### Formula

Calculating the amount of the amortization of a loan in accordance with the repayment terms is based on calculating logarithmic or exponential expressions.

If we use * Pmt* to represent the amount of the monthly payments to calculate for the repayment in

*monthly payments of a loan*

**n***that earns interest at a fixed annual rate of*

**C**_{o}*(in decimal notation and not as a percentage), then the value of*

**t***is given by:*

**Pmt**\(Pmt=\dfrac{C_0\space×\space r\space×\space(1\space+\space{r})^n}{(1\space+\space{r})^n\space-\space{1}}\)

or \(Pmt=\dfrac{C_0\space×\space{r}}{1\space-\space(1\space+\space{r})^{-n}}\)

where *r* = \(\dfrac{t}{12}\).

### Example

A household wants to borrow $25 000 and pay back this loan over a period of 5 years, which equals 60 months. If the loan earns interest at 4% annually, what will be the amount of the monthly payments to make?

Here, *r* = 0.04 ÷ 12 = 0.00333… or \(\frac{1}{300}\), and *n* = 60

By applying the formula above, we find:

\(M=\dfrac{25\space000\space×\space{0.00333}}{1\space-\space(1+0.00333)^{-60}}\) = \(\dfrac{83.333}{1\space{-1.00333}^{-60}}\) = \(\dfrac{83.333}{1\space-\space{0.8192}}\)

Therefore: *M* ≈ 461, so the monthly payments will be around $461.