Mortgage Equations

Mortgage Equations

Equations used to derive common measures used in the mortgage market, such as monthly payment, balance, and APR.

Fully Amortizing Payment: The following formula is used to calculate the fixed monthly payment (P) required to fully amortize a loan of L dollars over a term of n months at a monthly interest rate of c. [If the quoted rate is 6%, for example, c is .06/12 or .005.]

P = L[c(1 + c)n] / [(1 + c)n - 1]

Balance After a Specified Period: The next formula is used to calculate the remaining loan balance (B) of a fixed payment loan after p months.

B = L[(1 + c)n - (1 + c)p] / [(1 + c)n - 1]

Annual Percentage Rate (APR): The APR is what economists call an “internal rate of return” (IRR), or the discount rate that equates a future stream of dollars with the present value of that stream. In the case of a home mortgage, the formula is

L - F = P₁ + P₂ / (1 + i)2 +… (P_n + B_n) / (1 + i)n
where:

i = IRR

L = Loan amount

F = Points and all other lender fees P = Monthly payment

n = Month when the balance is paid in full B_n= Balance in month n

This equation can be solved for i only through a series of successive approximations, which must be done by computer. Many calculators will also do it provided that all the values of P are the same.

The APR is a special case of the IRR, because it assumes that the loan runs to term. In the equation, this means that n is equal to the term and B_n is zero.

Note that on ARMs, the payments used to calculate the APR are those that would occur under the assumption that the index rate does not change over the life of the loan.

Future Values: Many of my calculators measure financial results in terms of “future values”—the borrower's net wealth at the end of a specified period.

The future value of a single sum today is: FV_n = S(1 + c)n

where:

FV_n is the value of the single sum after n periods S is the amount of the single sum now
c is the applicable interest rate
n is the length of the period

The future value of a series of payments of equal size, beginning after one period, is:

FV_n = P[(1+c)n - 1]/c

where P is the periodic payment and the other terms are as defined above.