Question $1$

Five years ago, Bob purchased an apartment in Tai Po of $$2,000,000.$ He made a down payment

of $30\%$ and financed the balance with a $20$ years' conventional mortgage to be amortized through

monthly payments. The term is adjustable$-$rate mortgage $($ARM$)$ with interest of $8\%$ per year

compounded monthly. Currently, the interest rate for his ARM is $5\%$ per year compounded

monthly. Bob is thinking of refinancing his property by paying an extra amount of $$500,000$ now

and to amortize the rest of the loan in the next $8$ years.

$($a$)$ What was Bob's monthly mortgage payment for his first $5$ years? $($interest is $8\%)$

$($b$)$ What is Bob's outstanding principal now if he does not refinance his apartment? $($interest is

$8\%$

$($c$)$ What is Bob's outstanding principal now if he refinances his apartment by paying the extra

payment?

$($d$)$ What will Bob's new monthly mortgage payment be $($after the reset$)$ if he refinances his

apartment? $($interest is $5\%)$