$6.$ Brick House Financial offers a zero$-$coupon mortgage that has a $5-$year term where the interest rate accrues annually and is paid upon maturity along with the principle $($much like a $5-$year zero$).$ You are looking to buy a new home in two years for $$500,000$ and will finance the entire purchase price using this special $5-$year zero mortgage. Current interest rates are low and you would like to lock in the $5-$year zero mortgage today instead of risking that the $5-$year rate will rise significantly over the next two years. Current spot rates are:

a$.2-$year spot $=1\mathrm{.}50\%,$

b$.5-$year spot $=2\mathrm{.}75\%$

c$.7-$year spot $=2\mathrm{.}5\%$

a$)$ Compare the current $5-$year spot rate to the $5-$year forward rate $2-$years from now and determine the amount that the current $5-$year spot rate would need to rise to make it worthwhile to lock in the rate today. In other words, compare r$0,5$ to r$2,7.$

b$)$ Draw a timeline and set up the series of transactions that would enable you to lock in the $5-$year borrowing rate $2$ years from today.

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