## Saturday, April 20, 2013

### Interest expenses over the life of a mortgage

Question:    A person is purchasing a house and is considering two financing options.  The first option is a 30-year loan at 3.43%.  The second option is a 15-year loan at 2.65%.  What is the total interest expense over the life of the loan for the two options?

What is the outstanding balance on the loan if you make the monthly payment consistent with a 15-year loan at 2.65% but you are actually charged 3.43% interest?

Answer:  The formulas used in this problem are identical to the ones used in the April 16, 2013 mortgage math blog.   The link is at the bottom of this post.

First, we calculate monthly payments for the two mortgages.  Inputs for the 30-year FRM are r=0.343/13, N=360, and P=\$200,000.  Inputs for the 15-year FRM are r=0.0265, N=180, and P=\$200,000.  We get \$890.29 for the 30-year FRM and \$1347.75 for the 15-year FRM.

Second, we calculate total payments over the life of the loan (360 x \$890.20  = \$320,505) for the 30-year loan and  (180 x \$1346.75 = \$242,594.40) for the 15-year loan.

Third, we subtract the \$200,000 initial loan balance from the total payment calculations and get total interest payments of \$120,505 for the 30-year FRM and \$42,594 for the 15-year FRM.

These calculations are laid out in the table below.

 Total Interest Costs Over the Life of the Loan 30 year 15 year Interest Rate 0.0343 0.0265 # of Monthly Payments 360 180 Loan Balance 200000 200000 Monthly Payment (\$890.29) (\$1,347.75) Total Payments over the life of the loan (\$320,505.37) (\$242,594.40) Total Interest Payments (\$120,505.37) (\$42,594.40)

Now for the second part of the problem:

A person chooses to take out the 30-year FRM because he is fearful that he cannot maintain the higher payment on the 15-year loan.    He continues to make the 15-year payment on time for 15 years but he is charged the 30-year rate of 3.43%.   What is the outstanding loan balance after 15 years of payments?

We use the FV function in Excel with inputs of r=0.0343/12 (the 30-year rate) N=180 (the number of months in a 15-year loan) and \$1347.75 (the required payment for a 15-year FRM assuming r=2.65%).

After 15 years on this payment plan the borrower would owe \$17,658.

 Loan balance with 15-year payment but 30-year interest rate 30-year interest Rate 0.0343 # of Months 180 Actual Payment on 15-year FRM (\$1,347.75) Loan amount \$200,000 Future Value of Loan (\$17,657.49)

Interested readers may want to look at the previous mortgage math problem

Also, for interesting financial advice you may want to examine www.financememos.com.