Please show work for all math done:

Consider the following two$-$factor model for the

returns of three stocks. Assume that the factors and

epsilons have means of zero. Also, assume the

factors have variances of $\mathrm{.}01$ and are uncorrelated

with each other.

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$(tilde(lon{)}_A)=\mathrm{.}01$var$(tilde(lon{)}_B)=0\mathrm{.}4$var$(tilde(lon{)}_C)=\mathrm{.}02,$

a$.$Write out the factor betas, factor equations, and

expected returns of the following portfolios:

$(1)$ A portfolio of the three stocks in exercise $6\mathrm{.}3$

with $$20,000$ invested in stock $A,-$$$20,000$

invested in stock B$,$ and $$10,000$ invested in

stock C$.$

$(2)$A portfolio consisting of the portfolio formed

in part $(1)$ of this exercise and a $$3,000$ short

position in stock $C$ of exercise $6\mathrm{.}3.$

b$.$ How much should be invested in each of the stocks

in exercise $6\mathrm{.}3$ to design two portfolios? The first

portfolio has the following attributes:

factor $1$ beta $=1$

factor $2$ beta $=0$

The second portfolio has the attributes:

factor $1$ beta $=0$

factor $2$ beta $=1$

Compute the expected returns of these two

portfolios. Then compute the risk premiums of

these two portfolios assuming the risk$-$free rate is the "zero$-$beta rate" implied by the factor equations for the $3$ stocks above. This is the expected return of a portfolio with factor betas of $0.$