Investment Growth Calculator

An investment projection is arithmetic on two building blocks.

A lump sum left to compound:

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

where $P$ is the starting amount, $r$ the nominal annual rate as a decimal, $n$ the number of compounding periods per year, and $t$ the number of years.

A stream of equal deposits (an ordinary annuity), one at the end of each period:

$FV = PMT \cdot \frac{(1+i)^N - 1}{i}$

where $i = r/n$ is the periodic rate and $N = nt$ the number of deposits. This is a geometric series: each deposit compounds for a different length of time, and the closed form just sums them.

When you both start with a balance and keep contributing, add the two results.

A projection is arithmetic on assumptions, not a forecast. Real returns vary, and AI-Math states no rate for you — enter your own and the solver shows every step of the calculation.

Effective annual rate

A "6% nominal rate compounded monthly" does not grow money by 6% in a year. The effective annual rate does:

$EAR = \left(1 + \frac{r}{n}\right)^{n} - 1$

EAR is the only fair way to compare offers with different compounding frequencies. Compounding more often always raises the EAR, but with sharply diminishing returns — monthly to daily is a tiny move.

Solving for the rate or the time

Rearranging the lump-sum formula gives both inverses:

$r_{\text{eff}} = \left(\frac{A}{P}\right)^{1/t} - 1, \qquad t = \frac{\ln(A/P)}{\ln(1 + r/n) \cdot n}$

The first is the compound annual growth rate (CAGR) — the single annual rate that would have produced the observed result.

Real (inflation-adjusted) return

Subtracting inflation from the nominal rate is only an approximation. The exact relation is

$1 + r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + \text{inflation}}$

Project with the real rate when you want the answer expressed in today's purchasing power.

• Mismatching $i$ and $N$: if the rate is per month, the count must be in months. A monthly rate with a count in years is the single most common error.
• Percent left as a whole number: $r = 0.07$, not $7$. Using $7$ raises the result by an absurd factor.
• Using the lump-sum formula for contributions: deposits made over time do not each compound for the full term. Use the annuity formula, or sum them individually.
• Ordinary annuity vs annuity-due: the formula above assumes deposits at the end of each period. For deposits at the start, multiply by $(1+i)$.
• Comparing nominal rates across compounding frequencies: convert both to EAR first, otherwise you are comparing different things.
• Reporting nominal growth as though it were spending power: over 20 or 30 years the gap between the nominal and real figure is large. Say which one you mean.
• Ignoring fees and taxes: they change the effective rate that belongs in the formula, and both depend on your account and jurisdiction.