Resumen de Una aportación al análisis de solvencia: la teoría del valor extremo
- español
La Tesis, tras revisar los fundamentos de la teoría del Valor Extremo, busca la aplicabilidad de la distribución Generalizada de Pareto para la modelización de los extremos que exceden un determinado umbral elevado. La metodología establecida y las conclusiones obtenidas, a partir de la ampliación empírica a las observaciones de dos entidades aseguradoras, pueden ser inductivamente aplicadas a otros ramos u otras entidades. A través de aproximaciones asintóticas este trabajo obtiene un modelo de los grandes siniestros con el objeto de inferir futuras pérdidas. El problema se encuentra en que la dispersión de los datos es tan elevada que la inferencia puede alcanzar valores nunca observados aunque su ocurrencia es probable. A través de los modelos obtenidos es posible establecer relaciones con los requerimientos de capital de solvencia (SCR) determinados por Solvencia II, permitiendo calcular medidas como el Valor en Riesgo (VaR) o el Valor en Riesgo en la cola (TVaR). Otras aplicaciones de los modelos se encuentra en el Reaseguro de Exceso de Perdida, donde la Teoría del Valor Extremo puede apoyar las decisiones de la entidad aseguradora para determinar el pleno de retención y tomar decisiones financiero - actuariales óptimas y a la vez que permite el estudio de la esperanza de siniestralidad por encima de una prioridad, fundamental para establecer una prima de riesgo desde el punto de vista del Reasegurador de los eventos extremos.
- English
The main concern of this investigation is solvency in insurance companies and the Extreme Value Theory is the tool to achieve the objective. As far as these events can put in danger the whole stability of a company, the unusual behaviour of a random event might have more interest than the normal behaviour, which has been long studied by the classic risk theory. Even more, the classic theory does not consider the outliers, not to raise the variance of central observations. This means that the law of large numbers and the central limit theorem are not adequate for modelling a complete loss distribution. In the future this will lead to mixtures of distributions. The frame of this investigation is the Solvency II project. The proposals are based on the distribution of large claims to describe fat tails. After a revision of the origin of the theory, the fundamentals of the extreme value theory are summarized. Generalized Extreme Value Distribution and Generalized Pareto Distribution are the main models that describe large events. After the theoretical background we make an empirical study using two claim portfolios of two insurance companies based over a ten year period. Tendencies, seasonality and mean times are studied to describe the behaviour of large claims. Through asymptotic approaches this study gets an appropriate model for large claims in order to determine future losses and probabilities. A Generalized Pareto distribution is obtained to shape the excesses over a threshold. Each company has its own model with different layers. Because extreme values are scarce, the inference can reach levels never observed but likely to happen. Through these models we find relations with the solvency capital requirements determined by Solvency II, calculating measures like Value at Risk or Tail Value at Risk. Other applications to Excess Loss Reinsurance are developed from the angle of the reinsurer and from the point of view of the cedent company. Further investigations will be aimed at asymptotic mathematics and its application to insurance and other fields such as financial market or natural disasters.
