Maths SolverCompound interest is what you get when interest gets added to your balance and then itself starts earning interest. After enough compounding periods the difference from simple interest is dramatic; in finance, in inflation, in any growth process, this is the mechanism that does the work.
The formula you will see at school is
$$A = P\left(1 + \tfrac{r}{n}\right)^{nt},$$
which packs four parameters into one expression. Once the symbols are unpacked the formula is straightforward; this article walks through each letter, the special case of continuous compounding (where the formula simplifies into something involving $e$), and a calculator that lets you compare different compounding frequencies side by side.
What each letter means
- $P$ — the principal, the amount you start with.
- $r$ — the annual interest rate, written as a decimal. So $5\%$ is $r = 0.05$.
- $n$ — the compounding frequency per year. $n = 1$ for annual, $n = 12$ for monthly, $n = 365$ for daily.
- $t$ — the time in years. Six months is $t = 0.5$.
- $A$ — the final amount, including both principal and accumulated interest.
The interest earned alone (no principal) is $A - P$.
A worked example
$\pounds 1000$ at $5\%$ annual interest, compounded monthly, for $3$ years.
$P = 1000$, $r = 0.05$, $n = 12$, $t = 3$. Plug in:
$$A = 1000\left(1 + \tfrac{0.05}{12}\right)^{12 \cdot 3} = 1000(1.004167)^{36} \approx 1161.47.$$
The interest earned is $\pounds 161.47$. By comparison, simple interest would have given $1000 \cdot 0.05 \cdot 3 = \pounds 150$. The extra $\pounds 11.47$ is the effect of compounding.
Continuous compounding
What happens if you compound more often? Quarterly is more than annually; monthly is more than quarterly; daily, hourly, every second. The interest accumulated keeps going up, but it keeps going up by less and less each time. In the limit $n \to \infty$, the formula approaches a clean expression involving the number $e$:
$$A = P\,e^{rt}.$$
This is what mathematicians call continuous compounding, and it is the form you see almost exclusively in finance theory and in calculus problems involving exponential growth or decay (population models, radioactive decay, Newton’s law of cooling). The formula is shorter, has no $n$ in it, and behaves nicely under differentiation.
For our example: $A = 1000\, e^{0.05 \cdot 3} = 1000 \cdot 1.16183 = \pounds 1161.83$. The continuous figure is just $36$ pence more than the monthly figure — a useful check on the intuition that compounding more often than once a month gains you very little.
Solving for the other variables
You can rearrange the formula for any of $P$, $r$, $t$, but $t$ is the one that requires logarithms.
Solving for $t$ (the “how long” question)
“How long for $\pounds 1000$ at $5\%$ continuously compounded to double?” You want $A = 2000$, so $2000 = 1000\, e^{0.05 t}$, that is $e^{0.05 t} = 2$. Take the natural log of both sides: $0.05 t = \ln 2$, so $t = \ln 2 / 0.05 \approx 13.86$ years.
This is where the “rule of 72” comes from: $72 / r\%$ is a quick mental approximation to the doubling time at rate $r\%$. For $r\% = 5$, the rule gives $72/5 = 14.4$ years, which is close to the exact $13.86$. The rule works because $\ln 2 \approx 0.693 \approx 70/100$, and $72$ rounds nicely against more interest rates.
The mistakes I see most often
1. Using the percentage value instead of the decimal
$r$ in the formula is a decimal: $5\%$ is $0.05$, not $5$. Using $5$ gives an interest rate of $500\%$, and the answer balloons. Always divide the percentage by 100 before plugging in.
2. Forgetting that $n$ must match the units of $t$
$n$ is the number of compoundings per year and $t$ is in years, so $nt$ is the total number of compoundings. If you use $n = 12$ (monthly) but want the value after $6$ months, $t$ must be $0.5$, not $6$.
3. Confusing simple and compound interest
Simple interest is $A = P(1 + rt)$ — linear in time. Compound interest is $A = P(1 + r/n)^{nt}$ — exponential in time. The two agree for the first compounding period; they diverge sharply over long horizons. Use the right one.
4. Confusing the “nominal” rate with the “effective” rate
A nominal rate of $5\%$ compounded monthly produces an effective annual rate of $(1 + 0.05/12)^{12} - 1 \approx 5.116\%$. When comparing two loans or savings products, always compare effective rates — the nominal numbers are not directly comparable if the compounding frequencies differ.
Where this comes up later
The continuous-compounding form $A = Pe^{rt}$ is the simplest example of an exponential-growth model. The same equation describes population growth at constant per-capita birth rate, the spread of an epidemic in its early phase, the decay of a radioactive isotope (with negative $r$), and the behaviour of capacitor charging in basic circuits. The finance interpretation is just one application of a much wider pattern.
References
- Wikipedia: Compound interest, Rule of 72, Effective interest rate.
- Khan Academy: Interest and debt.