The Mathematics of Money

102 Chapter 3 Compound Interest

Beginning Interest End of Month

Month Balance Earned Balance

1 $5,000.00 $33.33 $5,033.33

2 $5,033.33 $33.56 $5,066.89

3 $5,066.89 $33.78 $5,100.67

4 $5,100.67 $34.01 $5,134.68

5 $5,134.68 $34.23 $5,168.91

6 $5,168.91 $34.46 $5,203.37

7 $5,203.37 $34.69 $5,238.06

8 $5,238.06 $34.92 $5,272.98

9 $5,272.98 $35.15 $5,308.13

10 $5,308.13 $35.39 $5,343.52

11 $5,343.52 $35.62 $5,379.14

12 $5,379.14 $35.86 $5,415.00

Notice that this is $15.00 more than we had at the end of the first year with annual

compounding. Not an enormous amount more, but more nonetheless. As we expected

would happen, more compounding results in more interest.

The Compound Interest Formula for Nonannual Compounding

The previous example shows that monthly compounding doesn’t work all that differently

from annual compounding. Using the same reasoning as in Section 3.1, we can observe that

crediting the first month’s interest is the same as multiplying the principal by (1  0.08/12),

and that crediting 12 months’ interest is the same as multiplying by (1  0.08/12)^12.

Still, it might come as a bit of a surprise that we can use the same compound interest

formula as before. But in fact, even though we didn’t take note of it in Section 3.1, this

was already built into the compound interest formula. Recall that i is the interest rate per

time period, and n is the number of time periods. While in Section 3.1 the time periods

were always years, there is no reason that the time periods couldn’t be months, or days, or

whatever.

We repeat it here:

FORMULA 3.1 (AGAIN)

The Compound Interest Formula

FV 
 PV(1  i)n

where

FV represents the FUTURE VALUE (the ending amount)

PV represents the PRESENT VALUE (the starting amount)

i represents the INTEREST RATE (per time period)

and

n represents the NUMBER OF TIME PERIODS

Since interest rates are usually given as annual rates, when using other compounding peri-

ods we will have to divide. If the term is stated in years, as it often is, we will usually have

to multiply. The following examples will illustrate.

Example 3.2.1 Find the future value of $2,500 at 6% interest compounded monthly

for 7 years.

The PV is $2,500, but the values of i and n require some work.

Since the interest is compounded monthly, the 6% (per year) needs to be divided by 12 (since

each month is 1/12 of a year) to make it monthly. By the same token, the term of 7 years