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# 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
1 year, 1 month ago