Compound Interest Explained: Formula, Rule of 72 and Examples

The compound interest formula, term by term

The formula is A = P × (1 + r/n)^(nt). P is the principal you start with, r the annual rate as a decimal (8 percent = 0.08), n how many times interest is added per year and t the time in years. The bracket grows the money by a slice of the rate each period, and the exponent counts how many periods pass. Interest is added to the balance, and the next round of interest is calculated on that bigger balance, which is the entire trick.

A quick base case: 1,000 at 8 percent, compounded monthly, for 10 years. The monthly rate is 0.08 ÷ 12 = 0.006667, the number of periods is 120, so A = 1,000 × (1.006667)^120 = about 2,220. The same 1,000 at simple interest would earn exactly 80 per year and reach 1,800; compounding quietly adds the extra 420 without you doing anything. That gap widens every single year the money stays invested.

Reading the exponent matters when you compare products: 10 years at 12 percent (120 monthly periods at 1 percent each) beats 10 years at 24 percent compounded once a year, even though the headline rate is double. Always reduce an offer to its rate per period and its number of periods before judging it.

The rule of 72: doubling time without a calculator

Divide 72 by the annual rate and you get the approximate number of years for money to double: 72 ÷ 8 = 9 years at 8 percent, 72 ÷ 6 = 12 years at 6 percent, 72 ÷ 3 = 24 years at 3 percent. The shortcut comes from solving (1 + r)^t = 2 and approximating the logarithm, and it stays within about a year of the exact answer for rates between 4 and 12 percent.

The rule runs in reverse and on inflation too. At 3 percent inflation, 72 ÷ 3 = 24 years is how long prices take to double, which is the same as your cash losing half its purchasing power. It also exposes fees instantly: a fund charging 2 percent per year instead of 0.5 percent stretches the doubling time at 8 percent gross from about 9.6 years to nearly 12, a gap most investors never notice on a statement.

For mental math beyond doubling, remember that two doublings multiply by four and three by eight: 1,000 at 8 percent for 27 years is close to 1,000 × 2 × 2 × 2 = 8,000, since 27 years is three doublings of 9 years each. The rule of 72 turns the formula into arithmetic you can do while walking down the street.

Does compounding frequency matter?

The same 10,000 at 8 percent for 10 years lands at noticeably different places depending on how often interest is added, as the table shows.

The jump from annual to monthly compounding is worth 607, but monthly to daily adds only 57. Frequency has diminishing returns because the rate per period shrinks as n grows; the mathematical ceiling is continuous compounding at Pe^(rt) = 10,000 × e^0.8 = about 22,255. The honest ranking of what moves outcomes: time invested first, contributions second, fees and rate third, frequency a distant fourth.

Banks advertise APY, the annual percentage yield, precisely to flatten this comparison: it restates any compounding schedule as the single yearly rate that produces the same result. An 8 percent nominal rate compounded monthly equals an APY of about 8.30 percent, so when two offers quote APY you can compare them directly with no further math.

And if you wonder why everyone does not chase daily compounding: fees dissolve the difference. A daily compounding fund charging 1.5 percent a year loses to a monthly compounding fund charging 0.3 percent fast, because half a percent of fees erases what all that extra frequency earned.

Adding monthly contributions to the math

Regular deposits follow their own term: FV = PMT × ((1 + i)^N - 1) ÷ i, where PMT is the payment, i the rate per period and N the number of periods. The formula sums every deposit grown at its own number of compounding periods, so deposits made early do the most growing. That single fact is why starting early beats depositing more later.

Check it with 100 per month at 8 percent compounded monthly for 10 years: i = 0.006667 and N = 120, so FV = 100 × ((1.006667)^120 - 1) ÷ 0.006667 = 100 × 182.95 = about 18,295. You deposited 12,000 of that; compounding contributed the remaining 6,295 without a single extra deposit from you.

Two habits sharpen this further. Increase the contribution with inflation or salary each year, and growth compounds on rising inputs instead of a flat 100. And match deposit timing to compounding: payments at the start of each month earn one extra period of interest versus end of month deposits, worth roughly one period of the final figure.

Consistency also beats size at the start: 50 a month begun today outgrows 100 a month begun five years later over a long horizon, because every five years of delay deletes a whole growth cycle from your oldest money, and those are the most expensive cycles you own.

Full worked example: 1,000 plus 100 a month for 10 years

Assemble the pieces. Start: 1,000. Monthly deposit: 100. Rate: 8 percent compounded monthly. Horizon: 10 years, so 120 periods at 0.6667 percent each. The principal grows to 1,000 × (1.006667)^120 = about 2,220. The deposits grow to about 18,295. Total: about 20,515, of which 13,000 is money you paid in and 7,515 is interest the account earned by itself.

Stretch the same plan to 20 years and the curve bends upward: the principal part becomes 1,000 × (1.006667)^240 = about 4,927, the deposits become about 58,902, and the total is roughly 63,829 against 25,000 deposited, meaning interest added about 38,800 on its own. Most of that arrives in the final years, which is exactly why quitting at year 12 is so expensive.

Run the same 10 year plan at 5 percent instead of 8 percent and it ends near 17,175 instead of 20,515. Three percentage points cost about 3,340 on these small numbers; over a 25 year horizon on larger balances the same gap changes lives, which is why fee negotiation is a form of earning.

Keep the verification sequence simple: compute the principal part alone, then the deposits alone, then add. If any step disagrees with a tool, check the rate per period and the period count first, because those cause most errors, not the formula itself.

Inflation: the silent counterparty

Nominal growth is not the same as getting richer. At 3 percent inflation, prices rise about 34 percent over 10 years, so the 20,515 from the worked example buys roughly what 15,300 buys today. The real return is approximated by subtraction: 8 percent nominal minus 3 percent inflation is about 5 percent real, and you can run the whole formula again with 0.05 to see the honest picture.

This reframes three common decisions. Cash savings at 1 percent lose to 3 percent inflation in real terms no matter how long they compound. Fees act like negative compounding, dragging every future year down from a smaller base. And long horizons are the strongest inflation defence, because assets have decades to outgrow the erosion. Taxes are the final bite: interest is often taxed as income, so compare post tax rates rather than headline ones.

A worked real return: 10,000 growing at 8 percent for 10 years ends near 21,590 annually compounded. Deflate by 1.03^10 = 1.344 and the purchasing power is about 16,060 in present money, a real multiple of 1.606, which is close to running the formula at 4.85 percent. Deflation math and growth math use the same exponent, just pointed the other way.

Simple versus compound interest

Simple interest pays only on the original principal: 1,000 at 8 percent simple earns exactly 80 every year and reaches 1,800 in ten. Compound interest at the same rate pays 80 the first year, then 86.40 the second, because the second year is calculated on 1,080. The difference starts tiny and grows exponentially; by year 10 the compounded balance is about 2,160 versus 1,800.

Knowing which applies to you matters in both directions. Most savings accounts, investments and credit card balances compound, which is why unpaid card debt snowballs at the same mathematics that grows an index fund. Some loans and bonds use simple interest, which makes them cheaper than a compound quote at the same headline rate. Ask one question of any financial product: is interest charged or paid on interest?