The balance at payment N is:
N B
0 P
1 P * r - m
2 (P * r - m) * r - m = P * r ^ 2 - r * m - m
3 (P * r ^ 2 - r * m - m) * r - m = P * r ^ 3 - m * r ^ 2 - m * r - m
n (P * r ^ n) - m * (1 + r + r ^ 2 + … + r ^ (n - 1))
= (P * r ^ n) - m * (1 - r ^ n) / (1 - r)
Here:
P = original principal of the loan: the home price minus downpayment
r = the per-pay-period interest rate - e.g., 3% APR would be 1.0025
m = the fixed monthly payment amount towards interest + principal
n = an arbitrary integer
To find P from the other variables we should fix the value n to N, the number of payments (the loan term) and then require that the balance after this payment be zero:
(P * r ^ N) - m * (1 - r ^ N) / (1 - r) = 0
P * r ^ N = m * (1 - r ^ N) / (1 - r)
P = m * (1 - r ^ N) / [(r ^ N)(1 - r)]
Notice that the original loan balance P is directly proportional to the amount m of your monthly payment. This makes a lot of sense if you consider it. Let's assume a middle-of-the-road monthly payment of $1000 and see what you can buy with a 3% APR on a 30-year (360-month) mortgage:
P = $1,000 * (1 - 1.0025 ^ 360) / [(1.0025 ^ 360)(1 - 1.0025)]
~ $1,000 * (-1.4568422115) / [(2.4568422115)(-0.0025)]
~ $1,000 * (-1.4568422115) / (-0.0061421055)
~ $237,190
