Kurtosis And Semi-Kurtosis For Portfolio Selection With Fuzzy Returns

The literature on portfolio analysis assumes that the securities returns are random variables with fixed expected returns and variances values (see Bachelier [1], Briec et al. [4] and Markowitz [10]). However, since investors receive efficient or inefficient information from the real world, ambiguous factors usually exist in it. Consequently, we need to consider not only random conditions but also ambiguous and subjective conditions for portfolio selection problems. A recent literature has recognized the fuzziness and the uncertainty of portfolios returns. As discussed in [6], investors can make use of fuzzy set to reflect the vagueness and ambiguity of securities (i.e. incompleteness of information due to the lack of data). Therefore, the probability theory becomes difficult to used. For example, some authors such as Tanaka and Guo [11] quantified mean and variance of a portfolio through fuzzy probability and possibility distributions, Carlsson et al.[2]-[3] used their own definitions of mean and variance of fuzzy numbers. In particular, Huang [7] quantified portfolio return and risk by the expected value and variance based on credibility measure. Recently, Huang [7] has proposed the mean-semivariance model for portfolio selection and, Li et al.[5], Kar et al.[8] introduced mean-variance-skewness for portfolio selection with fuzzy returns. Different from Huang [7] and Li et al.[5], after recalling the definition of mean, variance , semi-variance and skewness, this paper considers the Kurtosis and semi-Kurtosis for portfolio selection with fuzzy risk factors (i.e. returns). Several empirical studies show that portfolio returns have fat tails. Generally investors would prefer a portfolio return with smaller semi-kurtosis (or Kurtosis) which indicates the leptokurtosis (fat-tails or thin-tails) when the mean value, the variance and the asymmetry are the same. Our main objective is to contribute to a sound formal foundation of statistics and finance built upon the theory of fuzzy set.