An FI has a $$210$ million asset portfolio that has an average duration of $6\mathrm{.}6$ years. The average duration of its $$170$ million in liabilities is $4\mathrm{.}1$ years. Assets and liabilities are yielding $11$ percent. The FI uses put options on T$-$bonds to hedge against unexpected interest rate increases. The average delta $(\backslash$delta $)$ of the put options has been estimated at $0\mathrm{.}2$ and the average duration of the T$-$bonds is $7\mathrm{.}1$ years. The current market value of the T$-$bonds is $$95,000.$ Put options on T$-$bonds are selling at a premium of $$1\mathrm{.}00$ per face value of $$100.$

a$.$ What is the modified duration of the T$-$bonds if the current level of interest rates is $11$ percent? $($Do not round intermediate calculations. Round your answer to $4$ decimal places. $($e$.$g$.,32\mathrm{.}1616))$

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? $($Do not round intermediate calculations. Negative value should be indicated by a minus sign. 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$.,32\mathrm{.}16))$