Originally posted by geepamoogle
M = P * i / [1 - (1+i)^-n]
M = monthly payment
P = Current principle
i = monthly interest rate (In this case, 5% / 12)
n = number of monthly payments
If we let x = 1+i, then the expression becomes:
M = P * (x-1) / [1 - x^n]
Rearranging, we have:
M/P * [1 - x^n] = (x-1)
Multiplying both sides by x^n and factoring out (x-1) from the left hand side, we have:
M/P * (x-1) * [x^n - x^(n-1) + x - 1] = x^n * (x-1)
Cancelling (x-1) from both sides, we're left with:
M/P [x^n - x^(n-1) + x -1] = x^n
Rearranging into polynomial form, we have:
(1-M/P) * x^n + x^(n-1) - x + 1 = 0
Now, the fundamental theorem of algebra states that "every non-constant single-variable polynomial with complex coefficients has at least one complex root." However, the Abel-Ruffini theorem also states that "there is no general solution in radicals to polynomial equations of degree five or higher."
http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
http://en.wikipedia.org/wiki/Abel's_impossibility_theorem
So, the solution must exist, and a specific equation may exist to solve it, but there is no general formula for producing the result (provided n>4, which is usually the case for a mortgage 😉). I suppose this is just a long-winded way of saying approximation is the best method in this case.