1. Section 8.5 Home Ownership Math in Our World

2. Learning Objectives  Find a monthly mortgage payment using a payment table.  Find the total interest on a home loan.  Compare two mortgages with different lengths.  Find a monthly mortgage payment using a formula.  Make an amortization schedule for a home loan.

3. Mortgage A mortgage is a long-term loan where the lender has the right to seize the property purchased if the payments are not made. There are several types of mortgages. A fixed-rate mortgage means that the rate of interest remains the same for the entire term of the loan. The payments (usually monthly) stay the same. An adjustable-rate mortgage means that the rate of interest may fluctuate (i.e., increase and decrease) during the period of the loan.

4. Monthly Payments & Interest One way to find the monthly payments for a fixed- rate mortgage is to use a table. The table displays the monthly payment required for each $1,000 of a mortgage, which includes principal and interest.

5. Finding the Monthly Payment Step 1 Find the down payment. Step 2 Subtract the down payment from the cost of the home to find the principal of the mortgage. Step 3 Divide the principal by 1,000. Step 4 Find the number in the table that corresponds to the interest rate and the term of the mortgage. Step 5 Multiply that number by the number obtained in step 3 to get the monthly payment.

6. EXAMPLE 1 Finding Monthly Mortgage Payments The Petteys family plans to buy a home for $174,900, and have been offered a 30-year mortgage with a rate of 5.5% if they make a 20% down payment. What will their monthly payment be with this loan? SOLUTION Step 1 Find the down payment. 20% of $174,900 = 0.20 x $174,900 = $34,980 Step 2 Subtract the down payment from the cost of the home to get the principal. $174,900 – $34,980 = $139,920

7. EXAMPLE 1 Finding Monthly Mortgage Payments SOLUTION Step 3 Divide by 1,000. Step 4 Find the value in the table for a 30-year mortgage at 5.5%. It is $5.68. Step 5 Multiply the value from step 3 by $5.68. 139.92 x $5.68 ≈ $794.75 (Monthly payment with 30-year term.)

8. EXAMPLE 2 Finding Total Interest on a Mortgage Find the total amount of interest the Petteys family would pay if they take the loan in Example 1. SOLUTION On a 30-year mortgage, there are 30 x 12 = 360 payments. We found that the monthly payment would be $794.75. $794.75 x 360 = $286,110 This is the total of payments. We subtract the amount financed from Example 1: $286,110 – $139,920 = $146,190 (Interest on the loan.) The interest paid exceeds the principal of the loan by over $6,000!

9. EXAMPLE 3 Comparing Mortgages with Different Terms Suppose that the Petteys family from Examples 1 and 2 is also offered a 15-year mortgage with the same rate and down payment. Find the difference in monthly payment and interest paid between the 15- and 30-year mortgages. SOLUTION We essentially need to rework Examples 1 and 2 with a 15-year mortgage, then compare the results. Fortunately, some of the work we did carries over. We know that the principal is $139,920, and the principal divided by 1,000 is 139.92.

10. EXAMPLE 3 Comparing Mortgages with Different Terms SOLUTION This time we use the 15-year column and 5.5% row in the table to get $8.17. Now we multiply that by 139.92: 139.92 x $8.17 ≈ $1,143.15 (Monthly payment with 15-year term.) The difference in monthly payments is $1,143.15 – $794.75 = $348.40 (Recall that $794.75 was payment for 30 years.)

11. EXAMPLE 3 Comparing Mortgages with Different Terms SOLUTION With a monthly payment of $1,143.14 for 15 years (which is 180 months) the total payments are $1,143.15 x 180 = $205,767.00 and the interest paid is $205,767 – $139,920 = $65,847 The interest paid on the 30-year mortgage was $146,190: $146,190 – $65,847 = $80,343 If the Petteys family can manage an extra $348.40 a month, they will save over $80,000 in interest!

12. Computing Monthly Payments Formula for Computing Monthly Payments on a Mortgage R = regular monthly payment P = amount financed, or principal r = rate written as a decimal n = number of payments per year t = number of years

13. EXAMPLE 4 Finding a Monthly Payment Using the Formula After one hit single, a young singer unwisely decides that she needs a $2.2 million dollar mansion. With some of the proceeds from her CD, she puts down $500,000, leaving $1,700,000 to finance at 6% for 30 years. Find her monthly payment.

14. EXAMPLE 4 Finding a Monthly Payment Using the Formula SOLUTION In the formula, use P = 1,700,000, r = 0.06, n = 12, and t = 30. The monthly payment is $10,192.36.

15. Amortization Schedule After securing a mortgage, the lending institution will prepare an amortization schedule. This schedule shows what part of the monthly payment is paid on the principal and what part of the monthly payment is paid in interest.

16. Amortization Schedule Procedure for Computing an Amortization Schedule Step 1 Find the interest for the first month. Use I = Prt, where t = 1/12. Enter this value in a column labeled Interest. Step 2 Subtract the interest from the monthly payment to get the amount paid on the principal. Enter this amount in a column labeled Payment on Principal.

17. Amortization Schedule Procedure for Computing an Amortization Schedule Step 3 Subtract the amount of the payment on principal found in step 2 from the principal to get the balance of the loan. Enter this in a column labeled Balance of Loan. Step 4 Repeat the steps using the amount of the balance found in step 3 for the new principal.

18. EXAMPLE 5 Preparing an Amortization Schedule Compute the first two months of an amortization schedule for the loan in Example 1. SOLUTION The value of the mortgage is $139,920, the interest rate is 5.5%, and the monthly payment is $794.75. Step 1 Find the interest for month 1. Enter this in a column labeled Interest.

19. $153.45$139,766.55 Payment NumberInterestPayment on PrincipalBalance of Loan 1$641.30 EXAMPLE 5 Preparing an Amortization Schedule SOLUTION Step 3 Subtract principal payment from principal. $139,920 – $153.45 = $139,766.55 This goes into the Balance of Loan column. Now we repeat steps 1–3 using the balance of $139,766.55. Step 2 Subtract the interest from the monthly payment. $794.75 – $641.30 = $153.45 This goes into the Payment on Principal column.

20. $640.60 $154.15$139,612.40 Payment NumberInterestPayment on PrincipalBalance of Loan 1$641.30$153.45$139,766.55 2 EXAMPLE 5 Preparing an Amortization Schedule SOLUTION Step 5 $794.75 – $640.60 = $154.15 Step 6 $139,766.55 – $154.15 = $139,612.40 Step 4