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Consider the following balance sheet $($in millions$)$ for an FI:

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Duration $=10$ years $,$$$950$ Duration $=2$ years $,$ Equity $,960$

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a$.$ What is the Fl$\text{'}$s duration gap? $($Do not round intermediate calculations. Round your answer to $2$ decimal places. $($e$.$g$.,32\mathrm{.}16))$

b$.$ What is the Fl$\text{'}$s interest rate risk exposure?

c$.$ How can the FI use futures and forward contracts to create a macrohedge?

d$.$ What is the impact on the Fl$\text{'}$s equity value if the relative change in interest rates is an increase of $1$ percent? That is$,\frac{R}{1+R}=0\mathrm{.}01.($Negative amount should be indicated by a minus sign. Do not round intermediate calculations. Enter your answer in dollars not in millions.$)$

e$.$ Suppose that the FI in part $($c$)$ macrohedges using Treasury bond futures that are currently priced at $96.$ What is the change in value per futures contract used to hedge if the relative change in all interest rates is an increase of $1$ percent? That is$,\frac{R}{1+R}=0\mathrm{.}01.$ Assume that the deliverable Treasury bond has a duration of nine years. The bonds underlying the futures contract have a par value of $$100,000.($Negative amount should be indicated by a minus sign. Enter your answer in dollars not in millions.$)$

f$.$ If the FI wants to macrohedge, how many Treasury bond futures contracts does it need? $($Negative value should be indicated by a minus sign. Do not round intermediate calculations. Round down your answer to the nearest whole number.$)$

$\backslash$table$[[,],[$a$.,$Duration gap,$8\mathrm{.}19,$years$],[$b$.,$The FI would be hurt by$,$increasing,interest rates.$],[$c$.,$The FI could hedge its interest rate risk by$,$selling,futures or forward contracts.$],[$d$.,$Impact on the Fl$\text{'}$s equity value,$$(77,800,000),],[$e$.,$Change in value per futures contract,$$(8,640)+-0\mathrm{.}1\%,],[$f$.,$Number of Treasury bond,$(9,004)\frac{+}{-}0\mathrm{.}1\%,]]$

Explanation

Some values below may show as rounded for display purposes, though unrounded numbers should be used for the actual calculations.

a$.$

The duration gap is $10-(\frac{860}{950})(2)=8\mathrm{.}19$ years.

b$.$

The $FI$ is exposed to interest rate increases. The market value of equity will decrease if interest rates increase.

c$.$

The FI can hedge its interest rate risk by selling future or forward contracts.

d$.$

$E=-8\mathrm{.}19(950,000,000)(0\mathrm{.}01)=-$$$77,800,000$

e$.$

$F=-9(96,000)(0\mathrm{.}01)=-$$$8,640$ per futures contract. Since the macrohedge is a short hedge, this will be a profit of $$8,640$ per contract.

f$.$

To macrohedge, the Treasury bond futures position should yield a profit equal to the loss in equity value $($for any given increase in interest rates$).$ Thus, the number of futures contracts must be sufficient to offset the $$77,800,000$ loss in equity value. This will necessitate the sale of $$77,800,\frac{000}{8},640=-9004\mathrm{.}630$ contracts. Rounding down, to construct a macrohedge requires the $FI$ to sell $-9,004$ Treasury bond futures contract

Why did we say Sell in part C$?$