How the result is calculated
Put money in an account that pays interest and, at the end of the first period, you have your deposit plus a little interest. In the second period the interest is calculated on that slightly larger balance, so it is slightly larger too. Repeat for years and the small differences stack into a curve that bends upward. That is compounding, and it is why time in the market matters so much more than most people expect.
FVP × (1 + r)n + c × ((1 + r)n − 1) ÷ r
- FV
- the final balance
- P
- the starting amount
- c
- the deposit made at the end of every period
- r
- the rate per period: the annual rate divided by the periods a year
- n
- the total number of periods: years × periods a year
When deposits are made at the start of each period rather than the end, the second term is multiplied by (1 + r), because every deposit earns one extra period of interest. When the deposit schedule and the compounding schedule differ, daily compounding with monthly deposits, say, there is no single r that fits both, so the calculator converts the compounding rate into an equivalent monthly growth rate and steps through the projection a month at a time. Whenever the two schedules match, that stepping reproduces the formula exactly.
Banks advertise a nominal rate; what you actually receive over a full year is the annual percentage yield, shown beside the result with its own working:
APY(1 + r ÷ m)m − 1
- m
- the number of compounding periods a year
Worked example
| Item | Amount | How it is obtained |
|---|---|---|
| Starting amount, P | $10,000.00 | Starting amount |
| Rate per period, r | 0.5% | 6% ÷ 12 |
| Periods, n | 240 | 20 years × 12 |
| Growth factor, (1 + r)n | 3.3102 | 1.005240 |
| Lump sum grown | $33,102.04 | 10,000 × 3.3102 |
| Deposits grown | $46,204.09 | 100 × (3.3102 − 1) ÷ 0.005 |
| Final balance | $79,306.13 | 33,102.04 + 46,204.09 |
| Paid in | $34,000.00 | 10,000 + 240 × 100 |
| Interest earned | $45,306.13 | 79,306.13 − 34,000.00, more than you contributed |
The APY for this account is (1 + 0.06 ÷ 12)^12 − 1 = 6.168%.
| Item | Amount | How it is obtained |
|---|---|---|
| Rate per period, r | 0.58333% | 7% ÷ 12 |
| Periods, n | 360 | 30 years × 12 |
| Growth factor, (1 + r)n | 8.1165 | 1.0058333360 |
| Final balance | $609,985.50 | 500 × (8.1165 − 1) ÷ 0.0058333 |
| Paid in | $180,000.00 | 360 × 500 |
| Interest earned | $429,985.50 | 609,985.50 − 180,000.00 |
In the final year alone the balance grows by about $40,000, more than five years of deposits.
Reading the result
The headline is the balance at the end of the term, with the amount paid in and the interest earned beneath it. The stacked chart separates three layers: the starting amount (flat, since it never changes), cumulative deposits (a straight ramp) and cumulative interest (the curve). The year in which the interest layer becomes the thickest is the point where the account is doing more work than you are.
The table gives the same information year by year. Start is the balance carried in, Deposits what you added during the year, Interest what the account credited and End balance the sum of the three. The rows always add up, and the last end balance is the headline figure.
The projection steps a month at a time, so switching the table to Every month shows the calculation itself: each month's deposit, the interest credited at the effective monthly rate, and the balance it leaves. A year row is twelve of those months added up. Whichever way you enter the regular deposit, it is restated per month and per year under the result, so a monthly plan and an annual one can be compared on the same footing.
What changes the result
Why the deposits matter more than the lump sum
In the opening example the $100 monthly deposits end up worth $46,204 while the $10,000 lump sum grows to $33,102. Over twenty years the steady stream of small deposits beats the larger initial amount, partly because $24,000 of contributions is more money than $10,000, but also because each deposit still has years to compound.
Nominal rate versus APY
The compounding schedule moves the needle less than most people expect. At a 6% nominal rate:
| Compounding | Periods a year | APY |
|---|---|---|
| Yearly | 1 | 6.000% |
| Half-yearly | 2 | 6.090% |
| Quarterly | 4 | 6.136% |
| Monthly | 12 | 6.168% |
| Daily | 365 | 6.183% |
| Continuous | ∞ | 6.184% |
Two lessons follow. When comparing accounts, compare APYs, not nominal rates: the APY already contains the compounding schedule. And the choice between monthly and daily compounding is worth a few dollars a year on a typical balance, while a half-point difference in the rate itself is worth far more.
Time and rate
Time and rate are the two levers that matter. Doubling the term more than doubles the interest, because the second decade compounds on everything the first one built. A one-point change in the rate, held for twenty years, moves the final balance by more than a tenth.
What is not included
- Tax. Interest in an ordinary account is usually taxed every year, which slows compounding. In a tax-sheltered account the projection holds as shown.
- Inflation. A balance of $79,306 in twenty years will not buy what $79,306 buys today. Run the result through the inflation calculator, or use the retirement calculator, which shows real values.
- Changing rates. Savings rates move with central-bank policy. The projection assumes the rate you enter holds for the whole term; try a low and a high case to see the range.
- Fees. Account or fund fees reduce the effective rate. If you pay 0.5% a year in fees on a 7% return, enter 6.5%.
Using the result
Use the calculator to answer concrete questions: how much will I have if I keep this up, how much sooner do I reach a target if I add $50 a month, and what does a one-point rate difference cost over the long run. If you have a specific target in mind, the savings goal calculator inverts the same formula to tell you the monthly deposit you need.