Description: This paper develops credibility using a longitudinal data framework. In a longitudinal data framework, one might encounter data from a cross-section of risk classes towns with a history of insurance claims available for each risk class. For the marginal claims distributions, we use generalized linear models, an extension of linear regression that also encompasses Weibull and Gamma regressions. Copulas are used to model the dependencies over time specifically, this paper is the first to propose using a t-copula in the context of generalized linear models. The t-copula is the copula associated with the multivariate t-distribution like univariate t-distributions, it seems especially suitable for empirical work. Moreover, we show that the t-copula gives rise to easily computable predictive distributions that we use to generate credibility predictors. Like Bayesian methods, our copula credibility prediction methods allow us to provide an entire distribution of predicted claims, not just a point prediction.

Description: The two-parameter Pareto distribution is a commonly used model in reliability and risk modeling. Minimum variance unbiased estimates of the parameters of Pareto distribution are not known. In this paper, we propose and investigate a method of estimation of the parameters of the Pareto distribution. This method is based on minimization of the Kolmogorov-Smirnov distance between the empirical cumulative distribution function [cdf] and the cdf of the Pareto distribution.

Description: In this paper, we introduce a family of leptokurtic symmetric distributions represented by the difference of two gamma variates. Properties of this family are discussed. The Laplace, sums of Laplace and normal distributions all arise as special cases of this family. We propose a two-step method for fitting data to this family. We develop a statistical test for goodness-of-fit and introduce a test of normality of the data. A simulation study is provided to illustrate the theory.

Description: Exploration of sources of volatility in retiree health valuations, noting how volatility arises and might be mitigated, with discussion of valuation parameters more problematic for retiree health benefits than for pensions. Three significant influences on retiree health valuation volatility are covered-- changes in benefit level, fluctuations in cost level, and alterations in eligibility for, or duration of, benefit – and attention focused on the last two. Specific conclusions result in recommendations, which are supported by practical examples contrasting volatile measurement results with those likely from the recommended practices. Theoretical discussion includes health claim variation, health trend usage, financial economics, option pricing, discount rates, and deterministic vs. stochastic modeling.

Description: We consider a class of compound renewal risk process with claim waiting times have a discrete Km distribution. The classical compound binomial risk model is a special case when m = 1. Both recursive and explicit formulas are derived for the expected discounted penalty function due at ruin, for the surplus before ruin and the deficit at ruin. Many ruin related quantities can be analyzed through the penalty function, e.g., ruin probability, the p.g.f. of the time of ruin, joint and marginal distributions of the surplus before ruin and the deficit at ruin, claim causing ruin, as well as their moments. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution have a finite support.

Description: In this Chapter we present a novel way for estimating aggregated EC figures based on Bayesian copula estimation. Contrary to the classic approach of using a single inter-risk- correlation matrix we derive a probability distribution of possible correlation matrices that enables us to tackle the important issue of parameter uncertainty. Presented at 2010 Enterprise Risk Management Symposium, Society of Actuaries, April 12-15, 2010.

Description: This is a summary of the presentation given during the ARC Conference. Its purpose was to give a brief introduction to subexponential behavior and to show that the family of subexponential distributions provides ideal characteristics to model insurance risks. Although the literature related to this topic accumulated quickly during the last years, a textbook explaining the applications of extreme value theory with intuition as well as insight was still to be written. Embrechts et al, {I 997} satisfies this need and most of the results presented here can be found in this new classic in the actuarial literature.

Description: Quantitative operational risk assessment is essentially based on stochastic scenario modeling of operational loss sequences. Given the lack of reliable historical data in most cases, mathematical methods should meet even stronger requirements in terms of results received and include specific features, namely: rare event analysis, uncertainty analysis and human factor analysis, etc. In this paper, the analytical-statistical simulation approach ASSA is considered as the most flexible approach to stochastic scenario loss sequence modeling and compared with Markov chain, fault/event tree and Monte Carlo, in terms of scenario loss sequence model adequacy and calculation cost. Probabilistic risk analysis software PRAISE and some applications are given. Presented at 2010 Enterprise Risk Management Symposium, Society of Actuaries, April 12-15, 2010.

Description: This paper, originally published in 1992 in the Transactions of Society of Actuaries Vol. 44, deals with a topic that is becoming increasingly important to actuaries in most fields of practice. “Without relying on formulas, this paper presents the important concepts underlying the theory of arbitrage-free pricing of interest-rate-contingent cash flows.” This paper, by James A. Tilley, won the 1992-93 Annual Prize, and the Halmstad Prize for 1992. The paper includes written discussion and the author’s review of the discussions.

Description: In the present paper, we propose and investigate a numerical method of computing the probability distribution of S. Inversion of the Laplace transform moment generating function is shown to be a better approach than the existing recursive method for find the distribution of sum of several independent random variables. In the continuous case, in many situations, numerical inversion needs to be used to compute the probability distribution function of the sum.