The assumption of the Arbitrage Pricing Theory can be formulated in terms of the variance matrix V of the returns on a finite or infinite set of asset. This paper shows that the properties of this variance matrix are not robust properties of the space of portfolios spanned by the assets. Suppose V and V* are positive semi-definite matrices with the same rank. If a space of portfolios spanned by a set of assets with variance V or V* includes a safe portfolio, any space spanned by set of assets who returns have variance V, is also spanned by a set of portfolios whose returns have variance V*. If a space of portfolios spanned by a set of assets with variance V or V* does not include a safe portfolio, and space spanned by a set of assets whose returns have variance matrix which is scalar multiple of V*. The matrices V and V* may or may not have factor structures and they have factor structures with different numbers of factors.
The paper uses diagrams in which random variables are represented by vectors, whereby the length of a vector is the standard deviation, and the cosine of the angle between two vectors is the correlation, between the corresponding random variables. These diagrams are equivalent to a Hilbert Space approach to the mathematics.
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