Capital Recovery Factor

Suppose we make a loan.

The way a loan is calculated is that the present value of all the discounted future payments is equal to the loan amount.

That is

\textrm{Loan Amount} = \sum_{n=1}^{N} \frac{\textrm{Payment}}{(1+i)^n}

Where i is the interest rate, and N is the total number of payments.

Note that the present value of each payment get smaller as the payments are farther out, even though the same amount is paid off each period.

This formula allows us to calculate this payment.

\textrm{CRF} = \frac{\textrm{Payment}}{\textrm{Loan Amount}} = \frac {i(1+i)^n}{(1+i)^n-1}

Monthly vs. Yearly Payments

  • Many types of loans advertise a yearly interest rate, but require monthly payments and charge interest monthly.
  • The yearly interest rate is the APR or annual percentage rate
  • To find the monthly rate divide this by twelve
  • i_y is the annual percentage rate
  • n_y is the number of periods in years

The monthly numbers (i_m and n_m) are:

i_{m} = i_{y}/12 n_{m} = n_{y}\cdot 12

Cash Flow

This graph shows the value of the loan and payments at the time they were made. The sum of these loan payments will exceed the loan amount.

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This graph shows the value of the payments at the time of the loan (the present). The sum of these loan payment present values will equal the loan amount and value.

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Calculator syntax

Which of these is will be evaluated correctly by a calculator or spreadsheet?

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

a) i*(1+i)^n/(1+i)^n-1
b) (i*(1+i)^n)/(1+i)^n-1
c) i*(1+i)^n/((1+i)^n-1)
d) i*(1+i)^n/((1+i)^n)-1