You’ve decided to buy a house that is valued at $1 million. You have $200,000 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 $800,000 mortgage, and is offering a standard 30-year mortgage at a 9% fixed nominal interest rate (called the loan’s annual percentage rate or APR). Under this loan proposal, your mortgage payment will be ____ per month.

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, $800,000 loan at a fixed nominal interest rate of 9% (APR), then the difference in the monthly payment of the 15-year mortgage and 30-year mortgage will be:

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?

a. $1,010,987.89
b. $856,769.40
c. $1,182,341.77
d. $1,096,664.83

n is the total number of periods and is calculated as the product of the number of periods in a year times the number of years (different for each of the options given)

Quesiton 1. Mortgage payment for a stardard 30-year mortgage.

n = 30

[tex]Payment = L\times \frac{i}{1-(1 + i)^{-n}}\\\\Payment = \$ 800,00\times \frac{0.0075}{1-(1 + 0.0075)^{-(30\times 12)}}\\\\Payment=\$ 6436.98[/tex]

Answer: $6,436.98 ← answer

Question 2. Difference in the monthly payment

[tex]Payment = L\times \frac{i}{1-(1 + i)^{-n}}\\\\Payment = \$ 800,00\times \frac{0.0075}{1-(1 + 0.0075)^{-(15\times 12)}}\\\\Payment=\$ 8114.13[/tex]

Difference in the monthly payment of the 15-year mortgage and 30-year mortgage will be $8,114.13 - $6,436.98 = $1,677.15 ← answer

Question 3. How much more interest will you pay over the life of the loan if you take out a 30-year mortage instead of a 15-year mortgage?

i. Interest over the life of the 30-years mortgage

Interest = Total payment - loan amount

Total payment = $6436.9 × 30 × 12 = $2,317,312.80

Interest = $2,317,312.80 - $800,000.00 = $1,517,312.80

ii) Interest over the life of the 15-years mortgage

Total payment = $8,114.13 × 15 × 12 = $1,460,543.40

Interest = $1,460,543.40 - $800,000,.00 = $660,543.40

iii) Difference = $1,517,312.80 - $660,543.40 = $856,769.40

That is the option b. $856,769.40 ← answer