Compound Interest Calculator

Interest on interest

Compound interest is interest computed on a balance that already includes earlier interest. After the first period the balance is the principal plus one period's interest; the second period's interest is computed on that larger figure; and so on, so each period's growth is slightly larger than the one before. The calculator above runs exactly that process for a single lump sum, and every figure on this page is computed at build time by the same tested engine that runs the calculator — the copy cannot drift from the tool.

In the reference example — $10,000 at an assumed 6% rate for 10 years, compounded monthly — the balance grows to $18,193.97, of which $8,193.97 is interest. Simple interest, which computes every year's interest on the original principal only, would add $600.00 a year and $6,000.00 over the decade. The difference, $2,193.97, is interest earned by interest: the part of the result that exists only because earlier growth was left in the balance to grow again.

The same arithmetic runs inside every account type. As the companion guide 401(k), Roth IRA, and Taxable Accounts: How They Fit Together puts it, a dollar does not grow faster because of the label on the account — what differs between accounts is when the tax happens, and tax is something this calculator does not model at all.

The formula the calculator runs

FV = P·(1 + rate/m)^(m·t)

P is the starting principal, rate the stated annual rate as a decimal, m the number of compounding periods in a year, and t the number of years. The frequency selector sets m: 1 for annual, 2 for semiannual, 4 for quarterly, 12 for monthly, 365 for daily. The engine divides the annual rate by m to get the rate per period, multiplies m by t to get the number of periods, and raises the per-period growth factor to that power. Interest earned is FV − P. The result box reports the final balance, the interest, and the principal echoed back.

Two things happen when m increases, and they pull in opposite directions. The rate per period shrinks — a stated 6% a year is 0.5% a month or about 0.0164% a day — while the number of periods grows in step. If the two effects cancelled exactly, frequency would not matter at all. They almost cancel: the expression (1 + rate/m)^m rises as m rises, but slowly, and by a smaller amount at each step. That is the whole story of compounding frequency, and the tables below put numbers on it.

This is the general form of the site's core formula. The contribution-based calculators here — Investment, Roth IRA, Savings Goal, Retirement, and the Retirement Planner — fix m at 12 (monthly compounding at the annual rate ÷ 12 is the house convention) and add a second term for a stream of contributions; the 401(k) Calculator reaches the same monthly compounding by stepping through each month instead; the Inflation Calculator compounds once a year; and the CD Calculator, like this one, lets you choose the frequency. The methodology page sets all of this out. A 0% rate collapses the growth factor to 1, so the balance simply equals the principal. A fractional number of years is accepted and applied as a fractional exponent: unlike the monthly-compounding tools, which round the term to whole months, this function uses t exactly as entered.

Frequency: real but modest

The reference deposit — $10,000 at an assumed 6% for 10 years — at each posting schedule the form offers:

Reading down the last column: the step from annual to semiannual is worth $152.64, semiannual to quarterly $79.07, quarterly to monthly $53.78, and monthly to daily only $26.32. Each step at least doubles m — the last two steps multiply it by 3 and then by 30.4 — yet each step buys less than the one before. That is the shrinking gap the formula predicts: the sequence (1 + rate/m)^m has a ceiling — the continuous-compounding limit, where interest is added at every instant — and it closes most of the distance to that ceiling in the first few steps. By monthly compounding, little of the achievable gain is left; daily posting collects 9.2% of what the annual-to-monthly move collected.

Whether the gap is "small" is a matter of scale. Over the decade the whole spread from annual to daily is $311.81, or 1.7% of the annual-compounding balance. That is real money, and it is also less than a single extra year at monthly compounding would add on the same deposit — the section on time below makes that comparison directly.

The same ladder at a higher rate and a longer horizon

The dollar gaps scale with the balance, the rate, and the horizon. A second example uses $50,000 at an assumed 8% for 20 years:

Here the annual-to-monthly step is worth $13,292.28 and the monthly-to-daily step $1,268.07 — far larger in dollars than in the reference example, because the balance is larger, the rate higher, and the compounding runs twice as long. The whole spread is 6.2% of the annual-compounding balance, up from 1.7% in the reference case. The shape does not change, though: the second step is 9.5% of the first, almost the same proportion as the 9.2% in the reference example. The frequency gap grows with the inputs; the drop-off from one frequency to the next is a property of the formula and stays put.

APY versus the stated rate

A stated rate — the number entered in the rate field, sometimes called the nominal rate — says nothing about frequency on its own. The annual percentage yield (APY) folds frequency in: it is the growth the formula produces in exactly one year, expressed as a percentage of the principal, APY = (1 + rate/m)^m − 1. At a stated 6%:

The APY equals the stated rate only for annual compounding, and it rises with m by the same shrinking steps as the balances above, because it is the same expression with t = 1. It is never below the stated rate. APY is the frequency-free description of growth: compounding once a year at the monthly-compounding APY (6.168%, carried unrounded in the engine) for 10 years gives $18,193.97 — the monthly figure exactly — because (1 + rate/m)^(m·t) is ((1 + rate/m)^m)^t. The identity holds for the unrounded yield; a hand check that starts from the printed 6.168% lands at $18,194.34, $0.37 away — the same kind of rounding-before-exponentiation effect the methodology page describes for the monthly rate. Two offers with different posting schedules are therefore comparable on APY where their stated rates are not. A hypothetical stated 6.15% compounded annually is an APY of 6.150%; a stated 6% compounded daily is 6.183% — the lower stated rate carries the higher yield. The CD Calculator reports the APY a deposit's rate and frequency imply alongside its maturity value; for the larger example above, 8% compounded monthly is an APY of 8.300%.

Time is the real multiplier

Because each period builds on the last, the growth curve steepens the longer a balance is left alone. The reference deposit at an assumed 6%, compounded monthly, across three horizons:

Doubling the horizon does not double the interest; it multiplies it by 2.82×. Tripling it multiplies the interest by 6.13×. The reason is the exponent: each additional year applies the growth factor to a balance that already contains every earlier year's interest, so the last decade of the 30-year run adds $27,123.71 while the first decade added $8,193.97 — same deposit, same rate, same posting schedule.

Set against this, frequency is a small lever. On the reference inputs, switching from monthly to daily compounding adds $26.32; leaving the balance in place for an eleventh year at monthly compounding adds $1,122.16. The rough check described in the guides — the Rule of 72, an approximation that divides 72 by the annual growth rate to estimate the years to double, 72 ÷ 6 = 12 years at 6% — lands close to the model: at 12 years the monthly-compounded balance is $20,507.51, just past double the deposit.

How to read the output, and where the model stops

The result box reports three figures: the final balance, the interest earned (the final balance minus the principal), and the principal echoed back. Three conventions from the methodology page apply to all of them. Nominal dollars: the balance is in future dollars; to see purchasing power in today's dollars, enter a real (after-inflation) rate, or run the balance through the Inflation Calculator. Constant rate: one steady rate for the whole term — real returns vary year to year, and the model does not attempt to simulate that. Lump sum only: no contributions, no withdrawals, no fees, no taxes. A starting amount plus a monthly contribution stream is the Investment Calculator's job — it runs the same compounding with the contribution term added — and the Savings Goal Calculator runs that equation in reverse to solve for the contribution that reaches a target.

Four common misreadings of the output:

• Treating frequency as the lever. On the reference deposit, daily instead of annual compounding is worth $311.81 over the decade; the same deposit left for a second decade earns a further $14,908.08. The rate and the horizon set the result; the frequency fine-tunes it.
• Reading the stated rate as the yield. At any frequency above annual, the one-year growth is the APY, not the stated rate — 6.168% rather than 6% for the monthly reference case.
• Comparing stated rates across frequencies. A lower stated rate compounded daily can carry a higher APY than a slightly higher rate compounded annually, as the 6.15% example above shows. The APY column, not the rate, is the like-for-like figure.
• Extending one year's interest in a straight line. Interest is not a fixed amount per year. In the reference case the first year adds $600.00 at annual compounding, while the thirtieth year of the monthly run adds a much larger amount on a much larger base; the 6.13× figure in the table above is the cumulative result.

The engine validates every input before it computes: a principal above zero, a stated rate from 0% to 100%, a term of up to 80 years, and a frequency from 1 to 365 periods a year. Out of range, the result box shows the reason instead of a number.

Compounding is the accumulation half of a retirement projection. The other half — which account holds the balance, when it can be drawn, and what a sustainable withdrawal looks like — is covered in the site's guides, starting with 401(k), Roth IRA, and Taxable Accounts: How They Fit Together, which walks through combining this site's calculators into a single multi-account picture.

Frequently asked questions

Not financial advice: a general educational estimate. Real savings and investment returns vary and are not guaranteed. Values are processed locally in your browser and never transmitted. See the methodology page.