Modelización de riesgos dependientes con marginales Lomax inversa

• Sarabia Alegría, José María ^[1] ; Prieto Mendoza, Faustino ^[1] ; Jordá Gil, Vanesa ^[1] ; Remuzgo Pérez, Lorena ^[1]
  1. [1] Universidad de Cantabria

     Universidad de Cantabria

     Santander, España

     
• Localización: Anales de ASEPUMA, ISSN-e 2171-892X, Nº. 23, 2015
• Idioma: español
• Enlaces
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• Resumen
  • español

    En este trabajo se presenta una clase general de riesgos dependientes, donde los riesgos individuales siguen distribuciones de tipo Lomax inversa. La distribuci´on Lomax inversa posee colas pesadas y se modeliza por medio de dos par´ametros, uno de forma y otro de escala. Se obtiene la funci´on de densidad conjunta de los riesgos dependientes y se consideran varios submodelos relevantes. Se proporcionan expresiones cerradas tanto para la funci´on de densidad como para la funci´on de distribuci´on del riesgo total. Por otro lado, se obtienen f´ormulas para diferentes medidas de riesgo, incluyendo el valor en riesgo VaR, el valor en riesgo en la cola TVaR, as´ı como otras medidas basadas en la cola de la distribuci´on, tanto para los riesgos individuales como para el riesgo total. Se proponen m´etodos de estimaci´on seg´un el tipo de informaci´on disponible. Los modelos desarrollados se aplican a varios conjuntos de datos relativos a riesgos dependientes, incluyendo riesgos operacionales y datos de p´erdidas y ALAE.

  • English

    In this paper, we present a general class of dependent risks, where the individ-ual risk are modeled according to inverse Lomax distributions. The inverse Lomax distribution has heavy tails and depends on two parameters, a shape and a scale parameter. We obtain the joint density function of dependent risks and several rel-evant submodels are considered. Closed expressions for the probability density and the cumulative distribution functions of the total risk are provided. Furthermore, formulas for different risk measures are obtained, including value at risk VaR, the value at risk in the tail TVaR and other measures based on the tail of the distribu-tion, both for individual risks and for risk total. Estimation methods are proposed according to the type of information available. The developed models are applied to several sets of data for dependent risks, including operational risks and loss data and ALAE.

• Referencias bibliográficas
  • Acerbi, C., Tasche, D. (2002). “On the Coherent of Expected Shortfall”. Journal of Banking and Finance, 26, pp. 1487-1503.
  • Albrecher, H., CONSTANTINESCU, C., LOISEL, S. (2011). “Explicit Ruin Formulas for Models with Dependence Among Risks”. Insurance: Mathematics...
  • Asimit, A.V., Vernic, R., ZITIKIS, R. (2013). “Evaluating Risk Measures and Capital Allocations Based on Multi-Losses Driven a Heavy-Tailed...
  • Benkhelifa, L. (2014). “Kernel-type estimator of the reinsurance Premium for heavy-tailed loss distributions”. Insurance Mathematics and Economics,59,...
  • Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R. (2005). “Actuarial Theory for Dependent Risks. Measures, Orders and Models”. John Wiley, Chichester.
  • Guillen, M., Sarabia, J.M., Prieto, F. (2013). “Simple Risk Measure ´ Calculations for Sums of Positive Random Variables”. Insurance: Mathematics...
  • Klugman, S., Parsa, R. (1999). “Fitting Bivariate Loss Distributions with Copulas”. Insurance: Mathematics and Economics, 24, 139-148.
  • Klugman, S., Panjer, H.H., Willmot, G.E. (2004). “Loss Models. From Data to Decisions”, Second Edition. John Wiley, New York.
  • Kolev, N., Anjos, U., Mendes, B.M. (2006). “Copulas: a Review and Recent Developments”. Stochastic Models, 22, 617-660.
  • Markovich, N. (2005). “Nonparametric Analysis of Univariate HeavyTailed data. Research and Practice”. John Wiley, New York.
  • Mcdonald, J. B. (1984). “Some Generalized Functions for the Size Distribution of Income”. Econometrica, 52, 647-663.
  • Moschopoulos, P.G. (1985). “The Distribution of the Sum of Independent Gamma Random Variables”. Annals Instit. Statist. Math., 37, 541-544. Nelsen,...
  • Sarabia, J.M., Gómez Déniz, E. (2008). “Construction of Multivariate Distributions: a Review of Some Recent Results (with discussion)”. SORT,...
  • Sarabia, J.M., Guillen, M. (2008). “Joint Modelling of the Total Amount ´and the Number of Claims by Conditionals”. Insurance: Mathematics...
  • Sarabia, J.M., Prieto, F. (2011). “Sobre una Clase de Riesgos Dependientes”. Anales del Instituto de Actuarios Españoles, 2011, 31-50.
  • Sun, J., Frees, E.W., Rosenberg, M.A. (2008). “Heavy-tailed longitudinal data modeling using copulas”. Insurance Mathematics and Economics,...
  • Valdez, E.A., Dhaene, J., Maj, M., Vanduffel, S. (2009).“Bounds and Approximations for Sums of Dependent log-elliptical Random Variables”....