Take a sum today and ask what equal amount, every period, would pay it back with interest. That question has one answer and two names, and knowing they are the same thing is worth more than either formula alone.

The engineering name is the capital recovery factor: $\mathit{CRF} = \dfrac{i(1+i)^{n}}{(1+i)^{n} - 1}$, written (A/P, i, n) in the tables and read A given P. $i$ is the rate per period as a decimal, $n$ the number of periods, and CRF itself is a bare fraction per year — no currency on it. Multiply a first cost by CRF and you have that cost as an annual charge.

The finance name is the loan payment: $M = \dfrac{P\,i}{1 - (1+i)^{-n}}$, where $M$ is the level payment per period, $P$ the principal borrowed, and $i$ and $n$ are per PAYMENT period — monthly on a monthly loan. Multiply the top and bottom of CRF by $(1+i)^{-n}$ and the two expressions are identical. The mortgage on your house and the annualised capital cost of a pump are the same arithmetic. One is quoted to a borrower, the other to a lifecycle sheet.

Two rails on the value. CRF is always larger than $i$, because the payment must cover the interest and then some, and always larger than $1/n$, because straight repayment with no interest is the cheapest schedule there is. CRF minus $i$ is the sinking-fund factor, the deposit that would replace the asset — the same family again, one subtraction apart.

And what a loan costs beyond the money itself: $I = M\,n - P$. Everything paid, less everything borrowed. It is the figure every finance conversation eventually reaches and it flatters nobody — a seven-year loan at 6% nominal costs about a fifth of the purchase price in interest alone, which is the number that decides whether a fleet is bought or leased.