Financial Institution XY has assets of $$1$ million invested in a $30-$year, $10$ percent semiannual coupon Treasury bond selling at par. The duration of this bond has been estimated at $9\mathrm{.}94$ years. The assets are financed with equity and a $$900,000,$ two$-$year, $7\mathrm{.}25$ percent semiannual coupon capital note selling at par.

a$.$ What is the leverage$-$adjusted duration gap of Financial Institution XY$?$

b$.$ What is the impact on equity value if the relative change in all market inter$-$ est rates is a decrease of $20$ basis points? Note: The relative change in interest

rates is $\backslash$Delta R$/(1+$ R$/2)=0\mathrm{.}0020.$

c$.$ Using the information in parts $($a$)$ and $($b$),$ what can be said about the desired

duration gap for the financial institution if interest rates are expected to

increase or decrease?

d$.$ Verify your answer to part $($c$)$ by calculating the change in the market value

of equity assuming that the relative change in all market interest rates is an

increase of $30$ basis points.

e$.$ What would the duration of the assets need to be to immunize the equity

from changes in market interest rates?