 # How to Calculate Amortizing Loans with Series Formulas

Theory

### AmortizingLoans

An amortizing loan is a loan where your monthly rate varies from month to month. Each monthly rate consists of a principal payment and an interest payment. In an amortizing loan the principal payment is fixed, but the interest payment decreases every month to account for the fact that you owe the bank less money than last month. Because you owe the bank more money early on than late, the interest payment is higher early and lower towards the end of the repayment. As the principal payment stays the same the whole time, you pay a higher monthly rate at the start of the loan period and a smaller rate towards the end.

An amortizing loan is cheaper in total than an annuity loan, because the total interest is higher with an annuity loan than with an amortizing loan. This is true, even when you account for inflation unless the inflation is unusually high.

You can use arithmetic series to find the annual remaining loan.

Example 1

You loan $\text{}150\phantom{\rule{0.17em}{0ex}}000\text{}\phantom{\rule{0.17em}{0ex}}\text{}$ over 20 years. The interest on the loan is $\text{}5\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$. The remaining loan for the first four years are $\text{}150\phantom{\rule{0.17em}{0ex}}000\text{}$, $\text{}142\phantom{\rule{0.17em}{0ex}}500\text{}$, $\text{}135\phantom{\rule{0.17em}{0ex}}000\text{}$ and $\text{}127\phantom{\rule{0.17em}{0ex}}500\text{}$. What is the remaining loan after 13 years, and how much do you pay in total interest?

The exercise tells you that the remaining loans can be expressed as a series, where every annual remaining loan is a term in the series.

$\begin{array}{llll}\hfill & 150\phantom{\rule{0.17em}{0ex}}000+142\phantom{\rule{0.17em}{0ex}}500+135\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}+127\phantom{\rule{0.17em}{0ex}}500+\cdots \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $150\phantom{\rule{0.17em}{0ex}}000+142\phantom{\rule{0.17em}{0ex}}500+135\phantom{\rule{0.17em}{0ex}}000+127\phantom{\rule{0.17em}{0ex}}500+\cdots$

You can then find the remaining loan in year $n$ by finding ${a}_{n}$. As this is an arithmetic series, you can use the formula to find the $n$th term in an arithmetic series, ${a}_{n}={a}_{1}+\left(n-1\right)d$. To do this, the first thing you need to find is $d$:
$\begin{array}{llll}\hfill d& ={a}_{n+1}-{a}_{n}=142\phantom{\rule{0.17em}{0ex}}500-150\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-7500\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $d={a}_{n+1}-{a}_{n}=142\phantom{\rule{0.17em}{0ex}}500-150\phantom{\rule{0.17em}{0ex}}000=-7500$

Then you can find an expression for ${a}_{n}$: $\begin{array}{llll}\hfill {a}_{n}& =150\phantom{\rule{0.17em}{0ex}}000+\left(n-1\right)\left(-7500\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =157\phantom{\rule{0.17em}{0ex}}500-7500n\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

To find the remaining loan after 13 years, you insert $n=13$ into the expression for ${a}_{n}$:

 ${a}_{13}=157\phantom{\rule{0.17em}{0ex}}500-7500\cdot 13=60\phantom{\rule{0.17em}{0ex}}000$

To find the total paid interest for the whole loan, you can look at each individual remaining loan. If you find the interest cost for each remaining loan and add them together, you will find the total amount of interest you’ve paid. This gives you a new arithmetic series, where the interest for the first remaining loan is ${r}_{1}=150\phantom{\rule{0.17em}{0ex}}000\cdot 0.05$, the interest for the second remaining loan is ${r}_{2}=142\phantom{\rule{0.17em}{0ex}}500\cdot 0.05$ and the interest of the $n$th remaining loan is ${r}_{n}={a}_{n}\cdot 0.05$

That makes the new arithmetic series look like this:

 $7500+7125+6750+\cdots +{r}_{n}$

As the loan is paid over 20 years, you need to find the cost over 20 terms. You can find this by using the formula for the sum of an arithmetic series:

 ${S}_{n}=\frac{{r}_{1}+{r}_{n}}{2}\cdot n$

You know that ${r}_{1}=7500$ and $n=20$, and you need to find

 ${r}_{20}={a}_{20}\cdot 0.05.$

First, you can find ${a}_{20}$:

 ${a}_{20}=157\phantom{\rule{0.17em}{0ex}}500-7500\cdot 20=7500$

Then you can find ${r}_{20}$:

 ${r}_{20}={a}_{20}\cdot 0.05=7500\cdot 0.05=375$

That makes the total amount of interest you have paid

 ${S}_{20}=\frac{\left(7500+375\right)}{2}\cdot 20=78\phantom{\rule{0.17em}{0ex}}750.$

The total interest on a loan of $150\phantom{\rule{0.17em}{0ex}}000$ \$ with $5$ % interest is $78\phantom{\rule{0.17em}{0ex}}750$ \$ when the loan is repaid over 20 years.

You are paying back a total of

 $150\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}\text{}+78\phantom{\rule{0.17em}{0ex}}750\phantom{\rule{0.17em}{0ex}}\text{}=228\phantom{\rule{0.17em}{0ex}}750\phantom{\rule{0.17em}{0ex}}\text{}$

over 20 years.