Question $5$

A bank longs a $3-$year BBB$-$rated $($defaultable$)$ zero coupon bond with face value $$2$ million and current yield of $5\mathrm{.}5\%.$ The bank wants to hedge the risk of a downgrade of the bond rating to $BB$ with a yield of $6\mathrm{.}5\%$ in one year, using credit spread forwards. The current risk$-$free yield is $4\mathrm{.}25\%$ and remains unchanged. The contractual amount of the credit spread forward is $$10,000,$ and the contractual credit spread equals the current credit spread of this bond.

$($a$)(3$ marks$)$ What is the current credit spread of this zero coupon bond?

$($b$)(7$ marks$)$ What number of credit spread forward contracts should the bank buy or sell currently in order to hedge the downgrade risk?