Dudley Hill Bank has the following balance sheet:

Assets $($in millions$)$

A $$425$

$$425$

Liabilities and Equity $($in millions$)$

L $$380$

E $45$

$$425$

Further,

DA $=6$ years

DL $=2$ years

The interest rate on both the assets and the liabilities is $8$ percent. The bank manager receives information from an economic forecasting unit that interest rates are expected to rise from $8$ to $9$ percent over the next six months.

a$.$ Calculate the potential loss to Dudley Hill$$s net worth $($E$)$ if the forecast of rising rates proves to be true.

b$.$ Suppose the manager of Dudley Hill Bank wants to hedge this interest rate risk with T$-$bond futures contracts. The current futures price quote is $122\mathrm{.}03125$ per $100$ of face value for the benchmark $20-$year, and the minimum contract size is $100,000,$ so PF equals $122,031\mathrm{.}25.$ The duration of the deliverable bond is $14\mathrm{.}5$ years. That is$,$ DF $=14\mathrm{.}5$ years. How many futures contracts will be needed? Should the manager buy or sell these contracts? Assume no basis risk.

c$.$ Verify that selling T$-$bond futures contracts will indeed hedge the FI against a sudden increase in interest rates from $8$ to $9$ percent, a $1$ percent interest rate shock.

d$.$ If the bank had hedged with Eurodollar futures contracts that had a market value of $$98$ per $$100$ of face value, how many futures contracts would have been necessary to fully hedge the balance sheet?

e$.$ How would your answer for part $($b$)$ change if the relationship of the price sensitivity of futures contracts to the price sensitivity of underlying bonds were br $=1\mathrm{.}15?$

f$.$ Verify that selling T$-$bond futures contracts will indeed hedge the FI against a sudden increase in interest rates from $8$ to $9$ percent, a $1$ percent interest rate shock. Assume the yield on the T$-$bond underlying the futures contract is $8\mathrm{.}45$ percent as the bank enters the hedge and rates rise by

$1\mathrm{.}154792$ percent.