Consider the following securities and their sensitivities to two factors $($the factors have zero means$)$:

Stock A: $r_{A,t}=8+5F_{1,t}+6F_{2,t}+e_{A,t}$

Stock B: $r_{B,t}=6+4F_{1,t}+1F_{2,t}+e_{B,t}$

Riskfree: $r_f=1$

a$.$ Construct a portfolio out of stocks A and B which is riskless in terms of factor $2.$ You may sell short either A or B if necessary.

$($i$)$ What are $w_A$ and $w_B$ for this portfolio?

$($ii$)$ How sensitive is this portfolio to factor $1($that is$,$ how many units of factor $1$ risk$)?$

$($iii$)$ Given your answers to a$)$ and b$),$ what is the risk premium per unit of factor $1$ risk, ${}_1?$

b$.$ Construct a portfolio out of stocks A and B which is riskless in terms of factor $1.$ You may sell short either A or B if necessary.

$($i$)$ What are $w_A$ and $w_B$ for this portfolio?

$($ii$)$ How sensitive is this portfolio to factor $2($that is$,$ how many units of factor $2$ risk$)?$

$($iii$)$ Given your answers to a$)$ and b$),$ what is the risk premium per unit of factor $2$ risk, ${}_2?$

c$.$ Given your answers to a$.($iii$)$ and b$.($iii$),$ what does the APT predict the returns would be on the above securities$?($Hint: I$\text{'}$m not looking for the intercept a$).$

d$.$ What does APT predict the return should be for stock C$?$

Stock C: $r_{C,t}=5+3F_{1,t}+4F_{2,t}+e_{C,t}$

e$.$ Is stock $C$ underpriced or overpriced? How would you exploit this arbitrage opportunity? Explain your strategy