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Wang Paper Provides Tools for Modeling and Combining Correlated Risks

by James C. Sandor

Every actuary who does any kind of modeling for multiline contracts should be aware of "Aggregation of Correlated Risk Portfolios: Models and Algorithms" by Shaun S. Wang published in the 1998 Proceedings. This paper, which is a practical and theoretical guide to modeling dependent risks, is the result of a research project commissioned by the CAS Committee on Theory of Risk.

Wang's paper is an outstanding introduction to aggregate modeling, and it also (as the title suggests) provides models and algorithms to model correlated aggregate loss distributions. Correlation and the more general topic of dependence are also explained in this paper. As the author points out in the introduction, they are not the same.

Section 2, Probability Generating Function and Fast Fourier Transforms (FFT), and Section 3, Aggregate Loss Models and the FFT Method, are essential reading. The FFT method is an extremely flexible and powerful tool for generating aggregate loss distributions. This method has not enjoyed widespread use to date, largely because a concise, understandable explanation has been absent from the actuarial literature. Actuaries who have shied away from the FFT method because it is too difficult to explain to others should read Dr. Wang's description.

Section 3 also contains a simple, innovative method for calculating the sum of multiple lines of business with correlated frequencies. This method can be implemented quite easily in a spreadsheet or statistical package via the FFT procedure or through simulation.

Dr. Wang devotes the center section of the paper to measures of dependence and copulas. These two topics may not be familiar to many actuaries; however, they are critically important to the understanding of dependence modeling beyond simple correlation. Kendall's Tau and Spearman's Rank correlation are two of the alternate measures of dependence that are discussed.

Anyone doing dependence modeling should be aware of what copulas are and how they work. Simply stated, copulas are dependence models. Although they are not covered on the CAS Syllabus, they are a valuable tool for creating dependence structures beyond simple Pearson linear correlation.

The final sections of the paper cover alternate sources of dependence, including Common Mixture Models, Component Models, and the Distortion Method. Also included is a second method for modeling the sum of multiple risk portfolios with correlated frequencies, based on the characteristic function of the multivariate negative binomial distribution. This method can also be easily programmed into a spreadsheet or statistical package. The paper concludes with an example using the two methods assuming correlated frequency.

Dr. Wang has given the CAS an exceptional paper on topics and algorithms related to the modeling of dependent risks. Actuaries who are doing any type of aggregate modeling should read it and apply the principles to their real-world problems. The paper is available in the 1998 Proceedings, on the CAS Web Site at www.casact.org under publications.

Interested users can find a downloadable spreadsheet FFTCalc, built by Glenn Meyers, that demonstrates Dr. Wang's methods. Look in the "Downloadable Programs and Spreadsheets" section of the CAS Web Site.