Mortgage Months Ratio

The first step to understanding how monthly mortgage payments are calculated is converting from an annual percentage rate (APR) to a monthly millage rate (I). For small percentages, the monthly millage rate can be approximated by dividing by 12 (months in a year) and multiplying by 10 because mills are multiplied by 1/1000^th instead of 1/100^th for percent.

I \approx \frac{1000 ‰}{100 %} * \frac{year}{12 months} * APR%

When written in decimal form instead of percentage, an APR means that total the amount owed scales each year by an annual multiplying factor apr = 1 + APR. The "equivalent" monthly interest rate has to found through this multiplicative factor because it is compounded twelve times for every single annual multiplicative factor.

apr = 1 + APR

i = {apr}^{1 / 12}

I = i - 1

Combining these into a single equation for the monthly millage rate:

I = (\sqrt[12]{1 + \frac{APR}{100 %}} - 1) * 1000 ‰

Fixed rate mortgage payments are based on a finite geometric series. For reference later in this article, the open and closed-form solutions for a finite geometric series are shown below.

\sum_{k = 0}^{n - 1} r^{k} = 1 + r + r^{2} + r^{3} + \dots + r^{n - 1}

\sum_{k = 0}^{n - 1} r^{k} = \frac{r^{n} - 1}{r - 1} for r \neq 1

Let P_m be the principal remaining after the payment at the end of the m^th month. The change in principal each month is due to growth in principal from the monthly interest factor (i) minus the monthly payment (p). Starting from the initial loan size:

P_{1} = i P_{0} - p

P_{2} = i P_{1} - p = i (i P_{0} - p) - p = i^{2} P_{0} - p (i + 1)

P_{3} = i P_{2} - p = i (i^{2} P_{0} - p (i + 1)) - p = i^{3} P_{0} - p (i^{2} + i + 1)

...

P_{m} = i^{m} P_{0} - p \sum_{k = 0}^{m - 1} i^{k}

P_{m} = {\begin{matrix} i^{m} P_{0} - p \frac{i^{m} - 1}{i - 1} & , & i \neq 1 \\ P_{0} - p m & , & i = 1 \end{matrix}

For an N-month mortgage, the principal remaining after the payment at the end of the N^th month is zero. This allows us to calculate the required monthly mortgage payment for a given loan size.

0 = i^{N} P_{0} - p \frac{i^{N} - 1}{i - 1}

\frac{i^{N} P_{0}}{p} = \frac{i^{N} - 1}{i - 1}

\frac{P_{0}}{p} = \frac{1 - i^{- N}}{i - 1} or p = P_{0} (\frac{i - 1}{1 - i^{- N}})

We call this value, the coefficient that is independent of mortgage size, the mortgage-months-ratio (R_N).

R_{N} = {\begin{matrix} \frac{1 - i^{- N}}{i - 1} months & , & i \neq 1 \\ N months & , & i = 1 \end{matrix}

This ratio relates the initial loan size to the monthly mortgage payment and depends only on interest rates and mortgage term.

\frac{P_{0}}{p} = R_{N}

Substituting this result back into the equation for principal remaining in a given month:

\frac{P_{m}}{p} = i^{m} \frac{P_{0}}{p} - \frac{i^{m} - 1}{i - 1}

\frac{P_{m}}{p} = i^{m} R_{N} - \frac{i^{m} - 1}{i - 1}

\frac{P_{m}}{p} = \frac{i^{m} (1 - i^{- N})}{i - 1} - \frac{i^{m} - 1}{i - 1}

\frac{P_{m}}{p} = \frac{1 - i^{m - N}}{i - 1}

Once again combining with the zero percent interest rate case:

\frac{P_{m}}{p} = {\begin{matrix} \frac{1 - i^{m - N}}{i - 1} & , & i \neq 1 \\ N - m & , & i = 1 \end{matrix}