How Compound Interest Works

Compound interest is interest calculated on both the original principal and the interest already accumulated. This creates exponential growth — the longer the time horizon, the more powerful the effect. This page explains the formula, compounding frequencies, and real-world applications.

The Core Formula

A = P (1 + \frac{r}{n})^{n t}

Symbol	Variable	Example value
A	Final balance	$7,449.23
P	Principal	$5,000
r	Annual rate as decimal	0.08 (= 8%)
n	Compounding periods per year	12 (monthly)
t	Time in years	5

Why Compounding Produces Exponential Growth

Each compounding period, interest is added to the balance. The next period's interest is calculated on this larger balance — so you earn interest on your interest. Over time this creates a snowball effect.

Year	Opening Balance	Interest (8%)	Closing Balance
1	$1,000.00	$80.00	$1,080.00
2	$1,080.00	$86.40	$1,166.40
3	$1,166.40	$93.31	$1,259.71
4	$1,259.71	$100.78	$1,360.49
5	$1,360.49	$108.84	$1,469.33

ℹSimple interest on the same $1,000 at 8% for 5 years = $400 total. Compound interest = $469.33. The $69.33 difference comes entirely from interest-on-interest.

Worked Example — Monthly Compounding

P = $5,000, R = 8% per year, t = 5 years, n = 12 (monthly)

Convert rate: r = 8 / 100 = 0.08
Monthly rate = 0.08 / 12 = 0.006667
Total compounding periods = 12 × 5 = 60
A = 5,000 × (1.006667)^60

A = 5000 \times (1.006667)^{60} \approx 5000 \times 1.48985 \approx $7, 449.23

APY vs APR

APR (Annual Percentage Rate) is the nominal rate before compounding effects. APY (Annual Percentage Yield) reflects the true annual return after compounding:

A P Y = (1 + \frac{r}{n})^{n} - 1

For 6% APR compounded monthly:

A P Y = (1 + \frac{0.06}{12})^{12} - 1 = (1.005)^{12} - 1 \approx 6.168%

💡When comparing savings accounts, use APY — not APR. APY tells you the actual annual return including compounding. A 6% APR compounded monthly has a higher APY than 6% compounded annually.

The Rule of 72

A quick mental shortcut: divide 72 by the annual interest rate to estimate how many years it takes to double your money.

Annual Rate	Rule of 72 (years to double)	Exact years
4%	72 ÷ 4 = 18 years	17.67 years
6%	72 ÷ 6 = 12 years	11.90 years
8%	72 ÷ 8 = 9 years	9.01 years
12%	72 ÷ 12 = 6 years	6.12 years

Continuous Compounding

As n approaches infinity (compounding every instant), the formula approaches:

A = P \cdot e^{r t}

For P = $1,000, r = 8%, t = 5 years: A = 1,000 × e^(0.08 × 5) = 1,000 × e^0.4 ≈ $1,491.82. This is the theoretical maximum for continuous compounding — only marginally higher than daily compounding.

Compound Interest Calculator — Calculate CI with year-by-year breakdown
Compound Interest Formula — Full formula reference with worked examples

Frequently Asked Questions

How does more frequent compounding affect returns?

More frequent compounding slightly increases returns, but the effect diminishes as frequency increases. Going from annual to monthly compounding makes a meaningful difference; going from monthly to daily compounding makes only a tiny difference.

Does compound interest apply to debt?

Yes — and it works against borrowers. Credit card balances compound daily or monthly, so unpaid balances grow rapidly. This is why paying off high-interest debt is often the best 'investment' available.

What is the difference between compound interest and compound annual growth rate (CAGR)?

CAGR describes the smoothed annual growth rate of an investment over a period, assuming profits are reinvested each year. It uses the same underlying math as compound interest: CAGR = (End Value / Start Value)^(1/t) − 1.

How long does it take $1,000 to grow to $10,000 at 7% compounded annually?

Rearrange A = P(1+r)^t for t: t = log(A/P) / log(1+r) = log(10) / log(1.07) ≈ 34 years. The Rule of 72 would estimate 10 doublings at ~10.3 years each ≈ 103 years to reach 1000× — but 10× only requires about 34 years.