The concept
What is compound interest?
Compound interest is interest calculated on an initial amount and on interest added during earlier periods. Unlike simple interest, the base used for the calculation can grow over time.
The result depends on every assumption: the starting amount, contribution size and timing, assumed annual rate, compounding frequency and length of time. A calculator can show how those variables interact; it cannot determine what will happen. If you are comparing a quoted rate with an annual percentage rate, see APR vs interest rate; for a longer walk-through of the idea itself, read compound interest explained. Savings accounts usually quote an annual percentage yield (APY), which already includes the effect of compounding; the high-yield savings vs money market comparison covers how two common interest-bearing cash accounts differ.
The method
How the calculation works
For a starting principal without recurring contributions, the standard compound-interest formula is:
- A
- future value
- P
- starting principal
- r
- nominal annual rate as a decimal
- n
- compounding periods per year
- t
- time in years
With continuous compounding the growth factor becomes ert, so A = P·ert.
Regular contributions are added using the future value of an annuity. If an amount C is added k times a year (12 for monthly, 1 for annually), the nominal rate is first converted into the equivalent rate per contribution period, i = (1 + r/n)n/k − 1 (or er/k − 1 when continuous), and the contributions are worth:
- C
- contribution each period
- i
- equivalent rate per contribution period
- N
- number of contributions, k × t
- × (1 + i)
- applied when deposits are made at the start of each period
- i = 0
- the term is simply C × N
Using the equivalent periodic rate means a monthly deposit earns interest for the fraction of a compounding period it is held, even under annual or quarterly compounding. Some accounts credit nothing until the period closes, so a real statement can differ slightly.
Rounding rule
Every balance is evaluated directly from the formula for its own point in time, at full precision. It is not built by repeatedly adding interest to a running total, so no rounding error accumulates over long or daily-compounded periods. Values are rounded to the nearest cent (or whole yen) only when displayed. In the year-by-year table, each year's interest is the rounded closing balance minus the rounded opening balance minus that year's contributions, so every row reconciles exactly.
Worked example
A neutral example
A starting amount of $10,000 at a 5% nominal annual rate, compounded monthly for 10 years with no contributions, grows to $10,000 × (1 + 0.05/12)120 ≈ $16,470.09. Adding $100 at the end of every month puts in $12,000 of deposits that are worth about $15,528.23 by the end, for a total of about $31,998.32. If those deposits are made at the start of each month instead, their value rises to about $15,592.93, because each one earns one extra month of interest.
This example explains the method. It is not a recommendation and the assumed rate is not a prediction.
Frequently asked
Compound interest questions
Does a higher compounding frequency always make a large difference?
Not necessarily. The effect depends on the rate, time period and how the quoted rate is defined. At 5% on $10,000 over 10 years, annual compounding gives about $16,288.95 and continuous compounding about $16,487.21, a gap of roughly $198 on more than $6,000 of growth. Differences are smaller for shorter periods or lower rates.
Should contributions be at the start or the end of each period?
Choose whichever matches when the money is actually added. A deposit at the start of a period earns interest for that whole period, so the start-of-period result is exactly one period of growth larger than the end-of-period result for the same inputs.
What is continuous compounding?
It is the limit of compounding more and more often. Instead of (1 + r/n)^(nt), growth is e^(rt). Few savings products credit interest this way; it is included as the mathematical upper bound for a given nominal rate.
Does this calculator include tax or fees?
No. It is a simplified mathematical illustration. Taxes, fees and changing rates can materially change real outcomes.
Can the estimated growth be negative?
Yes. If you enter a negative assumed rate, the calculated value can decline. Contributions may still make the ending balance larger than the starting balance.