Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market

Abstract

The purpose of this paper is to demonstrate that a portfolio optimization model using the L 1 risk (mean absolute deviation risk) function can remove most of the difficulties associated with the classical Markowitz's model while maintaining its advantages over equilibrium models. In particular, the L 1 risk model leads to a linear program instead of a quadratic program, so that a large-scale optimization problem consisting of more than 1,000 stocks may be solved on a real time basis. Numerical experiments using the historical data of NIKKEI 225 stocks show that the L 1 risk model generates a portfolio quite similar to that of the Markowitz's model within a fraction of time required to solve the latter.portfolio optimization, L1 risk function, linear programming, Markowitz's model, single-factor model

Robust portfolios: contributions from operations research and finance

“…Portfolio optimization models based on alternative risk measures have been introduced in the literature such as semivariance (Markowitz 1959), lower partial moment (Bawa 1975), mean absolute deviation (Konno and Yamazaki 1991), minimax (Young 1998), VaR (Ahn et al 1999;Basak and Shapiro 2001), CVaR (Rockafellar and Uryasev 2000), coherent risk measures (Artzner et al 1999), convex risk measures Frittelli and Rosazza Gianin 2002), generalized deviation measures (Rockafellar et al 2006), proper and ideal risk measures (Stoyanov et al 2007;Rachev et al 2008a), and have been widely applied in practice (Dembo and Rosen 2000;Ortobelli et al 2005). Among these, we address in this section the minimization of VaR and CVaR in portfolio selection.…”

Zastosowanie modelu MAD z dodatkowymi warunkami ograniczającymi / Application of the MAD model with additional constraints

“…Oprócz opracowania liniowego modelu wyboru optymalnego portfela inwestycyjnego autorzy wykazali również, że zaproponowany przez nich model MAD przy założeniu o normalnym rozkładzie stóp zwrotu jest ekwiwalentem kwadratowego modelu średnia-ryzyko. Średnie odchylenie bezwzględne dla danej spółki j definiowane jest jako suma iloczynów prawdopodobieństwa wystąpienia t-tej możliwej wartości stopy zwrotu (p t ) i wartości bezwzględnych różnic stopy zwrotu spółki j-tej w okresie t (r j,t ) oraz średniej stopy zwrotu spółki j-tej (r j ) [Konno, Yamazaki 1991]: …”

Simulated annealing for complex portfolio selection problems

“…This is the case for instance of Dembo, Mulvey and Zenios [10] (with network°ow models), Konno and Yamazaki [16] (with an absolute deviation approach to the measure of risk, embedded in linear programming models), Takehara [26] (with an interior point algorithm), and Bienstock [2] (with a 'branch and cut' approach). Dahl, Meeraus and Zenios [8], Takehara [26] and Hamza and Janssen [12] discuss some of this work.…”