Simple interest is calculated only on the original principal, whereas compound interest is calculated on the principal plus interest already earned. Therefore, with a positive rate over multiple periods, compound interest normally produces a larger final amount.
Let be the original principal, the interest rate per period as a decimal, and the number of periods.
| Feature | Simple interest | Compound interest |
|---|---|---|
| Amount after periods |
| Interest calculated on | Original principal only | Principal and accumulated interest |
| Growth pattern | Linear or arithmetic | Exponential or geometric |
| Interest added each period | Constant amount | Increasing amount |
Simple interest creates an arithmetic sequence because the same amount, (Pr), is added each period. Compound interest creates a geometric sequence because the current balance is repeatedly multiplied by .
For example, suppose \1{,}0005%$ per year for three years.
With simple interest:
With compound interest:
The compound investment is worth \7.63$ more because the interest earned in years 1 and 2 also earns interest later. This mechanism is called interest on interest.
If interest is compounded times per year at nominal annual rate , use
Here, divide the annual rate by and multiply the number of years by . For quarterly compounding, for example, .
A common misconception is that compound interest is always greater. After one compounding period, both methods give the same amount; the difference develops only across multiple periods when previously earned interest begins earning further interest.
This is Mathematics AA, SL 1.4: Financial applications, and may appear on Paper 1 or Paper 2. For calculate, show the formula, convert the percentage rate to a decimal, identify the correct number of compounding periods, and give an exact answer or 3 significant figures unless instructed otherwise. Do not attempt to derive the compound-interest formula; that derivation is excluded from examination questions.
