Consider a market where two factors are sufficient to describe the returns on common stocks. For an asset i$,$ the asset$$s expected return is given by E$($ri$)=$ rf $+\backslash$beta i$1\backslash$lambda $1+\backslash$beta i$2\backslash$lambda $2,$ where $\backslash$lambda $1$ and $\backslash$lambda $2$ are the factor premiums $($expected return of the factors in excess of the risk$-$free rate$).$ Both factors are independent. The following table gives the sensitivities of the stocks ABC and PQR to the two factors, as well as the expected returns of each stock $($Hint: Portfolio betas are weighted averages of individual stock betas. This is also true for betas for each factor$)$:

Security $\backslash$beta i$1\backslash$beta i$2$ E$($Ri$)$

ABC $0\mathrm{.}50\mathrm{.}816\mathrm{.}2\%$

PQR $1\mathrm{.}51\mathrm{.}421\mathrm{.}6\%$

Riskfree $005\mathrm{.}0\%$

a$)$ Consider a portfolio, C$,$ made up by selling short $$0\mathrm{.}50$ of security PQR and purchasing $$1\mathrm{.}50$ of ABC. How sensitive will this portfolio be to each of the two factors?

b$)$ Consider a portfolio, D$,$ made up by borrowing $$1\mathrm{.}00$ at the risk$-$free rate and investing $$1\mathrm{.}00$ in portfolio C$.$ How sensitive will this portfolio be to each of the factors?

c$)$ What combination of securities ABC, PQR$,$ and the riskless security will move on a one$-$to$-$one basis with factor $1$ and be insensitive to factor $2?$