Compound Interest
Value of principal compounded n times per year.
Formula
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Example
$1000 at 5% compounded yearly for 3 years grows to 1000·1.05³ ≈ $1157.63.
How it works
Compound interest models money that grows by earning interest on both the original amount and the interest already added in previous periods. The factor \(1 + \frac{r}{n}\) is the growth rate for one compounding period, and raising it to \(nt\) applies that growth repeatedly over all compounding periods in the time span. This is why compounding more often can lead to a slightly larger final amount than simple annual interest.
When to use it
Use this formula for savings accounts, investments, or loans when the interest is compounded at regular intervals such as monthly, quarterly, or yearly. It is especially useful whenever you know the principal, annual rate, number of compounding periods per year, and total time.
Worked example, step by step
Suppose you invest $2000 at 6% annual interest compounded quarterly for 2 years. P = 2000, r = 0.06, n = 4, t = 2 A = P\left(1 + \frac{r}{n}\right)^{nt} A = 2000\left(1 + \frac{0.06}{4}\right)^{4\cdot 2} A = 2000(1.015)^8 A \approx 2000(1.12649) A \approx 2252.98 Final answer: $2252.98
Common mistakes to avoid
- Using the annual rate 0.06 without dividing by the number of compounding periods per year
- Forgetting to convert the percent to a decimal before substituting into the formula
- Using t as the number of compounding periods instead of the number of years
Did You Know?
Einstein reputedly called compound interest the eighth wonder of the world — "he who understands it, earns it".
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