Compound Interest Formula: Variables, Variants and Limits

The standard compound interest formula answers a question few savers actually ask: what a single untouched sum becomes over time. Regular contributions require a different equation entirely.

Search for the compound interest formula and the same expression comes back everywhere: A = P × (1 + r/n)^(n×t). It is correct. It is also the formula for a case that almost nobody is in, since it assumes a sum deposited once and never touched again. The moment monthly contributions enter the picture, that equation stops describing the situation, and the gap between the two answers is where most projection errors originate.

The formula never errs. The rate you feed it does.

The base formula and what each variable conceals

In A = P × (1 + r/n)^(n×t), P is the principal, r the annual nominal rate expressed as a decimal, n the number of compounding periods per year, and t the number of years. The exponent n×t counts total compounding periods; the factor r/n splits the annual rate across them.

That split is a convention, not an equivalence. At a 5% annual rate, r/12 gives a monthly rate of 0.4167%, while the rate that genuinely compounds to 5% over twelve months is (1.05)^(1/12) − 1 = 0.4074%. The two differ by less than a tenth of a percentage point, which is why the convention survives, but it explains why a monthly calculation ends up slightly above its annual counterpart rather than exactly equal to it.

The practical consequence runs against intuition. On €10,000 at 5% over 20 years, annual compounding yields €26,533, monthly compounding €27,126, daily compounding €27,181. Moving from annual to monthly adds €593 across two decades. Moving from monthly to daily adds €55. Compounding frequency is a second-order variable, while the rate and the horizon are first-order ones.

The formula with regular contributions, the one most people need

When money arrives periodically, each payment compounds over a different remaining horizon. The first contribution works for the full term, the last for a single period. Summing that geometric series gives the future value of an ordinary annuity, which is added to the compounded opening balance:

FV = P × (1 + i)^N + PMT × [((1 + i)^N − 1) / i], where i is the periodic rate, N the number of periods, and PMT the amount paid each period.

Applied to €200 per month for 20 years at 5% annual (i = 0.4167%, N = 240), the contribution stream alone reaches €82,207 against €48,000 actually paid in, so €34,207 comes from compounding. Add a €10,000 opening balance and the total reaches €109,333, of which the opening balance accounts for €27,126. The stream contributes three times what the initial sum does, on a quarter of the invested capital per unit of time. An interactive compound interest calculator makes the substitution easier to test than the algebra does.

This is also where the ordinary annuity convention matters. The formula above assumes payment at the end of each period. Paying at the start of each period multiplies the whole annuity term by (1 + i), which adds roughly one extra period of compounding across the entire stream.

Continuous compounding and where the constant e appears

Increasing n without limit does not send the result to infinity. As n grows, (1 + r/n)^(n×t) converges toward e^(r×t), which gives A = P × e^(r×t). At 5% over 20 years on €10,000, that ceiling sits at €27,183, precisely €56 above monthly compounding.

Continuous compounding is therefore not a more aggressive assumption. It is the upper bound of the family, and its usefulness is analytical rather than commercial: it turns a discrete product into an exponential that differentiates cleanly, which is why it appears throughout derivatives pricing and rarely on a savings statement.

From nominal to real: divide, do not subtract

A nominal return net of inflation is not a subtraction. The exact relation, from Fisher, is real = (1 + r) / (1 + π) − 1, where π is the inflation rate over the same period.

With 5% nominal and 2% inflation, subtraction gives 3% while the exact figure is 2.9412%. The 0.06 percentage point looks negligible until it compounds: on €10,000 over 20 years, the approximate rate projects €18,061 against €17,856 for the exact one, a €205 overstatement. The error scales with both the horizon and the inflation level, which is why the shortcut becomes untenable precisely in the regimes where the real return question matters most. The mechanism behind that divergence is developed in our note on the nominal illusion in long-horizon savings.

What the formula assumes, and cannot capture

Every version above holds the rate constant. Real returns arrive as a sequence, and sequence changes the outcome even when the average does not. A portfolio gaining 50% then losing 50% has an arithmetic mean return of 0% and a terminal value 25% below its starting point. Its geometric mean, the rate that actually reproduces the observed path, is −13.40%. The compound interest formula only accepts geometric rates; feeding it an arithmetic average silently inflates every projection, a distortion examined in why average returns mislead on trajectory.

The rule of 72 belongs to the same family of shortcuts. Dividing 72 by the rate estimates the doubling time, and its accuracy is narrow: at 10% it gives 7.2 years against 7.27 exact, at 5% it gives 14.4 against 14.21, and at 2% it gives 36 against 35.0. The exact expression is ln(2) / ln(1 + r). The approximation is calibrated for rates near 8% and degrades on either side, which makes it useful for mental arithmetic and unusable for a projection anyone intends to rely on. The same caution applies to the erosion side of the calculation, where real return on savings after inflation sets out the arithmetic in the opposite direction.

Frequently asked questions

What is the compound interest formula with monthly contributions?

FV = P × (1 + i)^N + PMT × [((1 + i)^N − 1) / i], with i the monthly rate and N the number of months. The first term compounds the opening balance, the second sums each contribution compounded over its own remaining horizon. For €200 monthly at 5% annual over 20 years, it returns €82,207 on €48,000 paid in.

How do you calculate compound interest monthly?

Set n = 12 in the base formula, giving A = P × (1 + r/12)^(12×t). Note that r/12 is a convention rather than the exact monthly equivalent of the annual rate: at 5% annual, r/12 gives 0.4167% while the strictly equivalent monthly rate is 0.4074%. This is why monthly compounding produces slightly more than annual compounding at the same stated rate.

What separates simple interest from compound interest?

Simple interest applies the rate to the principal alone, so growth is linear: €10,000 at 5% over 20 years yields €10,000 in interest. Compound interest applies the rate to the accumulated balance, so growth is exponential and the same case yields €16,533. The gap comes entirely from interest earning interest, and it widens with the square of the horizon rather than proportionally.

Why does the formula give a different result depending on the source?

Almost always because the compounding convention differs. Annual, monthly, daily and continuous compounding produce four different figures from the same stated rate, and the presence or absence of contributions produces two more. Comparing outputs requires checking n, the timing of payments within each period, and whether the rate quoted is nominal or effective.

Compound interest is one of the rare places in finance where the arithmetic is settled and the disagreement is entirely about inputs. Which is a useful reminder that a projection is never more solid than the rate assumption sitting inside it.

Last updated — 13 August 2026