Capital recovery factor

A capital recovery factor is the ratio of a constant annuity to the present value of receiving that annuity for a given length of time. Using an interest rate i, the capital recovery factor is:

${\displaystyle CRF={\frac {i(1+i)^{n}}{(1+i)^{n}-1}}}$

where ${\displaystyle n}$ is the number of annuities received.^[1]

This is related to the annuity formula, which gives the present value in terms of the annuity, the interest rate, and the number of annuities.

If ${\displaystyle n=1}$, the ${\displaystyle CRF}$ reduces to ${\displaystyle 1+i}$. Also, as ${\displaystyle n\to \infty }$, the ${\displaystyle CRF\to i}$.

In the limit of zero interest rate, ${\displaystyle \lim _{i\rightarrow 0}CRF=1/n}$ (L'Hôpital's rule).

Example

With an interest rate of i = 10%, and n = 10 years, the CRF = 0.163. This means that a loan of $1,000 at 10% interest will be paid back with 10 annual payments of $163.^[2]

Another reading that can be obtained is that the net present value of 10 annual payments of $163 at 10% discount rate is $1,000.^[2]