?
d) $1,000 is to be received at the end of one year, $500 at the end of two years,
and $100 at the end of three years. What is the aggregate present value of these
receipts assuming a discount rate of (i) 4 percent? (ii) 25 percent?
e) Compare your solutions in Part (c) with those in Part (d) and explain the reason
for the differences.
3. Joe Hernandez has inherited $25,000 and wishes to purchase an annuity that will
provide him with a steady income over the next 12 years. He has heard that the local
savings and loan association is currently paying 6 percent compound interest on an
annual basis. If he were to deposit his funds, what year-end equal-dollar amount (to the
nearest dollar) would he be able to withdraw annually such that he would have a zero
balance after his last withdrawal 12 years from now?
4. You need to have $50,000 at the end of 10 years. To accumulate this sum, you have
decided to save a certain amount at the end of each of the next 10 years and deposit it
in the bank. The bank pays 8 percent interest compounded annually for long-term
deposits. How much will you have to save each year (to the nearest dollar)?

5. Same as Problem 4 above, except that you deposit a certain amount at the beginning
of each of the next 10 years. Now, how much will you have to save each year (to the
nearest dollar)?
6. Vernal Equinox wishes to borrow $10,000 for three years. A group of individuals
agrees to lend him this amount if he contracts to pay them $16,000 at the end of the
three years. What is the implicit compound annual interest rate implied by this contract
(to the nearest whole percent)?

7. You have been offered a note with four years to maturity, which will pay $3,000 at the
end of each of the four years. The price of the note to you is $10,200. What is the
implicit compound annual interest rate you will receive (to the nearest whole percent)?
8. Sales of the P.J. Cramer Company were $500,000 this year, and they are expected
to grow at a compound rate of 20 percent for the next six years. What will be the sales
figure at the end of each of the next six years?

9. The H & L Bark Company is considering the purchase of a debarking machine that is
expected to provide cash flows as follows:

If the appropriate annual discount rate is 14 percent, what is the present value of this
cash-flow stream?
10. Suppose you were to receive $1,000 at the end of 10 years. If your opportunity rate
is 10 percent, what is the present value of this amount if interest is compounded (a)
annually? (b) quarterly? (c) continuously?
11. In connection with the United States Bicentennial, the Treasury once contemplated
offering a savings bond for $1,000 that would be worth $1 million in 100 years.
Approximately what compound annual interest rate is implied by these terms?

12. Selyn Cohen is 63 years old and recently retired. He wishes to provide retirement
income for himself and is considering an annuity contract with the Philo Life Insurance
Company. Such a contract pays him an equal-dollar amount each year that he lives. For
this cash-flow stream, he must put up a specific amount of money at the beginning.
According to actuary tables, his life expectancy is 15 years, and that is the duration on
which the insurance company bases its calculations regardless of how long he actually
lives.
a) If Philo Life uses a compound annual interest rate of 5 percent in its calculations,
what must Cohen pay at the outset for an annuity to provide him with $10,000
per year? (Assume that the expected annual payments are at the end of each of
the 15 years.)
b) What would be the purchase price if the compound annual interest rate is 10
percent?
c) Cohen had $30,000 to put into an annuity. How much would he receive each
year if the insurance company uses a 5 percent compound annual interest rate in
its calculations? A 10 percent compound annual interest rate?
13. The Happy Hang Glide Company is purchasing a building and has obtained a
$190,000 mortgage loan for 20 years. The loan bears a compound annual interest rate
of 17 percent and calls for equal annual installment payments at the end of each of the
20 years. What is the amount of the annual payment?

14. Establish loan amortization schedules for the following loans to the nearest cent
(see Table 3.8 for an example):
a) A 36-month loan of $8,000 with equal installment payments at the end of each
month. The interest rate is 1 percent per month.
b) A 25-year mortgage loan of $184,000 at a 10 percent compound annual interest
rate with equal installment payments at the end of each year.
15. You have borrowed $14,300 at a compound annual interest rate of 15 percent. You
feel that you will be able to make annual payments of $3,000 per year on your loan.
(Payments include both principal and interest.) How long will it be before the loan is
entirely paid off (to the nearest year)?

16. Lost Dutchman Mines, Inc., is considering investing in Peru. It makes a bid to the
government to participate in the development of a mine, the profits of which will be
realized at the end of five years. The mine is expected to produce $5 million in cash to
Lost Dutchman Mines at that time. Other than the bid at the outset, no other cash flows
will occur, as the government will reimburse the company for all costs. If Lost Dutchman
requires a nominal annual return of 20 percent (ignoring any tax consequences), what is
the maximum bid it should make for the participation right if interest is compounded (a)
annually? (b) semiannually? (c) quarterly? (d) continuously?

17. Earl E. Bird has decided to start saving for his retirement. Beginning on his twenty-
first birthday, Earl plans to invest $2,000 each birthday into a savings investment
earning a 7 percent compound annual rate of interest. He will continue this savings
program for a total of 10 years and then stop making payments. But his savings will
continue to compound at 7 percent for 35 more years, until Earl retires at age 65. Ivana
Waite also plans to invest $2,000 a year, on each birthday, at 7 percent, and will do so
for a total of 35 years. However, she will not begin her contributions until her thirty-first
birthday. How much will Earl’s and Ivana’s savings programs be worth at the retirement
age of 65? Who is better off financially at retirement, and by how much?

18. When you were born, your dear old Aunt Minnie promised to deposit $1,000 in a
savings account for you on each and every one of your birthdays, beginning with your
first. The savings account bears a 5 percent compound annual rate of interest. You
have just turned 25 and want all the cash. However, it turns out that dear old (forgetful)
Aunt Minnie made no deposits on your fifth, seventh, and eleventh birthdays. How much
is in the account now – on your twenty-fifth birthday?
19. Assume that you will be opening a savings account today by depositing $100,000.
The savings account pays 5 percent compound annual interest, and this rate is
assumed to remain in effect for all future periods. Four years from today you will
withdraw R dollars. You will continue to make additional annual withdrawals of R dollars
for a while longer – making your last withdrawal at the end of year 9 – to achieve the
following pattern of cash flows over time. (Note: Today is time period zero; one year
from today is the end of time period 1; etc.) How large must R be to leave you with
exactly a zero balance after your final R withdrawal is made at the end of year 9? (Tip:
Making use of an annuity table or formula will make your work a lot easier!)
20. Suppose that an investment promises to pay a nominal 9.6 percent annual rate of
interest. What is the effective annual interest rate on this investment assuming that
interest is compounded (a) annually? (b) semiannually? (c) quarterly? (d) monthly? (e)
daily (365 days)? (f ) continuously? (Note: Report your answers accurate to four decimal
places – e.g., 0.0987 or 9.87%.)

21. “Want to win a million dollars? Here’s how. . . . One winner, chosen at random from
all entries, will win a $1,000,000 annuity.” That was the statement announcing a contest
on the World Wide Web. The contest rules described the “million-dollar prize” in greater
detail: “40 annual payments of $25,000 each, which will result in a total payment of
$1,000,000. The first payment will be made January 1; subsequent payments will be
made each January thereafter.” Using a compound annual interest rate of 8 percent,
what is the present value of this “million-dollar prize” as of the first installment on
January 1?
22. It took roughly 14 years for the Dow Jones Average of 30 Industrial Stocks to go
from 1,000 to 2,000. To double from 2,000 to 4,000 took only 8 years, and to go from
4,000 to 8,000 required roughly 2 years. To the nearest whole percent, what compound
annual growth rates are implicit in these three index-doubling milestones?
 Time Value of Money
(Myers)

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 2024
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 4 percent?
What if it was 8 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:

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?
c) Why do your answers to (a) and (b) differ?

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

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

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

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 Loan shark 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.

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 × $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 yearend) 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.
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?

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 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.

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 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 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?