$[$You must show all the formula and steps for this question$]$

An investor is evaluating two bonds with the same maturity but different default

risks. Bond A has a yield of $5\%,$ and Bond B has a yield of $8\%.$ The investor

believes that the probability of default on Bond B is $15\%$ over the holding

period and estimates that, in the event of default, they will recover $30\%$ of the

bond's face value $($and lose $70\%).$ The investor also has the option of investing

in a risk$-$free bond with a yield of $2\%.$

a$)$ Calculate the investor's expected return on Bond B $,$ considering the default

risk and recovery rate.

b$)$ Assuming the investor is risk$-$averse, under what conditions might the

investor prefer Bond A over Bond B $,$ even though Bond B has a higher yield?

Discuss the impact of risk premium and recovery rate $(30\%)$ on the investor's

decision.

c$)$ The investor expects inflation to increase by $2\%$ over the holding period.

Analyze how the real returns for Bond A and Bond B would change, and how

these adjusted real returns would influence the investor's portfolio choice.

d$)$ Suppose a financial analyst provides new information that the default

probability on Bond B has increased to $25\%.$ Recalculate the expected return on

Bond B with this new information and determine whether this makes Bond A or

the risk$-$free bond more attractive to the investor, assuming the investor requires

a minimum expected return of $4\%.$