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.
1 Introduction
In order to make insurance a trade at all, the common premium must be sufficient to compensate the common losses, to pay the expense of management, and to afford such a profit as might have been drawn from an equal capital employed in any common trade.
Adam Smith, The Wealth of Nations (Book 1, Ch X, Part I, 5th Edition, 1789)
1.1 Our Subject and Why It Matters
Pricing insurance risk is the last mile of underwriting. It determines which risks are accepted onto the balance sheet and makes an insurer’s risk appetite operational. It is critical to successful insurance company management.
As the last mile, pricing depends on all that has come before. Actuaries and underwriters have analyzed and classified the risk, trended and developed losses, and on-leveled premiums to pick a best-estimate prospective loss cost. Accountants have allocated fixed and variable expenses. Simulation models place the new risk within the context of the company’s existing portfolio. The mechanics of all this work is the subject of much of the actuarial education syllabus: experience and exposure rating, predictive analytics, and advanced statistical methods. That is not the subject of this book! All of that prior effort determines the expected loss, and we take it as a given. Pricing adds the risk margin—to afford capital a reasonable return. The risk margin is our subject.
Since risk margins are often small, how is it they deserve a whole book? Because risk considerations have an outsized market impact. True, personal property may only earn a single-digit margin. But that business often relies on reinsurance priced with margins of 50% or more. When the reinsurance markets fail or become stressed—as seen after Hurricane Andrew and the Northridge earthquake, for example—the tail of high-risk-margin business wags the dog of much larger property lines. Risk margins are critical to the functioning of the insurance market. Even for lines with thin margins, the collective risk and return decisions of firms have profound macro impacts over time such as the secular increases in homeowners pricing over the last twenty years.
We emphasize insurance risk. We do not discuss credit risk nor operational risk. We have only a little to say about asset risk and nothing about interest rate risk. Market risk, underwriting cycles, competitive threats? Sorry, all off-topic. We are focused on the risk of losses arising from insurance contracts. We lean heavily towards a property-casualty perspective and, within that, towards catastrophe risk; however, the principles we lay out apply to any insurance risk. This is not a book about Enterprise Risk Management (ERM) although we do have a few words to say about optimization and portfolio management.
The goal of this book is to demonstrate how to
1 compute a reservation price (technical premium, required premium) for the portfolio, and
2 allocate it to portfolio units (policies, lines of business, etc.) in a defensible manner
starting from a model of the insured risks. These pricing techniques have powerful applications. They allow us to assess the performance of different units, evaluate needed reinsurance, and optimize overall strategy.
1.2 Players, Roles, and Risk Measures
Figure 1.1 shows the participants in the insurance pricing problem. Insureds, left, pay premiums to the insurer and in turn receive loss payments. The regulator, on top, observing the risk that the insurer is taking on, imposes asset requirements. Investors, right, provide capital and in turn receive the residual value (remaining assets) after losses are paid.
Figure 1.1 Players and their roles. The regulator applies a capital risk measure to determine required insurer assets. The pricing risk measure gives the cost of investors’ capital. Assets in excess of losses are paid to investors as the residual value of the business.
Insureds buy insurance because of their aversion to risk and because they are required to do so to drive a car, buy a house with a mortgage, etc. Regulators play a social policy role, addressing three principal concerns. First, to ensure mandated third-party insurance provides effective protection. Second, to manage the externality of losses exceeding assets. And third, to prevent insureds being fleeced by excessive premiums. The first concern is present in any tort-based system. We loosely identify the second as European and the third as American. We focus on the second concern, asset adequacy. Our development of technical premiums naturally aligns with the third fairness consideration if we assume that capital markets require fair returns.
Investors indirectly determine premiums because premiums plus capital add up to and fund assets, Figure 1.2. Investors’ willingness to provide capital to insurers translates into a pricing risk measure, which the insurer applies to the covered risks. Premium and asset levels are separate problems and need separate tools.
