How Compound Interest Works (and Why Time Beats Amount)

The basic idea

Simple interest is paid only on your original deposit. Compound interest is paid on the deposit plus all the interest already earned — so your balance grows a little faster every period, and the curve bends upward over time.

With $1,000 at 5% for 10 years, simple interest gives you $1,500. Compounded yearly you get $1,628.89. The extra $128.89 is interest earned on interest — small here, but over 30–40 years of retirement saving it becomes the largest part of the balance.

The formula

A = P(1 + r/n)^(nt), where P is the principal, r the annual rate as a decimal, n how many times interest compounds per year, and t the years. Regular contributions add a second term: each deposit compounds for the time it is invested.

Try it with your own numbers in the Compound Interest Calculator, and use the Savings Calculator to add a monthly deposit.

The Rule of 72

Divide 72 by the annual return to estimate how many years money takes to double. At 6%, about 12 years; at 9%, about 8 years. The same rule works backwards for inflation: at 4% inflation, prices double in roughly 18 years.

Frequency matters less than time

Daily compounding beats yearly compounding, but only slightly — at 5% for 10 years the difference on $10,000 is about $198. Starting five years earlier is worth far more than any change in compounding frequency.

Worked example: working it out by hand

Take $1,000 at 5% compounded monthly for 10 years. Step 1: the periodic rate is r ÷ n = 0.05 ÷ 12 = 0.0041667 per month. Step 2: the number of periods is n × t = 12 × 10 = 120 months. Step 3: the growth factor is (1 + 0.0041667)^120 = 1.647009. Step 4: multiply by the principal — $1,000 × 1.647009 = $1,647.01, of which $647.01 is interest.

Compounded yearly instead, the same deposit gives $1,628.89. The by-hand steps are identical either way — only n changes — which is why the Compound Interest Calculator asks for the compounding frequency explicitly.

Monthly vs daily compounding, in numbers

For $10,000 at 5% over 10 years, the Compound Interest Calculator gives: yearly $16,288.95, semi-annually $16,386.16, quarterly $16,436.19, monthly $16,470.09 and daily $16,486.65. Monthly compounding captures most of the available gain — moving from monthly to daily adds only $16.56 over the whole decade.

Frequency is therefore a tie-breaker between two otherwise equal accounts, never the main event. The rate, the amount and above all the time invested move the final balance far more.

APY vs the stated rate

APY (annual percentage yield) folds compounding into a single effective yearly rate: APY = (1 + r/n)^n − 1. A stated rate of 5% compounded monthly is an APY of 5.12%; compounded daily it is 5.13%; compounded yearly the APY equals the stated rate, 5.00%.

That is why savings accounts are compared by APY, not by the nominal rate: two accounts both advertising 5% pay slightly different amounts if one compounds monthly and the other yearly, and the APY is the number that already accounts for it.