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.     

Principal Applications of Bayesian Methods in Actuarial Science

Abstract

Bayesian ideas were introduced into actuarial science in the late 1960s in the form of empirical credibility methods for premium setting. The advance of the Bayesian methodology was slow due to its subjective nature and to the computational difficulties associated with the full Bayesian analysis. This paper offers a brief survey of Bayesian solutions to some actuarial problems and discusses the current state of research.     

Sequential Quasi-Credibility for Scale Dispersion Models

The sequential approach to credibility, developed by Landsman and Makov [(1999a) On stochastic approximation and credibility. Scand. Actuarial J. 1, 15-31; (1999b) Sequential credibility evaluation for symmetric location claim distributions. Insurance: Math. Econ. 24, 291-300] is extended to the scale dispersion family, which contains distributions often used in actuarial science: log-normal, Weibull, Half normal, Stable, Pareto, to mention only a few. For members of this family a sequential quasi-credibility formula is devised, which can also be used for heavy tailed claims. The results are illustrated by a study of log-normal claims.     

Notes on some present day Questions regarding Mortality

Abstract

In the absence of any definite development of actuarial science in Great Britain which would be a suitable subject for this paper I considered that the most useful course would be to discuss some of the problems which are at present exercising my own mind, and no doubt the minds of many of my colleagues. Probably, many of these questions are akin to others which arise in other countries, so that their discussion at a meeting such as this may prove of mutual interest, while even if they are peculiar to ourselves, a disclosure of our doubts and difficulties will indicate the directions in which future developments may be expected. I must, however, emphasize that any views expressed are purely personal, and do not in any way pretend to represent a consensus of British actuarial opinion. Throughout this paper only Ordinary (as opposed to Industrial) business in considered.     

Extending composite loss models using a general framework of advanced computational tools

ABSTRACT

Composite models have a long history in actuarial science because they provide a flexible method of curve-fitting for heavy-tailed insurance losses. The ongoing research in this area continuously suggests methodological improvements for existing composite models and considers new composite models. A number of different composite models have been previously proposed in the literature to fit the popular data set related to Danish fire losses. This paper provides the most comprehensive analysis of composite loss models on the Danish fire losses data set to date by evaluating 256 composite models derived from 16 parametric distributions that are commonly used in actuarial science. If not suitably addressed, inevitable computational challenges are encountered when estimating these composite models that may lead to sub-optimal solutions. General implementation strategies are developed for parameter estimation in order to arrive at an automatic way to reach a viable solution, regardless of the specific head and/or tail distributions specified. The results lead to an identification of new well-fitting composite models and provide valuable insights into the selection of certain composite models for which the tail-evaluation measures can be useful in making risk management decisions.     

Credibility Using Copulas

Abstract

Credibility is a form of insurance pricing that is widely used, particularly in North America. The theory of credibility has been called a “cornerstone” in the field of actuarial science. Students of the North American actuarial bodies also study loss distributions, the process of statistical inference of relating a set of data to a theoretical (loss) distribution. In this work, we develop a direct link between credibility and loss distributions through the notion of a copula, a tool for understanding relationships among multivariate outcomes.

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 the univariate tdistributions, 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.

We present an illustrative example of Massachusetts automobile claims, and compare our new credibility estimates with those currently existing in the literature.     

Actuarial validity of the binomial distribution for large numbers of lives with small mortality probabilities

Abstract

In a previous paper (see reference [1]), the binomial distribution was shown to be valid under rather general conditions for the case of a large number of statistically independent lives with possibly unequal mortality probabilities. This note extends these results to some situations where the lives are not necessarily statistically independent. An analysis is presented which indicates that these situations include most actuarial applications involving a large number of lives.     

Representation of concave distortions and applications

ABSTRACT

A family of concave distortion functions is a set of concave and increasing functions, mapping the unity interval onto itself. Distortion functions play an important role defining coherent risk measures. We prove that any family of distortion functions which fulfils a certain translation equation, can be represented by a distribution function. An application can be found in actuarial science: moment-based premium principles are easy to understand but in general are not monotone and cannot be used to compare the riskiness of different insurance contracts with each other. Our representation theorem makes it possible to compare two insurance risks with each other consistent with a moment-based premium principle by defining an appropriate coherent risk measure.     

“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.     

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.     

Preferences representation in actuarial sciences

Abstract

The paper presents a generalization of Bernoulli's principle and of Von Neumann-Morgenstern's theorem with a view to lending more realism to the representation of preferences in actuarial sciences.     

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.     

Fuzzy Financial Pricing of Property-Liability Insurance

Abstract

This paper uses fuzzy set theory (FST) to solve a problem in actuarial science, the financial pricing of property-liability insurance contracts. The fundamental concept of FST is the alternative formalization of membership in a set to include the degree or strength of membership. FST provides consistent mathematical rules for incorporating vague, subjective, or judgmental information into complex decision processes. It is potentially important in insurance pricing because much of the information about cash flows, future economic conditions, risk premiums, and other factors affecting the pricing decision is subjective and thus difficult to quantify by using conventional methods. To illustrate the use of FST, we “fuzzify” a well-known insurance financial pricing model, provide numerical examples of fuzzy pricing, and propose rules for project decision-making using FST. The results indicate that FST can lead to significantly different decisions than the conventional approach.     

Some Nonlinear Threshold Autoregressive Time Series Models for Actuarial Use

Abstract

This paper introduces nonlinear threshold time series modeling techniques that actuaries can use in pricing insurance products, analyzing the results of experience studies, and forecasting actuarial assumptions. Basic “self-exciting” threshold autoregressive (SETAR) models, as well as heteroscedastic and multivariate SETAR processes, are discussed. Modeling techniques for each class of models are illustrated through actuarial examples. The methods that are described in this paper have the advantage of being direct and transparent. The sequential and iterative steps of tentative specification, estimation, and diagnostic checking parallel those of the orthodox Box-Jenkins approach for univariate time series analysis.     

Actuarial Issues in the Novels of Jane Austen

Abstract

The novels of Jane Austen have enjoyed a resurgence of popularity recently, and many new readers have come to appreciate the relevance of her stories to modern times. This relevance should be particularly evident to actuaries, however, because the novels deal quite explicitly with the issues of wealth, inheritance, mortality, and life expectancy that confronted the nonworking classes of the early nineteenth century.

This paper examines the six novels of Jane Austen from an actuarial perspective. It provides historical background on inheritances, clerical livings, and mortality, and it analyzes the way in which these issues are central to Austen’s novels. It uses a contemporary mortality table to assess the accuracy with which Austen’s characters estimate life expectancies and annuity calculations. It presents a close study of Sense and Sensibility, a novel in which a number of actuarial issues are central to the plot and are presented in great detail. Finally, it suggests that Austen’s own background and family life meant that actuarial issues were important in her life and therefore reflected in her novels.

This paper offers a new argument for the relevance of great literature, and it offers actuaries a new perspective from which to explore and understand the history of their profession.     

Contracting out of a state pension scheme

Abstract

The title of this paper is not common in actuarial literature, but the subject is one which has become familiar to every Briton in the last two years and which is essentially actuarial. The phrase "contracting out" has become English shorthand for an arrangement by which a private pension scheme may decide that it will not join a State pension scheme, usually because it already provides equal or better benefits. If the private scheme "contracts out" it will be excused the payment of contributions and will, of course, forgo the benefits.     

Life Expectancy in the Future

Abstract

The public responsibility of life insurance actuaries has changed from supervisory compliance with detailed state laws to certifying adherence to more general regulatory objectives complemented by actuarial standards of practice.     

Growth of the Public Responsibility of Actuaries in Life Insurance

Abstract

The public responsibility of life insurance actuaries has changed from supervisory compliance with detailed state laws to certifying adherence to more general regulatory objectives complemented by actuarial standards of practice.     

Bonus-Malus Systems

Abstract

Bonus-malus is a merit-rating technique used in most of Europe and Asia, and some Latin American and African countries. Policyholders from a given risk cell are subdivided into bonus-malus classes. Their claims histories then modify the class upon each renewal. Markov chain theory provides the tools for the design, evaluation, and comparison of these systems. In this article, definitions and examples of bonus-malus systems are provided (Section 2). The main actuarial tools for the study and design of bonus-malus systems are reviewed (Section 3). In the discussions that follow, Krupa Subramanian outlines a model for analyzing market shares in a competitive environment, a crucial research topic given current deregulation trends, and Pierre Lemaire compares actuarial with regulatory approaches to bonus-malus.     

The DB Underpin Hybrid Pension Plan

Abstract

In this paper the defined benefit underpin guarantee is valued as a financial option, within the traditional funding paradigms of actuarial science. Assuming fixed interest rates, and assuming that salaries can be treated as a tradable asset, we value the guarantee using fair value principles. Contribution rates are developed for the Entry Age Normal, Projected Unit Credit, and Traditional Unit Credit funding methods. In addition, for the accruals methods, we demonstrate the implied hedging strategy. The traditional unit credit offers the best method of these three, as it is consistent with the principles of financial economics, and the resulting contributions more naturally follow the cost of the emerging benefit, without creating expensive barriers to new hires. The method generates significant contribution volatility, and we demonstrate how this can be reduced with suitable benefit design and ongoing risk management.