Capital recovery factor

The capital recovery factor $(A/P,i,N)$ converts a present amount $P$ into the equal periodic payment $A$ that pays back $P$ in full (with interest) over $N$ periods at rate $i$:

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

This is the factor inside every amortising-loan calculation. A $300,000 mortgage at 5% over 25 years has annual payment 300{,}000 \cdot (A/P, 5\%, 25) \approx \21{,}300. A \30,000 car loan at 6% over 5 years has monthly payment 30{,}000 \cdot (A/P, 0.5\%, 60) \approx \580$(usingmonthly$i = 0.06/12$).

The same factor does the work in equivalent annual cost calculations. Given a project’s present worth $PW$, the equivalent annual cost over its $N$-year life at rate $i$ is

$EAC=PW\cdot (A/P,i,N).$

Same amortising-loan idea applied to project cash flows: convert a lump sum (good or bad) into the equal annual rate it represents.

Derivation. The present worth of an annuity of $A$ over $N$ periods is $P=A\cdot \frac{(1+i{)}^N-1}{i(1+i{)}^N}$ (the Series present worth factor). Solving for $A$ gives the capital recovery factor as its reciprocal.

An identity worth knowing:

$(A/P,i,N)=(A/F,i,N)+i.$

The amortisation payment can be split into the sinking-fund portion (the bit that accumulates to repay the principal) plus an interest-only term ($i\cdot P$, the interest cost of holding the principal during the period).

In project economics this drives the Annual worth method and Equivalent annual cost. Part of the Compound interest factor family.