Future Value of an Annuity Formula

An annuity is a series of equal payments at equal intervals. If each payment $C$ is made at the end of a period — an ordinary annuity — its accumulated value after $n$ periods is

$FV = C \cdot \frac{(1+r)^n - 1}{r}$

• $C$ — the payment per period
• $r$ — the interest rate per period, as a decimal
• $n$ — the total number of payments

The fraction is the future-value annuity factor, often written $s_{\overline{n}|r}$. It comes straight from a geometric series: the first payment compounds for $n-1$ periods, the second for $n-2$, and so on down to the last, which earns nothing. Summing $(1+r)^{n-1} + \cdots + (1+r)^0$ gives exactly $((1+r)^n - 1)/r$. The interest portion alone is $FV - nC$.

Annuity due. If payments arrive at the start of each period, every one compounds an extra period:

$FV_{\text{due}} = C \cdot \frac{(1+r)^n - 1}{r} \times (1+r)$

The same $(1+r)$ multiplier converts the present-value factor, $PV = C \cdot (1 - (1+r)^{-n})/r$.

Growing annuity. If the payment itself rises by $g$ each period, and $r \neq g$:

$FV = C \cdot \frac{(1+r)^n - (1+g)^n}{r - g}, \qquad PV = \frac{C}{r-g}\left[1 - \left(\frac{1+g}{1+r}\right)^n\right]$

Solving for the payment. Invert the factor:

$C = FV \cdot \frac{r}{(1+r)^n - 1} \qquad \text{or} \qquad C = PV \cdot \frac{r}{1 - (1+r)^{-n}}$

The second form is the loan-payment equation - the same algebra read from the other end.

• Using the annual rate for monthly payments: $r$ must be per period, so 6% a year with monthly payments is $r = 0.005$ and $n =$ months.
• Confusing $n$ with years: 10 years of monthly payments is $n = 120$.
• Applying the lump-sum formula $P(1+r)^n$ to a payment stream: each payment compounds for a different length of time.
• Mixing up due and ordinary: forgetting the $(1+r)$ multiplier understates an annuity due.
• Using the growing-annuity formula when $r = g$: the denominator vanishes; the value is then $FV = nC(1+r)^{n-1}$.
• Assuming $FV - nC$ is taxable or spendable interest: it is arithmetic on your inputs, and what any real account credits depends on its own terms.