Please solve all problems and equations

Problem Set $2$

Background Information

A family is looking to buy a home, which they intend to stay in for the next $5$ years, after which they plan to sell it$.$ They are deciding between taking an ARM or an FRM$.$ They think interest rates are falling, and so would prefer to have an ARM to benefit from the falling interest rates. But they are also risk averse, and prefer the idea of an FRM$,$ so that they know exactly how much they have to pay each month. They are doing the math to work out the best scenario. They have $3$ options they are considering:

$1.$ Take the ARM and hold it for $5$ years until they sell the property.

$2.$ Take the FRM and hold it for $5$ years until they sell the property.

$3.$ Take the FRM now, and if rates fall sufficiently, to refinance the FRM with another FRM at the end of the Y$3.$

Here are the terms of the ARM and FRM they are considering now.

Adjustable$-$Rate Mortgage:

$$ Amortization period: $30$ years

$80\%$ LTV

$5\mathrm{.}50\%$ teaser rate.

$$ Margin of $3\mathrm{.}00\%.$

$$ Their financial advisor predicts index rates for the ARM to change over the next $5$ as follows: Year $2$: $3\mathrm{.}00\%.$ Year $3$: $2\mathrm{.}75\%.$ Year $4$: $2\mathrm{.}65\%.$ Year $5$: $2\mathrm{.}60\%$

$$ Closing costs of $2\mathrm{.}0\%$

$$ Price of the property $$500,000$

Fixed Rate Mortgage:

$$ Amortization period: $30$ years

$80\%$ LTV

$7\mathrm{.}03\%$ interest rate

$$ Prepayment penalty of $2\%$ of the outstanding loan balance in the first $2$ years, stepping down to $1\%$ for the next $2$ years.

$$ Closing costs of $2\mathrm{.}5\%.$

$$ Price of the property $$500,000$

Questions

$1.$ Calculate the outstanding loan balance on the ARM at the end of year $5.$

$2.$ Calculate the interest paid on the ARM over the first $5$ years.

$3.$ Calculate the total cost paid $($closing costs $+$ interest cost$)$ on the ARM over the first $5$ years.

$4.$ Calculate the total cost paid $($closing costs $+$ interest cost$)$ on the single FRM over the first $5$ years.

$5.$ If the borrower chooses to refinance the FRM at the end of year $3,$ to a new $30-$year FRM$,$ what would the interest rate of the new FRM need to be in order for the borrower to pay the same total cost $($interest $+$ prepayment penalties $+$ closing costs$)$ as they would if they just stayed with the original FRM $($i$.$e$.$ if they chose to not refinance$).$ Assume that the new FRM has $2\mathrm{.}5\%$ closing costs.

$6.$ If the borrower decides to take the FRM and chooses to refinance at the end of year $3($as per questions $4),$ what would the interest rate of the new refinanced FRM need to be in order for the borrower to pay the same total cost $($interest $+$ prepayment penalties $+$ closing costs$)$ as if they had decided to take the ARM for $5$ years?