# 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 $N-1$).

Note by Mountassir Farid
3 years, 8 months ago

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