Question $2(20$ marks$)$

Consider a bank whose asset and liability both consist of bonds only. The asset

consists of a $6-$year coupon bond with face value $$80$ million, coupon rate $6\mathrm{.}5\%,$ and

its coupons are paid once per year. The liability consists of a $7-$year zero coupon

bond with face value $$100$ million. The current yield for all these bonds are $5\mathrm{.}5\%.$

$($a$)(10$ marks$)$ What is the bank's current market value of equity $($expressed in

million dollars$)?$

$($b$)(5$ marks$)$ What is the bank's current leverage$-$adjusted modified duration

gap?

$($c$)(5$ marks$)$ Assume parallel yield shift. The bank considers a scenario in which

the yield changes to $5\mathrm{.}5\%+R,$ where $R$ follows a normal distribution

$N(,{}^2)$ with mean $=50$ basis points and standard deviation $=5$ basis

points $(1$ basis point is equivalent to $0\mathrm{.}01\%).$ Denote $E$ as the change in the

bank's market value of equity predicted by the duration model that is caused

by the yield change $R.$ What is the $95-$th percentile of $E($expressed in

million dollars$)?$