ASTIN Bulletin

Modelling Adverse Selection in the Presence of a Common Genetic Disorder

poj@peeters-leuven.be — Wed, 09 Dec 2009 11:54:58 GMT

The cost of adverse selection in the life and critical illness (CI) insurance markets, brought about by restrictions on insurers� use of genetic test information, has been studied for a variety of rare single-gene disorders. Only now do we have a study of a common disorder (breast cancer) that accounts for the risk associated with multiple genes. Such a collection of genes is called a polygene. We take two approaches to modelling the severity of adverse selection which may result from insurers being unable to take account of tests for polygenes as well as major genes. First, we look at several genetic testing scenarios, with a corresponding range of possible insurance-buying behaviours, in a market model for CI insurance. Because a relatively large proportion of the population is exposed to adverse polygenic risk, the costs of adverse selection are potentially much greater than have been associated with rare single genes. Second, we use utility models to map out when adverse selection will appear, and which risk groups will cause it. Levels of risk aversion consistent with some empirical studies do not lead to significant adverse selection in our model, but lower levels of risk aversion could effectively eliminate the market.

Demand Elasticity, Risk Classification and Loss Coverage

poj@peeters-leuven.be — Wed, 09 Dec 2009 11:57:33 GMT

This paper investigates the effects of high or low fair-premium demand elasticity in an insurance market where risk classification is restricted. The effects are represented by the equilibrium premium, and the risk-weighted insurance demand or 'loss coverage'. High fair-premium demand elasticity leads to a collapse in loss coverage, with an equilibrium premium close to the risk of the higher-risk population. Low fair-premium demand elasticity leads to an equilibrium premium close to the risk of the lower-risk population, and high loss coverage � possibly higher than under more complete risk classification. The demand elasticity parameters which are required to generate a collapse in coverage in the model in this paper appear higher than the values for demand elasticity which have been estimated in several empirical studies of various insurance markets. This offers a possible explanation of why some insurance markets appear to operate reasonably well under community rating, without the collapse in coverage which insurance folklore suggests.

Quasi-Likelihood Estimation of Benchmark Rates for Excess of Loss Reinsurance Programs

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:00:02 GMT

In this paper a method for determining benchmark rates for the excess of loss reinsurance of a Motor Third Party Liability insurance portfolio will be developed based on observed market rates. The benchmark rates are expressed as a percentage of the expected premium income that is available to cover the whole risk of the portfolio. The rates are assumed to be based on a compound process with a heavy tailed severity, such as Burr or Pareto distributions. In the absence of claim data these assumptions propagate the theoretical benchmark rate component of the regression model. Given the whole set of excess of loss reinsurance rates in a given market, the unknown parameters are estimated within the framework of quasi-likelihood estimation. This framework makes it possible to select a theoretical benchmark rate model and to choose a parsimonious submodel for describing the observed market rates over a 4-years observation period. This method is applied to the Belgian Motor Third Party Liability excess of loss rates observed during the years 2001 till 2004.

Taylor Approximations for Model Uncertainty within the Tweedie Exponential Dispersion Family

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:08:35 GMT

The use of generalized linear models (GLM) to estimate claims reserves has become a standard method in insurance. Most frequently, the exponential dispersion family (EDF) is used; see e.g. England, Verrall [2].We study the so-called Tweedie EDF and test the sensitivity of the claims reserves and their mean square error of predictions (MSEP) over this family. Furthermore, we develop second order Taylor approximations for the claims reserves and the MSEPs for members of the Tweedie family that are difficult to obtain in practice, but are close enough to models for which claims reserves and MSEP estimations are easy to determine. As a result of multiple case studies, we find that claims reserves estimation is relatively insensitive to which distribution is chosen amongst the Tweedie family, in contrast to the MSEP, which varies widely.

Asymptotic Ruin Probabilities of the L�vy Insurance Model under Periodic Taxation

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:10:33 GMT

Recently, Albrecher and his coauthors have published a series of papers on the ruin probability of the L�vy insurance model under the so-called loss-carry-forward taxation, meaning that taxes are paid at a certain fixed rate immediately when the surplus of the company is at a running maximum. In this paper we assume periodic taxation under which the company pays tax at a fixed rate on its net income during each period.We devote ourselves to deriving explicit asymptotic relations for the ruin probability in the most general L�vy insurance model in which the L�vy measure has a subexponential tail, a convolution-equivalent tail, or an exponential-like tail.

On Parameter Estimation in Hierarchical Credibility

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:12:58 GMT

One can find in the literature three main sets of estimators for the variance components in the hierarchical credibility model. This paper presents these estimators in a unified notation, studies some of their properties important for numerical evaluation and compares their relative performance by simulation. The paper also demonstrates how function cm of the R package actuar can be used to fit hierarchical models to insurance data.

Option Pricing in a Jump-Diffusion Model with Regime-Switching

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:14:36 GMT

Nowadays, the regime switching model has become a popular model in mathematical finance and actuarial science. The market is not complete when the model has regime switching. Thus, pricing the regime switching risk is an important issue. In Naik (1993), a jump diffusion model with two regimes is studied. In this paper, we extend the model of Naik (1993) to a multi-regime case. We present a trinomial tree method to price options in the extended model. Our results show that the trinomial tree method in this paper is an effective method; it is very fast and easy to implement. Compared with the existing methodologies, the proposed method has an obvious advantage when one needs to price exotic options and the number of regime states is large. Various numerical examples are presented to illustrate the ideas and methodologies.

Stochastic Mortality

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:18:00 GMT

In this paper, we take the point of view of an insurer dealing with life annuities, which aims at building up a (partial) internal model in order to quantify the impact of mortality risks, namely process and longevity risk, in view of taking appropriate risk management actions. We assume that a life table, providing a best-estimate assessment of annuitants� future mortality is available to the insurer; conversely, the insurer has no access to data sets and the methodology underlying the construction of the life table. Nonetheless, the insurer is aware that, in the presence of mortality risks, a stochastic approach is required. The (projected) life table, which provides a deterministic description of future mortality, should then be used as the basic input of a stochastic model. The model we propose focuses on the annual number of deaths in a given cohort, which we represent allowing for a random mortality rate. To this purpose, we adopt the widely used Poisson model, first assuming a Gamma-distributed random parameter, and second introducing time-dependence in the parameter itself. Further, we define a Bayesian-inferential procedure for updating the parameters to experience in some situations. The setting we define does not demand advanced analytical tools, while allowing for process and longevity risk in a rigorous way. The model is then implemented for capital allocation purposes. We investigate the amount of the required capital for a given life annuity portfolio, based on solvency targets which could be adopted within internal models. The outcomes of such an investigation are compared with the capital required according to some standard rules, in particular those proposed within the Solvency 2 project.

Multi-Level Risk Aggregation

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:19:12 GMT

In this paper we compare the current Solvency II standard and a genuine bottom-up approach to risk aggregation. This is understood to be essential for developing a deeper insight into the possible differences between the diversification assumptions between the standard approach and internal models.

Continuous Monitoring

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:21:28 GMT

We present a model for pricing credit risk protection for a limited liability non-life insurance company. The protection is typically provided by a guaranty fund. In the case of continuous monitoring, i.e., where the market values of the company�s assets and liabilities are continuously observable, and where the market values of assets and liabilities follow continuous processes, regulators can liquidate the insurance company at the instant the market value of its assets equals the market value of its liabilities, implying that the credit protection is worthless. When jumps are included in the claims process, the protection provided by the guaranty fund has a strictly positive market value. The ability to continuously monitor asset prices with continuous sample paths eliminates economic losses from default. Our analysis suggests that economic losses from default stem from jumps in continuously observed asset prices and/or that asset prices are not continuously observed.

Sharing Risk

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:23:44 GMT

We revisit the relative retention problem originally introduced by de Finetti using concepts recently developed in risk theory and quantitative risk management. Instead of using the Variance as a risk measure we consider the Expected Shortfall (Tail-Value-at-Risk) and include capital costs and take constraints on risk capital into account. Starting from a risk-based capital allocation, the paper presents an optimization scheme for sharing risk in a multi-risk class environment. Risk sharing takes place between two portfolios and the pricing of risktransfer reflects both portfolio structures. This allows us to shed more light on the question of how optimal risk sharing is characterized in a situation where risk transfer takes place between parties employing similar risk and performance measures. Recent developments in the regulatory domain (�risk-based supervision�) pushing for common, insurance industry-wide risk measures underline the importance of this question. The paper includes a simple non-life insurance example illustrating optimal risk transfer in terms of retentions of common reinsurance structures.

The Application of Expected-Utility Theory to the Choice of Investment Channels in a Defined-Contribution Retirement Fund

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:25:16 GMT

This study examines the practical application of a system for the derivation of member utility functions for the purpose of recommending investment-channel choice to members of a defined-contribution retirement fund. The utility functions of post-retirement benefits from members of a defined-contribution fund are elicited. The risk aversion of each member is measured and the results are compared with a standard risk-tolerance assessment method.

Scenario Analysis for a Multi-Period Diffusion Model of Risk

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:26:46 GMT

This paper extends and develops the results of a previous paper Malinovskii (2007). Dealing with a simplistic diffusion multi-year model of insurance operations, this paper illustrates the adaptive control approach when the object of control is the balance of solvency and equity. Compared to the previous paper, a new element is the �scenario of nature�, or the incomplete knowledge of future risk, which is quite often the case in insurance. It introduces a new and inevitable randomness in the model and leads to a qualitative difference in its behavior.

A Bootstrap Estimate of the Predictive Distribution of Outstanding Claims for the Schnieper Model

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:28:30 GMT

This paper considers the bootstrapping approach for measuring reserve uncertainty when applying the model of Schnieper for reserves which separate Incurred But Not Reported (IBNR) and Incurred But Not Enough Reserved (IBNER) claims. The Schnieper method has been explored in Liu and Verrall (2009), and the Mean Square Errors of Prediction (MSEP) derived. This paper takes this further by deriving the full predictive distribution, using bootstrapping. Numerical examples are provided and the MSEP from the bootstrapping approach are compared with those obtained analytically.

New Goodness-of-Fit Tests for Pareto Distributions

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:30:15 GMT

A new approach to goodness-of-fit for Pareto distributions is introduced. Based on Euclidean distances between sample elements, the family of statistics and tests is indexed by an exponent in (0,2) on Euclidean distance. The corresponding tests are statistically consistent and have excellent performance when applied to heavy-tailed distributions. The exponent can be tailored to the particular Pareto distribution. The goodness-of-fit statistic measures all types of differences between distributions, hence it is also applicable as a minimum distance estimator. Implementation of the test statistics is developed and applied to estimation of the tail index in three well known examples of claims data, and compared with the classical EDF statistics.

A Note on Nonparametric Estimation of the CTE

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:34:32 GMT

The α-level Conditional Tail Expectation (CTE) of a continuous random variable X is defined as its conditional expectation given the event {X > q_α} where q_α represents its α-level quantile. It is well known that the empirical CTE (the average of the n(1 � α) largest order statistics in a sample of size n) is a negatively biased estimator of the CTE. This bias vanishes as the sample size increases, but in small samples can be significant. In this article it is shown that an unbiased nonparametric estimator of the CTE does not exist. In addition, the asymptotic behavior of the bias of the empirical CTE is studied, and a closed form expression for its first order term is derived. This expression facilitates the study of the behavior of the empirical CTE with respect to the underlying distribution, and suggests an alternative (to the bootstrap) approach to bias correction. The performance of the resulting estimator is assessed via simulation.

Asymptotics for Operational Risk Quantified with Expected Shortfall

poj@peeters-leuven.be — Wed, 09 Dec 2009 12:37:22 GMT

In this paper we estimate operational risk by using the convex risk measure Expected Shortfall (ES) and provide an approximation as the confidence level converges to 100% in the univariate case. Then we extend this approach to the multivariate case, where we represent the dependence structure by using a L�vy copula as in B�cker and Kl�ppelberg (2006) and B�cker and Kl�ppelberg, C. (2008). We compare our results to the ones obtained in B�cker and Kl�ppelberg (2006) and (2008) for Operational VaR and discuss their practical relevance.