Preface

Under Construction!!!

I am writing a book that I hope will be of interest to the actuarial community. This version of the book is reasonably complete. So, I am eager to receive any constructive suggestions for improvement, as well as typos, misstatements, or things that are simply incorrect. Please let me know by sending an email to .

Why help? This online version of the book is (and will be) open and freely available - a resource for our community. (Taylor and Francis / CRC will be publishing a printed version, available for purchase.)

Date: 20 August 2024

Firms face many risks, such as damage to their buildings due to fire, liability resulting from management misbehavior, and threats external to the organization like cyber attacks. Because the financial impact of these risks can be mitigated by using risk transfer tools like insurance, we think of them as insurable risks. Firms are financially responsible for these risks, essentially “owning” them. This naturally leads to the idea of managing them as a portfolio. This book shows how risk managers can manage a risk portfolio cost-effectively by adapting financial concepts of diversification to insurable risk management.

Who is this book for? It is written for financial analysts who advise risk managers, such as commercial insurers, risk brokers, and reinsurers, who are responsible for risk retention and risk portfolio management. Risk managers are accustomed to using their knowledge of industry practices and experience within the field to make retention recommendations. This book introduces techniques that provide an objective framework to supplement, and corroborate, an advisor’s thoughts. While no prior industry knowledge is required, familiarity with the applications will enhance the reading experience. As such, it is also suitable for students and other academics seeking an entry point into industry applications as well as industry analysts who would like an introduction to new developments. The book equips students, academics, and practitioners with quantitative tools for analyzing real-world risk portfolios. Further, it empowers financial analysts to advise risk managers with data-driven insights to enhance their advisory role.

What is this book about? It introduces a novel approach for constructing data-driven insurable risk portfolios. Drawing inspiration from Markowitz portfolio theory, it leverages techniques from probability, statistics, and optimization to build algorithms that create optimal risk insurance portfolios under budget constraints. As such, outputs of the algorithm include a frontier that portrays the uncertainty of a portfolio versus a cost of transferring risks. A visual display of this frontier, mirroring familiar Markowitz investment tools, allows for informed decision-making and seamless adoption by risk advisors.

The Table of Contents provides an overview of the topics covered, organized into four parts. The first four chapters comprise the foundations.

  • This first chapter sets the stage for the problem, emphasizing its importance, available resources such as data, and illustrating potential approaches through minimal scenarios.
  • The second chapter summarizes different contractual forms commonly used for decomposing a risk into that portion retained and that transferred. This chapter also introduces tools for summarizing the uncertainty of the risk retaining and the cost of that transferred.
  • The third chapter dives more deeply into the trade-off between the uncertainty of the risk retained and the cost of transferring risk. Different approaches historically used are summarized; the approach based on constrained optimization tools is emphasized. This approach will be used in the rest of the book.
  • To simplify the development, the first three chapters emphasize a single risk (as does most of the academic literature) whereas the fourth introduces multivariate risks. This is done in the context of reinsurance (where the risk owner is an insurance company), introducing copulas to capture dependence among risks, and describing numerical optimization techniques for selecting the best risk retention strategy subject to budget constraints. This chapter also sets the problem of “retain-transfer” risks in the more general context of risk exchanges.

The second part, chapters five through eight, constitutes the core material of the book. Specifically:

  • The fifth chapter delves into the details of managing two risks where notions of portfolio uncertainty and risk transfer costs, with their sensitivities, are examined with a fair bit of mathematical rigor.
  • The sixth chapter steps away from the mathematical development and seeks to provide context as to how quantitative liability portfolio tools could be applied. Risk management fundamentals are reviewed, together with case studies introduced in the first chapter and more recent developments such as robo-advising and peer-to-peer risk exchanges.
  • Extending the fifth chapter, the seventh chapter examines the multivariate risk retention problem using simulation techniques. This approach generates an optimal insurable risk portfolio – it provides the basic strategy advocated in the book.
  • To see how the seventh chapter strategy works with real applications, the eighth chapter practically applies the chapter seven strategy to the two case studies that were introduced in the introduction and developed further in chapter six.

The third part, chapters nine through twelve, examines the properties and characteristics of insurable risk portfolio strategies.

  • The ninth chapter focuses on the reliability of the optimal insurable risk portfolio results from three perspectives. First, reliability can be gauged by examining results under differing scenarios known as stress testing. Second, one can examine the “sensitivity” of optimization results to small changes in the input assumptions. Third, one can gauge the “robustness” of results by planning for the worst possible outcome within a broad region of an assumption and then trying to optimize results.
  • The tenth chapter further develops the notion of sensitivity by developing gradients that quantify the changes in the optimal decision variables as a result of a change in one of the inputs. Further, it uses this novel notion to quantify changes in optimal risk portfolio coefficients due to changes in the risk distribution parameters.
  • The eleventh chapter establishes conditions that must hold at the optimal strategy. This framework is used to establish conditions to achieve (1) a binding budget constraint, (2) a balance among retention parameters at the optimum, and (3) boundary constraints.
  • The twelfth chapter describes the impact of dependence on the composition of insurable risk portfolios.

The fourth part provides various materials that supplement the main body of the text.

Book Features

Practice Features

This book aims to develop the mathematical underpinnings for constructing optimal insurable risk portfolios in a manner that is both effective and aesthetically pleasing. Another key objective is the development of a suite of algorithmic tools that is potentially appealing to industry analysts.

To this end, this book dives deep into the Markowitz investment asset allocation problem as this approach is widely known to financial risk managers and will help them interpret results of liability portfolio risk retention. The case studies developed throughout the book provide further evidence as to how these tools might be used in practice. The Australian National University (ANU) case represents applications commonly confronted by risk management brokers and others who advise medium and large corporations on risk retention strategies. The Wisconsin property fund case represents applications commonly encountered by multi-line general insurers.

Important additional practice features include the accompanying statistical code and data, as follows.

Statistical Code and Data

With this text, you will be able to construct optimal insurable risk portfolios. Although mathematical justifications are important for understanding the “Why?”, for risk portfolio applications it is also crucial to be versed with the “How?” question.

You can do so by downloading the sample data of this text and practicing with illustrative statistical code. Many sets of code are available in an online supplement to the book. In the beginning chapters, I remind readers of their availability using a note such as:

Sample R Code Here

In later chapters, I assume interested readers will be used to looking for the code and so do not include explicit reminders. The findings and results are highlighted in the text, while the methods to achieve them are provided in the code. Shorter versions of the code are also available in an HTML supplement using convenient “show/hide” buttons. Even for analysts more interested in the mathematical development of insurable risk portfolio strategies, I believe that there are insights to be gained by working through the code. Certainly for analysts interested in business applications, the code provided here should be a useful springboard for additional investigations.

The data (and HTML versions of the code) are available at an online version of the book, Frees (2025).

A Variety of Mathematical Techniques

This book addresses real-world insurable risks with a flexible toolbox of mathematical methods. It is not a book that develops a set of techniques and then demonstrates their relevance on a variety of applications. Rather, this is a book focused on a set of problems and then brings to bear on them a variety of mathematical techniques including constrained optimization, simulation, probabilistic arguments, and empirical analytics. (As with the author, …) It is likely that one or more of these techniques will be new to readers and so you will learn about them in the context of insurable risks. For most sets of techniques, explanations and references are provided to get you started on a new area. Hopefully, you will see the opportunity to learn about a new technique, e.g., constrained optimization, as a benefit of this book.

The mathematical framework and supporting code are designed to allow practicing analysts and others beginning their research journey to be able to quickly determine optimal insurable risk portfolios that can be utilized in business settings. This mathematical framework was not created in a vacuum but rather carefully selected from an extensive academic literature. The chapter sections on “Further Resources and Readings” will help readers appreciate the connections between the underpinnings of this framework and the literature. Readers will find that much of this academic literature is based on advanced mathematical constructs which, although yielding deep insights, are not directed towards practical considerations. This book is meant to be a conduit between practical considerations and the academic literature; it also may help to re-direct the academic literature to place greater emphasis on practice features.

Technical Supplements

Curious about the mathematical intricacies behind insurable risk portfolio optimization? Detailed mathematical proofs and derivations are given in an online supplement to the book, indicated by the icon,

\(Under~the~Hood.\) Show Development of the Expected Shortfall.

For readers with the necessary mathematical background, the supplements provide additional material that can be useful when communicating ideas to technical audiences. Think of an analogy of driving a car. Most users do not need to understand the inner workings of an auto in order to drive a car. However, to advance the development of automobiles, engineers (a type of quantitative analyst) need to understand how a car engine works. To capture this perspective, an old saying is that one needs to understand “what is going on under the hood.”

Also available is an HTML supplement that employs convenient “show/hide” buttons. These technical supplements reinforce and extend the results in the main body of the text by giving a more formal, mathematical treatment of the material. This treatment is actually a supplement because the primary results and findings are described in the main body of the text. While readers may skip over these technical supplements during their first read-through, they can serve as valuable resources for understanding the inner workings of the mathematical models and answering more detailed questions.

Suggested Courses. This book provides a versatile framework for advanced courses in actuarial and financial analytics. For all audiences, the main emphasis is on five key chapters: 1, 4, 6, 7, and 8. This basic treatment should include considerable attention to the statistical implementation of the algorithms and resulting frontiers. Additionally:

  • Undergraduates and entry-level graduate students. Apart from the five key chapters, the foundational materials in Chapters 2, 3, and 5 should also be carefully explored, including the mathematical developments.
  • Advanced graduate students. This audience is likely to be interested in mathematical developments and should review the theoretical underpinnings provided in the supplements. In particular, time should be devoted to the mathematical developments in Chapters 5 and 9-11. However, the material in Chapters 2 and 3 can be treated as review and covered lightly.
  • Industry analysts. Alongside the basics, one could include a light introduction to ideas and notation from Chapters 2, 3, 5, and 9. Following Chapter 8, this audience will likely be interested in spending time exploring applications in industry.

Acknowledgements

I began writing this book six years ago, following my retirement from the University of Wisconsin-Madison. Six years is a long time for a book project; there are many who have helped me along the way that I wish to acknowledge. The ideas underpinning this book were conceived when I was writing an earlier paper, Frees (2017), and assisted Gee Lee with his doctoral thesis, Lee (2017). Gee (now at Michigan State) was not only there at the beginning but was also one of the prime collaborators in the development of the Wisconsin Property Fund data featured in this text. He also provided very helpful reviews of earlier drafts.

Soon after my retirement from the University of Wisconsin-Madison, I took on a part-time position at the Australian National University (ANU) and am grateful for their support of my work. In particular, I thank colleagues Adam Butt and Tim Higgins for their collaboration in developing the ANU data featured in this text and facilitating my presentation of a short course where I was able to sharpen my thoughts on quantifying insurable risk portfolios.

Over the years, I have had opportunities to present this stream of work at numerous conferences and university seminars. In addition, I am grateful for the guidance and critical feedback from many reviewers, including:

  • Vali Asimit (Bayes Business School)
  • Catalina Bolancé Losilla (Universitat de Barcelona)
  • Yiqing Chen (Drake University)
  • Tolulope (Tolu) Fadina (University of Illinois Champaign - Urbana)
  • Fei Huang (University of New South Wales)
  • Shimeng Huang (University of Wisconsin-Madison)
  • Hirokazu (Iwahiro) Iwasawa (Waseda University, Tokyo University, and the Institute of Actuaries of Japan)
  • Jacie Liu (Australian National University)
  • Ping Wang (Saint John’s University)
  • Shaun Shuxun Wang (Southern University of Science and Technology)
  • Emiliano Valdez (University of Connecticut)
  • Chengguo Weng (University of Waterloo)
  • Armando Zarruk Rivera (Universidad Nacional de Colombia)
  • Jinggong Zhang (Nanyang Technological University)
  • Wenjun Zhu (Nanyang Technological University).

In addition, I am pleased to acknowledge the contributions from my colleague Peng Shi (University of Wisconsin-Madison) and for his permission to include our joint work on sensitivity that is featured in Chapters 9 and 10. Peng and I have collaborated on insurance analytics for many years and much of this book indirectly reflects his thoughts. I thank him for his generosity in sharing his insights with me and the world.

Saving the most important for last, I thank my family for their support. Ten thousand thanks to my brothers Randy, Guy, and Joe; my wife Deirdre; our sons Nathan and Adam; and Adam’s wife, Claire, and their children, Frankie and Felicity.

Dedication

To my brother Joe and his wife Mary


In honor of their courageous stand against ALS. Their steadfastness and inspiring spirit in the face of adversity have touched the hearts of all who know them.

Constructing Insurable Risk Portfolios © 2024 by Edward Frees is licensed under CC BY-NC-ND 4.0

Video: Introducing Constructing Insurable Risk Portfolios

Under Construction!!!

Videos were done a couple of years ago and require some updating.

Data

So that you can practice constructing insurable risk portfolios, the book features three sets of data. The sources, and how to download these data, are described in this section.

Australian National University Case Study. These data are first described in Section 1.2.1 in the context of this book. In addition, another description is available from the open book Loss Data Analytics. You can download the data from this site.

Building the simulated ANU data distributions is realistic (and hence complex) and will be described in Sections 8.3.5 and 8.3.6. You will find useful the following data, in .csv format, that are available using these download buttons:


Wisconsin Property Fund Case Study. These data are first described in Section 1.2.2 in the context of this book. In addition, another description is available at the author’s web site. You can download the data from this site.

The data, in .csv format, are available using this download button:

The data, in .RData format, are available using this download button:

To retrieve these data in R, you load("~/dataout.RData"), that is, load the data.

Building the simulated Property Fund data distributions will be described in Section 8.3.4. You will find useful the following data, in .RData format, that are available using these download buttons:

Also, the illustratie code uses the following set up source code, in .R format (a text file):


Insurance Stock Returns. These data are first described in Example 1.3.1. They consist of daily stock prices extracted from Yahoo Finance and were converted to daily returns.

The data, in .RData format, are available using these download buttons:

Statistical Code

With the data, you will be able to replicate much of the analysis in this book by accessing the available statistical code. You can start by copying the following code:

Preliminary R Code