Assume that these $3$ stocks ahd factors and epsilons with a means of $0.$ Also assume the factors have variances of $\mathrm{.}01$ and are uncorrelated.

tilde$(r{)}_A=\mathrm{.}13+6$tilde$(F{)}_1+4$tilde$(F{)}_2+$tilde$(lon{)}_A$

tilde$(r{)}_B=\mathrm{.}15+2$tilde$(F{)}_1+2$tilde$(F{)}_2+$tilde$(lon{)}_B$

tilde$(r{)}_C=\mathrm{.}07+5$tilde$(F{)}_1-1$tilde$(F{)}_2+$tilde$(lon{)}_C$

If var$($Ea$)=\mathrm{.}01,$ var$($Eb$)=\mathrm{.}4,$ var$($Ec$)=,02,$ what are the variances of the returns of the $3$ stocks, covariances and correlations between them? Show your work.

Also, write out factor betas, factor equations, and expected returns of the folowing portfolios:

I$).$ A portfolio of $3$ stocks above with $$20,000$ invested in stock A$,$ $$20,000$ invested in stock B$,$ and $$10,000$ invested in stock C$.$

II$.$ A portfolio consisting of the portfolio formed in part I and $$3,000$ short position in stock C$.$

How much should be invested in each of the $3$ stocks above to design $2$ portfolios?

The first portfolio has the following attiributes:

factor $1$ beta $=1$

factor $2$ beta $=0$

The second portfolio has the attributes:

factor $1$ beta $=0$

factor $2$ beta $=1$

Find the expected returns of the $2$ portfolios, and the risk premium assuming the risk$-$free rate is the "zero$-$beta rate" implied by the factor equations for the $3$ stocks.