Michel Denuit

Actuarial theory for dependent risks : measures, orders and models

The increasing complexity of insurance and reinsurance products has seen a growing interest amongst actuaries in the modelling of dependent risks. For efficient risk management, actuaries need to be able to answer fundamental questions such as: Is the correlation structure dangerous? And, if yes, to what extent? Therefore tools to quantify, compare, and model the strength of dependence between different risks are vital. Combining coverage of stochastic order and risk measure theories with the basics of risk management and stochastic dependence, this book provides an essential guide to managing modern financial risk. * Describes how to model risks in incomplete markets, emphasising insurance risks. * Explains how to measure and compare the danger of risks, model their interactions, and measure the strength of their association. * Examines the type of dependence induced by GLM-based credibility models, the bounds on functions of dependent risks, and probabilistic distances between actuarial models. * Detailed presentation of risk measures, stochastic orderings, copula models, dependence concepts and dependence orderings. * Includes numerous exercises allowing a cementing of the concepts by all levels of readers. * Solutions to tasks as well as further examples and exercises can be found on a supporting website.

A Poisson log-bilinear regression approach to the construction of projected lifetables

This paper implements Wilmoth’s [Computational methods for fitting and extrapolating the Lee–Carter model of mortality change, Technical report, Department of Demography, University of California, Berkeley] and Alho’s [North American Actuarial Journal 4 (2000) 91] recommendation for improving the Lee–Carter approach to the forecasting of demographic components. Specifically, the original method is embedded in a Poisson regression model, which is perfectly suited for age–sex-specific mortality rates. This model is fitted for each sex to a set of age-specific Belgian death rates. A time-varying index of mortality is forecasted in an ARIMA framework. These forecasts are used to generate projected age-specific mortality rates, life expectancies and life annuities net single premiums. Finally, a Brass-type relational model is proposed to adapt the projections to the annuitants population, allowing for estimating the cost of adverse selection in the Belgian whole life annuity market.

The Concept of Comonotonicity in Actuarial Science and Finance: Applications

In an insurance context, one is often interested in the distribution function of a sum of random variables (rv’s). Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio, at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not a realistic one. In The Concept of Comonotonicity in Actuarial Science and Finance: Theory, we determined approximations for sums of rv’s, when the distributions of the components are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. Practical applications of this theory will be considered in this paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented.

Stochastic bounds on sums of dependent risks

There is a growing concern in the actuarial literature for the effect of dependence between individual risks Xi on the distribution of the aggregate claim S=X1+⋯+Xn. Recent work by Dhaene and Goovaerts (Dhaene, J., Goovaerts, M.J., 1996. ASTIN Bulletin 26, 201–212; Dhaene, J., Goovaerts, M.J., 1997. Insurance: Mathematics and Economics 19, 243–253) and Muller (Muller, A., 1997a. Insurance: Mathematics and Economics 21, 219–223; Muller, A., 1997b. Advances in Applied Probability 29, 414–428) has led, among other things, to the identification of the portfolio yielding the smallest and largest stop-loss premiums and hence to bounds on E{φ(S)} for arbitrary non-decreasing, convex functions φ in situations of dependence between the Xi’s. This paper extends these results by showing how to compute bounds on P(S>s) and more generally on E{φ(S)} for monotone, but not necessarily convex functions φ. Special attention is paid to the numerical implementation of the results and examples of application are provided.

Bootstrapping the Poisson log-bilinear model for mortality projection

This paper proposes bootstrap procedures for expected remaining lifetimes and life annuity single premiums in a dynamic mortality environment. Assuming a further continuation of the stable pace of mortality decline, a Poisson log-bilinear projection model is applied to the forecasting of the gender- and age-specific mortality rates for Belgium on the basis of mortality statistics relating to the period 1950–2000. Bootstrap procedures are then used to obtain confidence intervals on the aforementioned quantities.

Association and heterogeneity of insured lifetimes in the Lee-Carter framework

This paper is devoted to the study of some unexpected consequences of the Lee–Carter model for mortality projection. The fact that survival probabilities are governed by a stochastic process induces some positive dependence between insured lifetimes (namely, association). This, in turn, has an impact on solvency capital (as measured by distortion risk measures, for instance). Failing to take this dependence into account, by assuming falsely that the lifetimes are independent, leads to systematic underestimations of the risk capital. The heterogeneity between the policy benefits and the insured lifetimes is also studied (with the help of majorisation, Schur-increasingness and a frailty model).

Stochastic mortality under measure changes

We provide a self-contained analysis of a class of continuous-time stochastic mortality models that have gained popularity in the last few years. We describe some of their advantages and limitations, examining whether their features survive equivalent changes of measures. This is important when using the same model for both market-consistent valuation and risk management of life insurance liabilities. We provide a numerical example based on the calibration to the French annuity market of a risk-neutral version of the model proposed by Lee & Carter (1992).

Smoothing the Lee-Carter and Poisson log-bilinear models for mortality forecasting : a penalized log-likelihood approach

Mortality improvements pose a challenge for the planning of public retirement systems as well as for the private life annuities business. For public policy, as well as for the management of financial institutions, it is important to forecast future mortality rates. Standard models for mortality forecasting assume that the force of mortality at age x in calendar year t is of the form exp(αx + βxκt ). The log of the time series of age-specific death rates is thus expressed as the sum of an age-specific component αx that is independent of time and another component that is the product of time-varying parameter κt reflecting the general level of mortality, and an age-specific component βx that represents how rapidly or slowly mortality at each age varies when the general level of mortality changes. This model is fitted to historical data. The resulting estimated κt s are then modeled and projected as stochastic time series using standard Box–Jenkins methods. However, the estimated βxs exhibit an irregular pattern in most cases, and this produces irregular projected life tables. This article demonstrates that it is possible to smooth the estimated βxs in the Lee–Carter and Poisson log-bilinear models for mortality projection. To this end, penalized least-squares/maximum likelihood analysis is performed. The optimal value of the smoothing parameter is selected with the help of cross validation.

Does positive dependence between individual risks increase stop-loss premiums

Actuaries intuitively feel that positive correlations between individual risks reveal a more dangerous situation compared to independence. The purpose of this short note is to formalize this natural idea. Specifically, it is shown that the sum of risks exhibiting a weak form of dependence known as positive cumulative dependence is larger in convex order than the corresponding sum under the theoretical independence assumption.

Risk classification for claim counts: A comparative analysis of various zero-inflated mixed Poisson and hurdle models

Abstract This paper presents and compares different risk classification models for the annual number of claims reported to the insurer. Generalized heterogeneous, zero-inflated, hurdle, and compound frequency models are applied to a sample of an automobile portfolio of a major company operating in Spain. A statistical comparison between models is performed with the help of various specification tests (Score and Hausman tests for nested models, Vuong test or information criteria for nonnested ones). Interesting results about claiming behavior are obtained.