A company has one$-$ and two$-$year bonds outstanding, the one$-$year providing a coupon of $9\%$ and the two$-$year providing a coupon of $10\%,$ both payable annually. The yields on the bonds $($expressed with continuous compounding$)$ are $7\mathrm{.}0\%$ and $7\mathrm{.}6\%,$ respectively. Riskfree rates are $5\mathrm{.}5\%$ for all maturities. The recovery rate is $25\%.$ Defaults can take place halfway through each year.

a$)$ Compute the market price and the default$-$free price of both bonds.

$[5$ marks$]$

b$)$ Estimate the risk$-$neutral default rate each year.