5.3: Compound Interest
Most banks, loans, credit cards, etc. charge you compound interest, not simple interest. Compound interest is interest paid both on the original principal and on all interest that has been added to the original principal. Interest on a mortgage or auto loan is compounded monthly. Interest on a savings account can be compounded quarterly (four times a year). Interest on a credit card can be compounded weekly or daily!
| Compounding type | Number of compounding periods per year |
|---|---|
| Annually | 1 |
| Semiannually | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |
| Compound Interest: Interest paid on the principal AND the interest accrued. |
Suppose you invest $3000 into an account that pays you 7% interest per year for four years. Using compound interest, after the interest is calculated at the end of each year, then that amount is added to the total amount of the investment. Then the following year, the interest is calculated using the new total of the loan.
| Year | Interest Earned | Total of Loan |
|---|---|---|
| 1 | $3000*0.07 = $210 | $3000 + $210 = $3210 |
| 2 | $3210 *0.07 = $224.70 | $3210 + $224.70 = $3434.70 |
| 3 | $3434.70*0.07 = $240.43 | $3434.70 + $240.43 = $3675.13 |
| 4 | $3675.13 *0.07 = $257.26 | $3675.13 + $257.26 = $3932.39 |
| Total | $932.39 |
So, after four years, you have earned $932.39 in interest for a total of $3932.39.
\[F=P\left(1+\frac{r}{n}\right)^{nt} \nonumber \]
where
- F = Future value
- P = Present value
- r = Annual percentage rate (APR) changed into a decimal
- t = Number of years
- n = Number of compounding periods per year
Example \(\PageIndex{2}\): Comparing Simple Interest versus Compound Interest
Let’s compare a savings plan that pays 6% simple interest versus another plan that pays 6% annual interest compounded quarterly. If we deposit $8,000 into each savings account, how much money will we have in each account after three years?
6% Simple Interest : P = $8,000, r = 0.06, t = 3
Thus, we have $9440.00 in the simple interest account after three years.
6% Interest Compounded Quarterly: P = $8,000, r = 0.06, t = 3, n=4
So, we have $9564.95 in the compounded quarterly account after three years.
With simple interest we earn $1440.00 on our investment, while with compound interest we earn $1564.95.
Example \(\PageIndex{3}\): Compound Interest—Compounded Monthly
In comparison with Example \(\PageIndex{2}\) consider another account with 6% interest compounded monthly. If we invest $8000 in this account, how much will there be in the account after three years?
P = $8,000, r = 0.06, t = 3, n = 12
Thus, we will have $9573.44 in the compounded monthly account after three years.
Interest compounded monthly earns you $9573.44 - $9564.95 = $8.49 more than interest compounded quarterly.
Example \(\PageIndex{4}\): Compound Interest—Savings Bond
Sophia’s grandparents bought her a savings bond for $200 when she was born. The interest rate was 3.28% compounded semiannually, and the bond would mature in 30 years. How much will Sophia’s bond be worth when she turns 30?
P = $200, r = 0.0328, t = 30, n = 2
Sophia’s savings bond will be worth $530.77 after 30 years.
| Continuous Compounding: Interest is compounded infinitely many times per year. |
|
Continuous Compounding Interest Formula : where, F = Future value P = Present value r = Annual percentage rate (APR) changed into a decimal t = Number of years |
Example \(\PageIndex{5}\): Continuous Compounding Interest
Isabel invested her inheritance of $100,000 into an account earning 5.7% interest compounded continuously for 20 years. What will her balance be after 20 years?
P = $100,000, r = 0.057, t = 20
Isabel’s balance will be $312,676.84 after 20 years.
| Annual Percentage Yield (APY) : the actual percentage by which a balance increases in one year. |
Example \(\PageIndex{6}\): Annual Percentage Yield (APY)
Find the Annual Percentage Yield for an investment account with
a. 7.7% interest compounded monthly
b. 7.7% interest compounded daily
c. 7.7% interest compounded continuously.
To find APY, it is easiest to examine an investment of $1 for one year.
- P = $1, r = 0.077, t = 1, n = 12
The percentage the $1 was increased was 7.9776%. The APY is 7.9776%.
- P = $1, r = 0.077, t = 1, n = 365
The percentage the $1 was increased was 8.0033%. The APY is 8.0033%.
- P = $1, r = 0.077, t = 1
The percentage the $1 was increased was 8.0042%. The APY is 8.0042%.