Model misspecification,Bayesian versus credibility estimation,and Gibbs posteriors

ABSTRACT

In the context of predicting future claims, a fully Bayesian analysis – one that specifies a statistical model, prior distribution, and updates using Bayes's formula – is often viewed as the gold-standard, while Bühlmann's credibility estimator serves as a simple approximation. But those desirable properties that give the Bayesian solution its elevated status depend critically on the posited model being correctly specified. Here we investigate the asymptotic behavior of Bayesian posterior distributions under a misspecified model, and our conclusion is that misspecification bias generally has damaging effects that can lead to inaccurate inference and prediction. The credibility estimator, on the other hand, is not sensitive at all to model misspecification, giving it an advantage over the Bayesian solution in those practically relevant cases where the model is uncertain. This begs the question: does robustness to model misspecification require that we abandon uncertainty quantification based on a posterior distribution? Our answer to this question is No, and we offer an alternative Gibbs posterior construction. Furthermore, we argue that this Gibbs perspective provides a new characterization of Bühlmann's credibility estimator.     

Bayesian Risk Management for Equity-Linked Insurance

This paper describes how to apply Markov Chain Monte Carlo (MCMC) techniques to a regime switching model of the stock price process to generate a sample from the joint posterior distribution of the parameters of the model. The MCMC output can be used to generate a sample from the predictive distribution of losses from equity linked contracts, assuming first an actuarial approach to risk management and secondly a financial economics approach. The predictive distribution is used to show the effect of parameter uncertainty on risk management calculations. We also explore model uncertainty by assuming a GARCH model in place of the regime switching model. The results indicate that the financial economics approach to risk management is substantially more robust to parameter uncertainty and model uncertainty than the actuarial approach.     

Intrinsic objective Bayesian estimation of the mean of the Tweedie family

ABSTRACT

The Tweedie family, which is classified by the choice of power unit variance function, includes heavy tailed distributions, and as such could be of significant relevance to actuarial science. The class includes the Normal, Poisson, Gamma, Inverse Gaussian, Stable and Compound Poisson distributions. In this study, we explore the intrinsic objective Bayesian point estimator for the mean value of the Tweedie family based on the intrinsic discrepancy loss function – which is an inherent loss function arising only from the underlying distribution or model, without any subjective considerations – and the Jeffreys prior distribution, which is designed to express absence of information about the quantity of interest. We compare the proposed point estimator with the Bayes estimator, which is the posterior mean based on quadratic loss function and the Jeffreys prior distribution. We carry a numerical study to illustrate the methodology in the context of the Inverse Gaussian model, which is fully unexplored in this novel context, and which is useful to insurance contracts.     

Bayesian Premium Rating with Latent Structure

We propose a fully Bayesian approach to non-life risk premium rating, based on hierarchical models with latent variables for both claim frequency and claim size. Inference is based on the joint posterior distribution and is performed by Markov Chain Monte Carlo. Rather than plug-in point estimates of all unknown parameters, we take into account all sources of uncertainty simultaneously when the model is used to predict claims and estimate risk premiums. Several models are fitted to both a simulated dataset and a small portfolio regarding theft from cars. We show that interaction among latent variables can improve predictions significantly. We also investigate when interaction is not necessary. We compare our results with those obtained under a standard generalized linear model and show through numerical simulation that geographically located and spatially interacting latent variables can successfully compensate for missing covariates. However, when applied to the real portfolio data, the proposed models are not better than standard models due to the lack of spatial structure in the data.     

Accounting Year Effects Modeling in the Stochastic Chain Ladder Reserving Method

Abstract

In almost all stochastic claims reserving models one assumes that accident years are independent. In practice this assumption is violated most of the time. Typical examples are claims inflation and accounting year effects that influence all accident years simultaneously. We study a Bayesian chain ladder model that allows for accounting (calendar) year effects modeling. A case study of a general liability dataset shows that such accounting year effects contribute substantially to the prediction uncertainty and therefore need a careful treatment within a risk management and solvency framework.     

Do subjective expectations explain asset pricing puzzles?

The structural uncertainty model with Bayesian learning, advanced by Weitzman (AER 2007), provides a framework for gauging the effect of structural uncertainty on asset prices and risk premiums. This paper provides an operational version of this approach that incorporates realistic priors about consumption growth volatility, while guaranteeing finite asset pricing quantities. In contrast to the extant literature, the resulting asset pricing model with subjective expectations yields well-defined expected utility, finite moment generating function of the predictive distribution of consumption growth, and tractable expressions for equity premium and risk-free return. Our quantitative analysis reveals that explaining the historical equity premium and risk-free return, in the context of subjective expectations, requires implausible levels of structural uncertainty. Furthermore, these implausible prior beliefs result in consumption disaster probabilities that virtually coincide with those implied by more realistic priors. At the same time, the two sets of prior beliefs have diametrically opposite asset pricing implications.     

Modelling long-term investment returns via Bayesian infinite mixture time series models

This paper introduces the class of Bayesian infinite mixture time series models first proposed in Lau & So (2004) for modelling long-term investment returns. It is a flexible class of time series models and provides a flexible way to incorporate full information contained in all autoregressive components with various orders by utilizing the idea of Bayesian averaging or mixing. We adopt a Bayesian sampling scheme based on a weighted Chinese restaurant process for generating partitions of investment returns to estimate the Bayesian infinite mixture time series models. Instead of using the point estimates, as in the classical or non-Bayesian approach, the estimation in this paper is performed by the full Bayesian approach, utilizing the idea of Bayesian averaging to incorporate all information contained in the posterior distributions of the random parameters. This provides a natural way to incorporate model risk or uncertainty. The proposed models can also be used to perform clustering of investment returns and detect outliers of returns. We employ the monthly data from the Toronto Stock Exchange 300 (TSE 300) indices to illustrate the implementation of our models and compare the simulated results from the estimated models with the empirical characteristics of the TSE 300 data. We apply the Bayesian predictive distribution of the logarithmic returns obtained by the Bayesian averaging or mixing to evaluate the quantile-based and conditional tail expectation risk measures for segregated fund contracts via stochastic simulation. We compare the risk measures evaluated from our models with those from some well-known and important models in the literature, and highlight some features that can be obtained from our models.     

Bayesian Risk Measures for Derivatives via Random Esscher Transform

Abstract

This paper proposes a model for measuring risks for derivatives that is easy to implement and satisfies a set of four coherent properties introduced in Artzner et al. (1999). We construct our model within the context of Gerber-Shiu’s option-pricing framework. A new concept, namely Bayesian Esscher scenarios, which extends the concept of generalized scenarios, is introduced via a random Esscher transform. Our risk measure involves the use of the risk-neutral Bayesian Esscher scenario for pricing and a family of real-world Bayesian Esscher scenarios for risk measurement. Closed-form expressions for our risk measure can be obtained in some special cases.     

“The Future of Risk Classification in the Age of Predictive DNA-Based Testing”, Donald C. Chambers,January 1999

Abstract

The correlation among multiple lines of business plays an important role in quantifying the uncertainty of loss reserves for insurance portfolios. To accommodate correlation, most multivariate loss-reserving methods focus on the pairwise association between corresponding cells in multiple run-off triangles. However, such practice usually relies on the independence assumption across accident years and ignores the calendar year effects that could affect all open claims simultaneously and induce dependencies among loss triangles. To address this issue, we study a Bayesian log-normal model in the prediction of outstanding claims for dependent lines of business. In addition to the pairwise correlation, our method allows for an explicit examination of the correlation due to common calendar year effects. Further, different specifications of the calendar year trend are considered to reflect valuation actuaries’ prior knowledge of claim development. In a case study, we analyze an insurance portfolio of personal and commercial auto lines from a major U.S. property-casualty insurer. It is shown that the incorporation of calendar year effects improves model fit significantly, though it contributes substantively to the predictive variability. The availability of the realizations of predicted claims permits us to perform a retrospective test, which suggests that extra prediction uncertainty is indispensable in modern risk management practices.     

Eksamen i Forsikringsvidenskab og Statistik ved Köbenhavns Universitet. 2. Del. Den skriftlige Pröve

Abstract

This paper develops a Pareto scale-inflated outlier model. This model is intended for use when data from some standard Pareto distribution of interest is suspected to have been contaminated with a relatively small number of outliers from a Pareto distribution with the same shape parameter but with an inflated scale parameter. The Bayesian analysis of this Pareto scale-inflated outlier model is considered and its implementation using the Gibbs sampler is discussed. The paper contains three worked illustrative examples, two of which feature actual insurance claims data.     

Unconditional distributions obtained from conditional specification models with applications in risk theory

Bivariate distributions, specified in terms of their conditional distributions, provide a powerful tool to obtain flexible distributions. These distributions play an important role in specifying the conjugate prior in certain multi-parameter Bayesian settings. In this paper, the conditional specification technique is applied to look for more flexible distributions than the traditional ones used in the actuarial literature, as the Poisson, negative binomial and others. The new specification draws inferences about parameters of interest in problems appearing in actuarial statistics. Two unconditional (discrete) distributions obtained are studied and used in the collective risk model to compute the right-tail probability of the aggregate claim size distribution. Comparisons with the compound Poisson and compound negative binomial are made.     

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.     

Comparison of Some Bayesian Analyses of Heterogeneity in Group Life Insurance

Norberg (1989) analyses the heterogeneity in a portfolio of group life insurances using a parametric empirical Bayesian approach. In the present paper the model of Norberg is compared to a parametric fully Bayesian model and to a non-parametric fully Bayesian model.     

A Risk Model with Multilayer Dividend Strategy

Abstract

In recent years various dividend payment strategies for the classical collective risk model have been studied in great detail. In this paper we consider both the dividend payment intensity and the premium intensity to be step functions depending on the current surplus level. Algorithmic schemes for the determination of explicit expressions for the Gerber-Shiu discounted penalty function and the expected discounted dividend payments are derived. This enables the analytical investigation of dividend payment strategies that, in addition to having a sufficiently large expected value of discounted dividend payments, also take the solvency of the portfolio into account. Since the number of layers is arbitrary, it also can be viewed as an approximation to a continuous surplus-dependent dividend payment strategy. A recursive approach with respect to the number of layers is developed that to a certain extent allows one to improve upon computational disadvantages of related calculation techniques that have been proposed for specific cases of this model in the literature. The tractability of the approach is illustrated numerically for a risk model with four layers and an exponential claim size distribution.     

On the Joint Distributions of the Time to Ruin,the Surplus Prior to Ruin,and the Deficit at Ruin in the Classical Risk Model

Abstract

The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that utilize the ideas of Value-at-Risk and Tail Value-at-Risk. As is well known, calculation of these and related quantities requires knowledge of the associated probability distributions. In this paper we derive an explicit expression for the joint (defective) distribution of the time to ruin, the surplus immediately prior to ruin, and the deficit at ruin in the classical compound Poisson risk model. As a by-product, we obtain expressions for the three bivariate distributions generated by the time to ruin, the surplus prior to ruin, and the deficit at ruin. Finally, we consider mixed Erlang claim sizes and show how the joint (defective) distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin can be calculated.     

Ruin Problems for Phase-Type(2) Risk Processes

In this paper we consider a risk process in which claim inter-arrival times have a phase-type(2) distribution, a distribution with a density satisfying a second order linear differential equation. We consider some ruin related problems. In particular, we consider the compound geometric representation of the infinite time survival probability, as well as the (defective) distributions of the surplus immediately prior to ruin and of the deficit at ruin. We also consider explicit solutions for the infinite time ruin probability in the case where the individual claim amount distribution is phase-type.     

Analysis of a Generalized Penalty Function in a Semi-Markovian Risk Model

Abstract

In this paper an extension of the semi-Markovian risk model studied by Albrecher and Boxma (2005) is considered by allowing for general interclaim times. In such a model, we follow the ideas of Cheung et al. (2010b) and consider a generalization of the Gerber-Shiu function by incorporating two more random variables in the traditional penalty function, namely, the minimum surplus level before ruin and the surplus level immediately after the second last claim prior to ruin. It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation. Detailed examples are also considered when either the interclaim times or the claim sizes are exponentially distributed. Finally, we also consider the case where the claim arrival process follows a Markovian arrival process. Probabilistic arguments are used to derive the discounted joint distribution of four random variables of interest in this risk model by capitalizing on an existing connection with a particular fluid flow process.     

Portfolio selection using hierarchical Bayesian analysis and MCMC methods

This paper contributes to portfolio selection methodology using a Bayesian forecast of the distribution of returns by stochastic approximation. New hierarchical priors on the mean vector and covariance matrix of returns are derived and implemented. Comparison’s between this approach and other Bayesian methods are studied with simulations on 25 years of historical data on global stock indices. It is demonstrated that a fully hierarchical Bayes procedure produces promising results warranting more study. We carried out a numerical optimization procedure to maximize expected utility using the MCMC (Monte Carlo Markov Chain) samples from the posterior predictive distribution. This model resulted in an extra 1.5 percentage points per year in additional portfolio performance (on top of the Hierarchical Bayes model to estimate μ and Σ and use the Markowitz model), which is quite a significant empirical result. This approach applies to a large class of utility functions and models for market returns.     

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.