Capital Recovery Factor

The capital recovery factor converts a lump of capital into the equal annual payment that repays it with interest: $\mathit{CRF} = i(1+i)^n/((1+i)^n-1)$. At 10% over five years it is 0.2638, so every $100,000 of capital costs $26,380 a year. It is the same algebra as a loan payment, which it should be, since recovering capital and amortising a loan are the same problem wearing different hats.

The structure is easier to remember in the split form $\mathit{CRF} = i + i/((1+i)^n - 1)$: the interest, plus the sinking-fund deposit that rebuilds the principal. That decomposition also explains the floor. As $n$ grows the second term vanishes and CRF falls toward $i$ but never reaches it, which is why the solver refuses a CRF at or below the interest rate. Such a payment covers the interest and never touches the principal.

The single most common error in engineering economy is using $1/n$ instead of CRF, spreading a $100,000 machine over five years as $20,000 a year. That understates the annual cost by 32% here, and by more at higher rates or longer lives. Straight-line thinking is an accounting convention. CRF is what the money actually costs.