Introduction
Why does money grow faster with compound interest than simple interest?
Simple interest is interest earned on the principal invested amount only, whereas compound interest is interest earned on the principal amount plus interest already earned. So instead of the value of an investment increasing like a straight line, as with simple interest, it will exponentially grow something like this:
For example, consider a deposit of \$1000 in an online account for 2 years that pays 10\% p.a. simple interest. The interest earned in the 2 years is:\$1000 \times 10\% \times 2 = \$200
But suppose that, instead of simple interest, the interest is compounded annually. In this case, the interest earned in the first year would be:\$1000 \times 10\% = \$100
The new principal at the end of the first year would be \$1000 \times \$100 = \$1100. The interest earned in the second year would then be:\$1100 \times 10\% = \$110
So the total compound interest earned over the two years would be:\$100 \times \$110 = \$210
which is \$10 more than that earned with simple interest. Although a \$10 difference may not seem like much, think of how much the difference would have been if the amount invested was a million dollars instead of a thousand, or if the money had been invested for twenty years instead of two.
Effective interest rate
When studying compound interest, we looked at situations where an amount of money compounded more than once a year (e.g. monthly). Even though we divided the interest rate to match the time period, it was demonstrated that the more times interest is compounded in a year, the more interest will be earned.
The table below compares the effect of changing the number of compounding periods when \$1000 is invested for one year at a nominal rate of 5\% p.a. The final amount is calculated using the compound interest formula A=P\times \left(1+\dfrac{r}{n} \right)^{nt}.
| No. periods | Calculation | Final amount | Interest | Effective annual interest rate |
|---|---|---|---|---|
| 1 | 1000 \times \left(1+\frac{0.05}{1} \right)^{1} | \$1050 | \$50 | \dfrac{50}{1000} = 5\% |
| 4 | 1000 \times \left(1+\frac{0.05}{4} \right)^{4} | \$1050.95 | \$50.95 | \dfrac{50.95}{1000} =5.095\% |
| 365 | 1000 \times \left(1+\frac{0.05}{365} \right)^{365} | \$1051.27 | \$51.27 | \dfrac{51.27}{1000} = 5.127\% |
From the table we can see that the amount of interest earned increases when the number of compounding periods increases.
The effective annual interest rate has been calculated using the formula: \text{Effective interest rate}=\dfrac{\text{Interest earned in one year}}{\text{Balance at start of year }} \times 100\%
The published rate of 5\% per annum is called the nominal interest rate.
Note: If we only compound once per year then this nominal interest rate is the same as the effective interest rate.
The effective interest formula: i_{\text{effective}}=\left(1+\dfrac{i}{n} \right)^n-1 where i_{\text{effective}} is the effective interest rate per annum, expressed as a decimal, i is the nominal (or published) interest rate per annum, expressed as a decimal, and n is the number of compounding periods per annum.
Being able to calculate the effective interest rate can come in handy when we are choosing and comparing investments or loans. They allow us to more easily work out how much interest the investment or loan will actually earn and quickly compare rates that have different compounding periods.
When investing money, we want to have the highest possible effective interest rate.
When borrowing money, we want to have the lowest possible effective interest rate.
Examples
Example 1
The Bank of Hilo offers 2 investment opportunities to investors:
Option 1: 6.86\% p.a. simple interest for 10 years.
Option 2: 6.76\% p.a. compound interest for 6 years, compounding semiannually.
On an investment of \$1000 find the interest earned for Option 1.
Give the answer to the nearest cent.
On an investment of \$1000 find the interest earned for Option 2.
Write your answer to the nearest cent.
Find the effective annual interest rate for Option 2. Write your answer as a percentage to two decimal places.
Which of the two investment opportunities has the larger effective annual interest rate?
The effective interest formula: