Compound Interest Calculator
Calculate compound interest and total balance for annual, quarterly, monthly, daily, or continuous compounding. See the formula and step-by-step math.
What is compound interest?
Compound interest is interest that gets added back into the principal at the end of each compounding period, so future interest is calculated on a growing balance — "interest on interest." This is the mechanism behind exponential growth in savings accounts, 401(k)s, and long-term investments, and it's also what makes compounding debt (like credit card balances) grow faster than simple-interest debt if left unpaid.
Unlike simple interest, where the same principal earns the same interest every year, compound interest means each period's ending balance becomes the next period's starting principal.
The formula
- A — total amount after interest (principal + interest)
- P — principal (the amount deposited or borrowed)
- r — annual interest rate, as a decimal (rate ÷ 100)
- n — number of compounding periods per year (1 = annually, 12 = monthly, 365 = daily, …)
- t — time, in years
Interest earned is the difference between the total and the principal:
Worked example: monthly compounding
Say you deposit $10,000 at a 3.875% annual rate, compounded monthly, for 7.5 years.
- Rate as a decimal: r = 3.875 ÷ 100 = 0.03875.
- Periods per year: n = 12 (monthly).
- Total periods: nt = 12 × 7.5 = 90.
- Apply the formula: A = 10,000 × (1 + 0.03875/12)^90 ≈ $13,366.37.
- Interest earned: I = 13,366.37 − 10,000 = $3,366.37.
Worked example: comparing to simple interest
Deposit $10,000 at 5% for 10 years, compounded annually: A = 10,000 × (1.05)^10 ≈ $16,288.95. The same principal and rate under simple interest would only reach $15,000 (I = P × r × t = 10,000 × 0.05 × 10 = $5,000). The $1,288.95 gap is interest earned on interest already paid — and it grows larger the longer the money stays invested.
Continuous compounding
As the compounding frequency n grows without bound (compounding every instant rather than at fixed periods), the formula converges to:
where e ≈ 2.71828 is Euler's number. This is the theoretical ceiling on growth for a given nominal rate — no finite compounding frequency can exceed it.
Example: $1,000 at 5%, compounded continuously, for 2 years: A = 1,000 × e^(0.05 × 2) ≈ $1,105.17.
How compounding frequency changes the result
For the same $10,000 principal at a 5% annual rate over 10 years, a higher compounding frequency produces a slightly larger total:
| Compounding | n | Total after 10 years |
|---|---|---|
| Annually | 1 | $16,288.95 |
| Quarterly | 4 | $16,436.19 |
| Monthly | 12 | $16,470.09 |
| Daily | 365 | $16,486.65 |
| Continuous | ∞ | $16,487.21 |
The gains from more frequent compounding shrink as n grows — the jump from annual to monthly is much bigger than the jump from monthly to daily.
Common mistakes
- Forgetting to divide the rate by n. Each period applies r/n, not the full annual rate r — otherwise the total is wildly overstated.
- Mixing up the compounding period with the time unit. n is periods per year; t is always in years, even when compounding is monthly or daily.
- Confusing compound interest with simple interest. Compound interest reinvests prior interest; simple interest never does — the gap widens with time.
- Assuming continuous compounding for ordinary savings accounts. Most real accounts compound monthly, quarterly, or daily — continuous compounding is a mathematical limit, not a common bank policy.
Related calculators
To compare against linear growth, the simple interest calculator uses the same principal, rate, and time without reinvesting interest. If you're paying down debt in equal installments instead of growing a lump sum, the loan payment calculator models that amortization instead.
Frequently asked questions
- What is the formula for compound interest?
- Compound interest is A = P(1 + r/n)^(nt), where P is the principal, r is the annual rate as a decimal (rate ÷ 100), n is the number of compounding periods per year, and t is the time in years. Interest is I = A − P.
- How is compound interest different from simple interest?
- Compound interest adds each period's interest back into the principal, so later periods earn interest on interest already paid — growth is exponential. Simple interest is calculated only on the original principal, so growth is linear — see our simple interest calculator.
- What does the compounding frequency (n) actually change?
- A higher n applies interest more often at a smaller rate per period (r/n), which compounds faster and produces a larger total for the same nominal annual rate. Daily compounding always beats annual compounding at the same rate.
- What is continuous compounding?
- Continuous compounding is the limit as n approaches infinity — interest is applied at every instant rather than at discrete periods. The formula simplifies to A = Pe^(rt), using Euler's number e, and gives the theoretical upper bound on growth for a given rate.
- What happens if I enter a 0% interest rate?
- A 0% rate is a valid input, not an error. The interest is $0.00 and the total equals the principal, regardless of compounding frequency or time — useful as a baseline before comparing rates.
- Can compound interest work against me?
- Yes — on debt like credit card balances, compound interest works the same way but in the lender's favor, so unpaid interest gets added to the balance and starts earning its own interest. This is why carrying a balance on a compounding debt grows faster than a simple-interest loan.
- Why does monthly compounding use n = 12 and not the number of months elapsed?
- n is always the number of compounding periods per year (12 for monthly), while t is the total time in years. The formula multiplies them (nt) to get the total number of periods over the whole term — for 7.5 years monthly, that's 12 × 7.5 = 90 periods.
- How do I find the principal, rate, or time if I know the final amount?
- Rearranging A = P(1+r/n)^(nt) gives P = A/(1+r/n)^(nt) for principal, r = n·((A/P)^(1/nt) − 1) for rate, and t = ln(A/P) / (n·ln(1+r/n)) for time. This calculator solves for the total; solving for the other variables needs the rearranged formulas directly.