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Assume there are N assets in the world, whose gross returns are arranged into a column vector R . The mean vector and covariance matrix

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=-1ec1(0)--1c2(0)
where:
c1(0)=T-1(-e0)(T-1)(eT-1e)-(eT-1)2
c2(0)=eT-1(-e0)(T-1)(eT-1e)-(eT-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?
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