Heavy tails and mixtures of normal random variables

• Autor(a) principal: Rocha, Maria Luísa
• Data de Publicação: 2012
• Outros Autores: Pestana, Dinis, Menezes, António Gomes de
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.3/5046
• topic: Financial Series; Value-at-Risk

In recent research we can observe that statistical extreme value theory has been successfully used for modeling stock index prices log returns, since there is empirical evidence that all important samples exhibit heavy tail behaviour. However, the evidence for goodness-of-fit of an extreme value model is thin, and important empirical characteristics such as the V aR or the expected shortfall show that there may exist a aw in the reasoning leading to the preference for the classical long-tailed Gumbel or Fréchet extreme value distributions; this is not a big surprise since the iid hypothesis leading to those models doesn't apply. On the other hand, the classical normal model has very light tails, which clearly do not provide a good fit to the data. Therefore, the BASEL II recommendations show in general a shift from the normal towards more realistic models, keeping however an inverse square root scale when dealing with the value at risk at horizon h which is a remant of the normal modeling framework. We prove that scale mixtures of normal distributions, that can arise when dealing with maxima of non identical normal random variables, can indeed have a very heavy tail, and therefore that they may provide much better patterns to model log returns of stock index prices. We present empirical evidence, analyzing the PSI, which are the main basis for financial decisions in the Portuguese market.