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(Solved): Problem 2. (25p) (Revisiting Markowitz) Consider linear portfolios in the set M={L:L=\lambda ^(')x,\ ...



Problem 2. (25p) (Revisiting Markowitz) Consider linear portfolios in the set

M={L:L=\lambda ^(')x,\lambda inR^(d)}

where

x

is a fixed

d

-dimensional random vector of risk factors defined on some probability space

(\Omega ,F,P)

, and where

\lambda

represents portfolio weights. Denote by

L(\lambda )=\sum_(j=1)^d \lambda _(j)x_(j)

the loss generated by the portfolio weights

\lambda

and assume that

x

has an elliptical distribution with finite variance. Fix some

\alpha >0.5

and suppose that

VaR_(\alpha )

is used as a risk measure. Consider the problem of finding the

VaR_(\alpha )

-minimizing linear portfolio in the set of all portfolios whose expected loss is equal to a given constant

c

, that is consider the problem:

min_(\lambda inR^(d))VaR_(\alpha )(L(\lambda )), such that E[L(\lambda )]=c.

Explain why the solution

\lambda ^(**)

of this problem is identical to the classical Markowitz portfolio, that is to the solution of the problem:

min_(\lambda inR^(d))var(L(\lambda )) such that E[L(\lambda )]=c.


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