$$l_1$$ l 1 -Regularization for multi-period portfolio selection
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l1 -Regularization for multi-period portfolio selection Stefania Corsaro1 Francesca Perla1
· Valentina De Simone2
· Zelda Marino1
·
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this work we present a model for the solution of the multi-period portfolio selection problem. The model is based on a time consistent dynamic risk measure. We apply l1 regularization to stabilize the solution process and to obtain sparse solutions, which allow one to reduce holding costs. The core problem is a nonsmooth optimization one, with equality constraints. We present an iterative procedure based on a modified Bregman iteration, that adaptively sets the value of the regularization parameter in order to produce solutions with desired financial properties. We validate the approach showing results of tests performed on real data. Keywords Portfolio optimization · Time consistency · l1 norm · Constrained optimization Mathematics Subject Classification 91G10 · 90C30 · 65K05
1 Introduction In this work we focus on dynamic portfolio selection problem in a Markowitz framework. Dynamic portfolio selection arises in medium and long-term investments, in which one allows decisions to change over time by the end of the investment, taking into account the time evolution of available information. We consider dynamic decision problems formulated in a discrete multi-stage setting, with underlying time evolving continuously. This formulation is usually referred to as multi-period portfolio selection. A set of rebalancing dates is introduced,
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Francesca Perla [email protected] Stefania Corsaro [email protected] Valentina De Simone [email protected] Zelda Marino [email protected]
1
Department of Management and Quantitative Studies, University of Naples “Parthenope”, Naples, Italy
2
Department of Mathematics and Physics, University of Campania “Luigi Vanvitelli”, Caserta, Italy
123
Annals of Operations Research
which split the investment period into sub-periods; decisions are assumed at the rebalancing dates, and are kept within sub-periods. In order to ensure that investors preferences remain consistent over time, dynamic time consistent models should be defined. Different definitions of time consistency can be found in literature, either related to risk measures or investment policies (Chen et al. 2017). We consider the first case, in which one focuses on the properties of the multi-period risk measure employed for modelling the investment problem; this is time consistent if, according to it, the time evolving filtration related to the evaluation of a stochastic process does not modify decisions taken using values computed previously. Roughly speaking, if we today establish that two investments have the same level of risk, then the same level of riskiness should have been estimated for them yesterday. In Chen et al. (2013) authors show that a separable expected conditional mapping, obtained by summing single-period terms, is a time consistent
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