Understanding how risks change under aggregation is critical for many fields, from financial markets, where assets are securitized, to insurance, where firms hold a portfolio of many individual policies. When individual risks are independent and sufficiently thin-tailed, bundling can lower overall risk levels. This project centers on three characteristics of risks that threaten the benefits obtained through aggregation: fat tails, tail dependence, and micro-correlations. This research develops new approaches for detecting, measuring, and analyzing these phenomena, with the aim of improving catastrophe risk management across a range of applications.
Fat tailed distributions have been well studied since the early 1900s. Statistical estimators of tail indices, such as the Hill estimator, have been researched extensively, but have well known limitations, and variations continue to proliferate. We propose a new approach for measuring tail obesity based on the self-similarity of random aggregations. This has the benefit of focusing specifically on the properties of fat-tailed distributions that are of concern for securitization, rather than on hypothesized limit behavior. Tail dependence refers to the tendency for dependence between two variables to concentrate in the extreme values. While observable in datasets, such as damages from natural catastrophes, tail dependence is much less understood mathematically than fat tails. Empirically, tail dependence has been seen to grow with aggregation, raising concerns about securitizing or insuring tail dependent risks, but the theory lags far behind these observations. This research will provide risk managers with tools to understand and address tail dependence. Finally, global micro-correlations are small correlations which may be statistically insignificant when considered individually, but become dangerous when aggregated. Proof-of-concept analyses for detecting such correlations have been performed using flood and crop insurance claim data.
The first stage of this new research requires collecting and standardizing historical loss data. After developing the appropriate datasets, the researchers will begin the analytical work that involves developing scalar measures for tail obesity in finite data sets, studying the conditions under which aggregation amplifies tail dependence, studying micro-covariation in fat tail distributions and their behavior under aggregation, and studying empirical distributions of aggregated losses. Finally, the third stage of the research focusing on dissemination: preparing publications for peer-reviewed journals, developing a set of online educational materials, and hosting a workshop.
Tail Risk workshop www.rff.org/Events/Pages/Introduction-Climate-Change-Extreme-Events.aspx Ice Sheets on the Move expert elicitation and public event www.rff.org/Events/Pages/Ice-Sheets-on-the-Move.aspx The major results realized during this project were 1. Finding and analyzing data showing heavy tail behavior 2. Communicating to a wide scientific public regarding the impact of heavy tail phenomena 3. Developing a model for the utility of insurance in the presence of heavy tailed loss distributions and tail dependence. 4. Obtaining mathematical results on the amplification of dependence and of tail dependence under aggregation 5. Developing a new obesity index for finite data sets, and applying this to loss distributions 6. Writing a monograph making mathematical results in the fields of order statistics, records, majorization, and extreme value theory accessible for a non-specialist readership. When individual risks are independent and sufficiently thin-tailed, bundling can lower overall risk levels; diversification benefits are obtained. Our research for this project centered on three characteristics of risks that threaten the benefits obtained from aggregation: fat tails, tail dependence, and micro-correlations. We focused on three tasks: improving detection, measurement, and analysis of these three phenomena on actual datasets of catastrophic losses. The overall aim was to improve the management of catastrophic risks, particularly insurance, and some of our later work focused specifically on the difficult of insuring these types of risks. This research effort resulted in several peer-reviewed journal articles, book chapters, a monograph, and multiple publications and media products aimed at broader audiences. We held a workshop on these topics, bringing together leading thinkers in this field for in-depth discussions. Also as a result of our work, and discussed further below, we have developed a public website that includes background papers, presentations, educational materials, and public datasets on these topics. The website was first developed as resource for the workshop and has since been updated to serve as a continuing resource for students, researchers, and practitioners.
