$(12\%$;$2\%$ for each subquestion$)$ A firm has non$-$dividend$-$paying equity $(E_t$ : equity value at time $t)$ and zero$-$

coupon debt $(B_t$ : debt value at time $t$; promised payment at time $T$ is $800).$ The other parameters are: $A_t$ is the

asset value of the firm at time $t.,r_c=4\%($continuous time$),T=2$ years.

$($a$)$ If the initial asset value $A_0=1,000$ and asset volatility $=25\%,$ find the current market value of risky debt.

$($b$)$ In part $($a$),$ find the continuously compounded bond yield $(\%)$ and credit spread $(\%).$

$($c$)$ In part $($a$),$ if asset volatility $=35\%,$ find the continuously compounded bond yield $(\%)$ and credit spread

$(\%).$

$($d$)$ The distance to default $(D{D}_t)$ is defined as

$D{D}_t=\frac{ln(\frac{A_t}{B})+(-\frac{{}_A^2}{2})T}{{}_A\sqrt[\phantom{2}]{T}}$

$,$ where $$ is the expected continuously compounded return on A and ${}_A=25\%.$ If $=5\%,$ find the value of $D{D}_0.$

$($e$)$ In part $($d$),$ find the implied probability of default $($or the expected default frequency$),$ which is calculated

as $N(-D{D}_0).$

$($f$)$ In part $($a$),$ find the volatility of equity in this model, which is calculated as ${}_E=$ elasticity$*{}_A$

$=(\frac{A_0}{E_0})(del\frac{E}{d}$elA$)**{}_A=(\frac{A_0}{E_0})N(d_1)**{}_A.$ We use $r_c$ when calculating $d_1.$