The Mathematics of Money

132 Chapter 3 Compound Interest

We then subtract 1 from both sides to get i 
 0.0371372893. Moving the decimal, we

conclude that i 
 3.71%.

Remember that there is a particular need for caution when using fractional exponents with

your calculator due to a side effect of order of operations. If you enter 1.2^1/5 
 on most

calculators, the result will be 0.24, not the correct 1.0371372893. The reason is that under

order of operations, exponents take precedence over division, and so the calculator interprets

this entry as “fi rst raise 1.2 to the power 1, then divide the result by 5.” To obtain the correct

answer, you must use parentheses around the 1/5, and so instead enter 1.2^(1/5) 
 which

will give the correct result.

Solving for the Interest Rate (Nonannual Compounding)

Example 3.5.2 Kelli has a certifi cate of deposit on which the interest compounds

monthly. The balance in the account is $4,350.17 today, and the certifi cate will mature

2 years from today with a value of $4,715.50. What is the interest rate on her certifi cate?

We follow the same basic steps here as in the previous problem (being careful to note that

n must be in months):

FV 
 PV(1 + i)n

$4,715.50 
 $4,350.17(1 + i)^24

1.083980626 
 (1 + i)^24

1.083980626^1 ⁄^24 
 1 + i

1.003365652 
 1 + i

i 
 0.003365652

This gives i as a rate per month, however. Since i is the annual rate divided by 12, we must

multiply by 12 to get the annual interest rate: (0.003365652)(12) 
 0.040387829, and so

the interest rate to two decimal places is 4.04%.

Converting from Effective Rates to Nominal Rates

In Section 3.3 we discussed finding the effective rate for a given nominal rate. We did

not discuss going the other direction. We can, however, use a similar approach to that in

Example 3.5.2 to accomplish this.

Example 3.5.3 What quarterly compounded nominal interest rate is equivalent to

an effective rate of 17.4%?

We know that compounding at the unknown nominal rate for one year provides the same

result as using the effective rate for one year. Therefore:

1.174^1 
 (1 + i)^4

1.174¼ 
 1 + i

1.040919212 
 1  i

i 
 0.040919212

Multiplying this by 4 gives a nominal quarterly rate of 16.37%.

Solving for Time

Example 3.5.4 How long will it take for $100 to grow to $500 at 6% interest

compounded quarterly?

We begin once again by substituting into the formula, remembering to divide the rate by 4

because it is compounded quarterly:

FV 
 PV(1  i)n

$500 
 $100(1  .015)n

5 
 (1.015)n