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ple, the payments on a loan), it is easiest to use all nominal quantities.

Effective Annual Interest Rates
Thus far we have used annual interest rates to value a series of annual cash flows. But
interest rates may be quoted for days, months, years, or any convenient interval. How
should we compare rates when they are quoted for different periods, such as monthly
versus annually?
Consider your credit card. Suppose you have to pay interest on any unpaid balances
at the rate of 1 percent per month. What is it going to cost you if you neglect to pay off
your unpaid balance for a year?
Don™t be put off because the interest rate is quoted per month rather than per year.
The important thing is to maintain consistency between the interest rate and the num-
ber of periods. If the interest rate is quoted as a percent per month, then we must define
the number of periods in our future value calculation as the number of months. So if
you borrow $100 from the credit card company at 1 percent per month for 12 months,
you will need to repay $100 — (1.01)12 = $112.68. Thus your debt grows after 1 year to
$112.68. Therefore, we can say that the interest rate of 1 percent a month is equivalent
to an effective annual interest rate, or annually compounded rate of 12.68 percent.
In general, the effective annual interest rate is defined as the annual growth rate al-
Interest rate that is lowing for the effect of compounding. Therefore,
annualized using compound
(1 + annual rate) = (1 + monthly rate)12
When comparing interest rates, it is best to use effective annual rates. This compares
interest paid or received over a common period (1 year) and allows for possible com-
pounding during the period. Unfortunately, short-term rates are sometimes annualized
by multiplying the rate per period by the number of periods in a year. In fact, truth-in-
lending laws in the United States require that rates be annualized in this manner. Such
ANNUAL PERCENTAGE rates are called annual percentage rates (APRs).9 The interest rate on your credit card
RATE (APR) Interest loan was 1 percent per month. Since there are 12 months in a year, the APR on the loan
is 12 — 1% = 12%.
rate that is annualized using
simple interest. If the credit card company quotes an APR of 12 percent, how can you find the ef-
fective annual interest rate? The solution is simple:
Step 1. Take the quoted APR and divide by the number of compounding periods in a
year to recover the rate per period actually charged. In our example, the interest was
calculated monthly. So we divide the APR by 12 to obtain the interest rate per month:
APR 12%
Monthly interest rate = = = 1%
12 12

9 Thetruth-in-lending laws apply to credit card loans, auto loans, home improvement loans, and some loans
to small businesses. APRs are not commonly used or quoted in the big leagues of finance.

Step 2. Now convert to an annually compounded interest rate:
(1 + annual rate) = (1 + monthly rate)12 = (1 + .01)12 = 1.1268
The annual interest rate is .1268, or 12.68 percent.
In general, if an investment of $1 is worth $(1 + r) after one period and there are m
periods in a year, the investment will grow after one year to $(1 + r)m and the effective
annual interest rate is (1 + r)m “ 1. For example, a credit card loan that charges a
monthly interest rate of 1 percent has an APR of 12 percent but an effective annual in-
terest rate of (1.01)12 “ 1 = .1268, or 12.68 percent. To summarize:

The effective annual rate is the rate at which invested funds will grow over the
course of a year. It equals the rate of interest per period compounded for the
number of periods in a year.

The Effective Interest Rates on Bank Accounts
Back in the 1960s and 1970s federal regulation limited the (APR) interest rates banks
could pay on savings accounts. Banks were hungry for depositors, and they searched for
ways to increase the effective rate of interest that could be paid within the rules. Their
solution was to keep the same APR but to calculate the interest on deposits more fre-
quently. As interest is calculated at shorter and shorter intervals, less time passes before
interest can be earned on interest. Therefore, the effective annually compounded rate of
interest increases. Table 1.10 shows the calculations assuming that the maximum APR
that banks could pay was 6 percent. (Actually, it was a bit less than this, but 6 percent
is a nice round number to use for illustration.)
You can see from Table 1.10 how banks were able to increase the effective interest
rate simply by calculating interest at more frequent intervals.
The ultimate step was to assume that interest was paid in a continuous stream rather
than at fixed intervals. With one year™s continuous compounding, $1 grows to eAPR,
where e = 2.718 (a figure that may be familiar to you as the base for natural logarithms).
Thus if you deposited $1 with a bank that offered a continuously compounded rate of 6
percent, your investment would grow by the end of the year to (2.718).06 = $1.061837,
just a hair™s breadth more than if interest were compounded daily.

A car loan requiring quarterly payments carries an APR of 8 percent. What is the ef-
Self-Test 17
fective annual rate of interest?

TABLE 1.10
Compounding Periods Per-Period Growth Factor of Effective
Compounding frequency and
Period per Year (m) Interest Rate Invested Funds Annual Rate
effective annual interest rate
(APR = 6%) 1 year 1 6% 1.06 6.0000%
Semiannually 2 3 1.032 = 1.0609 6.0900
Quarterly 4 1.5 1.0154 = 1.061364 6.1364
Monthly 12 .5 1.00512 = 1.061678 6.1678
Weekly 52 .11538 1.001153852 = 1.061800 6.1800
Daily 365 .01644 1.0001644365 = 1.061831 6.1831
The Time Value of Money 69

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 ex-
ample, 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
Related Web and future value
Links www.bankrate.com/brm/default.asp Different interest rates for a variety of purposes, and some
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

nominal interest rate
future value annuity
Key Terms
perpetuity real interest rate
compound interest
annuity factor effective annual interest rate
simple interest
present value (PV) annuity due annual percentage rate (APR)
discount rate inflation
discount factor real value of $1

Quiz 1. Present Values. Compute the present value of a $100 cash flow for the following combina-
tions 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 Vic-
torian 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 inter-
est. How much was each entitled to if the interest rate was 5 percent? What if it was 10 per-
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
$400 11 $684
$183 4 $249
$300 7 $300

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.

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?
The Time Value of Money 71

Year Cash Flow
1 $200
2 $400
3 $300
10. Number of Periods. How long will it take for $400 to grow to $1,000 at the interest rate
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% 1 month
8% 3 months
10% 6 months
12. Calculating Interest Rate. Find the APR (the stated interest rate) for each case:
Effective Annual Compounding
Interest Rate Period
10.00% 1 month
6.09% 6 months
8.24% 3 months

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 per-
cent 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 com-
pound 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 ac-
counts. 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 de-
posit $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
19. Calculating Interest Rate. A zero-coupon bond which will pay $1,000 in 10 years is sell-
ing 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 mil-
lion a year for 5 years. A less famous receiver signed a $14 million 5-year contract provid-
ing $4 million now and $2 million a year for 5 years. Who is better paid? The interest rate
is 12 percent.


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