# Amortization

## Amortization

In financial operations, this word designates the repayment terms for the capital borrowed without accounting for the interest charges. These payments occur regularly in equal instalments called annuities (annual payments) or monthly payments.

### 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 n monthly payments of a loan Co that earns interest at a fixed annual rate of t (in decimal notation and not as a percentage), then the value of Pmt is given by:

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