An actuarial analysis of the French bonus-malus system

The bonus-malus system in force in France differs from most of those used in industrialized countries around the world. Policyholders do not move inside a scale but their premium is obtained with the help of multiplicative CRM coefficients (CRM stands for the acronym of the French coefficient de réduction-majoration). The French bonus-malus system has been the topic of very few scientific investigations in the actuarial literature. This paper purposes to analyze this bonus-malus system in details. Despite its apparent simplicity, it will be seen that it leads to nontrivial mathematical problems. The financial equilibrium of the bonus-malus system is also investigated thanks to the multivariate De?Pril's algorithm for the convolution of independent and identically distributed random vectors.     

Asymptotic Fairness of Bonus-Malus Systems and Optimal Scales of Premiums

In this paper we try to evaluate the asymptotic fairness of bonus-malus systems, assuming the simplest case when there is no hunger for bonus. The asymptotic fairness has to be understood as the bonus-malus system ability in assessing the individual risks in the long run. Firstly we define the asymptotic fairness of a bonus-malus system following an expression that can be found in J. Lemaire [1985]: Automobile Insurance. Actuarial Models. Dordrecht: Kluwer-Nijhoff Publishing, p. 168. Secondly, we define a measure of the global asymptotic fairness considering the structure function of the risk group. Finally we try to calculate, for each set of transition rules and a given structure function, the scale of premiums that brings the global asymptotic fairness closest to the ideal situation where each insured pays in the long run a premium corresponding to its own claim frequency. This is possible thanks to the application of a multiobjective optimization technique named Goal Programming.     

Second-order Bayesian revision of a generalised linear model

It is well known that the exponential dispersion family (EDF) of univariate distributions is closed under Bayesian revision in the presence of natural conjugate priors. However, this is not the case for the general multivariate EDF. This paper derives a second-order approximation to the posterior likelihood of a naturally conjugated generalised linear model (GLM), i.e., multivariate EDF subject to a link function (Section 5.5). It is not the same as a normal approximation. It does, however, lead to second-order Bayes estimators of parameters of the posterior. The family of second-order approximations is found to be closed under Bayesian revision. This generates a recursion for repeated Bayesian revision of the GLM with the acquisition of additional data. The recursion simplifies greatly for a canonical link. The resulting structure is easily extended to a filter for estimation of the parameters of a dynamic generalised linear model (DGLM) (Section 6.2). The Kalman filter emerges as a special case. A second type of link function, related to the canonical link, and with similar properties, is identified. This is called here the companion canonical link. For a given GLM with canonical link, the companion to that link generates a companion GLM (Section 4). The recursive form of the Bayesian revision of this GLM is also obtained (Section 5.5.3). There is a perfect parallel between the development of the GLM recursion and its companion. A dictionary for translation between the two is given so that one is readily derived from the other (Table 5.1). The companion canonical link also generates a companion DGLM. A filter for this is obtained (Section 6.3). Section 1.2 provides an indication of how the theory developed here might be applied to loss reserving. A sequel paper, providing numerical illustrations of this, is planned.     

Financial Economics and Actuarial Practice

Abstract

Starting in the United Kingdom and continuing through the U.S. and Canadian actuarial professions, proponents of financial economics have been forcefully promoting a review of traditional actuarial practices and training. In particular, the financial theories first proposed by Modigliani and Miller and subsequently developed by others have been used to highlight serious weaknesses in typical actuarial thinking. In summary, it is claimed that much actuarial advice wrongly specifies value, that guidelines and standards need radical revision, and that traditional actuarial intuition suffers in comparison to newer modes of thought adopted by other professions.

This paper examines concepts from both financial economics and actuarial science as applied to defined benefit schemes using a simple discounted cash-flow framework as a reference point. The general finding is that many standard modes of actuarial thought are, in fact, indefensible when examined with the tools and techniques of financial economics. The call for revision of actuarial training and practices is credible and necessary.

However, the paper also touches upon areas where a heavy-handed application of finance theory could be misguided due to limitations in the simple financial economic models presented. It concludes that financial economics should be carefully integrated into actuarial thought rather than appended to existing actuarial theory or inserted as a wholesale replacement.     

Actuarial Modeling with MCMC and BUGs

Abstract

In this paper, the author reviews some aspects of Bayesian data analysis and discusses how a variety of actuarial models can be implemented and analyzed in accordance with the Bayesian paradigm using Markov chain Monte Carlo techniques via the BUGS (Bayesian inference Using Gibbs Sampling) suite of software packages. The emphasis is placed on actuarial loss models, but other applications are referenced, and directions are given for obtaining documentation for additional worked examples on the World Wide Web.     

S -Convex Extrema,Taylor-Type Expansions and Stochastic Approximations

The present work studies s -convex orders using a remarkable probabilistic generalization of Taylor's theorem obtained by Massey & Whitt (1993) and further discussed by Lin (1994). We propose two methods for approximating a given risk with known first moments by means of s -convex extremal distributions. The goodness of those approximations is explored using stop-loss distances. Several applications show the interest of this approach in actuarial sciences.     

Fair valuation of insurance liabilities via mean-variance hedging in a multi-period setting

A general class of fair valuations which are both market-consistent (mark-to-market for any hedgeable part of a claim) and actuarial (mark-to-model for any claim that is independent of financial market evolutions) was introduced in Dhaene et al. [Insurance: Mathematics & Economics, 76, 14–27 (2017)] in a single period framework. In particular, the authors considered mean-variance hedge-based (MVHB) valuations where fair valuations of insurance liabilities are expressed in terms of mean-variance hedges and actuarial valuations. In this paper, we generalize this MVHB approach to a multi-period dynamic investment setting. We show that the classes of fair valuations and MVHB valuations are equivalent in this generalized setting. We derive tractable formulas for the fair valuation of equity-linked contracts and show how the actuarial part of their MVHB valuation decomposes into a diversifiable and a non-diversifiable component.     

Basic concepts in life assurance mathematics

Abstract

1. Introduction.

In this paper the basic concepts of the life insurance mathematics will be discussed. Due to the fact that the importance of the probability calculus as a hasis for the actuarial science has repeatedly been disclaimed in recent literature (See e.g. Ernst ZWinggi (1]), the present author feels that there is a justification for reconsidering the fundamental ideas of the actuarial science.     

Probability of ruin with variable premium rate

Abstract

The probability of ruin is investigated under the influence of a premium rate which varies with the level of free reserves. Section 4 develops a number of inequalities for the ruin probability, establishing upper and lower bounds for it in Theorem 4. Theorem 5 gives an expression for the ruin probability, and it is seen in Section 5 that this amounts to a generalization of the ruin probability given by Gerber for the special case of a negative exponential claim size distribution. In that same section it is shown the Lundberg's inequality is not derivable from the generalized theory of Section 4, and this is seen as a drawback of the methods used there. Sections 6 and 7 deal with some special cases, including claim size distributions with monotone failure rates. Section 8 shows that, in contrast with the result for a constant premium that the probability of ruin for zero initial reserve is independent of the claim size distribution, the same result does not hold when the premium rate is allowed to vary. Section 9 gives some comments on the possible effect of “dangerousness” of a claim size distribution on ruin probability.     

Author’s Reply: Skewness and Stock Option Prices - Discussion by Terence Chan,Michel Jacques,Heinz H. Mu¨ller,Geérard Pafumi,Elias S.W. Shiu,and Serena Tiong,July 1997

Abstract

In recent years, the combined effects of deregulation in financial services, along with advances in telecommunications and information technology, are forcing far-reaching changes upon the insurance industry. The result is the industry is becoming more competitive. The emerging role of electronic commerce (e-commerce) is particularly important and interesting to study.

I offer a brief survey of the role of e-commerce in the insurance industry. The paper is organized in the following manner: Section 1 summarizes Internet trends and discusses various related public policy issues; Section 2 addresses online insurance supply and demand; Section 3 discusses the economics of disintermediation and reintermediation and explains how this applies to e-commerce in the insurance industry. Finally, Section 4 offers a set of concluding remarks.     

Case Studies Using Panel Data Models

Abstract

In this paper, we examine case studies from three different areas of insurance practice: health care, workers’ compensation, and group term life. These different case studies illustrate how the broad class of panel data models can be applied to different functional areas and to data that have different features. Panel data, also known as longitudinal data, models are regression-type models that have been developed extensively in the biological and economic sciences. The data features that we discuss include heteroscedasticity, random and fixed effect covariates, outliers, serial correlation, and limited dependent variable bias. We demonstrate the process of identifying these features using graphical and numerical diagnostic tools from standard statistical software.

Our motivation for examining these cases comes from credibility rate making, a technique for pricing certain types of health care, property and casualty, workers’ compensation, and group life coverages. It has been a part of actuarial practice since Mowbray’s (1914) fundamental contribution. In earlier work, we showed how many types of credibility models could be expressed as special cases of panel data models. This paper exploits this link by using tools developed in connection with panel data models for credibility rate-making purposes. In particular, special routines written for credibility rate-making purposes are not required.     

Optimal proportional reinsurance with a loss-dependent premium principle

ABSTRACT

Empirical studies suggest that many insurance companies recontract with their clients on premiums by extrapolating past losses: a client is offered a decrease in premium if the monetary amounts of his claims do not exceed some prespecified quantities, otherwise, an increase in premium. In this paper, we formulate the empirical studies and investigate optimal reinsurance problems of a risk-averse insurer by introducing a loss-dependent premium principle, which uses a weighted average of history losses and the expectation of future losses to replace the expectation in the expected premium principle. This premium principle satisfies the bonus-malus and smoothes the insurer's wealth. Explicit expressions for the optimal reinsurance strategies and value functions are derived. If the reinsurer applies the loss-dependent premium principle to continuously adjust his premium, we show that the insurer always needs less reinsurance when he also adopts this premium principle than when he adopts the expected premium principle.     

Research Into the Aging Process: A Survey

Abstract

We are in the midst of a revolution in biological knowledge. Although research into the aging process was begun long before the Human Genome Project, it has benefited greatly from the powerful tools and techniques spun off from that endeavor. Current research is providing knowledge about life processes that may offer the prospect of slowing the aging process. Dr. Francis Collins, Director of the National Human Genome Research Institute, has predicted that “By 2030, major genes responsible for the aging process in humans will likely have been identified, and clinical trials with drugs to retard the process may well be getting underway” (Collins 2000). A growing number of scientists recognize extension of the maximum life span as a possibility.

The actuarial profession cannot lay claim to expertise in the area of mortality while ignoring scientific research into the causes of aging. This paper provides a brief overview of the subject and a bibliography for those interested in pursuing the matter further. It offers a brief historical perspective, a survey of current research, and a glimpse of future possibilities.?     

Stochastic Models for Continuing Care Retirement Communities

Abstract

Continuing care retirement communities (CCRCs) offer housing and a variety of services, including long-term care. Typically, the cost of this long-term care is wholly or partially covered by entry and/or periodic fees. Thus, CCRCs provide a long-term-care insurance benefit. For this and other reasons, actuarial involvement in the financial management of CCRCs is desirable. To carry out actuarial analyses of CCRCs, appropriate models are required to describe the status of individual residents and the CCRC population.

This paper presents models that assume that, at any time, a resident is in a given “state,” which is determined by the individual’s care requirements. The resident may make “transitions” between states at various times, and randomness is associated with both the transition times and the states entered. Actuarial calculations using such a model are discussed, and numerical illustrations are provided. A simple model is examined first; then generalizations are considered. The model for an individual resident can be embedded in a model for a CCRC population. This is explored with particular attention given to the “high-demand” situation in which potential residents are always waiting to enter the community. With this model, the goal is to analyze the future care requirements of the CCRC population.     

Measuring Terminable Postretirement Obligations

Abstract

New approaches are needed to value benefit plans subject to unilateral changes or termination. The paper focuses on postretirement health benefits, but the thesis may be relevant to any flow not guaranteed by law or accumulating funds.

Retiree health benefits have usually been extended to participating active employees only in concert with a reserved right by the plan sponsor to control the design and, by implication, the cash flow. Over the course of the last fifteen years, this reserved extension of benefits has almost invariably led to reductions in benefits, when compared to the plan of benefits at an earlier period. In most cases, such reductions were anticipated under the circumstances that came to prevail (high health care cost increases), but were not taken into account by most of the projection and discounting methods of the time.

Current actuarial and accounting methods generate present values for terminable retiree health plans that have little credibility as measures of the beneficiary’s asset or the sponsor’s liability. Improvements are needed that will expand the actuarial toolbox and provide solutions to economic and accounting problems.

This paper provides a basis for discussion of assumptions that are appropriate when the plan sponsor can unilaterally and dramatically change future cash flows. The paper discusses how actuaries might best approach measurement situations where further plan reductions, or outright terminations, are to be anticipated. It introduces refinements and briefly discusses how each would fit with the usual actuarial model and how differences might affect behavior. These ideas are related to financial economics and the Bader-Gold paper “Reinventing Pension Actuarial Science.” Before concluding, the wider topic of discount rate selection is briefly addressed.     

“Economic Capital Allocation Derived from Risk Measures”, Eddy Van den Borre,January 2003

Abstract

The sustained reduction in mortality rates and its systematic underestimation has been attracting the significant interest of researchers in recent times because of its potential impact on population size and structure, social security systems, and (from an actuarial perspective) the life insurance and pensions industry worldwide. Despite the number of papers published in recent years, a comprehensive review has not yet been developed.

This paper attempts to be the starting point for that review, highlighting the importance of recently published research—most of the references cited span the last 10 years—and covering the main methodologies that have been applied to the projection of mortality rates in the United Kingdom and the United States. A comparative review of techniques used in official population projections, actuarial applications, and the most influential scientific approaches is provided. In the course of the review an attempt is made to identify common themes and similarities in methods and results.

In both official projections and actuarial applications there is some evidence of systematic overestimation of mortality rates. Models developed by academic researchers seem to reveal a trade-off between the plausibility of the projected age pattern and the ease of measuring the uncertainty involved. The Lee-Carter model is one approach that appears to solve this apparent dilemma.

There is a broad consensus across the resulting projections: (1) an approximately log-linear relationship between mortality rates and time, (2) decreasing improvements according to age, and (3) an increasing trend in the relative rate of mortality change over age. In addition, evidence suggests that excessive reliance on expert opinion—present to some extent in all methods—has led to systematic underestimation of mortality improvements.     

Översikt av utländska aktuarietidskrifter

Abstract

In this paper a continuous-time model of a reinsurance market is presented, which contains the principal components of uncertainty transparent in such a market: Uncertainty about the time instants at which accidents take place, and uncertainty about claim sizes given that accidents have occurred.

Due to random jumps at random time points of the underlying claims processes, the absence of arbitrage opportunities is not sufficient to give unique premium functionals in general. Market preferences are derived under a necessary condition for a general exchange equilibrium. Information constraints are found under which premiums of risks are determined. It is demonstrated how general reinsurance treaties can be uniquely split into proportional contracts and nonproportional ones.

Several applications to reinsurance markets are given, and the results are compared to the corresponding theory of the classical one-period model of a reinsurance syndicate.

This paper attempts to reach a synthesis between the classical actuarial risk theory of insurance, in which virtually no economic reasoning takes place but where the net reserve is represented by a stochastic process, and the theory of partial equilibrium price formation at the heart of the economics of uncertainty.     

Weighted Pricing Functionals With Applications to Insurance

Abstract

We explore the role of weighted distributions in pricing insurance risks. In particular, we relate the distributions to actuarial and economic premium calculation principles and in this way provide a unifying methodology for constructing new principles and analyzing known ones.     

Ladislaus von Bortkiewicz

Summary

In the theory of Poisson processes and compound Poisson processes with time-dependent change variables some results are obtained by use of a transformed change variable, independent of time. The theorem presented below shows that this method can be used in a wide class of problems, many of which are of actuarial interest.     

Large-scale participatory co-shaping of technological developments: First experiments with PARDIZIPP

A future-oriented participatory procedure on the basis of the Delphi method was developed and empirically tested a first time with the goal to improve the shaping of technological developments. The technology under study here was micro-electronics or rather their relationship with labor and the test took place in NorthRhine–Westphalia.Today problems exist in all walks of life. There is a lot of talk about today's problems as if they were new, though one has heard similar arguments throughout history. How do we assess if we are really in danger of bringing the world to an end? Although this danger appears real, it would not be the first time in history that people have thought and felt like this—However, one thing that is new are the consequences of modern sciences and technology, which are not suited to given social and environmental requirements. They have given rise to questions concerning the quality of the decision-makers. The questioning of many of these decisions has increased for some time and is now getting more and more specific, with a demand for quality and information rather than managerial skills and competitiveness from the decision-makers. The term `decision-maker' describes those who determine the application of technology, science and technical equipment which has either existed for a long time already or has recently been developed.—It is not easy to change the structures and processes of decision-making so that new structures and processes will be more suited to social and environmental requirements. We have tested our ideas as to how this could be done, in an empirical project. Although we called it `Project NRW–2000', it would probably be better described as an experiment.²—We persuaded 90 ordinary people to participate in this project as `experts on daily life and work'. This group was asked to work in six regional sub-groups and discuss, with reference to three given normative societal scenarios for the year 2020, the relationship between microelectronics and labour markets of the year 2020, on the basis of a participatory Delphi procedure. Before we elaborate on the concept of our project in Section 3, we would like to outline it in terms of the mainstream of the sociology of technology as well as with research on `acceptance' in Section 1. In Section 2we will briefly illustrate the framework of the research programme `Socially Oriented Shaping of Technology' of the state of Northrhine–Westphalia, which funded our research project. Section 4particularly deals with the participatory elements of our project, while Section 5is devoted to the development of the scenarios. Section 6sums up the results of the `scenario-construction'. Regarding specific elements, we restrict ourselves to topics concerning technology, labour, and the relationship between women workers/employees and technology. As a final outlook we deal with the political implications of our approach. All that is left is to remind our readers that we regard this project as a first application or experiment within our overall approach.