Mortgage calculations

Mortgages are loans made to home buyers (borrower) by financial institutions such as banks (lender) with the purchased home as collateral for the loan. The borrower typically makes fixed or variable payments on the loan depending on the interest rate charged on the loan and the amount borrowed (loan or mortgage principal). The loan typically has a fixed term (tenor).

Calculations involving mortgages are based on present value analysis. The amount borrowed is equal to the present value of the future mortgage payments made by the borrower. For a fixed interest loan, the fixed monthly payments made by the borrower results in the mortgage analysis becoming one that is analogous to the present value of an annuity. An example will be used to illustrate the point.

Let us assume the following loan/mortgage terms:
Loan amount - $200,000
Term - 20 years (or 240 months)
Mortgage interest rate - 6% per annum (or 0.5% per month from 6%/12 months)

The amount you will be paying each month can be derived from the present value of an annuity formula given by:

A = (C/r)*[1 - {1/(1+r)^n}]         (Present value of an annuity formula)

where A is the present value of annuity (Loan amount or 200,000), C is the monthly payment to be calculated, r is the effective monthly interest rate (0.5% or 0.005 in decimal), and n is the number of payment periods (in this case, months which is 240).  A little algebra will suffice in showing that C can be calculated as:

C = (A*r) / [1 - {1/(1+r)^n}]  = (200,000*0.005) / [1 - {1/(1+0.005)^240}]
C = 1000 / 0.6979 = $1432.86

This analysis assumes that you have a fully amortizing loan where the amount borrowed is paid off at the end of the 240 months or 20 years. The first payment is also assumed to be made one month after the loan is taken. If the loan were an interest only loan, monthly payments would be $1000 (0.5% per month multiplied by $200,000), but the principal of $200,000 would still be outstanding at the end of the term.

The present value of an annuity formula can be rearranged to find C, r , n or used as it is to find A once the other 3 variables are known. It is somewhat more intricate to find r and n as the use of logarithms may be required to find n and iteration to find r as r appears twice in the formula.