Abstract
The typical behavior of optimal solutions to portfolio optimization problems with absolute deviation and expected shortfall models using replica analysis was pioneeringly estimated by S. Ciliberti et al. [Eur. Phys. B. 57, 175 (2007)]; however, they have not yet developed an approximate derivation method for finding the optimal portfolio with respect to a given return set. In this study, an approximation algorithm based on belief propagation for the portfolio optimization problem is presented using the Bethe free energy formalism, and the consistency of the numerical experimental results of the proposed algorithm with those of replica analysis is confirmed. Furthermore, the conjecture of H. Konno and H. Yamazaki, that the optimal solutions with the absolute deviation model and with the mean-variance model have the same typical behavior, is verified using replica analysis and the belief propagation algorithm.
Highlights
Portfolio optimization is one of the most fundamental frameworks of risk diversification management
Ciliberti and Mézard assessed the typical behavior of optimal solutions to portfolio optimization problems, in particular those described by the absolute deviation and expected shortfall models, using replica analysis, one of the most powerful approaches in disordered systems
In order to confirm the effectiveness of our method, the numerical experimental results of the proposed algorithm and those of the replica analysis for the case of the Markowitz model are shown in Figs 1 and 2, where xkμ are independently and identically drawn from the normal distribution with mean and variance 0 and 1, respectively
Summary
Portfolio optimization is one of the most fundamental frameworks of risk diversification management. Its theory was introduced by Markowitz in 1959 and is one of the most important areas being actively investigated in financial engineering [1,2,3] In their theoretical research, Ciliberti and Mézard assessed the typical behavior of optimal solutions to portfolio optimization problems, in particular those described by the absolute deviation and expected shortfall models, using replica analysis, one of the most powerful approaches in disordered systems. Ciliberti and Mézard assessed the typical behavior of optimal solutions to portfolio optimization problems, in particular those described by the absolute deviation and expected shortfall models, using replica analysis, one of the most powerful approaches in disordered systems With this approach, they showed that the phase transitions of these optimal solutions were nontrivial [1]. This requires a rapid algorithm for resolving the optimal portfolio problem with respect to a large enough in-sample set
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