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Portfolio risk optimization by fuzzy approaches

posted on 28.02.2017, 01:00 authored by Nguyen, Thanh Thi
Due to the complexity and uncertainty in real world portfolio management, investors might be reluctant and sometimes unable to provide precise judgements regarding stock performance. In this context, analysts have long advocated use of fuzzy mathematics so that uncertainties and lack of precision can be acknowledged. This research therefore explores the applications of fuzzy sets in particular, or fuzzy logic in general for representing vague and imprecise financial data for portfolio risk optimization. Asset returns are uncertain and changeable over time so we model asset returns as fuzzy random variables and propose portfolio optimization models. Using fuzzy random variables, we introduce a new concept of financial risk, and the fuzzy Sharpe ratio contributing an important advancement in portfolio selection in the fuzzy environment. Two solution methods using a fuzzy approach and a genetic algorithm are applied to the proposed models. The proposed approach exhibits advantages over the so-called standard mean-variance optimization (MVO), throughout experimental results. The non-Gaussian distribution of asset returns has long been recognized, and the conventional MVO has been criticized as inadequate. Hence utilizing higher moments than variance, i.e. skewness, kurtosis soon emerged in portfolio selection. This research investigates the importance of higher moments in portfolio optimization through deploying fuzzy approaches. Marginal impacts of stocks on portfolio return and higher moment risks, are modelled by fuzzy numbers. The fuzzy models are constructed to optimize not only portfolio return and normal variance risk but also the portfolio higher moment risks. From the stock marginal impact modelling, two fuzzy approaches are used to derive optimal portfolio allocations. The first approach applies the constrained fuzzy analytic hierarchy process, whereas the second approach uses the fuzzy linear programming method. The efficiency of both approaches shows advantages of the proposed fuzzy models in portfolio selection. Going beyond the normal variance and higher moment risks, investors also should take into account downside risk measures. The downside risks are inspired by the principle of safety first in portfolio selection. The principle states that an investor would prefer the investment with the smallest probability of going below the target return. A fuzzy integrated framework is proposed accounting for portfolio return and six risk criteria including normal risk (volatility), asymmetric risk (skewness), "fat-tail" risk (kurtosis) and downside risks, i.e. semi-variance, modified Value-at-Risk, and modified Expected Shortfall. Fuzzy goals of portfolio's return and risks are constructed by bootstrapping, and kernel smoothing density estimate. A preselection process dealing with large datasets is also adopted to eliminate low diversification potential stocks before running the optimization model. Various investors’ risk preference schemes are implemented with both national and international experimental datasets. Results reported demonstrate the advantages of the proposed fuzzy framework compared to a conventional higher moment portfolio optimization model. The conclusion is that fuzzy modelling is efficient and competent in various portfolio selection formulations when uncertainty and vagueness are deemed present. When appropriately utilized, fuzzy approaches can bring superior investment outcomes compared to conventional non-fuzzy models prevalent in the literature.


Campus location


Principal supervisor

Lee Gordon-Brown

Year of Award


Department, School or Centre

Econometrics and Business Statistics


Doctor of Philosophy

Degree Type



Faculty of Business and Economics