An FI has a $$200$ million asset portfolio that has an average duration of $6\mathrm{.}5$ years. The average duration of its $$160$

million in liabilities is $4\mathrm{.}5$ years. Assets and liabilities are yielding $10$ percent. The FI uses put options on T$-$bonds to

hedge against unexpected interest rate increases. The average delta $()$ of the put options has been estimated at $-0\mathrm{.}3$

and the average duration of the T$-$bonds is seven years. The current market value of the T$-$bonds is $$96,000.$ Put options

on T$-$bonds are selling at a premium of $$1\mathrm{.}25$ per face value of $$100.$

a$.$ What is the modified duration of the T$-$bonds if the current level of interest rates is $10$ percent? $($Do not round

intermediate calculations. Round your answer to $2$ decimal places. $($e$.$g$.,32\mathrm{.}16))$

b$.$ How many put option contracts should the FI purchase to hedge its exposure against rising interest rates? The face

value of the T$-$bonds is $$100,000.($Do not round intermediate calculations. Round your answer to the nearest whole

number.$)$

c$.$ If interest rates increase $50$ basis points, what will be the change in value of the equity of the FI$?($Enter your answer

in dollars not in millions. Negative amount should be indicated by a minus sign. Do not round intermediate

calculations. Round your answer to the nearest dollar amount.$)$

d$.$ If interest rates increase $50$ basis points, what will be the change in value of the T$-$bond option hedge position? $($Enter

your answer in dollars not in millions. Do not round intermediate calculations. Round your answer to the nearest

dollar amount.$)$

e$.$ What must be the change in interest rates before the change in value of the balance sheet $($equity$)$ will offset the cost

of placing the hedge? $($Negative value should be indicated by a minus sign. Do not round intermediate calculations.

Round your answer to $2$ decimal places. $($e$.$g$.,32\mathrm{.}16))$

f$.$ How much must interest rates change before the payoff of the hedge will exactly cover the cost of placing the hedge?

$($Do not round intermediate calculations. Round your answer to $2$ decimal places. $($e$.$g$.,32\mathrm{.}16))$

Answer is not complete.