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Mortgage Math

Variables

Ok, so you want a mortgage, but don't know what you can afford. The math for figuring it out is a little complicated, but once you understand it, not difficult. First we need to define some variables:

l → the amount of the mortgage loan (the cost minus the downpayment)
T → the term of the loan in years (generally 15 or 30)
R → the percentage rate for the loan
p → the monthly payment on the loan

First it is important to get everything into the same units. The term and rate are both in terms of a year, so new terms need to be defined. Also, the rate is currently a percentage, so it needs to be scaled down by a factor of 100 to get a decimal to use in calculations:

$r = R \frac{1}{12} \frac{1}{100} = \frac{R}{1200}$ → the monthly interest rate
$t = 12 T$ → the term of the loan in months

The Mortgage

The loan value for the first month ( $l_{0}$ ) is simply the total loan amount. After that first month though, there is both accumulated interest and a payment. So, the value for the next month is:

l_{1} = l_{0} + l_{0} r - p = l_{0} (1 + r) - p

The next month then will be:

l_{2} = l_{1} (1 + r) - p = (l_{0} (1 + r) - p) (1 + r) - p = l_{0} {(1 + r)}^{2} - p (1 + r) - p

This continues to unfold and in a general form is:

\begin{matrix} l_{i} & = & l_{i - 1} (1 + r) - p \\ = & (l_{i - 2} (1 + r) - p) (1 + r) - p \\ = & l_{i - 2} {(1 + r)}^{2} - p - p (1 + r) \\ = & (l_{i - 3} (1 + r) - p) {(1 + r)}^{2} - p - p (1 + r) \\ = & l_{i - 3} {(1 + r)}^{3} - p - p (1 + r) - p {(1 + r)}^{2} \\ … \\ = & l_{0} {(1 + r)}^{i} - p - p (1 + r) - p {(1 + r)}^{2} \dots - p {(1 + r)}^{i - 1} \\ = & l_{0} {(1 + r)}^{i} - p [1 + (1 + r) + {(1 + r)}^{2} + \dots + {(1 + r)}^{i - 1}] \\ = & l_{0} {(1 + r)}^{i} - p \sum_{x = 0}^{i - 1} {(1 + r)}^{x} \end{matrix}

The payments are collecting according to the summation of a geometric series. That summation is generally:

S_{n} = \sum_{k = 0}^{n} r^{k} = \frac{1 - r^{n + 1}}{1 - r}

S_{n - 1} = \sum_{k = 0}^{n - 1} r^{k} = \frac{1 - r^{(n - 1) + 1}}{1 - r} = \frac{1 - r^{n}}{1 - r}

So, for the mortgage, the value any given month is:

l_{i} = l_{0} {(1 + r)}^{i} - p \frac{1 - {(1 + r)}^{i}}{1 - (1 + r)} = l_{0} {(1 + r)}^{i} - p \frac{1 - {(1 + r)}^{i}}{- r}

The Payment

Figuring out what your payment will be each month is based on the fact that at the end you will have paid off the loan, so $l_{t} = 0$ .

l_{t} = 0 = l_{0} {(1 + r)}^{t} - p \frac{1 - {(1 + r)}^{t}}{- r}

p = l_{0} {(1 + r)}^{t} \frac{- r}{1 - {(1 + r)}^{t}}

The Loan

The equation for $l_{t}$ is also easily solved for the loan amount:

l_{t} = 0 = l_{0} {(1 + r)}^{t} - p \frac{1 - {(1 + r)}^{t}}{- r}

l_{0} = p \frac{1 - {(1 + r)}^{t}}{- r {(1 + r)}^{t}}