Carman County Bank $($CCB$)$ has a $$5$ million face value outstanding adjustable$-$rate loan to a

company that has a leverage ratio of $80$ percent. The current risk$-$free rate is $6$ percent and the time

to maturity on the loan is exactly $(1)/(2)$ year. The asset risk of the borrower, as measured by the standard

deviation of the rate of change in the value of the underlying assets, is $12$ percent. The normal density

function values are given below.

h N$($h$)$ h N$($h$)$

$-2\mathrm{.}550\mathrm{.}00542\mathrm{.}500\mathrm{.}9938$

$-2\mathrm{.}600\mathrm{.}00472\mathrm{.}550\mathrm{.}9946$

$-2\mathrm{.}650\mathrm{.}00402\mathrm{.}600\mathrm{.}9953$

$-2\mathrm{.}700\mathrm{.}00352\mathrm{.}650\mathrm{.}9960$

$-2\mathrm{.}750\mathrm{.}00302\mathrm{.}700\mathrm{.}9965$

a$.$ Use the Merton option valuation model to determine the market value of the loan.

b$.$ What should be the interest rate for the last six months of the loan?