Compound interest: formula, calculator and worked example

What the symbols mean

$A$: the final amount (principal plus interest)
$P$: the principal: the starting amount
$r$: the nominal annual interest rate as a decimal (6% = 0.06)
$n$: the number of times interest is added per year (1 yearly, 4 quarterly, 12 monthly, 365 daily)
$t$: the time in years (years)

When to use it

Use it for savings accounts, deposits and debts where interest is added to the balance each period. Solving for t answers "how long until it doubles?", and solving for r gives the rate needed to reach a target. The calculator does not solve for n: it is a whole number of periods chosen by the bank, and solving for it rarely gives a whole number.

Worked example

5,000 is saved at 6% a year, compounded monthly (n = 12), for 5 years. What is it worth?

Start from the formula.
$A = P \cdot {(1 + \frac{r}{n})}^{n t}$
Substitute $P$ = $5,000.00$ , $r$ = $0.06$ , $n$ = $12$ , $t$ = $5$ and work it out.
$A = 5,000.00 \cdot {(1 + \frac{0.06}{12})}^{12 \cdot 5}$
$= 5,000.00 \cdot {(1 + 0.005)}^{60}$
$= 5,000.00 \cdot {1.005}^{60}$
$\approx 6,744.25$
Interest earned:
$I = A - P$
$\approx 6,744.25 - 5,000.00$
$\approx 1,744.25$

Answer A ≈ 6,744.25, so the interest earned is 1,744.25

Why it works

Each period the balance is multiplied by $1 + r/n$ (the balance plus that period's share of the annual rate). There are $n t$ periods in t years, so the starting amount is multiplied by $(1 + r/n)$ that many times.

Common mistakes

Dividing t by n instead of multiplying: the exponent is $n t$ , the total number of periods.
Using the annual rate as the periodic rate: monthly compounding at 6% uses $0.06/12 = 0.005$ per month.
Rounding $(1 + r/n)$ before raising it to a large power; keep full precision until the end.
Adding interest to A twice: A already includes the principal.

Compound interest

Calculator

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What the symbols mean

When to use it

Worked example

Why it works

Common mistakes