Preface

      Pricing Insurance Risk is a topic of great concern to actuaries, especially property-casualty actuaries, our primary audience. But it is also relevant to those working in other fields, including risk management and Enterprise Risk Management, capital modeling, catastrophe modeling, financial regulation, and solvency assessment. Insurance risk is managed through pooling, unlike financial risk that is managed through hedging. The title could have been Pricing Non-Hedgeable Risk.

      The book came about through a confluence of supporting factors. We had worked independently for many years on the problem of defining the value of risk management and risk transfer (especially in the context of property catastrophe risk) and “escaping the efficient frontier.” Don Mango brought us together to work with him and Jesse Nickerson to present a multipart tutorial on spectral risk measures at the Casualty Actuarial Society Spring 2018 meeting. The tutorial was so successful that we felt it deserved a wider audience and set about developing a monograph: “Spectral Risk Measures for the Working Actuary.” As we proceeded to refine our thinking and presentation, we realized there was so much more to be explained. Three and a half years and 1200 git commits later, we had this book.

      The literature is rich with good answers to many fundamental questions about insurance risk that are consistent with finance theory and relatively easy to apply. Much is known, in the sense of being out there in the literature, but too much is not widely known by people who would benefit from that knowledge. Actuarial education and practice in this area lags the state of the art. We have encountered actuaries struggling with ill-defined terminology and concepts with multiple names. We have seen confusion wrought by inappropriate application of finance theories (remember the underwriting beta?). Our newly minted US Fellows are often ignorant of the latest developments because they are not on the exam syllabus and there has not been an easy way to incorporate them.

      This book presents these good answers in one systematic and comprehensive source for the first time, making them much more accessible to actuaries and other practitioners. With this book we intend to raise the bar in actuarial education, enable clear communications, and improve the efficiency of actuaries everywhere by delivering a fresh map of the conceptual territory. We wish we could take credit for the theory we present, but most of it is around twenty years old. We are simply reporting the work of others.

Insurance pricing is multidisciplinary, combining actuarial science and risk theory, probability and statistics, finance and economics, accounting and law. As we organized and synthesized a body of literature as nearly as old as the industry itself to tell the story of insurance pricing, we tried to be sensitive to its historical development—a play some of which we watched unfold in real time. It is a story we both found fascinating. From defining underwriting profit and a reasonable target profit in the 1920s to arguments about investment income in the 1960s. From systematic risk and option pricing theory applications in the 1980s to a more insurance-specific model in the 1990s. And most recently to the introduction of coherent, convex, and even star-shaped risk measures. We hope the reader has time to appreciate the giants on whose shoulders we are lucky enough to stand and can join us in taking in the spectacular vistas of the meaning, quantification, and management of risk they have revealed.

      In putting together this book we tried to stay reasonably rigorous without getting lost in a theorem-proof wilderness. We feel strongly that knowing how to use a technique is not helpful if you are unsure that it is valid to use in the first place! We include technical remarks and provide pointers into the literature (about 300 bibliographic references) for those who want a more thorough understanding of “why.” For the practitioner, we included nearly 100 examples and 150 exercises. The Learning Objectives at the end of each chapter summarize what we hope the reader will take away from it. Bold words and phrases introduce terminology that is used throughout the book.

      We aimed this book primarily at property-casualty actuaries, at minimum two years of experience as a student actuary with basic knowledge of insurance coverage and structuring, and having passed the beginning mathematics exams. We expect readers with different backgrounds will still be able to get something from the book. A lot of the insurance and finance terminology is only an internet search away. Mathematics background should include calculus and basic probability—sample spaces, discrete vs. continuous random variables, normal and lognormal distributions, integration by parts, etc. Of course, for an in-depth understanding, more background, especially in probability theory, is better.

      The manuscript was prepared using free software. It was written in Markdown and converted to TeX using Pandoc. TikZ was used for the figures and diagrams, and all the graphs and plots were made using Python, Pandas, and Matplotlib. We used R for the statistical analysis and to double check Python (they always agreed). Spreadsheets were used for the discrete examples. We both remember when computers booted from (genuinely) floppy disks. The existence of so much free software, of such a high quality, is an unexpected joy.

      We owe a debt of gratitude to many people. In academia, keeping us accurate, we thank Dani Bauer, Stuart Klugman, Andreas Tsanakas, Ruodu Wang, Shaun Wang, and George Zanjani. In business, keeping us real, we thank Avi Adler, John Aquino, Neil Bodoff, Julia Chu, Andrew Cox (1978–2021), Dan Dick, Paul Eaton, Bryon Ehrhart, Kent Ellingson, Stephen Fiete, Bob Fox, Jonathan Hayes, Greg Heerde, Wouter Heynderickx, Rodney Kreps, Morton Lane, Mike McClane, Tessa Moulton, Parr Schoolman, Paul Schultz, Jason Trock, Gary Venter, Steve White, and Rebecca Wilkinson.

Special thanks go to Don Mango for starting this all; to Jesse Nickerson for his early involvement in the research and his comments on drafts; and to Yuriy Krvavych and Lawrence McTaggart for their comments on drafts. Richard Goldfarb stands out for particular thanks, having provided very detailed and pertinent feedback that resulted in numerous improvements. Stephen: I would like to recognize the influence of Glenn Meyers and Richard Derrig (1941–2018) early in my career—they taught me how to think about pricing insurance risk. I am enormously grateful to my wife, Helen, who started proofreading the manuscript at a late stage and found herself learning the material in a crash course. Her fresh perspective and unyielding commitment to clarity helped improve the presentation in uncountably many ways. John: I would like to thank Jack Caron, Bernie Shorr, and Aaron Stern for opening doors.