4.01 Compound interest

Since compound interest always calculates the interest based on the current total, we can think of it as a continuous application of simple interest, one period at a time, where we update the principal amount to match the new total after each application.

We can calculate compound interest in this way using a table like the one below. In this example, the starting principal amount is \$1000 with a compounding interest rate of 5\% per annum.

If we check, we will see that the new total at the end of a year is always equal to 1.05 times the principal amount used, which was equal to the total at the end of the previous year.

With each earning of interest the total increases, increasing the principal amount used which in turn increases the amount of interest earned each year. What this tells us is that compound interest results in a total that increases at an increasing rate.

When calculating compound interest using repeated applications of simple interest, we notice that the new total at the end of each year is simply a percentage increase of the previous year's total.

Using the example with \$1000 invested at an interest rate of 5\% p.a., compounding annually, our calculations take the form:

• \text{End of 1st year} = 1000 \times 1.05

• \text{End of 2nd year} = 1000 \times 1.05 \times 1.05

• \text{End of 3rd year} = 1000 \times 1.05 \times 1.05 \times 1.05

If we don't evaluate each year's total, we can see a pattern form. Since each percentage increase will be the same, we can just apply it as many times as there are periods of interest.

It is also worth noting that since we are multiplying by the same amount for each percentage increase when calculating compound interest, we could also compress these terms using indices, like so:

• \text{End of 1st year} = 1000 \times 1.05

• \text{End of 2nd year} = 1000 \times {1.05}^2

• \text{End of 3rd year} = 1000 \times {1.05}^3

Examples

Appreciation is when an item's value increases over time, usually by some fixed percentage of its current value. An example of this is how antiques or collectables become rarer and more expensive over time.

Depreciation is when an item's value decreases over time, again by some fixed percentage of its current value. An example of this is how a new car will lose value over time as it gets older and more out of date.

Since we often calculate appreciation and depreciation in terms of annual percentage increases or decreases, we can model their change in value the same way that we do with compound interest.

Looking at it one year at a time, appreciation is simply a repeated application of percentage increases, exactly like compound interest.

Examples

So far, the examples we have looked at have all had interest calculated annually. However, interest can be calculated over any period and the interest rate per period will need to match it.

To match the interest rate to the period duration, we can use interest rate conversions.

It is also important to match the total number of periods to the duration of the investment. To do this, we can simply divide the duration of the investment by the duration of the period.

Examples