Compound Interest Formula

Compound interest is interest that is calculated not only on the original amount of money deposited or borrowed (the principal) but also on the interest that has already been added to it. Each time interest is calculated, it gets folded into the balance, so the next round of interest is earned on a slightly bigger number. Because the amount grows on top of itself, compound interest is an example of exponential growth, the same pattern you see in other exponential models.

This is different from simple interest, which only ever calculates interest on the original principal. Over short periods the two look similar, but over many years compound interest pulls noticeably ahead.

The compound interest formula gives the future value \(A\) of an account after compounding for a number of years:

\(A = P\left(1 + \frac{r}{n}\right)^{nt}\)

Each letter stands for a specific quantity:

• \(A\) — the final amount in the account (principal plus all interest earned)
• \(P\) — the principal, the amount originally deposited or borrowed
• \(r\) — the annual interest rate, written as a decimal (5% becomes \(0.05\))
• \(n\) — the number of times interest is compounded per year (12 for monthly, 4 for quarterly, 365 for daily)
• \(t\) — the number of years the money is invested or borrowed for

Notice the exponent \(nt\): this is where the exponential behavior comes from. Getting the order of operations right when evaluating an expression like \(\left(1+\frac{r}{n}\right)^{nt}\) matters a lot here, so it helps to be comfortable with exponents in the order of operations before working through several compound interest problems.

Suppose \(1{,}000\) is deposited into an account earning \(5\%\) annual interest, compounded monthly, for \(10\) years. Identify the values first:

\(P = 1000\), \(r = 0.05\), \(n = 12\), \(t = 10\)

Substitute into the formula:

\(A = 1000\left(1 + \frac{0.05}{12}\right)^{12 \times 10}\)

\(A = 1000(1.0041\overline{6})^{120}\)

\(A \approx 1647.01\)

So the account grows from \(1{,}000\) to about \(1{,}647.01\) over ten years, with the extra \(647.01\) coming entirely from interest, including interest earned on interest.

For the same principal, rate, and time, a larger value of \(n\) produces a larger final amount, because interest gets added to the balance more often. The table below compares the same \(1{,}000\) deposit at \(5\%\) for \(10\) years under different compounding frequencies.

The differences look small here, but they grow with larger principals, higher rates, or longer time spans. Compounding more often always helps the saver and costs the borrower a little more.

Simple interest only ever grows in a straight line, since it is always a fixed percentage of the original principal. Compound interest curves upward because each period's interest becomes part of the base for the next period. That curved, ever-steepening shape is exactly what you see when you graph exponential functions: compound interest is one of the most common real-world uses of that shape.

Plotting the balance \(A\) against time \(t\) for the \(1{,}000\) deposit at \(5\%\) compounded monthly shows the classic exponential growth curve. The balance rises slowly at first and then climbs more steeply as the years pass, since the growth factor is applied again and again.

This behavior matches the pattern used whenever a quantity grows repeatedly by the same multiplying factor, as covered in exponential growth and decay by a factor. Recognizing the compound interest formula as an exponential model makes it much easier to predict how a balance will behave far into the future, without recalculating every single year by hand.

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