Actuarial Theory for Dependent Risks

Transcription

1 Actuarial Theory for Dependent Risks Actuarial Theory for Dependent Risks: Measures, Orders and Models M. Denuit, J. Dhaene, M. Goovaerts and R. Kaas 2005 John Wiley &Sons, Ltd. ISBN: X

2 Actuarial Theory for Dependent Risks Measures, Orders and Models M. Denuit Université Catholique de Louvain, Belgium J. Dhaene Katholieke Universiteit Leuven, Belgium and Universiteit van Amsterdam, The Netherlands M. Goovaerts Katholieke Universiteit Leuven, Belgium and Universiteit van Amsterdam, The Netherlands R. Kaas Universiteit van Amsterdam, The Netherlands

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4 Contents Foreword xiii Preface xv PART I THE CONCEPT OF RISK 1 1 Modelling Risks Introduction The Probabilistic Description of Risks Probability space Experiment and universe Random events Sigma-algebra Probability measure Independence for Events and Conditional Probabilities Independent events Conditional probability Random Variables and Random Vectors Random variables Random vectors Risks and losses Distribution Functions Univariate distribution functions Multivariate distribution functions Tail functions Support Discrete random variables Continuous random variables General random variables Quantile functions Independence for random variables Mathematical Expectation Construction Riemann Stieltjes integral 22

5 vi CONTENTS Law of large numbers Alternative representations for the mathematical expectation in the continuous case Alternative representations for the mathematical expectation in the discrete case Stochastic Taylor expansion Variance and covariance Transforms Stop-loss transform Hazard rate Mean-excess function Stationary renewal distribution Laplace transform Moment generating function Conditional Distributions Conditional densities Conditional independence Conditional variance and covariance The multivariate normal distribution The family of the elliptical distributions Comonotonicity Definition Comonotonicity and Fréchet upper bound Mutual Exclusivity Definition Fréchet lower bound Existence of Fréchet lower bounds in Fréchet spaces Fréchet lower bounds and maxima Mutual exclusivity and Fréchet lower bound Exercises 55 2 Measuring Risk Introduction Risk Measures Definition Premium calculation principles Desirable properties Coherent risk measures Coherent and scenario-based risk measures Economic capital Expected risk-adjusted capital Value-at-Risk Definition Properties VaR-based economic capital VaR and the capital asset pricing model 71

6 CONTENTS vii 2.4 Tail Value-at-Risk Definition Some related risk measures Properties TVaR-based economic capital Risk Measures Based on Expected Utility Theory Brief introduction to expected utility theory Zero-Utility Premiums Esscher risk measure Risk Measures Based on Distorted Expectation Theory Brief introduction to distorted expectation theory Wang risk measures Some particular cases of Wang risk measures Exercises Appendix: Convexity and Concavity Definition Equivalent conditions Properties Convex sequences Log-convex functions Comparing Risks Introduction Stochastic Order Relations Partial orders among distribution functions Desirable properties for stochastic orderings Integral stochastic orderings Stochastic Dominance Stochastic dominance and risk measures Stochastic dominance and choice under risk Comparing claim frequencies Some properties of stochastic dominance Stochastic dominance and notions of ageing Stochastic increasingness Ordering mixtures Ordering compound sums Sufficient conditions Conditional stochastic dominance I: Hazard rate order Conditional stochastic dominance II: Likelihood ratio order Comparing shortfalls with stochastic dominance: Dispersive order Mixed stochastic dominance: Laplace transform order Multivariate extensions Convex and Stop-Loss Orders Convex and stop-loss orders and stop-loss premiums Convex and stop-loss orders and choice under risk Comparing claim frequencies 154

7 viii CONTENTS Some characterizations for convex and stop-loss orders Some properties of the convex and stop-loss orders Convex ordering and notions of ageing Stochastic (increasing) convexity Ordering mixtures Ordering compound sums Risk-reshaping contracts and Lorenz order Majorization Conditional stop-loss order: Mean-excess order Comparing shortfall with the stop-loss order: Right-spread order Multivariate extensions Exercises 182 PART II DEPENDENCE BETWEEN RISKS Modelling Dependence Introduction Sklar s Representation Theorem Copulas Sklar s theorem for continuous marginals Conditional distributions derived from copulas Probability density functions associated with copulas Copulas with singular components Sklar s representation in the general case Families of Bivariate Copulas Clayton s copula Frank s copula The normal copula The Student copula Building multivariate distributions with given marginals from copulas Properties of Copulas Survival copulas Dual and co-copulas Functional invariance Tail dependence The Archimedean Family of Copulas Definition Frailty models Probability density function associated with Archimedean copulas Properties of Archimedean copulas Simulation from Given Marginals and Copula General method Exploiting Sklar s decomposition Simulation from Archimedean copulas 224

8 CONTENTS ix 4.7 Multivariate Copulas Definition Sklar s representation theorem Functional invariance Examples of multivariate copulas Multivariate Archimedean copulas Loss Alae Modelling with Archimedean Copulas: A Case Study Losses and their associated ALAEs Presentation of the ISO data set Fitting parametric copula models to data Selecting the generator for Archimedean copula models Application to loss ALAE modelling Exercises Measuring Dependence Introduction Concordance Measures Definition Pearson s correlation coefficient Kendall s rank correlation coefficient Spearman s rank correlation coefficient Relationships between Kendall s and Spearman s rank correlation coefficients Other dependence measures Constraints on concordance measures in bivariate discrete data Dependence Structures Positive dependence notions Positive quadrant dependence Conditional increasingness in sequence Multivariate total positivity of order Exercises Comparing Dependence Introduction Comparing Dependence in the Bivariate Case Using the Correlation Order Definition Relationship with orthant orders Relationship with positive quadrant dependence Characterizations in terms of supermodular functions Extremal elements Relationship with convex and stop-loss orders Correlation order and copulas Correlation order and correlation coefficients Ordering Archimedean copulas Ordering compound sums Correlation order and diversification benefit 294

9 x CONTENTS 6.3 Comparing Dependence in the Multivariate Case Using the Supermodular Order Definition Smooth supermodular functions Restriction to distributions with identical marginals A companion order: The symmetric supermodular order Relationships between supermodular-type orders Supermodular order and dependence measures Extremal dependence structures in the supermodular sense Supermodular, stop-loss and convex orders Ordering compound sums Ordering random vectors with common values Stochastic analysis of duplicates in life insurance portfolios Positive Orthant Dependence Order Definition Positive orthant dependence order and correlation coefficients Exercises 305 PART III APPLICATIONS TO INSURANCE MATHEMATICS Dependence in Credibility Models Based on Generalized Linear Models Introduction Poisson Credibility Models for Claim Frequencies Poisson static credibility model Poisson dynamic credibility models Association Dependence by mixture and common mixture models Dependence in the Poisson static credibility model Dependence in the Poisson dynamic credibility models More Results for the Static Credibility Model Generalized linear models and generalized additive models Some examples of interest to actuaries Credibility theory and generalized linear mixed models Exhaustive summary of past claims A posteriori distribution of the random effects Predictive distributions Linear credibility premium More Results for the Dynamic Credibility Models Dynamic credibility models and generalized linear mixed models Dependence in GLMM-based credibility models A posteriori distribution of the random effects Supermodular comparisons Predictive distributions On the Dependence Induced by Bonus Malus Scales Experience rating in motor insurance Markov models for bonus malus system scales Positive dependence in bonus malus scales 345

10 CONTENTS xi 7.6 Credibility Theory and Time Series for Non-Normal Data The classical actuarial point of view Time series models built from copulas Markov models for random effects Dependence induced by autoregressive copula models in dynamic frequency credibility models Exercises Stochastic Bounds on Functions of Dependent Risks Introduction Comparing Risks With Fixed Dependence Structure The problem Ordering random vectors with fixed dependence structure with stochastic dominance Ordering random vectors with fixed dependence structure with convex order Stop-Loss Bounds on Functions of Dependent Risks Known marginals Unknown marginals Stochastic Bounds on Functions of Dependent Risks Stochastic bounds on the sum of two risks Stochastic bounds on the sum of several risks Improvement of the bounds on sums of risks under positive dependence Stochastic bounds on functions of two risks Improvements of the bounds on functions of risks under positive quadrant dependence Stochastic bounds on functions of several risks Improvement of the bounds on functions of risks under positive orthant dependence The case of partially specified marginals Some Financial Applications Stochastic bounds on present values Stochastic annuities Life insurance Exercises Integral Orderings and Probability Metrics Introduction Integral Stochastic Orderings Definition Properties Integral Probability Metrics Probability metrics Simple probability metrics Integral probability metrics 389

11 xii CONTENTS Ideal metrics Minimal metric Integral orders and metrics Total-Variation Distance Definition Total-variation distance and integral metrics Comonotonicity and total-variation distance Maximal coupling and total-variation distance Kolmogorov Distance Definition Stochastic dominance, Kolmogorov and total-variation distances Kolmogorov distance under single crossing condition for probability density functions Wasserstein Distance Definition Properties Comonotonicity and Wasserstein distance Stop-Loss Distance Definition Stop-loss order, stop-loss and Wasserstein distances Computation of the stop-loss distance under stochastic dominance or dangerousness order Integrated Stop-Loss Distance Definition Properties Integrated stop-loss distance and positive quadrant dependence Integrated stop-loss distance and cumulative dependence Distance Between the Individual and Collective Models in Risk Theory Individual model Collective model Distance between compound sums Distance between the individual and collective models Quasi-homogeneous portfolios Correlated risks in the individual model Compound Poisson Approximation for a Portfolio of Dependent Risks Poisson approximation Dependence in the quasi-homogeneous individual model Exercises 421 References 423 Index 439

12 Foreword Dependence is beginning to play an increasingly important role in the world of risk, with its strong embedment in areas like insurance, financial activities, safety engineering, etc. While independence can be defined in only one way, dependence can be formulated in an unlimited number of ways. Therefore, the assumption of independence prevails as it makes the technical treatment easy and transparent. Nevertheless, in applications dependence is the rule, independence the exception. Dependence quickly leads to an intricate and a far less convenient development. The authors have accepted the challenge to offer their readership a survey of the rapidly expanding topic of dependence in risk theory. They have brought together the most significant results on dependence available up to now. The breadth of coverage provides an almost full-scale picture of the impact of dependence in risk theory, in particular in actuarial science. Nevertheless, the treatment is not encyclopaedic. In their treatment of risk, the emphasis is more on the ideas than on the mathematical development, more on concrete cases than on the most general situation, more on actuarial applications than on abstract theoretical constructions. The first three chapters provide in-depth explorations of risk: after dealing with the concept of risk, its measurement is covered via a plethora of different risk measures; its relative position with respect to other risks is then treated using different forms of stochastic orderings. The next three chapters give a similar treatment of dependence as such: modelling of dependence is followed by its measurement and its relative position within other dependence concepts. While illustrations come mainly from the actuarial world, these first two parts of the book have much broader applicability; they make the book also useful for other areas of risk analysis like reliability and engineering. The last three chapters show a stronger focus on applications to insurance: credibility theory is followed by a thorough study of bounds for dependent risks; the text ends with a treatment of risk comparison by using integral orderings and probability metrics. An asset of the book is that a wealth of additional material is covered in exercises that accompany each chapter. This succinct text provides a thorough treatment of dependence within a risk context and develops a coherent theoretical and empirical framework. The authors illustrate how this theory can be used in a variety of actuarial areas including among others: value-at-risk, ALAE-modelling, bonus-malus scales, annuities, portfolio construction, etc. Jozef L. Teugels Katholieke Universiteit Leuven, Belgium

13 Preface Traditionally, insurance has been built on the assumption of independence, and the law of large numbers has governed the determination of premiums. But these days, the increasing complexity of insurance and reinsurance products has led to increased actuarial interest in the modelling of dependent risks. In many situations, insured risks tend to behave alike. For instance, in group life insurance the remaining lifetimes of husband and wife can be shown to possess a certain degree of positive dependence. The emergence of catastrophes and the interplay between insurance and finance also offer good examples in which dependence plays an important role in pricing and reserving. Several concepts of bivariate and multivariate positive dependence have appeared in the mathematical literature. Undoubtedly, the most commonly encountered dependence property is actually lack of dependence, in other words mutual independence. Actuaries have so far mostly been interested in positive dependence properties expressing the notion that large (or small ) values of the random variables tend to occur together. Negative dependence properties express the notion that large values of one variable tend to occur together with small values of the others. Instances of this phenomenon naturally arise in life insurance (think, for instance, of the death and survival benefits after year k in an endowment insurance, which are mutually exclusive and hence negatively correlated), or for the purpose of competitive pricing. Note that, in general, a negative dependence results in more predictable losses for the insurance company than mutual independence. The independence assumption is thus conservative in such a case. Moreover, assuming independence is mathematically convenient, and also obviates the need for elaborate models to be devised and statistics to be kept on mutual dependence of claims. There is only one way for risks to be independent, but there are of course infinitely many ways for them to be correlated. For efficient risk management, actuaries need to be able to answer the fundamental question: is the correlation structure dangerous? And if it is, how dangerous is the situation? Therefore, tools to quantify, to compare and to model the strength of dependence between different risks have now become essential. The purpose of this book is to provide its readership with methods to: measure risk compare risks measure the strength of dependence compare dependence structures model the dependence structure. To illustrate the theoretical concepts, we will give many applications in actuarial science.

14 xvi PREFACE This book is innovative in many respects. It integrates the theory of stochastic orders, one of the methodological cornerstones of risk theory, the theory of risk measures, the very foundation of risk management, and the theory of stochastic dependence, which has become increasingly important as new types of risks emerge. More specifically, risk measures will be used to generate stochastic orderings, by identifying pairs of risks about which a class of risk measures agree. Stochastic orderings are then used to define positive dependence relationships. The copula concept is examined in detail. Apart from the well-known correlation coefficient, other measures of dependence are presented, as well as multivariate stochastic orderings, to evaluate the strength of dependence between risks. We also emphasize the numerous connections existing between multivariate and univariate stochastic orders. In the third part of the book, we discuss some applications in actuarial mathematics. We first review credibility models. In these models, past claims history not only of the risk itself, but also of related risks, is used to determine the future premium. This method is based on the serial correlation among the annual claim characteristics (frequencies or severities) induced by their sharing a common random effect, and on the correlation between related risks caused by a similar effect. We describe the kind of dependence induced by credibility models, and establish numerous stochastic inequalities showing that the classical credibility construction pioneered by Bühlmann produces very intuitive results. Secondly, we will derive bounds on actuarial quantities involving correlated risks whose joint distribution is (partially) unknown or too cumbersome to work with. Our focus will be on stop-loss premiums and Value-at-Risk. Next, we will present probabilistic distances, and show the close connection between this theoretical tool and stochastic orderings. In particular, the relevance of probabilistic distances for the analysis of dependent risks will be demonstrated. This book complements our Modern Actuarial Risk Theory (Kaas et al. 2001), which only scratches the surface of the material found here. Since the traditional actuarial risk theory assumes independence between the different random variables of interest, the present book may be thought of as an advanced course on risk theory dropping this hypothesis. The target audience of this book consists of academics and practitioners who are eager to master modern modelling tools for dependent risks. The inclusion of many exercises also makes the book suitable as the basis for advanced courses on risk management in incomplete markets, as a complement to Kaas et al. (2001). Sometimes, we will give proofs only under simplifying assumptions, in order to help the reader understand the underlying reasoning, bringing out the main ideas without obscuring them with mathematical technicalities. Some proofs are omitted. Appropriate references to the literature will guide the readers interested in a more thorough mathematical treatment of the topic. Insurance markets are prominent examples of incomplete markets, since the products sold by insurance companies cannot be replicated by some financial trading strategy. We firmly believe that this book should be of interest not only to actuaries but more generally to traders aware that perfect hedges do not exist in reality. The main effect of accounting for market incompleteness has indeed been to bring utility theory back into pricing. More generally, it should bridge quantitative finance and actuarial science. We would like to thank the Committee on Knowledge Extension Research of the American Society of Actuaries for financial support, under grant Actuarial Aspects of Dependencies in Insurance Portfolios. Thanks also to Virginia Young, the scientific referee of our project,

15 PREFACE xvii for her careful reading of earlier drafts of the manuscript, and her invaluable advice on matters mathematical and stylistic. We express our gratitude to Paul Embrechts, Christian Genest, Alfred Müller and Moshe Shaked for having read (parts of) the manuscript, and for the numerous remarks they made. We also thank Professors Frees and Valdez for kindly providing the loss ALAE data set used in Chapter 4, which were collected by the US Insurance Services Office (ISO). Michel Denuit is grateful for the financial support of the UCL Fonds Spéciaux de Recherche (under projects Tarification en assurance: Vers une nouvelle approche intégrée and Nouvelles méthodes de gestion des risques assurantiels et financiers ) and the Belgian Fonds National de la Recherche Scientifique (Crédit aux Chercheurs Dépendances entre risques actuariels et financiers ). This work was also partly supported by the contract Projet d Actions de Recherche Concertées nr 98/ from the Communauté Française Wallonie-Bruxelles. Jan Dhaene and Marc Goovaerts acknowledge the financial support of the Onderzoeksfonds K.U. Leuven (GOA/02 Actuariële, financiële en statistische aspecten van afhankelijkheden in verzekerings- en financiële portefeuilles ). Finally, we would like to express our gratitude and appreciation to all those with whom we have had the pleasure of working on problems related to this book: Hélène Cossette, Christian Genest, Hans Gerber, Angela van Heerwaarden, Bart Kling, Claude Lefèvre, Etienne Marceau, Alfred Müller, Marco Scarsini, Mhamed Mesfioui, Moshe Shaked, Sergei Utev, Shaun Wang and Virginia Young. Michel Denuit Jan Dhaene Marc Goovaerts Rob Kaas Louvain-la-Neuve, Leuven and Amsterdam Supplementary material for this book can be found at denuit/mdenuit.html