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Markowitz Problem for a Case of Random Environment Existence

  • Alexander Andronov
  • Tatjana Jurkina
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 231)

Abstract

Classical Markowitz model considers n assets with \(R_1, R_2,\ldots ,R_n\) random profitability and \(r_1, r_2,\ldots ,r_n\) relevant average, \(\sigma _1^2, \sigma _2^2,\ldots , \sigma _n^2\) variances, and \(\sigma _{\mu , v}\), \(\mu , v=1,\ldots ,n\) covariance. The portfolio is built of these assets, by using weighting coefficients \(\omega _1, \omega _2,\ldots , \omega _n\), where \(\omega _\mu \) is the share of asset cost \(\mu \) in the whole portfolio value. The profitability of such portfolio is a random value \(F(\omega )=\omega _1R_1+ \omega _2R_2+\ldots + \omega _nR_n\). The cumulative hazard of the portfolio at pre-assigned value of average profitability \(r*\) can be measured by dispersion \(DF(\omega )\). It is necessary to determine weighting coefficients by such a way, that minimizes dispersion \(DF(\omega )\) given assigned value of \(r*\). A more general supposition considered in this chapter: It is supposed that a random environment exists. The last is described by a continuous-time irreducible Markov chain with k states and known matrix of transition intensities \(\lambda = (\lambda _{i,j})_{k \times k}\). The reward rate depends on a state of the random environment. For this case, the parameters of Markowitz model are derived.

Keywords

Markov chain Continuous time Markovic problem 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Transport and Telecommunication InstituteRigaLatvia

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