$[$Question $2]$ Three securities are available for investment, and their expected rates of

return and covariance matrix are given as follows.

$=([{}_1],[{}_2],[{}_3])=([0\mathrm{.}12],[0\mathrm{.}02],[0\mathrm{.}02]),=\left[\begin{array}{ccc}{}_1^2& {}_{12}& {}_{13}\\ {}_{21}& {}_2^2& {}_{23}\\ {}_{31}& {}_{32}& {}_3^2\end{array}\right]=\left[\begin{array}{ccc}0\mathrm{.}04& 0\mathrm{.}00& -0\mathrm{.}01\\ 0\mathrm{.}00& 0\mathrm{.}00& 0\mathrm{.}00\\ -0\mathrm{.}01& 0\mathrm{.}00& 0\mathrm{.}09\end{array}\right]$

Recall the $i-$th element of $$ is the expected rate of return on secrutiry $i,$ whereas $(i,j)-$element

of matirix $$ represents covariance between security i and security j$\text{'}$s rates of return. All

numbers are expressed on the per annum simple interest basis. Answer the following

subquestions.

$(2$a$)$ Portfolio Q consists of the first and the second securities with the ratio of $3$:$1.$ Portfolio

$R$ consists of the first and the third securities with the ratio of $3$:$1.$ Calculate the variances of

rates of return on $Q$ and $R.[$Hint: use distributive property of covariance.$]$

$(2$b$)$ Calculate covariance and correlation coefficient between rates of return on Q and R$.$

$(2$c$)$ Portfilio $G$ is the portfolio of the first and the third securities with the smallest variance

possible. Assuming short$-$sales are not restricted, identify how to construct $G$ and calculate its

expected rate of return and variance.

$(2$d$)$ Calculate covariance between rates of return on $G$ and $R.$

$(2$e$)$ Depict investment opportunity set created by three securities in $($standard deviation,

expected rate of return$)-$space, assuming no short$-$sale restriction. Identify coordinates which

specify the opportunity set.

$(2$f$)$ Depict investment opportunity set created by three securities in $($standard deviation,

expected rate of return$)-$space, assuming short$-$sale is not allowed for any security. Identify

coordinates which specify the opportunity set.

$(2$g$)$ Your expected utility function is given by

where widetilde$(r_p)$ is the rate of return on the portfolio, and $a$ is the coefficient of risk aversion.

There is no restriction on shortselling securities in the market, and you have $100$

million today. If your risk aversion is $a=3,$ how much do you invest in each security

to maximize your expected utility?