$(3)$ Consider a bond with par value of $1,000$; interest rate $6\%$ p$,$a; YTM $8\%$ p$.$a; mature in $5$ years. The bond currently trades at $$920\mathrm{.}15$

$($a$)$ Calculate the promised yield to maturity rate.

b$.)$ Assume that there is a $20\%$ probability of default on the bond and if there is default the bondholder will only receive $60\%$ principal and interest owed to them. Calculate the YTM on the bond if there is default.

$($c$)$ Calculate the expected YTM