Quantifying Risk using Copulae and Kernel Estimators

Abstract

In this Thesis we addressed two very important themes related to quantitative risk management. On one hand, we provided relevant results about the analysis of extreme value distributions; on the other hand, we also presented different results concerning the dependence modelling between extreme value distributions. These results will be useful in the calculation of the capital requirement in the context of Solvency II in terms of quantifying the risk when the data contain extreme values. This can occur when we analyse operational risk and subscription risk, where the company could have losses that have a very low probability but can reach high values, i.e. "rare cases". Specifically, two lines of research were examined in this Thesis: the dependence between two random variables from the viewpoint of the copulae and the nonparametric methods to estimate the cumulative distribution function and quantile. In addition some questions related to the theory of extreme values were considered: extreme value copulae and maximum domain of attraction of extreme value mixture distributions. Inference on copulae was necessary for analysing the structure of dependence between variables. For this, using the definition of max-stable, we generalised the test of extreme value copula to cover a more extensive alternative hypothesis.

In the context of copulae, nonparametric estimation of the cdf was useful for obtaining the pseudo-observations and for estimating the marginals. We proposed the use of new nonparametric methods that improve the accuracy in the risk estimations.

To illustrate the usefulness of the methods analysed in this Thesis, we used data on the costs of accidents in auto insurance. Specifically, we used two databases, the first contains information from a sample of bivariate costs and the second contains information related to a sample of univariate cost for different types of policyholders.

From our data, we found that the Gumbel copula with DTKE (double transformed kernel estimation) marginals provides a good fit. With this copula we obtained a balanced risk estimation that guarantees that the risk is not underestimated and, where it is relevant, not overestimated in excess.

In a lot of analyses -in economics, finance, insurance, demography,...- the fit of cdf is very important for evaluating the probability of extreme situations. In these cases, the data are usually generated by a continuous random variable whose distribution may be the result of the mixture of different EVDs; then both the classical parametric models and the classical nonparametric estimates do not work for the estimation of the cdf. All those problems were addressed in last chapter. There we presented a method to estimate cdf that is suitable when the loss is a heavy tailed random variable. The proposed double transformation kernel using the bias-corrected technique, in general, provides good fit results for the Gumbel and Fréchet types of extreme value distributions, especially when the sample size is small.

We show, when the sample size is small, that our proposed BCDTKE (bias-corrected double transformed kernel estimator) improves the classical kernel estimator and bias-corrected classical kernel estimator of the cumulative distribution function when the distribution is a right extreme value distribution and the maximum domain of attraction is the one associated with a Fréchet type distribution. Finally we provided some theoretical results about the maximum domain of attraction of extreme value mixture distributions.We concluded that the heavier tail (Fréchet type) prevails over the lighter tails (Gumbel type).

El objetivo esencial en el campo actuarial y las finanzas es analizar la distribución asociada a la pérdida total generada por un vector aleatorio multivariante. Los componentes de este vector son en general pérdidas dependientes entre ellos o se llaman factores de riesgo. El objetivo se reduce a estimar el riesgo de pérdida total teniendo en cuenta la relación entre estos factores de riesgo. El análisis de riesgos puede enfrentarse a dos problemas: 1. ¿Cuáles son las cópulas que mejor reflejan la estructura de dependencia entre estos factores? 2. ¿Cómo se estima la función de distribución de los marginales y se inserta en la cópula? En esta Tesis se investiga cómo responder a las dos preguntas anteriores, es decir, cómo seleccionar la cópula y la forma de estimar distribuciones marginales cuando tenemos valores extremos o eventos raros.

Podemos decir que esta tesis se centra en dos aspectos fundamentales para la cuantificación del riesgo. El primero está relacionado con la teoría de las cópulas y estimación del riesgo de pérdida. En el segundo se analizan los métodos no paramétricos y semiparamétricos para estimar la función de distribución y los cuantiles.

Para ilustrar la aplicabilidad de los métodos propuestos descritos en nuestro trabajo de investigación utilizamos una muestra bivariante de las pérdidas de una base de datos real extraída de las reclamaciones de una compañía de seguros de automóviles.