Question: A person is purchasing a house and is
considering two financing options. The
first option is a 30year loan at 3.43%.
The second option is a 15year 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 15year 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 30year FRM are r=0.343/13,
N=360, and P=$200,000. Inputs for the
15year FRM are r=0.0265, N=180, and P=$200,000. We get $890.29 for the 30year FRM and
$1347.75 for the 15year FRM.
Second, we calculate total payments over the life of the
loan (360 x $890.20 = $320,505) for the
30year loan and (180 x $1346.75 =
$242,594.40) for the 15year 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 30year FRM and $42,594 for the 15year 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 30year FRM because he is
fearful that he cannot maintain the higher payment on the 15year loan. He continues to make the 15year payment on
time for 15 years but he is charged the 30year 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 30year rate) N=180 (the number of months in a 15year loan) and $1347.75
(the required payment for a 15year FRM assuming r=2.65%).
After 15 years on this payment plan the borrower would owe
$17,658.
Loan balance with 15year
payment but 30year interest rate


30year interest Rate

0.0343

# of Months

180

Actual Payment on 15year
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
http://dailymathproblem.blogspot.com/2013/04/amortgagemathproblem.html
Also, for interesting financial advice you may want to examine www.financememos.com.
Also, for interesting financial advice you may want to examine www.financememos.com.
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