The compound interest formula is:
A = P × (1 + r ÷ n)^(n × t)
It answers one question: if an amount grows by a percentage each period, and each period's growth is calculated on the new total rather than the original, what is it worth after a given time? This guide defines every symbol, works two examples you can re-check digit for digit, and shows the two places people reliably get it wrong.
One thing up front: this explains how the number is computed, not what you should do with your money. Rates, fees, and terms vary by product and country; the arithmetic below is universal, but the figures that apply to you are yours to confirm.
The symbols, defined once
- A = the final amount — everything you have at the end, principal plus accumulated interest.
- P = the principal — the amount you start with.
- r = the annual interest rate as a decimal. 6% is 6 ÷ 100 = 0.06.
This is the single most common mistake: entering
6instead of0.06inflates the answer beyond all recognition. - n = the number of compounding periods per year. Annually = 1, semi-annually = 2, quarterly = 4, monthly = 12, daily = 365.
- t = the time in years. Eighteen months is t = 1.5, not 18.
The interest earned on its own is simply A − P.
Why the formula looks like that
You don't need the derivation, but one sentence makes it stick.
Each compounding period, the balance is multiplied by (1 + the period's rate). The period's rate is the annual rate divided by the number of periods per year, which is the r ÷ n part. Over the whole term there are n × t of those periods, so you apply that multiplier n × t times — which is what raising it to the power of (n × t) means.
That's the whole idea. Compound interest is repeated multiplication, where simple interest is repeated addition.
Worked example: $5,000 for 3 years, compounded monthly
Take a principal of $5,000 at an example annual rate of 6%, compounded monthly, for 3 years.
Step 1 — convert the inputs.
- P = 5000
- r = 6 ÷ 100 = 0.06
- n = 12
- t = 3
Step 2 — find the periodic rate. r ÷ n = 0.06 ÷ 12 = 0.005 (0.5% per month).
Step 3 — count the periods. n × t = 12 × 3 = 36.
Step 4 — raise the growth factor to that power. (1 + 0.005)^36 = 1.005^36 = 1.196681 (kept to 6 decimal places — round only at the end).
Step 5 — multiply by the principal. A = 5000 × 1.196681 = $5,983.40 (rounded to 2 decimal places).
Interest earned: A − P = 5983.40 − 5000 = $983.40.
The same money at simple interest
Simple interest never re-invests what it earns, so its formula is just:
Interest = P × r × t
With the same numbers: 5000 × 0.06 × 3 = $900.00, for a total of $5,900.00.
Compounding monthly produced $983.40 in interest instead of $900.00 — $83.40 more, from nothing but the timing of when interest starts earning interest. That gap widens the longer the term runs, because each period's head start compounds too.
How compounding frequency changes the answer
Same rate, same term, different n. Here is $1,000 at an example 10% annual rate for exactly 1 year:
| Compounding | n | Growth factor | Final amount | Effective annual rate |
|---|---|---|---|---|
| Annually | 1 | 1.100000 | $1,100.00 | 10.0000% |
| Semi-annually | 2 | 1.102500 | $1,102.50 | 10.2500% |
| Quarterly | 4 | 1.103813 | $1,103.81 | 10.3813% |
| Monthly | 12 | 1.104713 | $1,104.71 | 10.4713% |
| Daily | 365 | 1.105156 | $1,105.16 | 10.5156% |
| Continuously | — | 1.105171 | $1,105.17 | 10.5171% |
Figures rounded to the places shown; re-check any row with (1 + 0.10 ÷ n)^n.
Two things are worth noticing. More frequent compounding always gives more, but the gains shrink fast — the jump from annual to monthly is far bigger than the jump from monthly to daily. And there is a ceiling: as n grows without limit, the formula converges on continuous compounding, A = P × e^(r × t), where e is the mathematical constant ≈ 2.718282. That last row is the most any 10% nominal rate can produce in a year.
Nominal rate vs effective rate (the APR/APY trap)
The nominal rate is the headline annual number — the 10% above. The effective annual rate is what you actually gain over a year once compounding is included:
Effective rate = (1 + r ÷ n)^n − 1
At 10% nominal compounded monthly: (1 + 0.10 ÷ 12)^12 − 1 = 1.104713 − 1 = 0.104713, i.e. 10.4713%.
This is why two products quoting the same headline rate can pay or cost different amounts. Comparing a nominal rate against an effective one is comparing different quantities, and it is the money-side twin of the base-mismatch errors in our measurement mistakes guide. Names for these figures vary by product and jurisdiction, so read which rate a quote is stating rather than assuming.
Working the formula backwards
Two rearrangements cover most remaining questions.
How much do I need to start with? Divide instead of multiply:
P = A ÷ (1 + r ÷ n)^(n × t)
To reach $5,983.40 in 3 years at 6% compounded monthly, you'd need 5983.40 ÷ 1.196681 = $5,000.00 — the first example in reverse.
How long until it doubles? The quick estimate is the Rule of 72: divide 72 by the interest rate written as a percent. At 6%, 72 ÷ 6 = 12 years. The exact answer, for interest compounded once per period at rate i, is ln(2) ÷ ln(1 + i) — at 6% annually that's 11.90 years. The rule of 72 is an approximation and it is closest around the 6–10% range; at 8% it gives 9 years against an exact 9.01.
Four mistakes to avoid
- Using the percent instead of the decimal. r is 0.06, never 6.
- Mismatching n and t. n is periods per year and t is years. If you count 36 months, that's t = 3 with n = 12 — not t = 36.
- Rounding in the middle. Keep the growth factor to at least six decimal places and round only the final currency figure. Rounding early distorts the answer, and the error grows with the term.
- Comparing a nominal rate with an effective one. Convert both to the same basis before deciding anything.
FAQ
What is the compound interest formula?
A = P(1 + r/n)^(nt), where A is the final amount, P the principal, r the annual rate as a decimal, n the compounding periods per year, and t the time in years. The interest alone is A − P.
What's the difference between simple and compound interest?
Simple interest is always calculated on the original principal (P × r × t). Compound interest is calculated on the running balance, so earlier interest earns interest too. On $5,000 at 6% for 3 years, that difference is $900.00 versus $983.40 with monthly compounding.
Does compounding more often always give more?
Yes, but with rapidly diminishing returns, and there's a hard ceiling. Going from annual to monthly compounding matters far more than going from monthly to daily, and continuous compounding — A = P × e^(rt) — is the theoretical maximum for a given nominal rate.
How do I calculate compound interest for months instead of years?
Keep t in years and express the months as a fraction: 18 months is t = 1.5. Alternatively, work directly in periods — use the periodic rate (r ÷ n) and the total number of periods (n × t) as the exponent. Both give the same answer.
Is the Rule of 72 accurate?
It's a close approximation for mid-range rates, not an exact result. At 8% it gives 9 years against an exact 9.01; at very high or very low rates the error grows. Use ln(2) ÷ ln(1 + i) when you need the precise doubling time.
Compound interest is one formula and four inputs — get P, r, n, and t right and the arithmetic never surprises you. The same repeated-multiplication logic drives loan repayments too, which our EMI formula explainer works through end to end, and the wider money maths sits in the business calculations guide. When you'd rather check a figure than work it by hand, the free calculators on Wapuula Tools show the exact formula and inputs behind every result.
