We use the mean-variance model to study a portfolio problem characterized by an investment in two different types of asset. We consider m logically independent risky assets and a risk-free asset. We analyze m risky assets coinciding with m distributions of probability inside of a linear space. They generate a distribution of probability of a multivariate risky asset of order m. We show that an m-dimensional linear manifold is generated by m basic risky assets. They identify m finite partitions, where each of them is characterized by n incompatible and exhaustive elementary events. We suppose that it turns out to be n > m without loss of generality. Given m risky assets, we prove that all risky assets contained in an m-dimensional linear manifold are related. We prove that two any risky assets of them are conversely αorthogonal, so their covariance is equal to 0. We reinterpret principal component analysis by showing that the principal components are basic risky assets of an m-dimensional linear manifold. We consider a Bayesian adjustment of differences between prior distributions to posterior distributions existing with respect to a probabilistic and economic hypothesis
A reinterpretation of principal component analysis connected with linear manifolds identifying risky assets of a portfolio / Angelini, Pierpaolo. - In: INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS. - ISSN 1311-1728. - 33:4(2020), pp. 709-731. [10.12732/ijam.v33i4.14]
A reinterpretation of principal component analysis connected with linear manifolds identifying risky assets of a portfolio
angelini pierpaoloPrimo
2020
Abstract
We use the mean-variance model to study a portfolio problem characterized by an investment in two different types of asset. We consider m logically independent risky assets and a risk-free asset. We analyze m risky assets coinciding with m distributions of probability inside of a linear space. They generate a distribution of probability of a multivariate risky asset of order m. We show that an m-dimensional linear manifold is generated by m basic risky assets. They identify m finite partitions, where each of them is characterized by n incompatible and exhaustive elementary events. We suppose that it turns out to be n > m without loss of generality. Given m risky assets, we prove that all risky assets contained in an m-dimensional linear manifold are related. We prove that two any risky assets of them are conversely αorthogonal, so their covariance is equal to 0. We reinterpret principal component analysis by showing that the principal components are basic risky assets of an m-dimensional linear manifold. We consider a Bayesian adjustment of differences between prior distributions to posterior distributions existing with respect to a probabilistic and economic hypothesisFile | Dimensione | Formato | |
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