Series present worth factor

The series present worth factor $(P/A,i,N)$ converts an annuity of $A$ per period into the equivalent present-worth lump sum $P$ over $N$ periods at rate $i$:

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

It’s the factor for “what’s a stream of equal payments worth today?” If I’m offered $10,000/year for 20 years at a discount rate of 6%, the present-value lump sum that’s equivalent is

$10,000\cdot \frac{(1.06{)}^{20}-1}{0.06\cdot (1.06{)}^{20}}=10,000\cdot 11.47\approx \$114,700.$

So you should be roughly indifferent between accepting $114,700 today or $10,000/year for 20 years (at the 6% discount rate).

Derivation. The series present worth is a sum of discounted single payments:

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

The sum is a finite geometric series with first term $A/(1+i)$ and ratio $1/(1+i)$.

Special case: as $N\to \infty$, the factor approaches $1/i$ (the limit is finite because each term shrinks faster than linearly). The result $P_{\infty }=A/i$ is the capitalised value of a perpetual annuity — see Annuity (engineering economics).

Inverse direction (present → annuity) is the Capital recovery factor $(A/P,i,N)$.

For applications, see Present worth method and Annual worth method. For the broader family, see Compound interest factor.