A$)$

During your co$-$op term at RBC$,$ you are asked to assess the credit risk of an $8-$year

corporate bond that provides a coupon of $4\mathrm{.}4\%$ per year payable annually and has a

yield of $6\mathrm{.}25\%($continuous compounding$).$ The yields for all maturities on risk$-$free

bonds are $3\mathrm{.}85\%$ per year $($continuous compounding$).$ Assume that defaults can take

place every $6$ months, either in the middle of the year or at the end of the year

immediately before a coupon payment. The recovery rate is $41\%.$ Assuming that the

unconditional default probabilities are the same on each possible default date. What is

the approximate default probability? What is the more accurate default probability?

B$)$

Repeat question A$)$ by assuming that the $41\%$ recovery rate is applied to the market

value of the risk$-$free bond as opposed to par. Compare the default probability with

that in A$)$ and explain the difference. Note: in practice, the recovery rate is applied to

the par. Only in this question, we make the special assumption in order to derive

some insights.