The compound interest formula, letter by letter
For a lump sum left alone, FV = P × (1 + r/n)^(n×t): P is what you start with, r the nominal annual rate as a decimal (5% → 0.05), n the number of times a year interest is added, and t the term in years. Continuous compounding is the limit of that expression, FV = P × e^(r×t). Put in 10,000 at 5% for 10 years compounded annually and you get 10,000 × 1.05^10 = 16,288.95.
Add regular deposits and a second term appears: FV = P(1+i)^N + PMT × ((1+i)^N − 1) ÷ i, where i is the rate for one deposit period and N the number of deposits. Depositing at the start of each period (an annuity due) multiplies the second term by one extra (1+i). When the rate is exactly 0 the division breaks down, so the tool falls back to P + PMT × N.
Compounding and deposits do not have to line up. If interest compounds daily but you deposit monthly, the tool converts the daily rate into the rate for one month — i = (1 + r/n)^(n/g) − 1, where g is deposits per year — and then steps through the schedule one deposit period at a time. Because the balance is accumulated period by period rather than from a closed form, the year-by-year table always adds up to the headline figure.
How much compounding frequency is actually worth
10,000 at 5% for 10 years, with nothing added, ends at:
- Annually — 16,288.95 (effective yield 5.0000%)
- Monthly — 16,470.09 (5.1162%)
- Daily — 16,486.65 (5.1267%)
- Continuously — 16,487.21 (5.1271%)
Almost all of the benefit arrives in the first step. Going from annual to monthly is worth 181.14 over the decade; going from monthly all the way to continuous adds 17.12 more. So a deposit account advertising “daily compounding” is not meaningfully better than a monthly one at the same nominal rate — the rate itself is what matters. The tool offers nine frequencies (annual, semi-annual, quarterly, monthly, semi-monthly, biweekly, weekly, daily, continuous) so you can see this for your own numbers.
Nominal rate vs effective yield (APY) — which one is advertised
The nominal rate is the number in the contract; the effective annual yield is what a year actually pays after compounding: APY = (1 + r/n)^n − 1, or e^r − 1 when compounding is continuous. A 5% nominal rate compounded monthly is a 5.1162% yield, which is why two products quoting 5% and 5.12% can be identical. The tool shows the effective yield next to the result so you can compare like with like.
US deposit accounts must publish APY under the Truth in Savings Act, so a US bank’s headline number already includes compounding. Korean 예금·적금 advertise the 연이율 (nominal), and an 적금 pays the quoted rate only on the money’s actual time on deposit — the first monthly instalment of a 12-month 적금 earns 12 months of interest, the last one earns a single month, so the realised yield lands near half the quoted rate. To model an 적금 here, enter 0 as the initial amount and put the instalment in the contribution field.
No tax is deducted anywhere in this calculator. Korean interest income is withheld at 15.4% (14% income tax + 1.4% local surtax), so a 5% deposit nets 4.23%; US interest is ordinary income at your marginal rate. To see the after-tax path, enter the after-tax rate instead of the headline one.
Regular deposits: beginning vs end of period
Deposit 500 a month for 20 years at 7% compounded monthly, starting from nothing, and the balance is 260,463.33 at the end. You put in 120,000; the other 140,463.33 is interest. Interest first exceeds the money you deposited in month 217 — a little over 18 years in — which is the real reason the advice is always “start early”: the curve is nearly flat for the first decade and then bends.
Switching the timing to the beginning of each period (an annuity due) gives every deposit one extra period of interest and ends at 261,982.70 — 1,519.37 more for the same money. Payroll deductions and standing orders on the 1st behave like beginning-of-period deposits; interest credited before a month-end sweep behaves like end-of-period. The difference is small in absolute terms but it is free, and it grows with the rate.
A negative contribution is a withdrawal, so the same fields model drawdown: enter the starting balance, a negative monthly amount and an expected return to see how long the pot lasts. If the balance ever goes below zero the tool stops and reports the year the money ran out, showing the table up to that point instead of printing a meaningless negative balance.
Doubling time: the Rule of 72 and the exact answer
The Rule of 72 says a balance doubles in about 72 ÷ (rate in per cent) years. The exact figure is ln(2) ÷ ln(1 + effective annual yield). The tool shows both, because the shortcut is only a good approximation in the middle of the range:
- 2% — rule says 36.0 years, the exact answer is 35.0
- 7% — rule says 10.29 years, exact 10.24 (annual compounding) or 9.93 (monthly)
- 20% — rule says 3.6 years, exact 3.80
The rule is at its best between roughly 6% and 10%, where it is within about half a per cent of the truth; below 3% and above 15% it drifts noticeably. Note also that the exact figure uses the effective yield, so a higher compounding frequency shortens the doubling time even when the nominal rate is unchanged.
What the balance is worth in today’s money
Fill in an inflation rate and the tool divides every balance by (1 + inflation)^years to express it in today’s purchasing power. 100,000 twenty years from now, with 3% inflation, buys what 55,367.58 buys today. The 260,463.33 from the 20-year savings plan above is worth 158,953.20 in today’s money at 2.5% inflation.
The rough version of this — real return ≈ nominal rate − inflation — is fine for a quick sanity check but overstates the result over long terms. The exact relationship (the Fisher equation) is (1 + nominal) ÷ (1 + inflation) − 1: 7% nominal with 3% inflation is a 3.88% real return, not 4%. The year table has a separate “in today’s money” column so you can watch both curves at once.
What this calculator does not model
It assumes one constant rate for the whole term. Real deposits reset, bonus rates expire after an introductory period, and market returns are not a straight line — a 7% average with volatility ends below a steady 7% for the same arithmetic mean. For a variable-rate product, run the calculation twice, once at the floor and once at the promotional rate, and treat the pair as the range.
It also ignores taxes, account fees, deposit-protection limits and any penalty for early withdrawal. Input limits: rate within ±1000%, term up to 100 years plus 0–11 months, at most 200,000 compounding periods (which is what stops a 100-year daily-compounding run), and an inflation rate no lower than −50%.