$($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.$

$1.$ What will be your monthly mortgage amount if you decide to take the bank's offer to renegotiate at $4\%?$

$(3$ marks$)$

$2.$ 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$)$