Compound Interest Formula: A Worked Example by Hand

Compound interest calculators are convenient, but understanding the formula behind them means you can sanity-check any number they produce — including this site’s own compound interest calculator. This guide walks the formula step by step with round numbers, so you can follow the arithmetic by hand and see exactly where each figure comes from.

The formula

The standard compound interest formula for future value is:

FV = P × (1 + r/n)^(n×t)

where P is the starting principal, r is the nominal annual rate as a decimal (6% is 0.06), n is the number of compounding periods per year, and t is time in years. The bracket (1 + r/n) is the growth factor for one period; raising it to the power n×t applies that growth once per period, for every period across the whole time horizon.

Worked example: $1,000 at 6%, compounded annually

Take the simplest case — annual compounding, so n = 1and each “period” is one year. Starting principal P = $1,000, rate r = 6%. Each year, multiply the balance by 1.06:

Each row is the previous balance times 1.06 — that’s the entire mechanism. Using the formula directly for year 3: 1000 × (1.06)^3 = 1000 × 1.191016 = $1,191.02, matching the hand calculation exactly. The ^3 is shorthand for “multiply by 1.06, three times in a row” — the formula just skips the intermediate rows.

Switching to monthly compounding

Now compound the same $1,000 at 6% monthly instead of annually, over 3 years. The periodic rate becomes r/n = 0.06/12 = 0.005 (0.5% per month), and the exponent becomes n×t = 12×3 = 36 periods:

FV = 1000 × (1.005)^36 = $1,196.68

That’s $5.66 more than the $1,191.02 from annual compounding on the same rate and time — a small but real difference from letting interest compound monthly instead of once a year. See how much compounding frequency actually matters for the pattern at larger scales.

Adding a monthly contribution

Most real savings involve regular deposits, not just a lump sum. The extended formula adds a contribution term:

FV = P(1 + r/n)^(nt) + C × (((1 + r/n)^(nt) − 1) / (r/n))

Take the same $1,000 at 6% monthly for 3 years, now adding C = $50 at the end of every month. The growth factor (1.005)^36 = 1.19668 is the same as before. Plug it into the contribution term: 50 × ((1.19668 − 1) / 0.005) = 50 × 39.336 = $1,966.79. Add that to the principal’s own growth, 1000 × 1.19668 = $1,196.68, and the total is $3,163.49. Total contributions over 36 months are 1000 + 50×36 = $2,800, so interest earned is $3,163.49 − $2,800 = $363.49 — money the balance generated on top of what was actually deposited.

Checking your work

Three sanity checks catch most arithmetic mistakes. First, the balance should never go below your total contributions (unless the rate is negative, which this formula doesn’t model). Second, at r = 0 the formula collapses to FV = P + C×n — no growth, just addition — which is a useful check on the contribution term specifically. Third, doubling the number of years should more than double the interest earned, never less, because later periods compound on a larger base. If any of these checks fail, re-verify that the rate was converted to a decimal and divided by the right n before it went into the exponent.