The formula and its parts
Compound interest is interest paid on the principal and on the interest already earned.[4] The formula that captures it has four inputs:[1]
| Symbol | Meaning | Example |
|---|---|---|
| A | Amount at the end (principal + interest) | $8,051.62 |
| P | Principal, the starting amount | $5,000 |
| r | Annual interest rate as a decimal | 0.06 for 6% |
| n | Compounding periods per year | 4 (quarterly) |
| t | Time in years | 8 |
Interest earned is A − P. The quantity r/n is the rate for one period and n·t is the number of periods.
How to calculate compound interest step by step
- Divide the rate by the periods per year: 0.06 ÷ 4 = 0.015.
- Add 1: 1.015.
- Count the periods: 4 × 8 = 32.
- Raise to that power: 1.01532 = 1.610324.
- Multiply by the principal: 5,000 × 1.610324 = $8,051.62.
- Subtract the principal for the interest: $8,051.62 − $5,000 = $3,051.62.
In a spreadsheet: =5000*(1+0.06/4)^(4*8). The compound interest calculator does the same arithmetic and shows every year.
Compound interest examples, year by year
The SEC's investor.gov illustrates compounding with $100 at 5%, which becomes $105 after one year and $110.25 after two, because the second year's interest is paid on $105.[5] With $1,000 at 5% compounded yearly:
| Year | Start of year | Interest that year | End of year |
|---|---|---|---|
| Year 1 | $1,000.00 | $50.00 | $1,050.00 |
| Year 2 | $1,050.00 | $52.50 | $1,102.50 |
| Year 3 | $1,102.50 | $55.13 | $1,157.63 |
| Year 4 | $1,157.63 | $57.88 | $1,215.51 |
| Year 5 | $1,215.51 | $60.78 | $1,276.28 |
Each year's interest is larger than the last. Over 5 years simple interest would pay $250; compounding pays $276.28. The difference grows quickly with time; see compound vs simple interest.
Compound interest formula with monthly contributions
Regular deposits add the future value of an annuity to the formula:[3]
PMT is the deposit each period, made at the end of the period; multiply the second term by (1 + r/n) for deposits at the start. $2,000 plus $150 a month at 5% compounded monthly for 15 years: $4,227.41 + $40,093.34 = $44,320.75. The formula assumes deposits line up with compounding; when they do not, convert the rate to the deposit period, as the savings calculator does.
Daily and continuous compounding formulas
Daily compounding is the same formula with n = 365. Continuous compounding is the limit as n grows without bound:[2]
$1,000 at 10% for one year: compounded yearly, $1,100; daily, $1,105.16; continuously, 1,000 × e0.1 = $1,105.17. The jump from yearly to daily is far bigger than from daily to continuous.
Solving the formula for rate or time
- Rate: r = n × [(A ÷ P)1/(nt) − 1]. Use the interest rate calculator.
- Time: t = ln(A ÷ P) ÷ [n × ln(1 + r/n)]. Doubling at 6% compounded yearly takes ln 2 ÷ ln 1.06 = 11.90 years, close to the rule of 72's estimate of 12.
- Principal: P = A ÷ (1 + r/n)nt, the present value; see the present value calculator.
The formula and APY(1 + r/n)n − 1 is the effective annual yield. Banks quote it as APY, so with an APY you can skip n and use A = P(1 + APY)t.[6] 6% compounded quarterly is 6.136% APY.
Related calculators
- Compound interest calculatorDeposits, any compounding, inflation, goal seek, chart and CSV
- Compound vs simple interestThe same deposit both ways, year by year
- Interest rate calculatorThe rate behind a growth figure or a loan payment
- APY calculatorAPR to APY and back, for any compounding
Frequently asked questions
What is the formula for compound interest?
A = P(1 + r/n)nt, where P is the principal, r the annual rate as a decimal, n the compounding periods per year and t the years. Interest = A − P.
How do you calculate compound interest monthly?
Use n = 12: A = P(1 + r/12)12t. $10,000 at 5% for 10 years = 10,000 × 1.0041667120 = $16,470.09.
What is compound interest in simple words?
Interest that earns interest. Each period the interest is added to your balance, so the next period's interest is calculated on a bigger amount.
What is the continuous compounding formula?
A = Pert. $1,000 at 10% for a year gives $1,105.17.
Sources
- Contemporary Mathematics, 6.4 Compound Interest
- College Algebra 2e, 6.1 Exponential Functions (continuous compounding)
- Contemporary Mathematics, 6.6 Methods of Savings
- Compound Interest Calculator
- What is compound interest? (includes the Rule of 72)
- Appendix A to Part 1030 (Regulation DD), Annual Percentage Yield Calculation