A REINTERPRETATION OF PRINCIPAL COMPONENT
ANALYSIS CONNECTED WITH LINEAR MANIFOLDS
IDENTIFYING RISKY ASSETS OF A PORTFOLIO
Pierpaolo Angelini
Department of Statistical Sciences
Sapienza University of Rome
5 Aldo Moro Square, Rome - 00185, ITALY

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 $\alpha$-orthogonal, 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.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 4
Year: 2020

DOI: 10.12732/ijam.v33i4.14

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