The use of flexible quantile-based measures in risk assessment [WP]

A new family of distortion risk measures -GlueVaR- is proposed in Belles- Sampera et al. -2013- to procure a risk assessment lying between those provided by common quantile-based risk measures. GlueVaR risk measures may be expressed as a combination of these standard risk measures. We show here that this relationship may be used to obtain approximations of GlueVaR measures for general skewed distribution functions using the Cornish-Fisher expansion. A subfamily of GlueVaR measures satisfies the tail-subadditivity property. An example of risk measurement based on real insurance claim data is presented, where implications of tail-subadditivity in the aggregation of risks are illustrated

Quantitative risk assessment, aggregation functions and capital allocation problems

[eng] This work is focused on the study of risk measures and solutions to capital allocation problems, their suitability to answer practical questions in the framework of insurance and financial institutions and their connection with a family of functions named aggregation operators. These operators are well-known among researchers from the information sciences or fuzzy sets and systems community. The first contribution of this dissertation is the introduction of GlueVaR risk measures, a family belonging to the more general class of distortion risk measures. GlueVaR risk measures are simple to understand for risk managers in the financial and insurance sectors, because they are based on the most popular risk measures (VaR and TVaR) in both industries. For the same reason, they are almost as easy to compute as those common risk measures and, moreover, GlueVaR risk measures allow to capture more intricated managerial and regulatory attitudes towards risk. The definition of the tail-subadditivity property for a pair of risks may be considered the second contribution. A distortion risk measure which satisfies this property has the ability to be subadditive in extremely adverse scenarios. In order to decide if a GlueVaR risk measure is a candidate to satisfy the tail-subadditivity property, conditions on its parameters are determined. It is shown that distortion risk measures and several ordered weighted averaging operators in the discrete finite case are mathematically linked by means of the Choquet integral. It is shown that the overall aggregation preference of the expert may be measured by means of the local degree of orness of the distortion risk measure, which is a concept taken over from the information sciences community and brung into the quantitative risk management one. New indicators for helping to characterize the discrete Choquet integral are also presented in this dissertation. The aim is complementing those already available, in order to be able to highlight particular features of this kind of aggregation function. Following this spirit, the degree of balance, the divergence, the variance indicator and Rényi entropies as indicators within the framework of the Choquet integral are here introduced. A major contribution derived from the relationship between distortion risk measures and aggregation operators is the characterization of the risk attitude implicit into the choice of a distortion risk measure and a confidence or tolerance level. It is pointed out that the risk attitude implicit in a distortion risk measure is to some extent contained in its distortion function. In order to describe some relevant features of the distortion function, the degree of orness indicator and a quotient function are used. It is shown that these mathematical devices give insights on the implicit risk behavior involved in risk measures and entail the definitions of overall, absolute and specific risk attitudes. Regarding capital allocation problems, a list of key elements to delimit these problems is provided and mainly two contributions are made. Firstly, it is shown that GlueVaR risk measures are as useful as other alternatives like VaR or TVaR to solve capital allocation problems. The second contribution is understanding capital allocation principles as compositional data. This interpretation of capital allocation principles allows the connection between aggregation operators and capital allocation problems, with an immediate practical application: Properly averaging several available solutions to the same capital allocation problem. This thesis contains some preliminary ideas on this connection, but it seems to be a promising research field.[spa] Este trabajo se centra en el estudio de medidas de riesgo y de soluciones a problemas de asignación de capital, en su capacidad para responder cuestiones prácticas en el ámbito de las instituciones aseguradoras y financieras, y en su conexión con una familia de funciones denominadas operadores de agregación. Estos operadores son bien conocidos entre los investigadores de las comunidades de las ciencias de la información o de los conjuntos y sistemas fuzzy. La primera contribución de esta tesis es la introducción de las medidas de riesgo GlueVaR, una familia que pertenece a la clase más general de las medidas de riesgo de distorsión. Las medidas de riesgo GlueVaR son sencillas de entender para los gestores de riesgo de los sectores financiero y asegurador, puesto que están basadas en las medidas de riesgo más populares (el VaR y el TVaR) de ambas industrias. Por el mismo motivo, son casi tan fáciles de calcular como estas medidas de riesgo más comunes pero, además, las medidas de riesgo GlueVaR permiten capturar actitudes de gestión y regulatorias ante el riesgo más complicadas. La definición de la propiedad de la subadditividad en colas para un par de riesgos se puede considerar la segunda contribución. Una medida de riesgo de distorsión que cumple esta propiedad tiene la capacidad de ser subadditiva en escenarios extremadamente adversos. Con el propósito de decidir si una medida de riesgo GlueVaR es candidata a satisfacer la propiedad de la subadditividad en colas se determinan condiciones sobre sus parámetros. Se muestra que las medidas de riesgo de distorsión y varios operadores de medias ponderadas ordenadas en el caso finito y discreto están matemáticamente relacionadas a través de la integral de Choquet. Se muestra que la preferencia global de agregación del experto puede medirse usando el nivel local de orness de la medida de riesgo de distorsión, que es un concepto trasladado des de la comunidad de las ciencias de la información hacia la comunidad de la gestión cuantitativa del riesgo. Nuevos indicadores para ayudar a caracterizar las integrales de Choquet en el caso discreto también se presentan en esta disertación. Se pretende complementar a los existentes, con el fin de ser capaces de destacar características particulares de este tipo de funciones de agregación. Con este espíritu, se presentan el nivel de balance, la divergencia, el indicador de varianza y las entropías de Rényi como indicadores en el ámbito de la integral de Choquet. Una contribución relevante que se deriva de la relación entre las medidas de riesgo de distorsión y los operadores de agregación es la caracterización de la actitud ante el riesgo implícita en la elección de una medida de riesgo de distorsión y de un nivel de confianza. Se señala que la actitud ante el riesgo implícita en una medida de riesgo de distorsión está contenida, hasta cierto punto, en su función de distorsión. Para describir algunos rasgos relevantes de la función de distorsión se usan el indicador nivel de orness y una función cociente. Se muestra que estos instrumentos matemáticos aportan información relativa al comportamiento ante el riesgo implícito en las medidas de riesgo, y que de ellos se derivan las definiciones de les actitudes ante el riego de tipo general, absoluto y específico. En cuanto a los problemas de asignación de capital, se proporciona un listado de elementos clave para delimitar estos problemas y se hacen principalmente dos contribuciones. En primer lugar, se muestra que las medidas de riesgo GlueVaR son tan útiles como otras alternativas tales como el VaR o el TVaR para resolver problemas de asignación de capital. La segunda contribución consiste en entender los principios de asignación de capital como datos composicionales. Esta interpretación de los principios de asignación de capital permite establecer conexión entre los operadores de agregación y los problemas de asignación de capital, con una aplicación práctica inmediata: calcular debidamente la media de diferentes soluciones disponibles para el mismo problema de asignación de capital. Esta tesis contiene algunas ideas preliminares sobre esta conexión, pero parece un campo de investigación prometedor

Beyond Value-at-Risk : GlueVaR Distortion Risk Measures

We propose a new family of risk measures, called GlueVaR, within the class of distortion risk measures. Analytical closed-form expressions are shown for the most frequently used distribution functions in financial and insurance applications. The relationship between Glue-VaR, Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) is explained. Tail-subadditivity is investigated and it is shown that some GlueVaR risk measures satisfy this property. An interpretation in terms of risk attitudes is provided and a discussion is given on the applicability in non-financial problems such as health, safety, environmental or catastrophic risk managemen

The use of fexible quantile-based measures in risk assessment

A new family of distortion risk measures GlueVaR is proposed in Belles-Sampera et al. (2014) to procure a risk assessment lying between those provided by common quantile-based risk measures. GlueVaR measures may be expressed as a combination of these standard risk measures. We show here that this relationship may be used to obtain approximations of GlueVaR measures for general skewed distribution functions using the Cornish-Fisher expansion. A subfamily of GlueVaR measures satisfies the tail-subadditivity property. An example of risk measurement based on real insurance claim data is presented, where implications of tail-subadditivity in the aggregation of risks are illustrated

"The connection between distortion risk measures and ordered weighted averaging operators"

Distortion risk measures summarize the risk of a loss distribution by means of a single value. In fuzzy systems, the Ordered Weighted Averaging (OWA) and Weighted Ordered Weighted Averaging (WOWA) operators are used to aggregate a large number of fuzzy rules into a single value. We show that these concepts can be derived from the Choquet integral, and then the mathematical relationship between distortion risk measures and the OWA and WOWA operators for discrete and nite random variables is presented. This connection oers a new interpretation of distortion risk measures and, in particular, Value-at-Risk and Tail Value-at-Risk can be understood from an aggregation operator perspective. The theoretical results are illustrated in an example and the degree of orness concept is discussed.Fuzzy systems; Degree of orness; Risk quantification; Discrete random variable JEL classification:C02,C60

What attitudes to risk underlie distortion risk measure choices?

Understanding the attitude to risk implicit within a risk measure sheds some light on the way in which decision makers perceive losses. In this paper, a two-stage strategy is developed to characterize the underlying risk attitude involved in a risk evaluation, when executed by the family of distortion risk measures. First, we show that aggregation indicators defined for Choquet integrals provide information about the implicit global risk attitude of the agent. Second, an analysis of the distortion function offers a local description of the agent's stance on risk in relation to the occurrence of accumulated losses. Here, the concepts of absolute risk attitude and local risk attitude arise naturally. An example is provided to illustrate the usefulness of this strategy for characterizing risk attitudes in an insurance company

What attitudes to risk underlie distortion risk measure choices? [WP]

Understanding the attitude to risk implicit within a risk measure sheds some light on the way in which decision makers perceive losses. In this paper, a two-stage strategy is developed to characterize the underlying risk attitude involved in a risk evaluation, when executed by the family of distortion risk measures. First, we show that aggregation indicators defined for discrete Choquet integrals provide informa- tion about the implicit global risk attitude of the agent. Second, an analysis of the distortion function offers a local description of the agent's stance on risk in relation to the occurrence of accumulated losses. Here, the concepts of absolute risk attitude and local risk attitude arise naturally. An example is provided to illustrate the usefulness of this strategy for characterizing risk attitudes in an insurance company

GlueVaR risk measures in capital allocation applications

GlueVaR risk measures defined by Belles-Sampera et al. (2014) generalize the traditional quantile-based approach to risk measurement, while a subfamily of these risk measures has been shown to satisfy the tail-subadditivity property. In this paper we show how GlueVaR risk measures can be implemented to solve problems of proportional capital allocation. In addition, the classical capital allocation framework suggested by Dhaene et al. (2012) is generalized to allow the application of the Value-at-Risk (VaR) measure in combination with a stand-alone proportional allocation criterion (i.e., to accommodate the Haircut allocation principle). Two new proportional capital allocation principles based on GlueVaR risk measures are defined. An example based on insurance claims data is presented, in which allocation solutions with tail-subadditive risk measures are discussed

Compositional methods applied to capital allocation problems

In this paper, we examine the relationship between capital allocation problems and compositional data, ie, information that refers to the parts of a whole conveying relative information. We show that capital allocation principles can be interpreted as compositions. The natural geometry and vector space structure of compositional data are used to operate with capital allocation solutions. The distance and average that are appropriated in the geometric structure of compositions are presented. We demonstrate that these two concepts can be used to compare capital allocation principles and merge them. An illustration is provided to show how the distance between capital allocation solutions and average of these solutions can be computed, and interpreted, by risk managers in practice

Indicators for the characterization of discrete Choquet integrals

Ordered weighted averaging (OWA) operators and their extensions are powerful tools used in numerous decision-making problems. This class of operator belongs to a more general family of aggregation operators, understood as discrete Choquet integrals. Aggregation operators are usually characterized by indicators. In this article four indicators usually associated with the OWA operator are extended to discrete Choquet integrals: namely, the degree of balance, the divergence, the variance indicator and Renyi entropies. All of these indicators are considered from a local and a global perspective. Linearity of indicators for linear combinations of capacities is investigated and, to illustrate the application of results, indicators of the probabilistic ordered weighted averaging -POWA- operator are derived. Finally, an example is provided to show the application to a specific context