A company is considering its optimal capital structure. The firm currently has $1$ million shares outstanding at $ $20$ per share $($tax rate $=40\%)$ and a debt balance of $$5$ million. Currently, its $($levered$)$ beta is $1\mathrm{.}5$ and its ERP is $5\mathrm{.}5\%.$ The current risk$-$free rate is $5\%.$

Your research indicate the following ratings and pre$-$tax cost of debt across the different debt ratios:

D$/($D$+$E$)$

Rating

Pre$-$tax cost of debt

$0\%$

AAA

$10\%$

$10\%$

AA

$10\mathrm{.}5\%$

$20\%$

A

$11\%$

$30\%$

BBB

$12\%$

$40\%$

BB

$13\%$

$50\%$

B

$14\%$

$60\%$

CCC

$16\%$

$70\%$

CC

$18\%$

$80\%$

C

$20\%$

$90\%$

D

$25\%$

a$.$ Using the optimal WACC approach, what is the firm's optimal debt ratio?

b$.$ Calculate the company's unlevered value, assuming that the probability of default is $5\%$ and the company loses $30\%$ of its value in the event of a default.