The Mathematics of Money

120 Chapter 3 Compound Interest

Using Effective Rates

Though interest may be actually credited to an account by using the nominal rate and

compounding, we can also use effective rates, since they are by definition equivalent to

the nominal rate and compounding. The important thing to remember is that the effective

rate is treated as though the compounding is annual regardless of the actual compounding

frequency used with the nominal rate.

Example 3.3.8 Tw elve Corners Federal Credit Union compounds interest on all of

its accounts daily. The credit union is offering an effective rate of 7.33% on its 5-year

certifi cates of deposit. If someone put $20,000 into one of these CDs, how much would

the certifi cate be worth at maturity?

Even though we know that the interest will compound daily, we do not know the nominal rate

being used with that daily compounding. We do have the effective rate, though, which we

can use just as well since it is equivalent to the nominal rate. The effective rate uses annual

compounding though. So:

FV 
 PV(1  i)n

FV 
 $20,000(1.0733)^5

FV 
 $28,486.27

It is critically important to be clear about whether a given rate is nominal or effective. Even

though we know that the interest in the previous example will actually be compounded daily,

it will not be compounded daily with a 7.33% rate. While it would be an easy and entirely

understandable error to use that 7.33% with daily compounding, this would be incorrect. The

nominal compounding frequency is used with the nominal rate; the effective rate is always

treated as annual compounding regardless of how frequently interest actually compounds.

Using Effective Rate with “Messy” Terms

In the previous example, annual compounding made matters quite a bit simpler, since it

meant we didn’t have to divide the rate by anything to get i, or do anything to the term to

get n. This will not always work out so well, though.

Example 3.3.9 Twelve Corners FCU also allows its customers to set up a CD with

whatever term suits them. For CDs with terms between 2 and 5 years, the effective

rate currently being offered is 6.25%. If a customer deposits $20,000 to a CD with a

term of 1,000 days, how much will it be worth a maturity?

Here, since the effective rate is annual, n must be in terms of years. In the past, whenever

we have had to calculate n, we have multiplied. To convert years into days, for example, we

have multiplied by 365 (or 360 if bankers’ rule is being used.) Here we want to go the other

direction, and so we instead divide by 365 to get n 
 1,000/365 
 2.739726027. While

it may seem strange to have a fraction or decimal in an exponent, this is actually perfectly

permissible mathematically. Marching forward:

FV 
 PV(1  i)n

FV 
 PV(1.0625)2.739726027

FV 
 $23,613.70

Typing the entire decimal into the calculator is annoying, and can be avoided just by

entering the fraction as the exponent. In that case, though, parentheses need to be put

around the fraction. Using this with the previous example, we would enter:

Operation Result

1+.0625


^(1000/365)


*20000

1.0625

1.180685089

23613.70

cb