$[8$ points$]$ Jack and Diane are considering purchasing a new home. They have identified a suitable

option, a two bedroom townhouse that will cost $$212,400$ to purchase. They are now trying to

select the best method of financing and have identified a couple of suitable options. As their most

trusted friend, they have asked for your help in selecting the best option for their needs. They plan

to live in this townhouse for only $10$ years, after which they anticipate needing a larger home, with

a backyard, to accommodate their growing family.

Option A: Conventional $30$ year mortgage with an interest rate of $4\mathrm{.}375\%$ APR with monthly

payments. If Jack and Diane choose this option they would need to make a $20\%$ down payment

and would owe an additional $$5250$ in closing costs and fees. They will make the down

payment from their savings, however the lender will allow them to include the closing costs

and fees in the loan finance amount $($i$.$e$.$ the amount borrowed$).$

Option B: A special $25$ year mortgage with an interest rate of $4\mathrm{.}76\%$ compounded monthly with

monthly payments. This loan begins with a $5$ year period in which Jack and Diane would only

pay interest on the loan. If they choose this option they will make an $18\%$ down payment and

would owe an additional $$4750$ in closing costs and fees. Once again closing costs and fees can

be included into the loan amount. After the interest only period is over, Jack and Diane's

monthly payment would be applied to principal and interest.

Option C: A special $15$ year fixed rate mortgage, with bi$-$m payments $($i$.$e$.$ every $2$ weeks or two

times per month$).$ The loan's interest rate of $3\mathrm{.}94\%$ compounded monthly. Because of the short

term $($i$.$e$.$ length$)$ of this loan the bank will only require a $10\%$ down payment, which will be

paid from their savings. Jack and Diane would owe an additional $$3200$ in closing costs and

fees, which can once again be included in the loan finance amount.

a$)[2$ points$]$ Determine the monthly payments that Jack and Diane will make for each

period of each loan in the first ten years of ownership.

b$)[3$ points$]$ How much will Jack and Diane still owe $($i$.$e$.$ what would the remaining

balance be$)$ at the end of the ${10}^{th}$ year for each loan?

c$)[3$ points$]$ Which of these option should Jack and Diane want to choose, if their

objective is to pay the least for their home over the first ten years of purchase. Support

your decision by comparing the total interest paid over this period of time.

$[4$ points$]$ Jack and Diane's have received one more offer for a home mortgage. This option $-$

Option C$,$ is described below. The purchase cost $($$$212,400)$ of the condo will remain the same and

the time they expect to remain in the home $(10$ years$)$ remain the same. Details of this new loan

option are described below.

Option D: A $30$ year adjustable rate mortgage $($ARM$)$ in which the initial interest rate of $4\mathrm{.}25\%$

APR for the first three years of the loan. After this initial fixed rate period the interest rate will

change as described below. This option will require a $22\%$ down payment, which will again be

paid from their savings. The $$5500$ in closing costs and fees can once again be included in the

loan finance amount. Payments will once again be made on a monthly basis.

a$)[3$ points$]$ Find the monthly payment for each of the $10$ years of this adjustable rate mortgage.

b$)[1$ points$]$ Is this a better choice for Jack and Diane, relative to Options A$,$ B$,$ and C outlined in

Question $1?$ Once again, support your choice by calculating the total interest paid over the first ten

years of scenario.

Solve mathematically