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WILEY SERIES IN PROBABILITY AND STATISTICS

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Reinsurance: Actuarial and Statistical Aspects

Hansjörg Albrecher

University of Lausanne
Switzerland

Jan Beirlant

KU Leuven, Belgium and University of the Free State
South Africa

Jozef L. Teugels

KU Leuven
Belgium

 

 

 

 

 

 

 

 

 

 

 

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Preface

Reinsurance is a fascinating field. Several of the challenges of classical insurance are amplified for reinsurance, particularly when it comes to dealing with extreme situations like large claims and rare events. This poses particular challenges for the modelling of claims and their occurrence, which often needs to be based on only few data points. In addition, in terms of better diversification of usual‐scale risk on the local and global level as well as in terms of the development of innovative and sustainable techniques to deal with risks of an unusual kind, reinsurers play a crucial role in the insurance process.

This also reflects on practitioners and researchers involved in such topics, as they have to rethink classical models in order to cope successfully with the respective challenges. Over the years, there has been enormous research activity on problems connected to reinsurance. Close to 40% of the references in our literature list have appeared over the last 10 years, with a steep upward gradient over the last 3–4 years. While there exist some excellent classical textbooks on reinsurance either from the academic or the practitioner’s side, our impression was that there was no modern reference book available that gave an overview of the academic research landscape in this field and also puts it in perspective with the practical viewpoint. The main reason for writing this book was to try to address this gap, at least for actuarial and statistical matters. As all the authors are from academia, there naturally remains a bias towards the academic angle. However, numerous and enlightening discussions with insurance and reinsurance practitioners over the last few years have motivated us to produce the current account, hoping and trying to further bridge the two worlds. The focus of the book is on modelling together with the statistical challenges that go along with it. We illustrate the discussed statistical approaches alongside six case studies of insurance loss data sets, ranging from MTPL over fire to storm and flood loss data. Some of the presented material also contains new results that have not yet been published in the research literature. We hope that the material presented can trigger new research questions and foster the communication between (re)insurance practitioners and academics working in these fields. One of our main goals was to give an up‐to‐date overview of the relevant research literature and to frame it to questions that matter in reinsurance practice. Since this a vast topic, we naturally had to take various compromises and we apologize for possible omissions on either side.

The book is written for researchers with an interest in reinsurance problems, for graduate students with a basic knowledge of probability and statistics as well as for practitioners in the field.

We start with a general introduction to the field in Chapter 1, presenting some basic facts and motivations for reinsurance activities. We also introduce the six real‐life case studies that will accompany the considerations throughout the book. In Chapter 2, we discuss the most common reinsurance forms and their properties, together with some practical aspects of their implementation. Chapter 3 is dedicated to motivating and developing models for claim size distributions that are commonly used. Here we emphasize those aspects from actuarial mathematics that are relevant for reinsurance. Reinsurance is often invoked in the presence of large claims, therefore we need a thorough discussion of models capable of catching the essentials of what actuaries would call large. Chapter 4 contains detailed guidelines on how to proceed in the model choice when actually facing data. Throughout the text, we illustrate the presented procedures for our case studies. Chapter 5 proceeds with models for claim numbers, both from a conceptual and a practical viewpoint. We also provide guidelines for a statistical analysis of data sets in this context. The two ingredients (claim numbers and claim sizes) are then used in Chapter 6 for the aggregation of the claims. Emphasis is put on the aggregation of independent risks, and we describe both numerical and asymptotic methods in detail. The case of dependence in the aggregation process is also discussed briefly, although not in detail, as the results typically are very sensitive to the particular dependence structure used in the modeling process, and often the number of data points does not allow one such model to be decisively favored over another. It is beyond the scope of this book to discuss all such approaches. Chapter 7 treats important actuarial aspects of reinsurance pricing, once a distribution for the individual (or aggregate) risk is available (or, rather, decided upon). In Chapter 8 we discuss some guidelines on possible criteria for the choice of reinsurance forms and the respective consequences on the optimal choice of contracts. The identification of optimal reinsurance forms has been a very active research field recently and it is impossible to reflect all these contributions in one book chapter in an exhaustive way. We instead provide an overview of some of the main approaches and contributions alongside a structure in terms of decision criteria, with an emphasis on the intuition behind the results. Since stochastic simulation is an essential tool in many models relevant in reinsurance, we cover this topic in Chapter 9 and discuss some variance reduction techniques that can help to considerably speed up calculations. Chapter 10 then examines some further topics. We first provide more information on large claim analysis, and continue with an overview of alternative risk transfer products, which can serve as a complement to traditional reinsurance. We also highlight the role of finance in reinsurance and finish with a section on catastrophe insurance. Within the chapters and in particular at the end of the chapters we provide links for further reading.

Many of the topics dealt with in the book apply to both non‐life and life insurance. Even when there is a clear emphasis on non‐life insurance throughout, we hope that some of our attempts may help to also be of service to life insurance. As the title suggests, this book is about (traditional) actuarial as well as statistical aspects arising in reinsurance. As is outlined in Chapter 1, reinsurance also serves financial and management purposes in practice. Correspondingly, the role of capital is nowadays an important ingredient in managing and steering reinsurance companies, and financial pricing techniques for reinsurance contracts as well as general capital management tools eventually have to complement the actuarial approach. While we do consider such aspects when discussing the pricing and the possible choice of contracts in Chapters 7, 8 and 10, it is beyond the scope of this book to treat and reflect the merging of actuarial and financial principles in the amount of detail this may deserve from a general perspective.

The idea for writing this book was born in the legendary and productive environment of EURANDOM, Eindhoven. We would like to thank this institution for its continuing support over the years as well as the University of Lausanne and KU Leuven for generous support for extended research visits that enabled the book to progress. We also thank Sophie Ladoucette, MunichRe, and the Versicherungsverband Österreich for providing data for our case studies.

We would like to thank all the people with whom we had interesting discussions about the topic over the recent years, including the participants of the Summer School of the Swiss Actuarial Association in Lausanne in 2015, as well as short course participants in Paris, Johannesburg, Lisbon, Lyon, Luminy, Yerevan, Warsaw, and Hong Kong.

Particular thanks for stimulating discussions or advice in earlier and later stages of the book writing go to Jose Carlos Araujo Acuna, Katrien Antonio, Peiman Asadi, Alexandru Asimit, Anastasios Bardoutsos, Arian Cani, Michel Dacorogna, Dalit Daily‐Amir, Michel Denuit, François Dufresne, John Einmahl, Karl‐Theodor Eisele, Michael Fackler, Damir Filipović, Hans U. Gerber, Alois Gisler, William Guevara‐Alarcon, Jürgen Hartinger, Christian Hipp, Frans Koning, Yuriy Krvavych, Sandra Kurmann, Sophie Ladoucette, Stéphane Loisel, Franz Prettenthaler, Christian Y. Robert, Robert Schall, Matthias Scherer, Thorsten Schmidt, Wim Schoutens, Johan Segers, Wim Senden, Stefan Thonhauser, Joël Wagner, Roel Verbelen, Robert Verlaak, Leonard Vincent, Jean‐François Walhin, Gord Willmot, and Mario Wüthrich. Special thanks go to Tom Reynkens for his tremendous effort writing an R package with this book and producing the plots linked with the statistical procedures. Further thanks go to William Guevara‐Alarcon and Dominik Kortschak for help with the R codes underlying the illustrations in Chapter 9, and to Roel Verbelen and Tom Reynkens for their significant contribution to the splicing methods. We will maintain a webpage connected to the book at

http://www.hec.unil.ch/halbrech_files/reinsurance.html

where we also intend to keep a list of misprints and remarks. We are grateful to receive relevant material sent to us by email. The R package ReIns can be found at the CRAN page

cran.r‐project.org/package = ReIns

Hansjörg Albrecher, Jan Beirlant, and Jozef L. Teugels
Lausanne and Leuven,
December 2016