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How Is Fixed-rate Mortgage Calculated?

Theory

Fixed-rate Mortgage

A fixed-rate mortgage is a loan where the monthly rate stays the same for the entire repayment period of the loan. When the monthly rate stays the same, the ratio between the principal and interest payment will be different each term.

In the beginning, the principal payment will be relatively small, and the interests relatively high. As time passes this will change, and the principal payments will grow larger and the interest smaller.

An amortizing loan is cheaper in total than a fixed-rate mortgage, due to the cost of the total interest being higher for a fixed-rate mortgage than for an amortizing loan.

While calculating fixed-rate mortgages, geometric series are our most important tool.

Example 1

You get a loan of $$ 150 000$ today and will pay it back over 10 years, with annual payments. The first payment is in one year. The interest rate is $6 %$ . Find the value of the annual payment.

Here, it can be useful to make a timeline. It can look like this.

Timeline showing the payment plan for the fixed-rate mortgage

This gives us the geometric series

\begin{array}{l} \frac{x}{1.06} + \frac{x}{1.0 6^{2}} + \dots \\ + \frac{x}{1.0 6^{9}} + \frac{x}{1.0 6^{10}}, \end{array}

\frac{x}{1.06} + \frac{x}{1.0 6^{2}} + \dots + \frac{x}{1.0 6^{9}} + \frac{x}{1.0 6^{10}},

with

a_{1} = \frac{x}{1.06}, k = \frac{1}{1.06}, n = 10 .

a_{1} = \frac{x}{1.06}, k = \frac{1}{1.06}, n = 10 .

We can insert these numbers into the formula for the sum of a geometric series:

\begin{array}{l} \frac{x}{1.06} \cdot \frac{{(\frac{1}{1.06})}^{10} - 1}{\frac{1}{1.06} - 1} & = 150 000 \\ \frac{x}{1.06} \cdot 7.8 & = 150 000 | \cdot \frac{1.06}{7.8} \\ x & \approx 20 385 \end{array}

This makes the annual payment $ $20 385$ when you borrow $ $150 000$ and pay it back over 10 years.

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