What Is the Equation to Determine a Mortgage Payment?
A mortgage payment is composed of an interest portion, a principal portion, and an escrow portion to pay for property taxes and possibly insurance. The escrow portion is found by dividing the expected annual tax burden by 12. The interest portion is easily found from the current balance, but the principal payment must be found from the difference between the monthly payment and the interest payment.
Other People Are Reading
-
The Interest Payment
-
The interest payment is found by dividing the interest rate given on a mortgage statement by 12, then multiplying by the outstanding balance.
This may seem surprising to those familiar with compounding interest. After all, if the interest rate is 12%, (1 + 0.12/12 months)^12 > 1.12. So clearly the interest rate stated in mortgage statements and used in the standard calculation to find the monthly payment is not the annual effective interest rate, but a lower number presented for computational facility.
Finding the principal payment would require knowledge of the monthly payment. That shall be found next.
Statement of the monthly payment formula
-
The equation for the monthly payment is M = P * ( j / (1 - (1 + j) ^ -n)), where P is the outstanding balance, j is the interest rate divided by 12 months, and n is the remaining number of payments left.
-
Derivation of a useful formula
-
As a helpful aside, note that (1 - x) * (1 + x + x^2 + x^3 + ... + x^n) = 1 - x ^ (n+1). Why? Because all terms in (1 + x + x^2 + x^3 + ... + x^(n-1)) are multiplied by -x and (x + x^2 + x^3 + ... + x^n) is multiplied by 1. Adding these two gives zero. So all that remains is 1 - x ^ (n+1). This gives a helpful formula for later.
So now we have a simple formula for 1 + x + x^2 + x^3 + ... + x^n, which will be helpful, since n is usually 360.
Derivation of the payment formula
-
The starting mortgage principal, P, equals the sum of all monthly mortgage payments, before any of them accumulated interest. So the mortgage payments will total a larger amount than the starting mortgage principal, because they include interest accumulated. The clock doesn't stop ticking on interest accumulation on a given payment until the payment is made. For example, the first payment will have accumulated j interest by the time it is paid.
So we can write a formula removing all the interest from the monthly payments to get the original mortgage amount before interest began inflating payment amounts:
P = M/(1+j) + M/(1+j)^2 + ... + M/(1+j)^n
Pulling out a (1+j) in order to prepare this for use of the formula above, which has a leading 1,
P = M/(1+j) [1 + 1/(1+j) + ... + 1/(1+j)^(n-1)]
Using the polynomial identity from the previous section:
P = M/(1+j) [1 - 1/(1+j)^n]/[1 - 1/(1+j)]
Multiplying through with the 1+j under the M:
P = M [1 - 1/(1+j)^n]/[j]
And this gives the equation given above for M.
Why your mortgage statement differs
-
The above derivation assumes all payments were made on time, not early, and that the first payment started one month after closing. If closing occurred in the middle of the month, or if the mortgage company didn't require the first payment until two months after closing, then the monthly payments would differ from the payment amount indicated by this formula.
-
