Suppose the one year risk$-$free rate $($the yield on a one$-$year Treasury discount bond$)$ is

$5\%.$ There are two one$-$year corporate discount bonds with a face value of $1,000.$ The

first one is investment grade and trades at $$925\mathrm{.}38$ and the second bond is speculative

grade and trades at $$769\mathrm{.}23.$

$($a$)$ What is the price of an one$-$year Treasury discount bond with a face value of

$$1,000.$

$($b$)$ What is the yield spread of the investment grade bond?

$($c$)$ What is the yield spread of the speculative grade bond?

$($d$)$ Suppose both bonds default in one year. The recovery rate for the investment$-$

and speculative$-$grade bonds are $60\%$ and $50\%,$ respectively. Compute the realized

one$-$year return for both bonds and compare them.