Optimization of mean-CVaR model with non-convex transaction costs : Cutting Plane and Scenario-Reduction methods

The modern portfolio theory is spreading in the financial risk management and stochastic programming (Birge and Louveaux 1997) is being used for uncertainty in this criteria . The modern portfolio theory makes a decision by considering low-risk and high-return investments . In this paper we used mean-risk model and replacing the risk part by Conditional Value-at-Risk (CVaR; Rockafellar and Uryasev 2000 , 2002) .The CVaR as a risk measure has some properties which is important in optimization , such as convexity , stability , (see Rockafellar and Uryasev 2002) and coherence which is some desirable features , i.e. , positive homogeneity , translation invariance , monotonicity , subadditivity (see Artzner et al . 1999) .The CVaR is usually with discrete loss distribution for using scenario-based approximation which has been proposed by Uryasev and Rockafellar (2002) . Optimization of scenario-based CVaR can be formulated as a linear programming (LP) problem (Rockafellar and Uryasev 2000 , 2002) .Studying CVaR with nonconvex transaction costs has been proposed in a number of articles (see , e.g. , Kellerer et al . 2000; Yamamoto and Konno 2005 , 2006; Takano and Nanjo 2014) . Among them , Konno and Yamamoto (2005) represent piecewise linear transaction cost functions and by using this cost functions Takano and Nanjo (2014) proposed a Cutting-Plane method for solving mean-CVaR portfolio optimization with non-convex transaction costs.Scenario reduction techniques focus on decreasing the number of scenarios to avoid the complexity of the model without losing the stochastic information contained in the tail of the loss distribution (see , e.g. , Dupacova et al . 2003; Heitsch and Romisch 2007; Garcia-Bertland and Minguez 2012) . Garcia-Bertland and Minguez (2012) present a Scenario-Reduction method for optimizing CVaR as an objective function .We implemented Scenario Reduction method (Garcia-Bertland and Minguez 2012) on mean-CVaR model with non-convex transaction costs and compared the result with Cutting-Plane method (Takano and Nanjo 2014) . It is good to mention that the foundation of two methods which is mentioned above is Proposition 8 in Rockafellar and Uryasev (2002) .The rest of the paper is organized as follows : in section 2 , we formulate the mean-CVaR portfolio problem with non-convex transaction costs for Scenario-Reduction method . Section 3 provides some details about the Cutting-Plane and Scenario-Reduction methods . Section 4 will show the computational results of those two methods on mean-CVaR portfolio with non-convex transaction costs for a simple and clarifying example . Finally , conclusions are given in section 5 .