Compound interest factor

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

The standard notation is $(X/Y,i,N)$, read as ”$X$ given $Y$“. It’s a multiplier: starting with a known $Y$, you compute the equivalent $X$ by

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

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 also 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 that’s equivalent to a stream of annual payments $A$ over $N$ periods.
• Uniform series compound amount factor $(F/A)$ — the future-value lump sum that 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 small rounding errors propagate dramatically.
• The factors live in interest factor tables indexed by $i$ and $N$. Modern practice is to compute them inline in spreadsheets, but tables are still useful for exam settings and for quick sanity checks.

For each individual factor, see the dedicated notes (Sinking fund factor, Capital recovery factor, Series present worth factor, Uniform series compound amount factor). For applications, see Present worth method and Annual worth method.