The Mathematics of Money

44 Chapter 1 Simple Interest

1.5 Nonannual Interest Rates (Optional)

Back in Section 1.1, we noted that, unless it is clearly stated otherwise, an interest rate is

always assumed to be a rate per year. While this is the accepted convention, there is abso-

lutely no reason why an interest rate could not be stated as a rate per month, week, day, or

any other unit of time we want, and in fact this does happen sometimes. We assume annual

rates simply because it is customary to do so, and because having a common standard makes

it easier to compare different rates.

For example, take a look at this excerpt from a credit card’s monthly statement:

Purchases

Cash Advances

Avg. Daily

Balance

Daily Periodic

Rate

Nominal

Annual Rate

FINANCE

CHARGE

$882.71 0.05203% 18.99%

$0.00 0.06025% 21.99%

$14.23

$0.00

Notice the column that gives the daily periodic rate. Because the balance on a credit card can

change from one day to the next, it is reasonable to think of the interest as being calculated

on a daily basis. And so, it may make sense to talk about a rate per day rather than per year.

(Credit cards will be covered in much greater detail in Chapter 10.) This doesn’t really pose

any new difficulties, except that the units of time used in the interest rate must agree with the

units of time used when you are plugging in for T. The following examples will illustrate:

Example 1.5.1 Suppose that my credit card charges a daily simple interest rate of

0.05%. How much interest would I owe for 1 day, on which my balance was $1,800?

The simple interest formula still applies. However, since the rate is per day it stands to reason

that the time, T, should be expressed in terms of days as well. Thus

I  PRT

I  ($1,800)(0.0005)(1)

I  $0.90 or 90 cents

Example 1.5.2 For the same credit card used in Example 1.5.1, how much interest

would I owe for two weeks during which my balance was $2,000?

Since our interest rate is per day, the term must also be in days. Since 1 week is equal to

7 days, 2 weeks is 2(7)  14 days. Thus

I  PRT

I  ($2,000)(0.0005)(14)

I  $14.00

The interest rate being used in these two examples, 0.05%, is much lower than the rates

you are probably accustomed to seeing. It seems to run counter to the conventional wis-

dom about credit card rates—most people think of credit cards as having high rates. But

the number is actually a little deceptive; it is so small because it covers only a short period

of time. This points to another reason why someone might want to use a nonannual inter-

est rate: to make the interest look like less than it would if it were expressed as an annual

rate. What would the interest rate look like if it were expressed as a rate per year?

Converting to an Annual Simple Interest Rate

Looking back at the credit card statement excerpt, right next to the daily rate there is a

column for the nominal annual percentage rate. (For our purposes at the moment, the word

cf

cf