Capital recovery factor

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

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

It’s the factor that sits 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 is the workhorse of equivalent annual cost calculations. Given a project’s present worth $PW$, the equivalent annual cost over its $N$-year life at rate $i$ is

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

This is exactly the 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).

For applications in project economics, see Annual worth method, Equivalent annual cost, Replacement decision. For the broader family, see Compound interest factor and Annuity (engineering economics).