9 Financial Mathematics
9.2 Compound Interest
Learning Objectives
By the end of this section it is expected that you will be able to:
- Determine the compound amount (future value) of an investment or loan
- Determine the interest component of an investment or loan that involves compound interest
- Determine the present value of a compound amount
Compound Interest
We have seen that with simple interest an investment will earn interest on the original amount. For an investment of $100 earning 10% simple interest, the interest earned after one year will be $10 since 10% of $100 = $10. An investment will grow more quickly if the interest is calculated more often than once a year. Interest will not only be calculated on the principal amount but also on the previously earned interest. This process is referred to compounding.
Figure 1 illustrates the process of compounding or earning interest on interest. Consider an investment of $100 that earns 10%/year with interest being compounded semiannually. With semiannual compounding the interest on the investment will be calculated twice during the year.
Using the simple interest formula I = Prt, at the end of six months (half a year) interest will be calculated as follows:
I = $100 x 10% x 1/2 year = $5.
Adding this $5 to the principal of $100 you will have $105 at the end of the first six months. At the end of the year interest will be calculated again on the $105:
I = $105 x 10% x 1/2 year = $5.25.
Adding this $5.25 to $105 you will have $110.25 at the end of the year. In this case you would be earning interest not only on the original principal of $100, but also on the previously earned interest of $5. When interest is earned on interest, we say the interest is compounded. The total amount of principal and accumulated interest at the end of a loan or investment is called the compound amount.
Consider a $100 investment that earns 10%/year compounded annually. The table in Figure 2 shows how the value of the $100 investment will grow over a 6-year period.
| Year | Amount at the beginning of the year | Earned Interest | Year End Total |
|---|---|---|---|
| 1 | $100 | $10 | $110 |
| 2 | $110 | $11 | $121 |
| 3 | $121 | $12.10 | $133.10 |
| 4 | $133.10 | $13.31 | $146.41 |
| 5 | $146.41 | $14.64 | $161.05 |
| 6 | $161.05 | $16.11 | $177.16 |
Fig. 2
At the beginning of Year 1, $100 is invested, so the interest earned in the first year will be:
I = Prt = $100 × 0.10 × 1 = $10. This is added to the original $100 to result in $110 at the end of Year 1.
At the beginning of Year 2 the process will repeat but the principal P is now $110.
I = Prt = $110 × 0.10 ×1 = $11 in interest so at the end of Year 2 there will be:
$110 + $11 = $121 in the account.
Notice that the compound amount at the end of the six year period is $177.16. The investment has earned an accumulated $77.16 in interest. If the investment had earned simple interest as opposed to compound interest it would have only earned:
I = Prt = 100 × 0.10 × 6 =