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• W256824573 abstract "I. INTRODUCTION The empirical analysis of hedge fund returns has shown that the mean-variance approach is not adequate to study the risk and performance of these funds. Using monthly data, Fung and Hsieh (1997), Brooks and Kat (2001), Amenc et al. (2002, 2003) demonstrate that distributions of hedge funds returns have a negative skewness and an excess kurtosis. This leads to conclude that, for this type of fund, the challenge lies not only in the first two moments but also in higher moments. In this context, the selection of a family of distributions with sufficient flexibility to take account of the asymmetry and the kurtosis is necessary to study the behaviour of hedge funds returns. The Johnson distribution system (1949) satisfies this requirement. As mentioned in Passow (2004), this family of probability distributions can calibrate independently the four first moments, while disregarding potentially insignificant moments of order higher than 5. The introduction of Johnson distributions allows covering up to fourth-order moment patterns of (Log-) Normal, (2-sided) Student t and Weibull distributions. The family of Edgeworth expansions and twelve different types of Pearson distributions are also included. Note that these latter distributions include 1st and 2nd kind Beta and Gamma distributions. This is the reason why Johnson distributions can actually model the specific moment patterns. Since 2000, portfolio management theory has been based in particular on downside risk measures. In line with Basel II accords for banking regulations, these latter ones are related to determination of economical capital allocation (see Goovaerts et al., 2002). As illustrated also by Jarrow and Zhao (2006), there exist significant differences between mean-variance optimal portfolios and those based on lower partial moments, when asset returns are far from being Gaussian. The literature about financial risk and performance measurement has increased continuously. The first and basic performance measures for asset management are the Sharpe's ratio, the Treynor's ratio and the Jensen's Alpha. However, we must often deal with asymmetric return distributions having also fat tails. Consequently, it is necessary to introduce performance measures that are usually based on reward/risk ratios. Additionally, these risk measures must involve the whole return distribution, in particular the probabilities to get significant negative returns. For this purpose, downside risk measures have been introduced (see e.g. Pedersen and Satchell, 1998; Artzner et al., 1999; Szego, 2002). (1) Then, using the downside lower partial moment, Keating and Shadwick (2002) define a new performance measure, called the Omega measure. This performance measure is based on a gain-loss approach. It takes account of investor loss aversion, as in Tversky and Kahneman (1992). This performance measure is defined as the ratio of the expectation of gains above the threshold and the expectation of losses below the threshold. Farinelli and Tibiletti (2008) and Zakamouline and Koekebakker (2009) introduce generalized Sharpe ratios to evaluate portfolio performance. As proved by Pedersen and Satchell (1998, 2002), the Sortino ratio (which corresponds to Kappa (2)) is linked to utility function with lower risk aversion. Zakamouline (2010) generalizes this result by showing that Kappa measures correspond to performance measures associated to piecewise linear plus power utility functions. Kappa (n) measures have been used to examine performance of a large class of financial models, in particular to deal with hedge fund style or structured equity funds (see Bertand and Prigent, 2011). We examine how these measures behave according to the parameters of Johnson distributions. This paper is organized as follows. Section II provides general results about Kappa performance measures. Section III is a survey about Johnson distributions. We also develop an empirical illustration of such probability distributions and illustrate in particular the relationship between their four parameters and their four moments. …" @default.
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• W256824573 title "Kappa Performance Measures with Johnson Distributions" @default.
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