Question

Mortgages, loans taken to purchase a property, involve regular payments at fixed intervals and are treated as reverse annuities. Mortgages are the reverse of annuities, because you get a lump-sum amount as a loan in the beginning, and then you make monthly payments to the lender. You’ve decided to buy a house that is valued at $1 million. You have $300,000 to use as a down payment on the house, and want to take out a mortgage for the remainder of the purchase price. Your bank has approved your $700,000 mortgage, and is offering a standard 30-year mortgage at a 10% fixed nominal interest rate (called the loan’s annual percentage rate or APR). Under this loan proposal, your mortgage payment will be ???? per month. (Note: Round the final value of any interest rate used to four decimal places.)

Your friends suggest that you take a 15-year mortgage, because a 30-year mortgage is too long and you will pay a lot of money on interest. If your bank approves a 15-year, $700,000 loan at a fixed nominal interest rate of 10% (APR), then the difference in the monthly payment of the 15-year mortgage and 30-year mortgage will be ??? ?(Note: Round the final value of any interest rate used to four decimal places. ) It is likely that you won’t like the prospect of paying more money each month, but if you do take out a 15-year mortgage, you will make far fewer payments and will pay a lot less in interest. How much more total interest will you pay over the life of the loan if you take out a 30-year mortgage instead of a 15-year mortgage?

Answer 1

Calculating Monthly mortgage payment using EMI formula:-

EMI

P* r(1+r)" (1+r)n-1

where, P = Mortgage amount ;= $700,000

r = monthly interest rate ;= 0.10/12 ;= 0.0083

n = no of payments ;= 30 years*12 = 360

EMI

0.0083(1+0.0083) 350 = 700000 * [140.0083)360 -1.

= $ 6122.32

So, the mortgage payment will be $ 6122.32 for each month over 30 years.

-- If the mortgage loan is taken for 15 years rather than 30 years, monthly mortgage payment will be :-

EMI

P* r(1+r)" (1+r)n-1

where, P = Mortgageamount ;= $700,000

r = monthly interest rate ;= 0.10/12 ;= 0.0083

n = no of payments ;= 15 years*12 = 180

EMI

0.0083(1+0.0083) 180 = 700000 * [140.0083) 180 -1.

= $ 7505.12

The difference in the monthly payment of the 15-year mortgage and 30-year mortgage is $ 1382.8 ($7505.12-$6122.32)

-- Total Interest paid if mortgage is taken for 30 years = ($ 6122.32*360) - $ 700,000

= $ 1,504,035.2

Total Interest paid if mortgage is taken for 15 years = ($7505.12*180) - $ 700,000

= $ 650,921.6

So, more total interest will be paid over the life of the loan if we take out a 30-year mortgage instead of a 15-year mortgage is $ 853,113.6 ($1504035.2 - $650921.6)

If you need any clarification regarding this solution, then you can ask in comments

If you like my answer then please Up-vote as it will be motivating.