$40.$ HomeEquity Bank$$s average asset duration is $8$ years and average liability duration is $4\mathrm{.}25$ years. Suppose the size of its assets is $$965$ million and its liabilities are $$950$ million. If R is $14\mathrm{.}5\%$ and the bank is expecting a $125$ basis point increase in interest rates, how many T$-$bond futures contracts are required to fully hedge the equity value if the Treasury bond futures are selling for $94\mathrm{.}5\%$ of $$100,000$ face value with duration of $4\mathrm{.}50$ years?

How many contracts are needed to offset risk on the balance sheet?

a$.$ Long $4000$

b$.$ Short $350$

c$.$ Long $5150$

d$.$ Short $8660$

$41.$ Help Bank has assets of $$225$ million and liabilities of $$180$ million. The asset duration is $5$ years and the duration of the liabilities is $4$ years. Market interest rates are $10$ percent. Help Bank wishes to hedge the balance sheet with Treasury bond futures contracts, which currently have a price quote of $$96\mathrm{.}75$ per $$100$ face value for the benchmark $20-$year market yield of $6\mathrm{.}985$ percent, and duration of $13\mathrm{.}50$ years.

How many contracts are necessary to fully hedge the bank if the relationship of the price sensitivity of futures contracts to the price sensitivity of underlying bonds were br $=0\mathrm{.}98?$

a$.$ Long $($buy$)720$

b$.$ Short $($sell$)316$

c$.$ Long $($buy$)525$

d$.$None