Financial Data Analysis with Two Symmetric Distributions

Abstract

The normal inverted gamma mixture or generalized Student t and the symmetric double Weibull, as well as their logarithmic counterparts, are proposed for modeling some loss distributions in non-life insurance and daily index return distributions in financial markets. For three specific data sets, the overall goodness-offit from these models, as measured simultaneously by the negative log-likelihood, chi-square and minimum distance statistics, is found to be superior to that of various "good" competitive models including the log-normal, the Burr, and the symmetric a-stable distribution. Furthermore, the study justifies on a statistical basis different important models of financial returns like the model of Black-Scholes (1973), the log-Laplace model of Hurlimann (1995), the normal mixture by Praetz (1972), the symmetric a-stable model by Mandelbrot (1963) and Fama (1965), and the recent double Weibull as limiting geometric-multiplication stable scheme in Mittnik and Rachev (1993). As an application, the prediction of one-year index returns from daily index returns is discussed. KEYWORDSClaim size data, financial market data, index return, normal inverted gamma mixture, generalized Student t, symmetric double Weibull, goodness-offit.

The General Segmented Distribution

“…We first consider the industrial fire insurance claim data used by Hürlimann (2001). The GSD is able to approximate these (grouped) data with arbitrary precision as we increase the number of segments used, and thus we show that the GSD has the capacity to outperform other recommended distributions.…”

“…As a first example, we examine the industrial fire insurance claim data originally appearing in Beard et al (1984) and later used by Hürlimann (2001) to assess the goodness-of-fit of competing distributions. The data consist of N = 8324 observations that are grouped into M = 29 classes.…”

A Mean-Variance Portfolio Optimal Under Utility Pricing

“…The mixture of a normal with inverted gamma variance yields the Pearson type VII distribution or generalised Student t [4][5][6] . It has been proposed to model financial returns by Praetz [7] , Blattberg and Gonedes [8] , Kon [9] , Taylor [10] , Hürlimann [11,12] . An actuarial application is found in Hürlimann [13] .…”

“…Since a lot of bivariate random models satisfy the required linear regression property, the displayed covariance identity has a wide application. Among the many multivariate models satisfying linear regression properties, let us mention the following few but important classes and families of multivariate distributions: * The class of symmetric elliptical distributions [3] * Bivariate and multivariate distributions of Pearson type [36,37] * Bivariate and multivariate Pareto distributions of the first kind [38] * Bivariate and multivariate distributions constructed from linear Spearman or Fréchet copulas with margins from location-scale families [11,12,37,39] Appendix B: Numerical evaluation of two special integral functions First, we show how to compute the integral (2.30), that is where the integrals can be calculated recursively as follows (use partial integration) :…”

Tabulations for value at risk and expected shortfall

“…But the parametric distributions used have been very limited. For modeling of VaR, only the normal distribution (de Moivre, 1738; Gauss, 1809), the Student's t distribution (Gosset, 1908), the Pareto positive stable distribution (Sarabia and Prieto, 2009;Guillen et al, 2011), the asymmetric Laplace distribution (Kotz et al, 2001;Trindade and Zhu, 2007), the ACCEPTED MANUSCRIPT alized logistic distribution for analysis of management fraud (Hansen et al, 1996); the Gumbel distribution for extreme stock market returns (Longin, 1996); the inverse gamma distribution for the payoff of Asian options (Milevskya and Posnera, 1998); the generalized Pareto distribution for estimation of tail-related risk measures for heteroscedastic financial time series (McNeil and Frey, 2000); the log Laplace and double Weibull distributions as models for industrial fire loss data (Hurlimann, 2001); the generalized gamma distribution for financial transaction data (Zhang et al, 2001); the Fréchet distribution for estimating financial risk under time-varying extremal return behavior (Wagner, 2003); the Burr distribution for financial duration models (Bauwens et al, 2004); the F distribution for traded volume in financial markets (Duarte Queiro, 2005); the generalized extreme value distribution for measuring financial risk (Gilli and Kellezi, 2006); the log-logistic distribution for stock prices of European call option (Al-Najjab and Thiele, 2007); the Dagum distribution to model daily returns of four Italian stocks (Domma and Perri, 2008); the logistic distribution for modeling the extreme share returns in Singapore (Tolikas and Gettinby, 2009); the Birnbaum-Saunders distribution to model unpaid credits in a commercial bank in the US (Ahmed et al, 2010); the Laplace distribution for evaluation of mutual funds (Zhao and Shi, 2010); the exponential and Gompertz distributions for estimating the probability of participating in a process of merging or acquisition for financial institutions in Colombia (Garca-Suaza and Gomez-Gonzalez, 2010); the lognormal distribution for the computation of operational risk capital (Guegan et al, 2011); the Pareto positive stable distribution for the analysis of motor insurance claims of a major ...…”