Compound Interest Calculator
Compound interest reinvests interest as it accrues, so the balance grows faster than under simple interest. The calculator uses A = P·(1 + r/n)^(nt) for discrete compounding or A = P·e^(rt) for continuous, returning both the final amount and the interest earned.
Discrete and continuous growth
Compound interest pays interest on interest. Under simple interest a balance grows by the same amount each period; under compounding, each period's interest joins the principal and earns in its turn, so growth accelerates.
This calculator takes a principal, an annual rate, a term in years and a compounding frequency, and returns the final balance together with the interest earned. It supports both discrete compounding a fixed number of times a year and the continuous limit.
Discrete compounding is what real accounts do: interest is credited monthly, quarterly or annually, and the frequency is a term of the product. Continuous compounding is the mathematical limit as that frequency grows without bound, and it is the form that appears throughout calculus and financial theory.
Having both in one tool makes the comparison easy, and the comparison is instructive: the gap between compounding monthly and compounding continuously is far smaller than most people expect.
How to use this calculator
- Enter the principal The starting balance. The calculator is currency-agnostic and performs no conversion, so the unit you enter is the unit you get back.
- Enter the annual rate as a percentage Enter 5 for five percent, not 0.05. The rate line in the output echoes both the percentage you typed and the decimal actually used, which is the quickest way to confirm this field.
- Enter the time in years Always years, whatever the compounding frequency. Eighteen months is 1.5, not 18 — the frequency is handled by n, not by changing the unit of t.
- Choose the compounding mode, and set n if discrete For discrete compounding n is the number of times interest is credited per year: 1 annual, 2 semi-annual, 4 quarterly, 12 monthly, 365 daily. It must be positive. In continuous mode the n field is ignored entirely.
The formula, and where it comes from
A = P·(1 + r/n)^(n·t) A = P·e^(r·t) (continuous) I = A − P
In the discrete formula the annual rate is divided by n to give the rate for one period, and the exponent n·t counts how many periods elapse in total. Each period multiplies the balance by the same factor, so the growth is geometric rather than linear, and the exponent is where the compounding effect lives.
The continuous formula is what the discrete one becomes as n grows without bound: (1 + r/n)^n tends to e^r, so the whole expression tends to P·e^(rt). That limit is the definition of e, which is why the exponential function appears in a finance formula at all.
The percentage you enter is divided by 100 before use, so 5 becomes r = 0.05. Interest earned is then simply the final amount minus the principal — the tool does not compute it independently, so it always reconciles exactly.
What each input means
- P Principal — form field “Principal P”
- The starting balance, before any interest. A negative value is arithmetically accepted but describes a debt rather than a deposit. Units: currency, unconverted.
- r Annual rate — form field “Annual rate r (%)”
- The nominal annual rate as a percentage. It is nominal rather than effective: the actual yearly growth depends on n as well, which is why two accounts quoting the same rate can pay differently. Units: percent per year.
- t Time — form field “Time t (years)”
- Term in years. Fractional values are meaningful and are handled directly by the exponent. Units: years.
- n Compoundings per year — form field “Compoundings per year n”
- How many times interest is credited annually. Must be positive; the solver rejects zero and negatives. Ignored in continuous mode, where the field can be left at any value. Units: times per year.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
Monthly compounding over a decade
1,000 at a nominal 5% a year, compounded monthly for ten years. This is the shape of an ordinary savings account.
Inputs Principal P = 1000, Annual rate r (%) = 5, Time t (years) = 10, Compoundings per year n = 12, Compounding mode = discrete
- Principal P = 1000.00
- Rate r = 5% = 0.05 per year
- Time t = 10 years
- Compoundings per year n = 12
- Formula A = P · (1 + r/n)^(n·t)
- Final amount A = 1000.00 · (1 + 0.05/12)^(12·10) = 1647.01
- Interest earned I = A − P = 647.01
Result A = 1647.01, I = 647.01
The balance grows by about 65% over the decade, well beyond the 50% that simple interest at the same rate would give. The extra comes entirely from interest earned on previously credited interest.
The formula line shows the rate divided by twelve and the exponent as 12 × 10 = 120 periods. Reading those two together is the clearest way to see that n does two opposing things: it shrinks each period's rate and multiplies the number of periods, and the second effect wins slightly.
The same deposit compounded continuously
Identical principal, rate and term, but with compounding taken to its limit. This is the upper bound on what any compounding frequency can achieve.
Inputs Principal P = 1000, Annual rate r (%) = 5, Time t (years) = 10, Compoundings per year n = 0, Compounding mode = continuous
- Principal P = 1000.00
- Rate r = 5% = 0.05 per year
- Time t = 10 years
- Compounding Continuous
- Formula A = P · e^(rt)
- Final amount A = 1000.00 · e^(0.05·10) = 1648.72
- Interest earned I = A − P = 648.72
Result A = 1648.72, I = 648.72
The continuous result exceeds the monthly one by only a couple of units on a thousand — a fraction of a percent over ten years. Compounding frequency matters far less than the rate or the term, which is worth knowing before paying attention to a product advertising daily compounding.
Note that n is entered as 0 here without causing an error. In continuous mode the field is never read, so its value is irrelevant; the same input would be rejected in discrete mode.
Reading the result
Nominal rate versus what you actually earn
The rate entered is nominal. The effective annual rate — what the balance actually grows by in a year — is (1 + r/n)^n − 1, which exceeds the nominal rate whenever n is greater than one. Comparing two products means comparing effective rates, not the headline figures.
What the result excludes
This is pure compound growth: no tax, no fees, no inflation, no deposits or withdrawals after the start. A real account will return less. For a series of regular payments rather than a single lump, the annuity calculation is the right model.
When you would use this
Projecting a deposit or a fixed-term investment
With principal, rate and term known, the final balance is a direct calculation. Varying the rate across a plausible range shows how sensitive the outcome is to an assumption you cannot control.
Comparing compounding frequencies fairly
Running the same principal and rate at n = 1, 12 and 365, then in continuous mode, quantifies exactly what a more frequent compounding schedule is worth — usually much less than the marketing suggests.
Assumptions and limitations
What this calculator assumes
- The rate is constant for the whole term and the balance is untouched: no further deposits, withdrawals or rate changes.
- The rate entered is nominal and annual, expressed as a percentage, and is divided by 100 internally.
- In discrete mode n must be positive; in continuous mode it is not read at all.
Where it stops being the right tool
- A single lump sum only. Regular contributions make the problem an annuity, which this tool does not model.
- No taxes, fees, inflation adjustment or currency conversion — all of which change a real outcome materially.
Common mistakes
Entering the rate as a decimal
Why it happens. The formula is written with r as a decimal, so 0.05 looks like the right thing to type. This field expects a percentage, so 0.05 is read as 0.05% — a hundredfold understatement that yields almost no growth.
How to avoid it. Enter 5 for five percent, and check the rate line, which shows both the percentage and the decimal the calculation actually used.
Converting the term to match the compounding frequency
Why it happens. Having set n = 12 for monthly compounding, it feels consistent to enter the term in months as well. The formula already multiplies t by n, so doing it yourself compounds the term twelve times over.
How to avoid it. Leave t in years always. If a ten-year projection returns an implausibly enormous balance, this is almost certainly why.
Comparing nominal rates across different frequencies
Why it happens. Two accounts quoting 5% look identical, but one compounding daily grows faster than one compounding annually. The headline rate hides the difference.
How to avoid it. Compute both here with their real n values and compare the final amounts, or convert each to its effective annual rate before comparing.
Frequently asked questions
What is the difference between discrete and continuous compounding?
Discrete compounding credits interest n times a year, using A = P(1 + r/n)^(nt). Continuous compounding is the limit as n grows without bound, where (1 + r/n)^n tends to e^r and the formula becomes A = P·e^(rt).
What value of n should I use?
The number of times interest is credited per year: 1 for annual, 2 for semi-annual, 4 for quarterly, 12 for monthly, 365 for daily. It must be positive, and it is ignored entirely in continuous mode.
How much does compounding frequency actually matter?
Less than most people expect. On 1,000 at 5% for ten years, monthly compounding and continuous compounding differ by only a couple of units — a fraction of a percent. The rate and the term dominate the outcome.
Can I use this for a regular monthly saving?
No. This models a single lump sum left untouched. A stream of equal payments is an annuity, with its own formula, and using compound interest on the total of your contributions will overstate the result substantially.