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: This paper presents the modeling of mortality risk from exposure to a potential future extreme event such as a natural disaster and a terrorist attack, and its impact on life insurance. With this new model, quantities associated with the total force of mortality such as survival function are derived and the impact of mortality risk on life insurance is examined.

Description: This is a study on the Nesbitt formula and the variants to this formula.

Description: It has been twenty years since DeWit 1982 first applied fuzzy logic FL to insurance. That article sought to quantify the fuzziness in underwriting. Since then, the universe of discourse has expanded considerably and now the FL methodologies include used fuzzy set theory, fuzzy numbers and fuzzy arithmetic, fuzzy inference systems, fuzzy clustering algorithms, fuzzy linear programming, fuzzy regression, and hybrids involving other soft computing technologies. This article presents an overview of these studies.

Description: In this essay, we focus on the “Prospective Solvency Assessment” aspect of the NAIC’s ORSA. This places a spotlight on an insurer’s strategic decision making from a risk and capital management perspective. It requires those in senior management to describe actions they would take to address possible adverse changes in their risk and solvency position, i.e., what they would do to reduce risk, increase capital, and/or adjust the business strategy. We contend that by applying a slightly broader scope than the specific requirements of the NAIC’s ORSA, insurers can develop this into a valuable process to support risk-based decision making. In this essay, we initially consider how to determine which metrics to include in such a process and then discuss some important business requirements and practical challenges for implementing a calculation approach to support it. Part of the essay collection "Risk Metrics for Decision Making and ORSA"

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: The costly, time-consuming and complicated process of enterprise risk management (ERM) can be improved in many companies and made less tedious for managers by using reasonable data and templates obtained from peer group entities. The models used to calculate economic capital (EC) often underestimate its value because they do not consider decision-maker perception about risk. We assume here that managers, as the decision-makers, have appropriate business understanding and they can provide substantial information about risk characteristics regarding all business processes. Therefore we are focused, in this paper, on collecting data from managers across the different businesses to derive the appropriate knowledge about risky events, the importance of particular types of risk, the relationship between the risk outcomes and the level of risk control in a particular industry. We conclude that the collected data has a high potential for use as a benc hmarking reference and analysis for improving ERM models for individual businesses.

Description: This paper investigates the optimal multi-period proportional reinsurance strategy that minimizes the ruin probability of an insurer. Some conditions under which it is optimal to have no reinsurance are derived. Based on these results, a new concept, capital threshold of proportional reinsurance, is introduced. In the case of two periods, a lower bound of the capital threshold of proportional reinsurance is provided. Some economic implications and properties of this new concept are also discussed.

Description: The paper o ers a simple framework for ranking the common reinsurance struc- tures in practice with the theory of stochastic orders. The basic idea is to slice the space of reinsurance structures into groups by expected loss cost to facilitate the comparisons within the group and between groups. Given the standard risk aversion assumption in economics, a spectrum of reinsurance structures with the same expected loss cost can be compared analytically with one another and sequenced based on their risk coverages under the convex order. The paper then expands the dimension of the comparison to groups of reinsurance structures with di erent expected loss costs, which can be ranked under the increasing convex order and the usual stochastic order. As such, the paper maps out the ordering for the entire space of reinsurance structures and presents it in a ma- trix format for quick reference. The implication

Description: This paper investigates the use of models of dependent mortality for determining annuity values. We discuss a broad class of parametric models using a bivariate survivorship function called a copula. Using data from a large insurance company to illustrate our methods, we calculate maximum likelihood estimates to calibrate the rnodel.