Question $4(20$ points$)$

$($a$)$ Assume that five years ago, you renewed your TD Bank mortgage that had $10$

years left at $6\%$ fixed rate for five years. The interest rate is compounded semi$-$

annually, and at an effective monthly rate $($EMR$)$ of $0\mathrm{.}4939\%,$ you have made

monthly payments $$1659\mathrm{.}765$ at the end of each month for the past five years.

$($i$)$ How much in interest and how much of the loan have you paid over the five

years?

$(6$ marks$)$

$($ii$)$ You are about to renew the mortgage for the remaining $5$ years. Fortunately for

you, interest rates have dropped to $5\%$ and you decide to refinance the mortgage.

What will be your new monthly payment for the remaining five years?

$(3$ marks$)$

$($iii$)$ Immediately after you signed the paperwork to refinance the remaining

mortgage, mortgage rates dropped further from $5\%$ to $4\%.$ Your bank has offered

you the opportunity to renegotiate the mortgage at the lower rate of $4\%$ for a

penalty of $$3,000.$

What will be your monthly mortgage amount if you decide to take the bank's

offer to renegotiate at $4\%?$

$(3$ marks$)$

Should you take this opportunity offered by the bank? Assume your

opportunity cost $($or discount rate$)$ equals the mortgage rate. $($Note: Estimate

the savings in mortgage payments and discount them using the lower

mortgage as the discount rate.

$(3$ marks$)$

$($b$)$ Suppose your mother is expecting to receive the following annual income from

one of her investments with RIBC: $$20,000$ in Year $1$; $$40,000$ in Years $2$ and $3,$

$$60,000$ in Years $4$ and $5,$ and $$80,000$ in Year $6.$ The cash flows occur at the end of

the year. She has sought your help in analyzing the cash flows. If her required

discount rate is $5\%$ annual compounding, what is the value of the cash flows at the

end of Year $4?$

$(5$ marks$)$