$($a$)$ The mean change in the value of a portfolio of trading assets has been estimated to be $0$ with a standard deviation of $30$ percent. Yield changes are assumed to be normally distributed.

What is the maximum yield change expected if a $90$ percent confidence $($one$-$tailed$)$ limit is used? $(5$ points$).$

$($b$).$ Bank of America has an inventory of AAA$-$rated, $25-$year zero$-$coupon bonds with a face value of $$800$ million. The bonds currently are yielding $10\mathrm{.}0$ percent in the over$-$the$-$counter market.

$($i$)$ What is the modified duration of these bonds? $(5$ points$).$

$($ii$)$ What is the price volatility if the potential adverse move in yields is $20$ basis points? $(10$ points$)$

$($iii$)$ What is the DEAR? $(10$ points$)$

$($iv$)$ What is the VAR assuming a $15-$day period? $(5$ points$).$

$($v$)$ If the price volatility is based on a $99\mathrm{.}5$ percent confidence limit and a mean historical change in daily yields of $0\mathrm{.}0$ percent, what is the implied standard deviation of daily yield changes expressed in basis points? $(5$ points$)$