How compound interest works and how to calculate it yourself with the formula

Compound interest is interest calculated on your balance including the interest you have already earned, not just on the money you first put in. The short answer to “how much will I have” is A = P(1 + r/n)^(nt). This guide shows how to use that formula by hand, why compounding frequency matters less than people expect, how regular deposits change the picture, and which details (fees, inflation, APR versus APY) quietly move the result.

What compounding actually does

Simple interest pays a fixed percentage of the original principal each year. Put 10,000 into an account paying 5% simple interest and you receive 500 a year, every year, so after ten years you hold 15,000.

Compound interest pays the percentage on the current balance instead. In the first year you still earn 500. In the second year the 5% is applied to 10,500, so you earn 525. The third year pays 551.25 on 11,025, and so on. After ten years the balance is 16,288.95 rather than 15,000, and the extra 1,288.95 is interest earned on earlier interest.

Over ten years the difference looks modest. Over forty it is enormous, because every year’s growth is a percentage of an ever-larger base, which is why starting early usually beats contributing more later.

The formula and how to use it

A = P(1 + r/n)^(nt)

Here A is the final amount, P is the principal, r is the annual nominal rate as a decimal (5% is 0.05), n is the number of compounding periods per year and t is the number of years. The bracket (1 + r/n) is the growth factor for a single period; raising it to the power nt applies that factor once for every period in the whole term.

Take 10,000 at 5% compounded monthly for 10 years. The periodic rate is 0.05 ÷ 12, and there are 12 × 10 = 120 periods. The growth factor is (1 + 0.05 ÷ 12)^120 = 1.64701, and 10,000 × 1.64701 = 16,470.09. Interest earned is always A − P, so 6,470.09 here. The only things to get right are the rate as a decimal, and the rate and the period count expressed in the same unit.

How much compounding frequency matters

Every account credits interest on some schedule: yearly, quarterly, monthly or daily. More frequent crediting means interest starts earning interest sooner, but the gain shrinks quickly as the frequency rises. The table uses the same 10,000 at 5% over 10 years and only changes n.

Compounding	n	Balance after 10 years	Interest earned	Effective annual rate (APY)
Annual	1	16,288.95	6,288.95	5.000%
Quarterly	4	16,436.19	6,436.19	5.095%
Monthly	12	16,470.09	6,470.09	5.116%
Daily	365	16,486.65	6,486.65	5.127%

Going from annual to daily compounding adds 197.70 over the decade, about 3% more interest. Compare that with one extra percentage point of rate: 10,000 at 6% compounded annually for 10 years ends at 17,908.48, more than 1,600 ahead of the 5% account. When you compare accounts, the headline rate and the fees deserve almost all of your attention. The compounding schedule is a tiebreaker at best.

APR versus APY

The nominal rate, commonly labelled APR, is the figure before compounding is taken into account. APY, also called the effective annual rate, is what you actually earn over a year once interest earns interest. The conversion is APY = (1 + r/n)^n − 1, which is the right-hand column in the table above. At 7% nominal, monthly compounding gives about 7.23% APY and daily compounding 7.25%.

The distinction matters in two directions. Banks normally advertise APY for savings accounts and APR for lending, so an 18% APR credit card that compounds daily really costs about 19.72% a year if you carry the balance. And if you take an APY from a savings account and type it into a calculator set to monthly compounding, you count the compounding twice and overstate your result. When your source figure is already an APY, use yearly compounding. The compound interest calculator shows the APY for whatever rate and frequency you enter, so you can check which figure you are looking at.

Adding regular contributions

Almost nobody saves as a single lump sum, and the plain formula says nothing about monthly deposits. The calculator handles them by simulating every period, and by default it assumes each deposit lands at the end of its period, so a deposit made this month starts earning interest next month. Here is the arithmetic for its default scenario: 10,000 to start, 200 a month, 7% compounded monthly, 20 years.

1. Monthly rate i = 0.07 ÷ 12 = 0.0058333. Because the deposits and the compounding are both monthly, no conversion is needed.
2. Number of months N = 20 × 12 = 240.
3. Growth factor (1 + i)^240 = 4.038739.
4. The lump sum grows to 10,000 × 4.038739 = 40,387.39.
5. A series of end-of-period deposits grows to D × ((1 + i)^N − 1) ÷ i, so 200 × (4.038739 − 1) ÷ 0.0058333 = 200 × 520.93 = 104,185.33.
6. Final balance = 40,387.39 + 104,185.33 = 144,572.72.

You contributed 10,000 + 240 × 200 = 58,000 of that, so 86,572.72 is interest. Switching the deposits to the start of each month, which is closer to what happens if you pay in on payday, gives every deposit one extra month of growth and lifts the result to 145,180.47. If your deposits run on a different rhythm from the compounding, weekly pay-ins into a daily-compounding account for example, the calculator converts the rate to an exact per-deposit rate through the APY.

The rule of 72, fees and inflation

For a quick mental estimate, divide 72 by the annual rate in percent to get the number of years a lump sum takes to double. At 7%, 72 ÷ 7 is about 10.3 years; the exact figure is 10.2. At 5% the rule says 14.4 years against an exact 14.2, and at 10% it says 7.2 against 7.3. The rule only describes a single lump sum with no further deposits, but it is accurate enough to sanity-check a calculator result or to see at a glance that a fund charging an extra 1% a year is costing you years of doubling.

Compound growth cuts both ways. A fund charging 1% a year turns a 7% market return into 6% in your hands. On 10,000 left for 20 years with annual compounding that is 38,696.84 versus 32,071.35, so a fee that sounds trivial removes about 17% of the final balance. The same logic runs against you on debt: an unpaid card balance compounds daily at the card’s APR, and for a loan with fixed repayments the loan amortization schedule shows exactly how much of each payment is interest.

Inflation is the other silent deduction. A 7% return in a year of 3% inflation is a real return of 1.07 ÷ 1.03 − 1 = 3.88%, roughly 4%. The calculator reports nominal figures before tax, so if you want purchasing power, enter the real rate instead of the headline one, and reduce the rate again if the account is taxable.

Common mistakes

• Entering a percentage where the formula wants a decimal (5 instead of 0.05).
• Feeding an APY into a calculator set to monthly or daily compounding, which double counts.
• Using years for t but a monthly rate for r, or the other way round.
• Assuming the rule of 72 covers regular deposits. It only describes a lump sum.
• Comparing final balances without allowing for fees, tax and inflation.
• Forgetting that a deposit at the end of the month earns nothing that month.