Modelling and constructing membership function for uncertain portfolio parameters: A credibilistic framework

Abstract

In this paper we attempt to automate the process of fitting the uncertain parameters of a multi-objective portfolio selection problem by generating L-R fuzzy numbers that belong to power reference function family. Such an approach is advantageous when the fuzzy parameters of the portfolio are best represented as general functional forms. We also propose four new portfolio selection models in a multi-criteria credibilistic framework. The key financial criteria considered are return, illiquidity (antagonistic to liquidity) and risk. In the absence of joint possibility distribution of these parameters, the return and illiquidity of the entire portfolio are considered as historical data set instead of return and illiquidity of the individual assets. In the process of fitting the most appropriate L-R fuzzy number, the vagueness within the information and deviation of the L-R fuzzy number from historical data is measured using entropy and cross-entropy respectively. These principles are embedded into the modelling process of proposed portfolio selection problems. One of the key contribution of this study is that we propose and design a sub-algorithm namely “Entropy-Cross Entropy (ECE) Algorithm” that is appended within an “MIBEX-SM” genetic algorithm and is used to solve the proposed portfolio optimization problems. This proposed solution methodology results in an automated system that is intelligent enough to extract information required for fitting of L-R fuzzy number from the historical data and does not need any human intervention in terms of stating the parameters of the problem. We also conduct an empirical study to demonstrate the impact of the solution approach and applicability of the proposed models in practical applications of portfolio selection. For this purpose we collected historical data from National Stock Exchange (NSE), Mumbai, India. The data for a period of 2008–2013 is first used to train the models. Then the data sets of one-year period of 2013−2014 and two-year period of 2013−2015 are used to test the performance of the models. We introduce a decision support system strategy of comparing the performance of models by proposing a modified version of Sharpe ratio in fuzzy context, named as “Credibilistic Sharpe Ratio (CrSR)”.