Principal component analysis is employed to construct a new formula defining ‘sports cars’, a classification variable commonly used by Belgian insurers in motor insurance. Five hundred and eighty-one different car models were used in the design of the formula. It is based solely on the cars' technical characteristics and hence does not rely on the subjective opinion of experts; the resulting classification is independent of units of measurement employed. Thus the definition is suitable for application world-wide.

While driver telematics has gained attention for risk classification in auto insurance, scarcity of observations with telematics features has been problematic, which could be owing to either privacy concerns or favorable selection compared to the data points with traditional features.

To handle this issue, we apply a data integration technique based on calibration weights for usage-based insurance with multiple sources of data. It is shown that the proposed framework can efficiently integrate traditional data and telematics data and can also deal with possible favorable selection issues related to telematics data availability. Our findings are supported by a simulation study and empirical analysis in a synthetic telematics dataset.

Information relating to the expected number of losses is of importance in automobile insurance systems. The distribution of risks by number of losses per year may be based on the following model

with λ representing the average number of losses per year. This distribution is the Poisson distribution. Tests of this model versus actual observations often indicate significant deviation. This discrepency can result from the constancy of λ which makes the model appropriate for an individual but would require an isohazardous population when applied to a group of individuals. In reality, however, λ will vary from individual to individual. A model accounting for this spread in λ is given in

where z(λ) is a distribution describing the spread of λ. The results of model (2) certainly will depend on the form of z(λ). It has been hypothesized that z(λ) can be represented by (3)

which is a Pearson Type III [1, 5, 7]. With this assumption model (2) becomes the negative binominal distribution with a mean of and a variance of

If the observed mean is and the observed variance it is possible to determine a and b by solving the above equations for mean and variance. Thus and

The results indicate an improved fit to actual observations [1, 5, 6].

This paper examines the balancing of the investment and liability portfolios of a (re)insurance firm operating in the international market. The model captures two effects which are ignored by traditional analysts.

(a) According to the Interest Parity Theorem, the expected changes in the exchange rates should already be reflected in the expected rate of return on foreign investments. Therefore, an insurer operating in a perfect market should be indifferent to the currency denomination of its financial assets.

(b) A second effect which has often been ignored is related to the additional risk resulting from the unexpected fluctuations of the exchange rates.

The multi-index model suggested in this paper is capable of capturing these effects. The model can be used to examine and analyze alternative policies of the firm operating in international markets. For example, the model can be used to examine whether an insurer should take positions in certain currencies, or rather take a “full-hedge” policy.

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.

Cash balance pension plans with crediting rates linked to long bond yields are relatively common in the United States, but their liabilities are proving very challenging to hedge. In this paper, we consider dynamic hedge strategies using the one-factor and two-factor Hull White models, based on results for the liability valuation from Hardy et al. (2014). The strategies utilise simple hedge portfolios combining one or two zero-coupon bonds, and a money market account. We assess the effectiveness of the strategies by considering how accurately each one would have hedged a 5-year cash balance liability through the past 20 years, using real-world returns and crediting rates, and assuming parameters calibrated using the information available at the time. We show that there is considerable impact of model and parameter uncertainty, with additional, less severe impact from discrete hedging error and transactions costs. Despite this, the dynamic hedge strategies do manage to stabilize surplus substantially, even through the turbulence of the past two decades.

Claims reserving models are usually based on data recorded in run-off tables, according to the origin and the development years of the payments. The amounts on the same diagonal are paid in the same calendar year and are influenced by some common effects, for example, claims inflation, that can induce dependence among payments. We introduce hierarchical generalized linear models (HGLM) with risk parameters related to the origin and the calendar years, in order to model the dependence among payments of both the same origin year and the same calendar year. Besides the random effects, the linear predictor also includes fixed effects. All the parameters are estimated within the model by the h-likelihood approach. The prediction for the outstanding claims and an approximate formula to evaluate the mean square error of prediction are obtained. Moreover, a parametric bootstrap procedure is delineated to get an estimate of the predictive distribution of the outstanding claims. A Poisson-gamma HGLM with origin and calendar year effects is studied extensively and a numerical example is provided. We find that the estimates of the correlations can be significant for payments in the same calendar year and that the inclusion of calendar effects can determine a remarkable impact on the prediction uncertainty.

A variable annuity is a modern life insurance product that offers its policyholders participation in investment with various guarantees. To address the computational challenge of valuing large portfolios of variable annuity contracts, several data mining frameworks based on statistical learning have been proposed in the past decade. Existing methods utilize regression modeling to predict the market value of most contracts. Despite the efficiency of those methods, a regression model fitted to a small amount of data produces substantial prediction errors, and thus, it is challenging to rely on existing frameworks when highly accurate valuation results are desired or required. In this paper, we propose a novel hybrid framework that effectively chooses and assesses easy-to-predict contracts using the random forest model while leaving hard-to-predict contracts for the Monte Carlo simulation. The effectiveness of the hybrid approach is illustrated with an experimental study.

Maximum likelihood estimation is derived for the Lagrangian Poisson distribution for a simple and a loglinear model and illustrated with real data.

A compound Poisson process, in this context abbreviated to cPp, is defined by a probability distribution of the number m of events in the interval (o, τ) of the original scale of the process parameter, assumed to be one-dimensional, in the following form.

where du shall be inserted for t, λτ being the intensity function of a Poisson process with the expected number t of events in the interval (O, τ) and U(ν, τ) is the distribution function of ν for every fixed value of τ, here called the risk distribution. If the inverse of is substituted for τ, in the right membrum of (1), the function obtained is a function of t.

If the risk distribution is defined by the general form U(ν, τ) the process defined by (1) is called a cPp in the wide sense (i.w.s.). In the sequel two particular cases for U(ν, τ) shall be considered, namely when it has the form of distribution functions, which define a primary process being stationary (in the weak sense) or non-stationary, and when it is equal to U1(ν) independently of τ. The process defined by (1) is in these cases called a stationary or non-stationary (s. or n.s.)cPp and a cPpin the narrow sense (i.n.s.) respectively. If a process is non-elementary i.e. the size of one change in the random function constituting the process is a random variable, the distribution of this variable conditioned by the hypothesis that such a change has occurred at τ is here called the change distribution and denoted by V(x, τ), or, if it is independent of τ, by V1(x). In an elementary process the size of one change is a constant, so that, in this case, the change distribution reduces to the unity distribution E(x — k), where E(ξ) is equal to I, o, if ξ is non-negative, negative respectively, and k is a given constant.

Plusieurs communications sont présentées au Colloque sur les Mathématiques de l'Assurance Automobile. Afin de pouvoir mieux situer chacune d'elles dans ce vaste sujet, il peut être bon de schématiser l'ensemble de la question. Il comprend:

1. la survenance des accidents

2. le tarif et la mesure de son efficacité par rapport à l'estimation optimale.

Groupons dans une classe les propriétaires de véhicules ayant un même ensemble de caractéristiques extérieures de risques: même modèle de voiture, même zone de garage habituel, même usage ou profession du propriétaire, etc. Les risques s de survenance d'accident ne sont pas identiques pour tous les propriétaires de la classe, mais sont distribués selon une fonction de répartition F(s).

Désignons par P(s′ ∣ s) la probabilité de survenance de s′ sinistres pour un assuré dont le risque individuel est s (des études antérieures ont montré que P(s′ ∣ s) est représentable par une loi de probabilité de Poisson). La probabilité qu'un assuré, pris au hasard dans la classe, ait s′ sinistres est:

s0 étant le risque minimum de la classe.

Désignons par g(X) la densité de probabilité de la distribution des couts X des sinistres. On peut admettre qu'à l'intérieur de la classe, au moins en première approximation, g(X) est indépendant de s, d'où la probabilité de survenance d'un sinistre de coût X est:

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.

In this paper, we present closed-type formulas for some multivariate compound distributions with multivariate Sarmanov counting distribution and independent Erlang distributed claim sizes. Further on, we also consider a type-II Pareto dependency between the claim sizes of a certain type. The resulting densities rely on the special hypergeometric function, which has the advantage of being implemented in the usual software. We numerically illustrate the applicability and efficiency of such formulas by evaluating a bivariate cumulative distribution function, which is also compared with the similar function obtained by the classical recursion-discretization approach.

We propose a new inference strategy for general population mortality tables based on annual population and death estimates, completed by monthly birth counts. We rely on a deterministic population dynamics model and establish formulas that link the death rates to be estimated with the observables at hand. The inference algorithm takes the form of a recursive and implicit scheme for computing death rate estimates. This paper demonstrates both theoretically and numerically the efficiency of using additional monthly birth counts for appropriately computing annual mortality tables. As a main result, the improved mortality estimators show better features, including the fact that previous anomalies in the form of isolated cohort effects disappear, which confirms from a mathematical perspective the previous contributions by Richards, Cairns et al., and Boumezoued.

In the context of life insurance with profit participation, the future discretionary benefits (FDB), which are a central item for Solvency II reporting, are generally calculated by computationally expensive Monte Carlo algorithms. We derive analytic formulas to estimate lower and upper bounds for the FDB. This yields an estimation interval for the FDB, and the average of lower and upper bound is a simple estimator. These formulae are designed for real world applications, and we compare the results to publicly available reporting data.

It has been argued in previous studies that the expected utility decision criterion provides useful insights for certain insurance problems, such as underwriting, reinsurance and portfolio optimization problems. In this study three new models are developed which extend and generalize previous results. The first model analyses modified stop-loss reinsurance. The second model analyses risk pooling where both inward and outward reinsurance occur. Expected utility calculations can be used to provide insight on the attractiveness of competing reinsurance and risk pooling options. The third model is for strategic planning, where risk/reward tradeoffs for all the insurer's business activities (underwriting, investment, reinsurance) can be considered in aggregate. The simpler models can often be solved analytically however the strategic planning model is relatively complex and uses Monte Carlo techniques to determine retained earnings distributions. The expected utility approach has been found to be powerful, flexible and comprehensive as a decision aiding mechanism. From a normative viewpoint, this approach accounts very well for all the important decision elements. Recent developments in decision support systems will allow these models to be made available to practitioners in user friendly forms.

Although loss reserving has been deeply studied in the literature, there are still practical issues that have not been addressed a lot. One of them is the estimation of reserves during the year, which is necessary for forecasts or closings during the year. We will study the following question: What can be done for forecasts and closings during the year that goes along with the reserving at year end? In order to make it not too complicated, we will focus on the Chain-Ladder method introduced by Mack (1993). We will describe several methods that are used in practice. We will discuss advantages and disadvantages of these methods based on a simple deterministic example. Roughly spoken, we will see that you may shift development or accident periods, or may split development periods, but should not split accident periods.