Assume there are $N$ assets in the world, whose gross returns are arranged into a

column vector $R.$ The mean vector and covariance matrix are $$ and $.$ Further

assume that $$ is not proportional to the unit vector $e,$ and $$ positive definite.

The optimal portfolio $($in a mean$-$variance efficient sense$)$ conditional on a

pre$-$specified expected return ${}_0$ is given by:

$w={}^{-1}e{c}_1({}_0)-{}^{-1}{c}_2({}_0)$

where:

$c_1({}_0)=\frac{{}^T{}^{-1}(-e{}_0)}{({}^T{}^{-1})(e^T{}^{-1}e)-(e^T{}^{-1}{)}^2}$

$c_2({}_0)=\frac{e^T{}^{-1}(-e{}_0)}{({}^T{}^{-1})(e^T{}^{-1}e)-(e^T{}^{-1}{)}^2}$

$(1)$ Please demonstrate how you can identify the entire efficient frontier.

$(2)$ Please explain the two$-$fund separation theorem by identifying the "two

funds".

$(3)$ Please write down the unique global minimum variance $($GMV$)$ portfolio

$($with the weight adding up to one$)$ and show why it has the minimum variance

among all the portfolios whose weights add up to one.

$(4)$ Please write down the gross return, expected gross return, and covariance

matrix of the GMV portfolio.

$(5)$ Describe how introducing a new asset would move the efficient frontier?

$(6)$ Now, we assume that $$ is not positive definite. Please identify the GMV

portfolio?

$(7)$ Answer the following questions in words: what is the tangency portfolio?

Does a tangency portfolio always exist? Why?