Uniform series compound amount factor

The uniform series compound amount factor $(F/A,i,N)$ converts an annuity of $A$ per period to the future value $F$ at the end of $N$ periods at rate $i$:

$F=A\cdot (F/A,i,N),(F/A,i,N)=\frac{(1+i{)}^N-1}{i}.$

It answers: “If I deposit $A$ at the end of every period into an account earning $i$ per period, how much do I have after $N$ periods?”

A worked example. Deposit $5,000/year into an RRSP for 30 years at 7% average return.

$F=5,000\cdot \frac{(1.07{)}^{30}-1}{0.07}=5,000\cdot 94.46\approx \$472,300.$

The future value of $5,000 × 30 = $150,000 in deposits grows to ~$472,300 through compounding over 30 years.

Derivation. Each annuity payment $A$ deposited at the end of period $t$ grows for the remaining $N-t$ periods, accumulating to $A(1+i{)}^{N-t}$. Summing over $t=1,\dots ,N$:

$F=A{\sum }_{t=1}^N(1+i{)}^{N-t}=A{\sum }_{k=0}^{N-1}(1+i{)}^k=A\cdot \frac{(1+i{)}^N-1}{i}.$

The last step uses the geometric-series sum formula.

The inverse direction (future → annuity) is the Sinking fund factor $(A/F,i,N)$, and the two are reciprocals.

For applications: retirement savings, college funds, bond sinking funds, future-value of any equal-deposit savings plan. For the broader factor family see Compound interest factor; for the corresponding present-value factor see Series present worth factor.