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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