Today is $1$ July $2021,$ William plans to purchase a corporate bond with a coupon rate of j$2=2\mathrm{.}96\%$ p$.$a$.$ and face value of $100.$ This corporate bond matures at par. The maturity date is $1$ January $2024.$ The yield rate is assumed to be j$2=3\mathrm{.}4\%$ p$.$a$.$ Assume that this corporate bond has a $9\mathrm{.}5\%$ chance of default in any six$-$month period during the term of the bond. Assume also that, if default occurs, William will receive no further payments at all. Calculate the purchase price for $1$ unit of this corporate bond. Round your answer to three decimal places.

a$.$

$61\mathrm{.}086$

b$.$

$99\mathrm{.}721$

c$.$

$98\mathrm{.}285$

d$.$

$55\mathrm{.}676$