This article introduces actuaries to the concept of "copulas," a tool for understanding relationships among multivariate outcomes. A copula is a function that links univariate marginals to their full multivariate distribution. Copulas were introduced in 1959 in the context of probabilistic metric spaces. Recently, there has been a rapidly developing literature on the statistical properties and applications of copulas. This article explores some of these practical applications, including estimation of joint life mortality and multidecrement models. In addition, we describe basic properties of copulas, their relationships to measures of dependence and several families of copulas that have appeared in the literature. An annotated bibliography provides a resource for researchers and practitioners who wish to continue their study of copulas. This article will also be useful to those who wish to use copulas for statistical inference. Statistical inference procedures are illustrated using insurance company data on losses and expenses. For this data, we (1) show how to fit copulas and (2) describe their usefulness by pricing a reinsurance contract and estimating expenses for pre-specified losses.

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Annuities are contractual guarantees, issued by insurance companies, pension plans, and government retirement systems, that offer promises to provide periodic income over the lifetime of individuals. It is well-known how to use univariate models of survivorship for valuing annuities.However, standard industry practice assumes independence of lives when valuing annuities where the promise is based on more than one life. 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.The estimation results show strong positive dependence between joint lives. This statistically significant result translates into real economic significance. That is, there is an approximate five percent reduction in annuity values when dependent mortality models are used compared to the standard models that assume independence. We show that the results are robust in terms of the choice of parametric family of distribution functions. 32Annuity Valuation with Dependent Mortality

During the 1980s and early 1990s, the world insurance market grew substantially. World insurance premiums in 1993 accounted for about 8 percent of world gross domestic product (GDP), compared to 4 percent in 1984. This article explains a substantial proportion of the variation in propertyliability insurance consumption across countries belonging to the Organization for Economic Cooperation and Development (OECD). The study focuses on two lines of insurance: motor vehicle and general liability. The authors' analysis indicates that economic conditions affect the demand for insurance differently across lines of coverage. In particular, the authors' results suggest that income has a far greater effect on motor vehicle insurance consumption than on general liability insurance consumption. The authors find evidence that several factors are important in explaining the purchase of both kinds of insurance. These factors include income, wealth, the percent of a country's insurance market controlled by foreign firms, and the form of the legal system in the country.

This work describes statistical modeling of detailed, micro-level automobile insurance records. We consider 1993-2001 data from a major insurance company in Singapore. By detailed micro-level records, we refer to experience at the individual vehicle level, including vehicle and driver characteristics, insurance coverage and claims experience, by year. The claims experience consists of detailed information on the type of insurance claim, such as whether the claim is due to injury to a third party, property damage to a third party or claims for damage to the insured, as well as the corresponding claim amount.We propose a hierarchical model for three components, corresponding to the frequency, type and severity of claims. The …rst is a negative binomial regression model for assessing claim frequency. The driver's gender, age, and no claims discount as well as vehicle age and type turn out to be important variables for predicting the event of a claim. The second is a multinomial logit model to predict the type of insurance claim, whether it is third party injury, third party property damage, insured's own damage or some combination. Year, vehicle age and vehicle type turn out to be important predictors for this component.Our third model is for the severity component. Here, we use a generalized beta of the second kind long-tailed distribution for claim amounts and also incorporate predictor variables. Year, vehicle age and a person's age turn out to be important predictors for this component. Not surprisingly, we show that there is a signi…cant dependence among the di¤erent claim types; we use a t-copula to account for this dependence.The three component model provides justi…cation for assessing the importance of a rating variable. When taken together, the integrated model allows an actuary to predict automobile claims more e¢ ciently than traditional methods. Using simulation, we demonstrate this by developing predictive distributions and calculating premiums under alternative reinsurance coverages.