What compounding actually does to a balance
Compound interest is interest that itself earns interest. Simple interest pays you the same dollar amount every period because it is always computed on the original principal; compound interest computes each period's interest on the balance you now hold, including everything credited before. That single difference is why the two methods look almost identical over one year and wildly different over thirty.
The gap grows because the balance follows a geometric sequence rather than an arithmetic one. Each period multiplies the balance by the same factor, so after many periods the balance is the starting amount times that factor raised to a power. Raising a number slightly above one to a large power is the entire mechanism — there is nothing else going on.
Two decisions dominate the result. The first is the rate. The second is time, because time sits in the exponent while the rate sits in the base, and an exponent is a far more powerful lever. Deposits matter enormously too, but in a different way: they raise the amount that gets multiplied rather than the multiplier. This calculator keeps those contributions visible as a separate output so you can always see how much of a projected balance is your own money and how much is growth.
The formula, term by term
The accumulation has two independent parts, and you can add them because compounding is linear in the amount invested. The first part is the starting balance P compounded forward: P(1 + r/n)^(nt). Divide the nominal annual rate r by the number of compounding periods per year n to get the rate credited each period, then raise one plus that rate to the total number of periods, n times t.
The second part is the stream of deposits. Each deposit is compounded from the moment it lands until the end of the horizon, so the deposit made first grows the longest and the last one does not grow at all. Summing that geometric series gives PMT · [((1 + i)^N − 1) / i], the future value of an ordinary annuity, where N is the number of deposits.
The subtlety most calculators get wrong is the rate i used in that annuity term. If you deposit monthly into an account that credits interest annually, the monthly rate is not the annual rate divided by twelve; it is the rate that compounds twelve times into exactly the annual factor. This page uses i = (1 + r/n)^(n/m) − 1, which reduces to r/n whenever your deposit frequency m equals the compounding frequency n, and stays correct when it does not.
Two special cases need their own branch. When the rate is exactly zero the annuity term divides by zero, so the code returns PMT × N instead — the balance is simply the money you put in. When deposits arrive at the start of each period rather than the end, every deposit earns one extra period of interest, so the whole annuity term is multiplied by (1 + i). That variant is called an annuity due, and it is the right choice for a payroll deduction that hits the account before the month's interest is credited.
Worked example: $10,000 plus $500 a month at 7% for 20 years
Take the default case — a $10,000 starting balance, $500 deposited at the end of every month, a 7% nominal annual return compounded monthly, over 20 years, with inflation assumed at 2.5%.
- Rate per period. r/n = 0.07 ÷ 12 = 0.00583333.
- Number of periods. n·t = 12 × 20 = 240.
- Growth factor. 1.00583333240 = 4.038738. Every dollar sitting there at the start becomes just over four dollars.
- Lump-sum part. $10,000 × 4.038738 = $40,387.38.
- Rate per deposit. Deposits are monthly and compounding is monthly, so i = 0.00583333 as well, and N = 240.
- Annuity part. (4.038738 − 1) ÷ 0.00583333 = 520.9265, and 520.9265 × $500 = $260,463.26.
- Ending balance. $40,387.38 + $260,463.26 = $300,850.64.
- Money you put in. $10,000 + 240 × $500 = $130,000, so interest earned is $300,850.64 − $130,000 = $170,850.64.
- In today's dollars. 1.02520 = 1.638614, and $300,850.64 ÷ 1.638614 ≈ $183,600.
Notice that growth ($170,851) exceeds contributions ($130,000) at this rate and horizon. That crossover is the single most useful thing to watch in the schedule table, because it tells you when the account starts doing more work than you do.
How to read the result
Read the ending balance last, not first. The three numbers that tell you whether the projection is honest are the total you put in, the interest earned, and the effective annual yield.
Total put in versus interest earned. Their ratio is a direct measure of how much of the answer depends on your rate assumption. If interest is a small fraction of the total, the projection is mostly arithmetic and fairly reliable. If interest is several times contributions, you are looking at a projection whose accuracy rests almost entirely on a rate you guessed, and a one-point error in that rate moves the answer by a large amount.
Effective annual yield. This is the number to compare across accounts. A 5.00% rate compounded daily and a 5.10% rate compounded annually are not directly comparable as quoted; their yields are 5.1267% and 5.1000%, and now they are. Under Regulation DD, U.S. depository institutions must disclose exactly this figure, computed on this basis, for deposit accounts.
Today's dollars. The real figure is the nominal balance divided by the inflation factor over the same horizon. It answers a different question: not how many dollars you will have, but roughly what those dollars will buy. Over long horizons the two diverge sharply, which is why retirement planning is normally done in real terms.
If you are comparing this projection against a competing use of the money — paying down a loan, or a project with irregular cash flows — a growth projection is the wrong tool. Use the net present value calculator or the internal rate of return calculator, which handle cash flows that vary in size and sign.
What one dollar becomes: growth factors with annual compounding
| Annual rate | 5 years | 10 years | 20 years | 30 years |
|---|---|---|---|---|
| 3% | 1.159274 | 1.343916 | 1.806111 | 2.427262 |
| 5% | 1.276282 | 1.628895 | 2.653298 | 4.321942 |
| 7% | 1.402552 | 1.967151 | 3.869684 | 7.612255 |
| 9% | 1.538624 | 2.367364 | 5.604411 | 13.267678 |
| 11% | 1.685058 | 2.839421 | 8.062312 | 22.892297 |
Each cell is (1 + r)^t with interest credited once a year. Monthly compounding raises these slightly: at 7% for 30 years the monthly factor is 8.116, not 7.612.
Mistakes that break a compounding projection
- Mixing up the nominal rate and the yield. A bank advertising 5.00% APY compounded daily has a nominal rate of about 4.879%, not 5.00%. Enter the nominal rate here and check that the APY output matches the advertised figure.
- Assuming the deposit rate is the annual rate over twelve. That is only correct when interest is credited monthly. For an annually compounded account the correct monthly rate is (1 + r)^(1/12) − 1, which this page computes for you.
- Projecting a volatile return as if it were a rate. A portfolio that returns +30% then −20% has grown 4% in total, not 10%. A single smoothed rate is the right input here, and the CAGR calculator is how you extract one from real start and end values.
- Ignoring fees. An expense ratio comes straight off the return. If a fund charges 0.60% and you expect 7% gross, enter 6.4%. Over decades that difference is larger than most people expect because it compounds too.
- Ignoring tax on interest. In a taxable account, interest and dividends are usually taxed in the year they are credited, which reduces the amount left to compound. Enter an after-tax rate, or model the account as tax-deferred and remember the projection is pre-tax.
- Confusing nominal growth with purchasing power. A balance that grows at 3% while inflation runs at 3% has bought you nothing. The today's-dollars output exists precisely to make that visible.
Where the APY convention comes from
The effective annual yield reported here follows the annual percentage yield definition used by Regulation DD, the rule implementing the U.S. Truth in Savings Act, which is codified at 12 CFR Part 1030. Its appendix defines APY for deposit accounts as the total interest that would be received on a $100 deposit over a 365-day year, expressed as a percentage, given the institution's compounding method. That is what makes accounts with different compounding frequencies comparable on a single number, and it is why the yield rather than the nominal rate is the figure to shop on.
Where this sits among the other time-value tools
Compounding forward and discounting back are the same operation run in opposite directions. This calculator answers what will I have. If you already know the target balance and want the amount needed today, that is the present value calculator. If your problem is stated in periods and a periodic rate rather than years and an annual rate — the way finance coursework and calculator keystroke problems are written — use the future value calculator, which takes i and n directly and reports the lump-sum and annuity components separately.
For a quick sanity check without any calculator, the rule of 72 is close enough for mental arithmetic: divide 72 by the percentage rate to get the approximate doubling time. At 7% that is about 10.3 years, and the exact answer from ln(2) ÷ ln(1.07) is 10.24 years. The approximation drifts at high rates, so treat it as a check on the order of magnitude rather than a substitute for the formula.
Where cash flows are irregular — a business investment, a property with different net income each year — a single compounding formula cannot represent them. Discount each flow on its own date instead, which is what the discounted payback period calculator and the payback period calculator do. And when the question is what rate a whole business must clear before an investment adds value, the relevant benchmark is its cost of capital, not a savings rate; the WACC calculator produces that figure.
One limitation is worth stating plainly. This model assumes a constant rate. Real markets deliver a sequence of different returns, and for a portfolio you are drawing down, the order of those returns changes the outcome even when their average is identical. A constant-rate projection is a planning tool and a comparison device. It is not a forecast, and it does not price risk.
Key terms
- Nominal annual rate
- The quoted yearly rate before the effect of compounding within the year. Sometimes labelled the stated rate or, on loans, the note rate.
- Annual percentage yield (APY)
- The rate that would produce the same one-year growth with a single annual crediting. It is the number that makes accounts with different compounding frequencies comparable.
- Ordinary annuity
- A stream of equal payments made at the end of each period. The default assumption on this page and in most textbook problems.
- Annuity due
- The same stream paid at the start of each period. Each payment earns one extra period of interest, so its future value is the ordinary annuity times (1 + i).
- Real return
- The nominal return adjusted for inflation. Precisely, (1 + nominal) ÷ (1 + inflation) − 1, though the difference of the two rates is a common approximation.