Problem $12-10$ APT

Assume that the returns on individual securities are generated by the following two$-$factor model:

Rit$=$E$($Rit$)+\backslash$beta ijF$1$t$+\backslash$beta i$2$F$2$t

Here:

Rit is the return on Security i at Time t$.$

F$1$t and F$2$t are market factors with zero expectation and zero covariance.

In addition, assume that there is a capital market for four securities$,$ and the capital market for these four assets is perfect in the sense that there are no transaction costs and short sales $($i$.$e$.,$ negative positions$)$ are permitted. The characteristics of the four securities follow:

Security $\backslash$beta $1\backslash$beta $2$ E$($R$)$

$11\mathrm{.}2001\mathrm{.}70025\%$

$2\mathrm{.}6002\mathrm{.}30025$

$31\mathrm{.}200\mathrm{.}60015$

$41\mathrm{.}800\mathrm{.}90010$

a$.$

Construct a portfolio containing $($long or short$)$ Securities $1$ and $2,$ with a return that does not depend on the market factor, F$1$t$,$ in any way $($Hint: Such a portfolio will have $\backslash$beta $1=0.)($A negative answer should be indicated by a minus sign. Do not round intermediate calculations.$)$

a$-1.$ Compute the expected return and $\backslash$beta coefficient for this portfolio. $($Do not round intermediate calculations and enter your expected return answer as a percent. Leave no cells blank $-$ be certain to enter $"0"$ wherever required.$)$

b$.$ Construct a portfolio containing Securities $3$ and $4,$ with a return that does not depend on the market factor, F$1$t$,$ in any way. $($A negative answer should be indicated by a minus sign. Do not round intermediate calculations and round your answer to $1$ decimal, e$.$g$.,32\mathrm{.}1.)$

b$-1.$ Compute the expected return and $\backslash$beta $2$ coefficient for this portfolio. $($Do not round intermediate calculations and enter your expected return answer as a percent. Leave no cells blank $-$ be certain to enter $"0"$ wherever required.$)$a$.$ Construct a portfolio containing $($long or short$)$ Securities $1$ and $2,$ with a return that

does not depend on the market factor, $F_1,$ in any way $($Hint$.$ Such a portfolio will have

${}_1=0.)($A negative answer should be indicated by a minus sign. Do not round

intermediate calculations.$)$

a$-1.$ Compute the expected return and $$ coefficient for this portfolio. $($Do not round

intermediate calculations and enter your expected return answer as a percent.

Leave no cells blank $-$ be certain to enter $"0"$ wherever required.$)$

b$.$ Construct a portfolio containing Securities $3$ and $4,$ with a return that does not

depend on the market factor, $F_1,$ in any way. $($A negative answer should be

indicated by a minus sign. Do not round intermediate calculations and round your

answer to $1$ decimal, e$.$g$.,32\mathrm{.}1.)$

b$-1.$Compute the expected return and ${}_2$ coefficient for this portfolio. $($Do not round

intermediate calculations and enter your expected return answer as a percent.

Leave no cells blank $-$ be certain to enter $"0"$ wherever required.$)$