A fixed-rate mortgage is a type of loan where the installments are equally sized, but the ratio between the interest and the principal payments vary over time.

Initially, the amount of your installment that goes to interest is high, and the principal payment is low. But as you pay down on your loan, you pay less in interest and more towards the principal. Fixed-rate mortgages are ultimately more expensive than serial loans.

Formula

$$\begin{array}{cc}\text{Installment}& \\ =& \\ \text{principalpayment}+\text{interest}& \end{array}$$

$$\text{Installment}=\text{principalpayment}+\text{interest}$$ |

Example 1

Steve Rogers and Stephen Strange want to buy an apartment in Manhattan, and they need to apply for a loan to pay for it. They know that the installments on a serial loan can be a bit high at first, so they are considering an fixed-rate mortgage instead. They have the ability to pay $$115\phantom{\rule{0.17em}{0ex}}000$ per yearly installment. They apply for a loan of $$1\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000$ at an interest rate of $4$ % per year. The first principal payment equates to $$83\phantom{\rule{0.17em}{0ex}}291$. What is the amount of interest on the first installment? Are they able to repay the loan?

The amount of interest on the first installment:

$$\begin{array}{cc}\text{Interest(firstinstallment)}& \\ =& \\ 1\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\cdot 0.04& \\ =& \\ \text{\$}40\phantom{\rule{0.17em}{0ex}}000& \end{array}$$

$$\begin{array}{llll}\hfill \text{Interest(firstinstallment)}& =1\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\cdot 0.04\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{\$}40\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

Now you can calculate the size of the installments and see whether Steve and Stephen are able to service the loan: $$\begin{array}{llll}\hfill \text{Installment}& =\text{\$}83\phantom{\rule{0.17em}{0ex}}291+\text{\$}40\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{\$}123\phantom{\rule{0.17em}{0ex}}291\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

$$\text{Installment}=\text{\$}83\phantom{\rule{0.17em}{0ex}}291+\text{\$}40\phantom{\rule{0.17em}{0ex}}000=\text{\$}123\phantom{\rule{0.17em}{0ex}}291$$ |

In the two following figures, you’ll see the difference between how a serial loan and a fixed-rate mortgage is paid off.

A serial loan has equally sized principal payments, but the amount of interest decreases as the total loan balance decreases. This leads to a high initial installment amount, but the installments reduce in size over time.

A fixed-rate mortgage has differently sized principal and interest payments over time. The amount contributed towards principal increases by exactly the same amount as the amount contributed towards interest decreases. This is why the installment amount stays the same over time—as the payment towards the principal goes up, the interest payment goes down by the exact same amount.

Sum of principal payments on serial loan: $$1\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000$

Sum of interests on serial loan: $$220\phantom{\rule{0.17em}{0ex}}000$

Sum of principal payments on fixed-rate mortgage: $$1\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000$

Sum of interests on fixed-rate mortgage: $$232\phantom{\rule{0.17em}{0ex}}909.44$