How to Calculate Compound Interest: Formula and Worked Examples

Compound interest is one of the most powerful concepts in finance. Unlike simple interest, which is calculated only on the initial principal, compound interest earns interest on both the principal and the accumulated interest from previous periods. Understanding how it works can help you make smarter decisions about savings, investments, and loans.

The Compound Interest Formula

The general formula for compound interest is:

Where each variable represents:

• A = the future value of the investment or loan, including interest
• P = the principal (the initial amount of money)
• r = the annual interest rate (expressed as a decimal, so 5% becomes 0.05)
• n = the number of times interest is compounded per year (e.g. 1 for annually, 12 for monthly, 365 for daily)
• t = the number of years the money is invested or borrowed

The total compound interest earned is simply .

Worked Example 1: Annual Compounding

Problem: You invest £5,000 at an annual interest rate of 6%, compounded once per year, for 10 years. What is the final amount?

Here, , , , and . Substituting into the formula:

After 10 years, your investment grows to £8,954.24, earning £3,954.24 in compound interest.

Worked Example 2: Monthly Compounding

Problem: You deposit £10,000 into a savings account that pays 4.8% per annum, compounded monthly. How much will you have after 5 years?

With monthly compounding, :

The final balance is £12,706.41. Notice that monthly compounding yields slightly more than annual compounding at the same nominal rate, because interest starts earning interest sooner.

Worked Example 3: Continuous Compounding

When interest is compounded continuously (infinitely many times per year), we use a special form of the formula involving Euler's number :

Problem: You invest £7,500 at 5% per annum, compounded continuously, for 8 years.

With continuous compounding, the investment grows to £11,188.78.

Comparing Compounding Frequencies

The more frequently interest compounds, the more you earn. Here is a comparison for £1,000 at 10% over 1 year:

• Annually ():
• Quarterly ():
• Monthly ():
• Continuously:

The difference between annual and continuous compounding on £1,000 is only £5.17 over one year. However, over longer periods and with larger sums, the gap widens significantly.

The Rule of 72

A handy shortcut for estimating how long it takes to double your money is the Rule of 72. Simply divide 72 by the annual interest rate:

For example, at 6% interest, your money doubles in approximately years.

Common Mistakes to Avoid

1. Forgetting to convert the rate to a decimal. An interest rate of 5% must be entered as 0.05 in the formula, not 5.
2. Confusing n and t. Remember that is the number of compounding periods per year, while is the total number of years.
3. Using simple interest instead of compound. Simple interest () will always underestimate the final amount when compounding is involved.
4. Ignoring the compounding frequency. A savings account advertising 5% compounded monthly is not the same as 5% compounded annually. Always check the effective annual rate (EAR).
5. Rounding intermediate steps. Keep full precision during calculations and only round the final answer to avoid accumulating rounding errors.

Calculating the Effective Annual Rate (EAR)

To compare accounts with different compounding frequencies, convert to the EAR:

For 5% compounded monthly: .

Try It Yourself

Ready to run your own compound interest calculations? Use our free Percentage Calculator to work through percentage-based problems, or plug in your own values using the formula above to see how your savings could grow over time. For tracking growth rates, our Exponent Calculator handles the power calculations that underpin compound interest.