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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.`