Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing.Dependence structure, Extremal values, Copula modeling, Copula review

Rational plunging and the option value of sequential investment : the case of petroleum exploration

Any investor in assets that can be exploited sequentially faces a tradeoff between diversification and concentration. Loading a portfolio with correlated assets has the potential to inflate variance, but also creates information spillovers and real options that may augment total return and mitigate variance. The task of optimal portfolio design is therefore to strike an appropriate balance between diversification and concentration. We examine this tradeoff in the context of petroleum exploration. Using a simple model of geological dependence, we show that the value of learning options creates incentives for explorationists to plunge into dependence; i.e., to assemble portfolios of highly correlated exploration prospects. Risk-neutral and risk-averse investors are distinguished not by the plunging phenomenon, but by the threshold level of dependence that triggers such behavior. Aversion to risk does not imply aversion to dependence. Indeed the potential to plunge may be larger for risk-averse investors than for risk-neutral investors. To test the empirical validity of our theory, we examine the concentration of bids tendered in petroleum lease sales. We find that higher levels of risk aversion are associated with a revealed preference for more highly concentrated (i.e., less diversified) portfolios

Set-valued shortfall and divergence risk measures

Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include set-valued versions of the entropic risk measure and the average value at risk. As a second step, the minimization of these risk measures subject to trading opportunities is studied in a general convex market in discrete time. The optimal value of the minimization problem, called the market risk measure, is also a set-valued risk measure. A dual representation for the market risk measure that decomposes the effects of the original risk measure and the frictions of the market is proved

The multivariate directional approach: high level quantile estimation and applications to finance and environmental phenomena

Mención Internacional en el título de doctorThe aim of this thesis is to introduce a directional multivariate approach to analyze extremes. The proposal point out the importance of two factors from the dimensional world we live in, the center of reference and the direction of observation. These factors are inherent to the multivariate setting and allow us to introduce manager preferences or external information available for the system of interest. The key definition in which is based this thesis is the notion of directional multivariate quantiles. It is introduced in Chapter 1 jointly with its properties which help to develop directional risk analysis. Besides, Chapter 1 describes the background and motivation for the directional multivariate approach. The rest of the chapters are devoted to the main contributions of the thesis. Chapter 2 introduces a directional multivariate risk measure which is a multivariate extension of the well-known univariate risk measure Value at Risk (VaR), which is defined as a quantile of the distribution of the random loss at level and it has become a benchmark in fields such as Economy, Insurance and Finance. Properties for the proposed multivariate risk measure are provided as extensions of the axiomatic for univariate risk measures given in the literature. We have also proved relationships between the univariate VaR evaluated on the marginal loss and the component associated to that marginal loss in our vector-valued proposal. Chapter 2 also highlights the importance of using directions thanks to a result providing a conservative bound (upper bound) of the total risk in a portfolio investment by using the direction of the weights of investment to analyze such loss. In the literature, copula models are frequently used to model the loss, thus solutions of our risk measure for some of these models are shown and a non-parametric approach to estimate the output in more general cases is also provided. Finally, a study of robustness in comparison with other vector-valued risk measure found in the literature is developed. Chapter 3 is focused on the formal definition and estimation of the directional multivariate extremes. Given that environmental science possesses different phenomena where join behavior of variables may cause disasters, two real cases of study are analyzed. In the literature, it is possible to find copula theory to model those dependencies, which leads us to introduce the directional approach to the copula framework. Thus, advantages and disadvantages between non-parametric approaches and theoretical copula approaches are highlighted in this chapter. Moreover, it is presented a proposal to choose a suitable direction of analysis by considering the direction of the maximum variability on the data, which links the use of Principal Component Analysis (PCA). Applications are performed on the real cases of study of flood risk at a dam (3 dimensional case) and sea storms (5 dimensional case). In extreme value theory, it is known that standard non-parametric methods can not be applied to estimate quantiles at high levels. Therefore, a different approach known as out-sample estimation must be considered. In this sense, Chapter 4 introduces the necessary background to face the multivariate extreme value theory. Then, results including the directional approach to the multivariate extreme value theory are given. An estimator of the directional multivariate quantiles is provided and its asymptotic normality is also proved. Finally, it is presented a nonparametric methodology to accomplish the goal of estimation, with an illustration using the multivariate distribution for which are known all the theoretical elements of the estimation process. Finally, Chapter 5 summarizes the conclusions of the thesis, open questions and future works are also commented.El objetivo de esta tesis es el de introducir aspectos direccionales a las metodologías multivariantes utilizadas para el análisis de extremos y problemas derivados. Se explica en el documento que la utilización de direcciones en determinadas situaciones posibilitan considerar información externa o preferencias particulares del analista. El elemento matemático clave en este proyecto es la definición de cuantil direccional multivariante. Las propiedades que satisface y otras nociones relacionadas con esta definición son las bases que fundamentan los desarrollos teóricos y sus aplicaciones al análisis de riesgo, las cuales constituyen las contribuciones de esta tesis. Después de una introducción de conceptos preliminares y motivaciones dadas en el Capítulo 1, los Capítulos 2 a 4 recogen las siguientes aportaciones: En el Capítulo 2, se introduce una extensión direccional multivariante del Value at Risk, el cual en dimensión uno es un referente en campos como economía, seguros y finanzas, y se define como un cuantil a nivel para la distribución de la variable de pérdidas. Nuestra propuesta describe una medida de riesgo de resultado vectorial basada en los cuantiles direccionales multivariantes. Se estudian sus propiedades como una extensión de la axiomática definida para medidas de riesgo univariantes y también se presentan relaciones entre el valor de la medida de riesgo univariante VaR, aplicada sobre las marginales del vector de pérdidas, y los valores de las correspondientes componentes de la medida de riesgo propuesta. En este Capítulo se fundamenta la importancia de las direcciones, gracias a la cota conservadora (cota superior) de pérdida total que permite establecer nuestra propuesta a través del análisis en la dirección del vector de pesos de la inversión Se analizan expresiones cerradas de solución para la medida de riesgo direccional multivariante en modelos de copula de alta aplicación en la teoría financiera y se presenta un método de estimación no-paramétrico para el resultado de dicha medida en ámbitos generales. Finalmente, se presenta un análisis de robustez sobre los resultados obtenidos para la medida propuesta ante presencia de atípicos en la muestra, obteniendo buen comportamiento especialmente en casos de alta presencia de atípicos, en comparación con la única medida de valor vectorial encontrada en la literatura a la fecha. El Capítulo 3 se ha centrado en la definición de extremos direccionales y en la descripción de una metodología de detección no-paramétrica de los mismos. Se presentan casos de estudio reales en el ámbito de la ingeniería ambiental, dado que en los fenómenos ambientales se requiere del análisis conjunto de variables cuya dependencia conlleva a resultados catastróficos en muchas situaciones. Debido a la necesidad de modelar estas dependencias, una de las herramientas más utilizadas en la literatura son las cópulas. Por tanto, en este Capítulo se presentan las ventajas y desventajas de los métodos de copula y direccional no-paramétrico, y se plantea la inclusión del enfoque direccional para las metodologías basadas en cópulas. Se presenta una interesante alternativa de dirección a través de la dirección de máxima variabilidad en los datos, lo cuál genera la inclusión de análisis de componentes principales a la metodología propuesta. Finalmente se analizan los casos reales de riesgo de inundación en una presa (en 3 dimensiones) y de tormentas costeras extremas (en 5 dimensiones), así como casos simulados que complementan la importancia del análisis direccional. Por otra parte, es bien conocido que las metodologías clásicas de estimación no paramétrica fallan cuando se desea realizar análisis para niveles altos del cuantil incluso en el caso univariante, es decir, para muy cercano a o , lo cual se conoce en la literatura como estimación out-sample y para abordarlo es necesario recurrir a resultados asintóticos de la teoría de valores extremos. Nuestra propuesta no se encuentra exenta de esta necesidad y en el Capítulo 4 se describen las hipótesis necesarias para introducir una metodología de estimación out-sample para los cuantiles multivariantes direccionales. Adicionalmente, se prueban resultados que incluyen el enfoque direccional en el marco de la teoría de valores extremos multivariante y se demuestra también la propiedad de normalidad asintótica para el estimador propuesto. Finalmente, se presenta el comportamiento del estimador a través de un ejemplo basado en la distribución multivariante, para la cual los resultados teóricos de los cuantiles direccionales son conocidos, así como los valores teóricos de los elementos necesarios para el proceso de estimación. Finalmente, en el Capítulo 5 se presentan las conclusiones de la tesis y problemas abiertos para futuros trabajos de investigación.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Ignacio Cascos Fernández.- Secretario: José María Fernández Ponce.- Vocal: Elena di Bernardin

Notions of Dependence with Applications in Insurance and Finance

Many insurance and finance activities involve multiple risks. Dependence structures between different risks play an important role in both theoretical models and practical applications. However, stochastic and actuarial models with dependence are very challenging research topics. In most literature, only special dependence structures have been considered. However, most existing special dependence structures can be integrated into more-general contexts. This thesis is motivated by the desire to develop more-general dependence structures and to consider their applications. This thesis systematically studies different dependence notions and explores their applications in the fields of insurance and finance. It contributes to the current literature in the following three main respects. First, it introduces some dependence notions to actuarial science and initiates a new approach to studying optimal reinsurance problems. Second, it proposes new notions of dependence and provides a general context for the studies of optimal allocation problems in insurance and finance. Third, it builds the connections between copulas and the proposed dependence notions, thus enabling the constructions of the proposed dependence structures and enhancing their applicability in practice. The results derived in the thesis not only unify and generalize the existing studies of optimization problems in insurance and finance, but also admit promising applications in other fields, such as operations research and risk management

Diversification Quotients: Quantifying Diversification via Risk Measures

To overcome several limitations of existing diversification indices, we introduce the diversification quotient (DQ). Defined through a parametric family of risk measures, DQ satisfies three natural properties, namely, non-negativity, location invariance and scale invariance, which are shown to be conflicting for any traditional diversification index based on a single risk measure. We pay special attention to the two most important classes of risk measures in banking and insurance, the Value-at-Risk (VaR) and the Expected Shortfall (ES, also called CVaR). DQs based on VaR and ES enjoy many convenient technical properties, and they are efficient to optimize in portfolio selection. By analyzing the popular multivariate models of elliptical and regular varying distributions, we find that DQ can properly capture tail heaviness and common shocks which are neglected by traditional diversification indices. When illustrated with financial data, DQ is intuitive to interpret, and its performance is competitive when contrasted with other diversification methods in portfolio optimization

Non-additive anonymous games

This paper introduces a class of non-additive anonymous games where agents are assumed to be uncertain (in the sense of Knight) about opponents’ strategies and about the initial distribution over players’ characteristics in the game. These uncertainties are modelled by non-additive measures or capacities. The Cournot-Nash equilibrium existence theorem is proven for this class of games. It is shown that the equilibrium distribution can be symmetrized under milder conditions than in the case of additive games. In particular, it is not required for the space characteristics to be atomless under capacities. The set-valued map of the Cournot-Nash equilibria is upper-semicontinuous as a function of initial beliefs of the players for non-additive anonymous games