Note: $(1),(2)$ & $(3)$ share the same assumptions.

$(1)$ Suppose the simple CAPM is valid and all portfolios are priced correctly. The expected return of market portfolio is $6\%.$ Please conclude whether the expected return of each portfolio in the below tables is reasonable and explain why? $($Consider each situation independently, and assume the risk$-$free rate is $1\%.)$

Portfolio Expected Return Beta

A $9\%1\mathrm{.}4$

B $-5\%-1\mathrm{.}2$

$(2)$ However, in reality, CAPM may not hold. List at least two reasons why CAPM does not hold.

$(3)$ When CAPM fails in the pricing of stocks, we can introduce additional factors $($besides market portfolio return premium$)$ to our pricing model. Now suppose there is an economic factors X which is independent from market factor. Portfolios A in the below table is well diversified.

Portfolio Beta on $($E$[$r$\_$M $]-$r$\_$f$)$ Beta on X factor return premium Expected Return $(\%)$

A $1\mathrm{.}72\mathrm{.}519\mathrm{.}5$

$($a$)$ What is the expected return$$beta relationship in this economy? $($In other words, write down the equation describing how portfolio expected returns depend on the value of betas$)($Hint: when you add a new factor to CAPM model, the new factor$$s risk premium is additive which means you simply add $\backslash$beta $\_2[$E$($r$\_$X$)-$r$\_$f$]$ to the standard CAMP.$)$

$($b$)($Bonus $2\%)$ Can you give an example of an additional factor and explain why you think it matters.