Consider the following two scenarios for the economy and the expected returns in each scenario

for the market portfolio, an aggressive stock $A,$ and a defensive stock $D.$

If each scenario is equally likely $($that is $p=0\mathrm{.}5),$ calculate the expected alpha on stock $A$ if the new

risk$-$free rate is $4\%$

This is an involved problem. Here are the steps.

You have two scenarios, boom and bust. For each of these scenarios, write the CAPM equation,

$r=r_{-}f+b(r_{-}m-r_{-}f)$

The thing is you don't know $r_{-}f($the risk free rate$)$ and you don't know $b($ beta$).$ But for the two

scenarios, you have $2$ equations and two unknowns. Solve for beta.

For instance, for the boom scenario, $38=r_{-}f+b(32-r_{-}f).$ And for the bust scenario $-10=r_{-}f+$

$-\{$:$r_{-}f).$

Calculate the expected return for stock A from the portfolio. $($Use our probability model$)$

Calculate the expected return from CAPM using $r_{-}A=r_{-}f+b($Expected $\{$:$r_{-}m-r_{-}f).$ The

expected return on the market Expected $r_{-}m$ is the return you calculated in the previous question.

Dont forget to use the new risk$-$free rate.

The alpha of the stock is the Expected $r_{-}A($calculated in step $2)-r_{-}A($from CAPM calculated in

step $3).$