Abstract: This monograph is written for the numerate nonspecialist, and hopes to serve three purposes. First it gathers mathematical material from diverse but related fields of order statistics, records, extreme value theory, majorization, regular variation and subexponentiality. All of these are relevant for understanding fat tails, but they are not, to our knowledge, brought together in a single source for the target readership. Proofs that give insight are included, but for most fussy calculations the reader is referred to the excellent sources referenced in the text. Multivariate extremes are not treated. This allows us to present material spread over hundreds of pages in specialist texts in twenty pages. Chapter 5 develops new material on heavy tail diagnostics and gives more mathematical detail. Since variances and covariances may not exist for heavy tailed joint distributions, Chapter 6 reviews dependence concepts for certain classes of heavy tailed joint distributions, with a view to regressing heavy tailed variables. Second, it presents a new measure of obesity. The most popular definitions in terms of regular variation and subexponentiality invoke putative properties that hold at infinity, and this complicates any empirical estimate. Each definition captures some but not all of the intuitions associated with tail heaviness. Chapter 5 studies two candidate indices of tail heaviness basedon the tendency of the mean excess plot to collapse as data are aggregated. The probability that the largest value is more than twice the second largest has intuitive appeal but its estimator hasvery poor accuracy. The Obesity index is defined for a positive random variable X as:Ob(X) = P (X1 + X4 > X2 + X3|X1 = X2 = X3 = X4) , Xi independent copies of X. For empirical distributions, obesity is defined by bootstrapping. This index reasonably captures intuitions of tail heaviness. Among its properties, if a > 1 then Ob(X) < Ob(Xa). However, it does not completely mimic the tail index of regularly varying distributions, or the extreme value index. A Weibull distribution with shape 1/4 is more obese than a Pareto distribution with tail index 1, even though this Pareto has infinite mean and the Weibull’s moments are all finite. Chapter 5 explores properties of the Obesity index. Third and most important, we hope to convince the reader that fat tail phenomena pose real problems; they are really out there and they seriously challenge our usual ways of thinking about historical averages, outliers, trends, regression coefficients and confidence bounds among many other things. Data on flood insurance claims, crop loss claims, hospital discharge bills, precipitation and damages and fatalities from natural catastrophes drive this point home. Whilemost fat tailed distributions are ”bad”, research in fat tails is one distribution whose tail will hopefully get fatter.

Abstract: The Oil Spill Commission’s chartered mission—to “develop options to guard against … any oil spills associated with offshore drilling in the future” (National Commission 2010)—presents a major challenge: how to reduce the risk of low-frequency oil spill events, and especially high-consequence events like the Deepwater Horizon accident, when historical experience contains few oil spills of material scale and none approaching the significance of the Deepwater Horizon. In this paper, we consider precursor analysis as an answer to this challenge, addressing first its development and use in nuclear reactor regulation and then its applicability to offshore oil and gas drilling. We find that the nature of offshore drilling risks, the operating information obtainable by the regulator, and the learning curve provided by 30 years of nuclear experience make precursor analysis a promising option available to theU.S. Bureau of Ocean Energy Management, Regulation and Enforcement (BOEMRE) to bring costeffective, risk-informed oversight to bear on the threat of catastrophic oil spills.

Abstract: Between 1990 and 2000 the U.S. Nuclear Regulatory Commission and the Commission of the European Communities conducted a joint uncertainty analysis of accident consequences for nuclear power plants. This study remains a benchmark for uncertainty analysis of large models involving high risks with high public visibility, and where substantial uncertainty exists. The study set standards with regard to structured expert judgment, performance assessment, dependence elicitation and modeling and uncertainty propagation of high dimensional distributions with complex dependence. The integratedassessment models for the economic effects of climate change also involve high risks and large uncertainties, and interest in conducting a proper uncertainty analysis is growing. This article reviews the EU-USNRC effort and extracts lessons learned, with a view toward informing a comparable effort for the economic effects of climate change.

Abstract: Recent events in the financial and insurance markets, as well as the looming challenges of a globally changing climate point to the need to re-think the ways in which we measure and manage catastrophic and dependent risks. Management can only be as good as our measurement tools. To that end, this paper outlines detection, measurement, and analysis strategies for fat-tailed risks, tail dependent risks, and risks characterized by micro-correlations. A simple model of insurance demand and supply is used to illustrate the difficulties in insuring risks characterized by these phenomena. Policy implicationsare discussed.

Abstract: Adapting to climate change will not only require responding to the physical effects of global warming, but will also require adapting the way we conceptualize, measure, and manage risks. Climate change is creating new risks, altering the risks we already face, and also, importantly, impacting the interdependencies between these risks. In this paper we focus on three particular phenomena of climate related risks that will require a change in our thinking about risk management: global micro-correlations, fat tails, and tail dependence. Consideration of these phenomena will be particularly important for natural disaster insurance, as they call into question traditional methods of securitization and diversification.