Compound interest factor

A compound interest factor is a precomputed multiplier that converts one type of cash flow into another at a given interest rate $i$ over $N$ periods. They turn long time-value derivations into one-line lookups.

Notation is $(X/Y,i,N)$, read ”$X$ given $Y$“. Starting from a known $Y$, the equivalent $X$ is

$X=Y\cdot (X/Y,i,N).$

The six standard factors are:

Factor	Name	Formula	Use
$(F/P,i,N)$	Single payment compound amount	$(1+i{)}^N$	$P$ → $F$
$(P/F,i,N)$	Single payment present worth	$1/(1+i{)}^N$	$F$ → $P$
$(F/A,i,N)$	Uniform series compound amount	$\frac{(1+i{)}^N-1}{i}$	annuity $A$ → $F$
$(A/F,i,N)$	Sinking fund	$\frac{i}{(1+i{)}^N-1}$	$F$ → annuity $A$
$(P/A,i,N)$	Series present worth	$\frac{(1+i{)}^N-1}{i(1+i{)}^N}$	annuity $A$ → $P$
$(A/P,i,N)$	Capital recovery	$\frac{i(1+i{)}^N}{(1+i{)}^N-1}$	$P$ → annuity $A$

The names matter for vocabulary:

• Sinking fund factor $(A/F)$ — the equal annual deposit needed to accumulate a future amount $F$. “Sinking fund” is the historical term for a savings plan that builds toward a future obligation.
• Capital recovery factor $(A/P)$ — the equal annual payment that pays back a present-day loan $P$ over $N$ periods. The basis of all amortising loan calculations.
• Series present worth factor $(P/A)$ — the present-value lump sum equivalent to a stream of annual payments $A$ over $N$ periods.
• Uniform series compound amount factor $(F/A)$ — the future-value lump sum an annuity of $A$ accumulates to over $N$ periods.

Two more for gradient series:

• $(P/G,i,N)$ — arithmetic gradient to present worth.
• $(A/G,i,N)$ — arithmetic gradient to annuity.

The arithmetic-gradient factors handle cash flows that start at zero and grow by a constant amount $G$ each period: $0,G,2G,3G,\dots ,(N-1)G$ at periods $1,2,3,\dots ,N$. See Arithmetic gradient series.

For geometric gradients (growth by a constant percentage each period), there’s a closed-form factor as well, but in practice geometric gradients are usually handled in spreadsheets rather than by lookup. See Geometric gradient series.

Two practical conventions:

• Carry four decimal places in factor lookups until the final answer, then round. The factors are exponentials of small numbers, so rounding errors propagate fast.
• The factors live in interest factor tables indexed by $i$ and $N$. Modern practice computes them inline in spreadsheets, but tables are still handy for exams and quick sanity checks.

Applications: Present worth method and Annual worth method.