A mortgage is nothing more than a debt instrument with a lien on the property being mortgaged, and mortgage payments are calculated as payments in an ordinary annuity. The formula to calculate mortgage payments is shown below:
Where:
- PMT = mortgage payment
- PV = present value (mortgage amount)
- i = period interest rate expressed as a decimal
- n = number of mortgage payments
Example
Suppose you wish to acquire a home that costs $550,000. Right now, you only have enough saved to be able to make a down payment of $100,000. The bank you are working with has offered you a fixed interest rate of 4.0% on a 15-year, $450,000 loan. You choose to make monthly payments.
We will use the ordinary annuity formula to calculate each monthly payment. The present value here is $450,000, which is the value of the loan. The annual mortgage rate is 4.0%, so the monthly rate is 4.0% divided by twelve. The number of mortgage payments is 180, which is twelve payments per year for fifteen years. The work to calculate monthly payments is shown below:
This means that every month you will pay $3,328.60.
What is PMI, and How is It Calculated
Private mortgage insurance, or PMI, is a type of insurance typically required by the mortgage lender when the borrower’s down payment on a home is less than 20% of the total cost of the home. Private mortgage insurance rates are typically 0.5% to 1.0% of the value of the mortgage. In the United States, the borrower can generally ask to stop PMI payments when the loan to value ratio reaches 80%. If the request is denied or never made, the payments will usually be stoped automatically by the lender when the loan to value ratio reaches 78%.
In our example above, the purchaser made a down payment of only 18.2% of the total cost of the home, so the lender of the mortgage could require PMI payments until the borrower reaches an equity stake in the home of 20%, which is the same as a loan to value ratio of 80%. If the lender required PMI of 1.0% of the value of the loan annually, then the borrower would have to pay 1.0% of $450,000, which is $4,500 per year. To make this a monthly value, divide $4,500 by twelve, which is $375 per month. This value would simply be added to the base mortgage payment.