Markovitz portfolio and change of cach system:

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

Let PminP_{\text{min}} be the portfolio in which we invest 1$1\$ at t=0t=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 PminP_{\text{min}}. For instance, if an asset costs 1$1\$ at t=0t=0 it costs 1010 in the new cach (we suppose that we have only two dates t=0t=0 and t=0t=0).

We note RminR_{\text{min}} the vector of assets's returns in the new cach PminP_{\text{min}}, Emin\mathcal{E}_{\text{min}} the vector of RminR_{\text{min}}'s means and Λmin\Lambda_{\text{min}} the matrix of RminR_{\text{min}}'s covariances.

Explain why the matrix Λmin\Lambda_{\text{min}} is not invertible (its rank is N1N-1).

Note by Mountassir Farid
4 years ago

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