An integrated fuzzy-grey relational analysis approach to portfolio optimization

This paper combines two approaches (Fuzzy set theory and Grey Relational Analysis) for modelling an investor's imprecise linguistic expectations and the uncertain returns of assets. We propose a novel maximization-type risk measure capable of incorporating the investor's individual preferences. The investor provides the expectations of what is considered the "ideal" return from the portfolio. We use Credibility theory to capture the investors' subjective and imprecise expectations in a precise mathematical form. We construct a portfolio return sequence using the assets' actual return data and an ideal sequence based on investors' preferences. Subsequently, we calculate the Grey similitude and the closeness incidence degree between the two sequences. The closer the portfolio return is to the ideal return, the better. In this manner, we develop a new risk measure that can quantify an investor's perception of risk. This measure is intuitive and easy to calculate. It does not involve estimating many parameters, something which would increase the estimation risk. We use a genetic algorithm to solve the resulting portfolio optimization model. We illustrate this method with two case studies: (i) a case study of 100 assets of the U.S. stock market's NASDAQ-100 index and (ii) a case study of 50 assets of the Indian stock market's NIFTY-50 index. We comprehensively analyze the model's out-of-sample performance and discuss its implications. The portfolios obtained using the proposed approach exhibit healthy growth outside the in-sample period. We also compare the out-of-sample performance of the proposed model with several approaches in the literature to establish its superiority.