Markovitz portfolio and change of cach system:

Consider a Financial market in which $N$ risky assets are listed. We note $R$ the vector of its return, $\mathcal{E}$ the vector of $R$ 's means and $\Lambda$ the matrix of $R$ 's covariances. We suppose that $\Lambda$ is not invertible and that the means of returns are not all equals.

Let $P_{\text{min}}$ be the portfolio in which we invest $1\$$ at $t=0$ and having a minimal variance. We do the following change of cach: instead of calculating the prices of an asset or a portfolio in dollars, we will evaluate the price in numbers of $P_{\text{min}}$ . For instance, if an asset costs $1\$$ at $t=0$ it costs $10$ in the new cach (we suppose that we have only two dates $t=0$ and $t=0$ ).

We note $R_{\text{min}}$ the vector of assets's returns in the new cach $P_{\text{min}}$ , $\mathcal{E}_{\text{min}}$ the vector of $R_{\text{min}}$ 's means and $\Lambda_{\text{min}}$ the matrix of $R_{\text{min}}$ 's covariances.

Explain why the matrix $\Lambda_{\text{min}}$ is not invertible (its rank is $N-1$ ).

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Markovitz portfolio and change of cach system:

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