# Compound interest, in plain arithmetic

How compound interest is calculated, why the period matters, and what a constant rate leaves out.

Compound interest means the return is calculated on a balance that already includes earlier returns. Add a contribution and the next period's return applies to that contribution too. That is the whole mechanism. The dramatic charts are just this sentence repeated for many years.

## The formula and the calculator

For a lump sum with no contributions, compounded once a year, the balance is principal × (1 + rate) ^ years. At 7% for 30 years, $10,000 becomes about $76,123. The Rule of 72 is a napkin version: 72 divided by the annual percent is a rough doubling time. At 7% that is a bit more than 10 years. It is an approximation, not the formula the calculator uses.

Contributions make a closed form uglier, so the calculator walks month by month. Each month it adds the contribution, and at the end of each compounding period it multiplies the balance by one plus the periodic rate. Periodic rate is the annual rate divided by the number of compounds per year. That is an nominal APR-style split, not a continuously compounded model.

With the default illustration — $10,000 to start, $500 a month, 7% compounded monthly, 30 years — the balance is about $695,000. Roughly $190,000 of that is money you put in. The rest is growth on those deposits. Change the rate before you treat that figure as a plan. A 7% constant return is an illustration, not a forecast of stocks or of a savings account.

## What moves the result

- Years. The exponent is why the last decade of a long chart adds more dollars than the first decade. Starting earlier is mostly this.
- The amount you add. On a short horizon, contributions dominate. On a long horizon, the rate dominates. Both matter, and the savings-rate chart on the calculator page shows how stark the contribution effect is when you start from zero.
- The rate, including fees. A 1% annual fee is not a small slice of the return. It is a slice of the balance, every year, and it compounds against you.
- Interruptions. The model never pauses contributions and never has a down year. A real account does both.

## Real versus nominal

A nominal 7% with 3% inflation is not a 7% improvement in what you can buy. A useful approximation of the real return is (1.07 / 1.03) − 1, about 3.9%. The Coast FIRE calculator lets you switch modes so a retirement target is not accidentally mixed with today's spending. The compound-interest chart on this site is nominal unless you lower the rate yourself to a real rate and read the result as today's buying power.

> The guide describes the calculator. It does not predict a market return and it does not subtract your taxes.

## Questions

### What is the difference between compound interest and simple interest?

Simple interest pays a rate on the original principal only. Compound interest pays a rate on principal plus interest already credited. A savings account, a bond fund's reinvested income, and a reinvested stock fund all compound in that everyday sense, even when the 'interest' is really a return that can also fall.

### How often should I assume compounding?

Use the schedule the account actually uses for a deposit product. For a diversified market portfolio, monthly or annual steps are both models. Monthly is a reasonable default for a contribution that lands each payday. The calculator lets you switch. The gap between monthly and annual is real and usually smaller than the gap created by fees or by saving less.

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