We consider variation of observed claim frequencies in non-life insurance, modeled by Poisson regression with overdispersion. In order to quantify how much variation between insurance policies that is captured by the rating factors, one may use the coefficient of determination, R2, the estimated proportion of total variation explained by the model. We introduce a novel coefficient of individual determination (CID), which excludes noise variance and is defined as the estimated fraction of total individual variation explained by the model. We argue that CID is a more relevant measure of explained variation than R2 for data with Poisson variation. We also generalize previously used estimates and tests of overdispersion and introduce new coefficients of individual explained and unexplained variance.

Application to a Swedish three year motor TPL data set reveals that only 0.5% of the total variation and 11% of the total individual variation is explained by a model with seven rating factors, including interaction between sex and age. Even though the amount of overdispersion is small (4.4% of the noise variance) it is still highly significant. The coefficient of variation of explained and unexplained individual variation is 29% and 81% respectively.

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.

In recent Solvency II considerations much effort has been put into the development of appropriate models for the study of the one-year loss reserving uncertainty in non-life insurance. In this article we derive formulas for the conditional mean square error of prediction of the one-year claims development result in the context of the Bayes chain ladder model studied in Gisler-Wüthrich. The key to these formulas is a recursive representation for the results obtained in Gisler-Wüthrich.

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.

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.

Frangos and Vrontos (2001) proposed an optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. In this paper, we introduce a generalized form of those obtained previously.

This contribution shows that for certain classes of insurance risks, pricing can be based on expected values under a probability measure ℙ* amounting to quasi risk-neutral pricing. This probability measure is unique and optimal in the sense of minimizing the relative entropy with respect to the actuarial probability measure ℙ, which is a common approach in the case of incomplete markets. After expounding the key elements of this theory, an application to a set of industrial property risks is developed, assuming that the severity of losses can be modeled by “Swiss Re Exposure Curves”, as discussed by Bernegger (1997). These curves belong to a parametric family of distribution functions commonly used by pricing actuaries. The quasi risk-neutral pricing approach not only yields risk exposure specific premiums but also Risk Adjusted Capital (RAC) values on the very same level of granularity. By way of contrast, the conventional determination of RAC is typically considered on a portfolio level only.

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.

In this paper, a long-term equilibrium model of a local market is developed. Subject to minor qualifications, the model is arbitrage-free. The variables modelled are the prices of risk-free zero-coupon bonds – both index-linked and conventional – and of equities, as well as the inflation rate. The model is developed in discrete (nominally annual) time, but allowance is made for processes in continuous time subject to continuous rebalancing. It is based on a model of the market portfolio comprising all the above-mentioned asset categories. The risk-free asset is taken to be the one-year index-linked bond. It is assumed that, conditionally upon information at the beginning of a year, market participants have homogeneous expectations with regard to the forthcoming year and make their decisions in mean-variance space. For the purposes of illustration, a descriptive version of the model is developed with reference to UK data. The parameters produced by that process may be used to inform the determination of those required for the use of the model as a predictive model. Illustrative results of simulations of the model are given.

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.

The α-level Conditional Tail Expectation (CTE) of a continuous random variable X is defined as its conditional expectation given the event {X > qα} where qα 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.

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 Lévy copula as in Böcker and Klüppelberg (2006) and Böcker and Klüppelberg, C. (2008). We compare our results to the ones obtained in Böcker and Klüppelberg (2006) and (2008) for Operational VaR and discuss their practical relevance.