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Modern portfolio theory
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==== Non-invertible covariance matrix ==== If the covariance matrix is not invertible, then there exists some nonzero vector <math>v</math>, such that <math>v^T R</math> is a random variable with zero variance—that is, it is not random at all. Suppose <math>\sum_i v_i = 0</math> and <math>v^T R = 0</math>, then that means one of the assets can be exactly replicated using the other assets at the same price and the same return. Therefore, there is never a reason to buy that asset, and we can remove it from the market. Suppose <math>\sum_i v_i = 0</math> and <math>v^T R \neq 0 </math>, then that means there is free money, breaking the ''no arbitrage'' assumption. Suppose <math>\sum_i v_i \neq 0 </math>, then we can scale the vector to <math>\sum_i v_i = 1</math>. This means that we have constructed a risk-free asset with return <math>v^T R </math>. We can remove each such asset from the market, constructing one risk-free asset for each such asset removed. By the no arbitrage assumption, all their return rates are equal. For the assets that still remain in the market, their covariance matrix is invertible.
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