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Fuzzy Optim Decis Making (2011) 10:167191 DOI 10.1007/s10700-011-9101-x
A portfolio selection model using fuzzy returns
Li Duan Peter Stahlecker
Published online: 17 March 2011 Springer Science+Business Media, LLC 2011
Abstract We study a static portfolio selection problem, in which future returns of securities are given as fuzzy sets. In contrast to traditional analysis, we assume that investment decisions are not based on statistical expectation values, but rather on maximal and minimal potential returns resulting from the so-called -cuts of these fuzzy sets. By aggregating over all -cuts and assigning weights for both best and worst possible cases we get a new objective function to derive an optimal portfolio. Allowing for short sales and modelling -cuts in ellipsoidal shape, we obtain the optimal portfolio as the unique solution of a simple optimization problem. Since our model does not include any stochastic assumptions, we present a procedure, which turns the data of observable returns as well as experts expectations into fuzzy sets in order to quantify the potential future returns and the investment risk.
Keywords Portfolio selection Fuzzy sets -cuts Minimum volume ellispoid
JEL Classication C61 G11
1 Introduction
In the majority of cases portfolio selection models are built on stochastic elements, i.e. probability theory is usually taken as the fundamental framework, whereby the rate of return of a security is considered as a random variable. Recently however the theory of fuzzy sets and possibility theory become another main approach for modeling portfolio selection problems. We just mention a few papers about this line of research. For
L. Duan P. Stahlecker (B)
Institute for Statistics and Econometrics, University of Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germanye-mail: [email protected]
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example Wang and Zhu (2002) provide an overview of different groups of models. A review of models using credibility measures as a closely related approach is given by Huang (2009). Considering the rates of return as fuzzy numbers Carlsson and Fullr (2001) introduce the possibilistic mean value and the variance of fuzzy numbers. A model which integrates both probability and possibility theory is analyzed by Tanaka et al. (2000). Another approach using interval numbers can be referred to Lai et al. (2002) and Li and Xu (2007). Finally Inuiguchi and Ramk (2000) compare the fuzzy...