Measures of concordance have been widely used in insurance and risk management to summarize nonlinear dependence among risks modeled by random variables, which Pearson’s correlation coefficient cannot capture. However, popular measures of concordance, such as Spearman’s rho and Blomqvist’s beta, appear as classical correlations of transformed random variables. We characterize a whole class of such concordance measures arising from correlations of transformed random variables, which includes Spearman’s rho, Blomqvist’s beta and van der Waerden’s coefficient as special cases. Compatibility and attainability of square matrices with entries given by such measures are studied—that is, whether a given square matrix of such measures of concordance can be realized for some random vector and how such a random vector can be constructed. Compatibility and attainability of block matrices and hierarchical matrices are also studied due to their practical importance in insurance and risk management. In particular, a subclass of attainable block Spearman’s rho matrices is proposed to compensate for the drawback that Spearman’s rho matrices are in general not attainable for dimensions larger than three. Another result concerns a novel analytical form of the Cholesky factor of block matrices which allows one, for example, to construct random vectors with given block matrices of van der Waerden’s coefficient.

Holtan (1994) suggests to replace traditional bonus-malus systems by a high deductible financed by a short-term loan. Practical consequences of this proposal are investigated here. Simulation is used to evaluate the efficiency of the Taiwanese Bonus-malus system and the variability of premiums of an average policyholder. Holtan's high deductible system is analysed under a compound Poisson assumption, with truncated exponential claims. It is shown that the introduction of a high deductible would increase the variability of payments and the efficiency of the rating system for most policyholders.

This paper concerns the Borch model of a reinsurance market seen as a model of an economy under uncertainty.

In a market of this type the goods traded are unit coverings contingent to a particular state of nature (n-tuple of claims).

Our idea is to regard the probability of a state of nature as a sort of intrinsic value of the related contingent covering. From this point of view we examine the role of the reinsurance market in modifying values in market equilibrium prices and other questions, related to this classical economic problem, in the particular case of a quadratic utility function for all companies.

In this paper non-normal distributions via scale mixtures are introduced into insurance applications. The symmetric distributions of interest are the Student-t and exponential power (EP) distributions. A Bayesian approach is adopted with the aid of simulation to obtain posterior summaries. We shall show that the computational burden for the Bayesian calculations is alleviated via the scale mixtures representations. Illustrative examples are given.

We analyze optimal consumption and pension insurance during the life time of a consumer using the life cycle model, when the consumer has recursive utility. The relationship between substitution of consumption and risk aversion is highlighted, and clarified by the introduction of this type of preferences. We illustrate how recursive utility can be used to explain the empirical consumption puzzle for aggregates. This indicates a plausible choice for the parameters of the utility function, relevant for the consumer in the life cycle model. Optimal life insurance is considered, as well as the portfolio choice problem related to optimal exposures in risky securities. A major finding is that it is optimal for the typical insurance buyer to smooth adverse shocks to the financial market, unlike what is implied by the conventional model. This has implications for what type of contracts the life and pension insurance industry should offer.

This report is a contribution to the discussion on the solvency problem, which has been taking place at ASTIN-meetings. In his report in Edinburgh 1964 Beard referred to many aspects which are closely connected with the problem. Such aspects are

1. the evaluation of liabilities;

2. the evaluation of assets;

3. the level of the premiums of long term policies and

4. reinsurance.

If all of these are not in order, there is no sense in speaking about solvency. E.g. a solvency margin defined as the difference between assets and the expected value of liabilities would not be a reliable measure of the financial state of an insurance company, if either of these—or maybe both—are not evaluated in a reliable way. The fixing of solvency margins is not an isolated problem, on the contrary it is only part of the security measures which must all be managed at the same time. The ultimate purpose of the security system prescribed by legislation must be to safeguard policyholders and claimants against losses.

Le principe de l'assurance contre les accidents d'automobiles est la formation d'une mutualité où les assurés mettent en commun leurs risques. Si les risques des assures ne sont pas tous égaux entre eux, il est normal de demander à chacun des assurés une prime proportionnelle au risque qu'il fait supporter à la mutualité. La tarification a pour objet l'estimation du risque propre à chaque assuré, afín de répartir équitablement la charge totale de la mutualité.

On groupe habituellement les véhicules ayant en commun plusieurs caractéristiques du risque telles que même modèle de voiture, même usage, même lieu de garage habituel. Nous désignerons un tel groupement par classe de tarif.

On observe les accidents relatifs aux véhicules appartenant à une même classe de tarif. Le quotient du coût de ces accidents au nombre de voitures-année donne la prime moyenne pure empirique. Cette prime moyenne subit alors quelques corrections pour tenir compte de la survenance au hasard des gros sinistres et du fait que la prime à indiquer dans le tarif estime le risque moyen futur et non pas le risque passé Cette prime moyenne sera demandée à toutes les voitures ayant les caractéristiques définies par la classe de tarif; elle suppose que tous les risques de cette classe sont égaux, c'est-à-dire que les seuls paramètres de tarification retenus sont exhaustifs pour caractériser le risque que l'assuré fera supporter à la mutualité.

Standard optimal portfolio choice models assume that investors maximise the expected utility of their future outcomes. However, behaviour which is inconsistent with the expected utility theory has often been observed.

In a discrete time setting, we provide a formal treatment of risk measures based on distortion functions that are consistent with Yaari’s dual (non-expected utility) theory of choice (1987), and set out a general layout for portfolio optimisation in this non-expected utility framework using the risk neutral computational approach.

As an application, we consider two particular risk measures. The first one is based on the PH-transform and treats the upside and downside of the risk differently. The second one, introduced by Wang (2000) uses a probability distortion operator based on the cumulative normal distribution function. Both risk measures rank-order prospects and apply a distortion function to the entire vector of probabilities.

In this paper we study the benefits derived from international diversification of equity portfolios from the German and the Hungarian points of view. In contrast to the German capital market, which is one of the largest in the world, the Hungarian Stock Exchange is an emerging market. The Hungarian stock market is highly volatile, high returns are often accompanied by extremely large risk. Therefore, there is a good potential for Hungarian investors to realise substantial benefits in terms of risk reduction by creating multi-currency portfolios. The paper gives evidence on the above mentioned benefits for both countries by examining the performance of several ex ante portfolio strategies. In order to control the currency risk, different types of hedging approaches are implemented.

Following the EU Gender Directive, that obliges insurance companies to charge the same premium to policyholders of different genders, we address the issue of calculating solvency capital requirements (SCRs) for pure endowments and annuities issued to mixed portfolios. The main theoretical result is that, if the unisex fairness principle is adopted for the unisex premium, the SCR at issuing time of the mixed portfolio calculated with unisex survival probabilities is greater than the sum of the SCRs of the gender-based subportfolios. Numerical results show that for pure endowments the gap between the two is negligible, but for lifetime annuities the gap can be as high as 3–4%. We also analyze some conservative pricing procedures that deviate from the unisex fairness principle, and find that they lead to SCRs that are lower than the sum of the gender-based SCRs because the policyholders are overcharged at issuing time.

We consider a modification to the Poisson common factor model and utilise a generalised linear model (GLM) framework that incorporates a smoothing process and a set of linear constraints. We extend the standard GLM model structure to adopt Lagrange methods and P-splines such that smoothing and constraints are applied simultaneously as the parameters are estimated. Our results on Australian, Canadian and Norwegian data show that this modification results in an improvement in mortality projection in terms of producing more accurate forecasts in the out-of-sample testing. At the same time, projected male-to-female ratio of death rates at each age converges to a constant and the residuals of the models are sufficiently random, indicating that the use of smoothing does not adversely affect the fit of the model. Further, the irregular patterns in the estimates of the age-specific parameters are moderated as a result of smoothing and this model can be used to produce more regular projected life tables for pricing purposes.

Given the rapid reductions in human mortality observed over recent decades and the uncertainty associated with their future evolution, there have been a large number of mortality projection models proposed by actuaries and demographers in recent years. Many of these, however, suffer from being overly complex, thereby producing spurious forecasts, particularly over long horizons and for small, noisy data sets. In this paper, we exploit statistical learning tools, namely group regularisation and cross-validation, to provide a robust framework to construct discrete-time mortality models by automatically selecting the most appropriate functions to best describe and forecast particular data sets. Most importantly, this approach produces bespoke models using a trade-off between complexity (to draw as much insight as possible from limited data sets) and parsimony (to prevent over-fitting to noise), with this trade-off designed to have specific regard to the forecasting horizon of interest. This is illustrated using both empirical data from the Human Mortality Database and simulated data, using code that has been made available within a user-friendly open-source R package StMoMo.

In this paper, following an open portfolio approach, we show how to estimate a Bonus-malus system evolution.

Considering a model for the number of new annual policies, we obtain ML estimators, asymptotic distributions and confidence regions for the expected number of new policies entering the portfolio in each year, as well as for the expected number and proportion of insureds in each bonus class, by year of enrollment. Confidence regions for the distribution of policyholders result in confidence regions for optimal bonus scales.

Our treatment is illustrated by an example with numerical results.

In this paper, we introduce the simplest exponential dispersion model containing the Pareto and exponential distributions. In this way, we obtain distributions with support (0, ∞) that in a long interval are equivalent to the Pareto distribution; however, for very high values, decrease like the exponential. This model is useful for solving relevant problems that arise in the practical use of extreme value theory. The results are applied to two real examples, the first of these on the analysis of aggregate loss distributions associated to the quantitative modelling of operational risk. The second example shows that the new model improves adjustments to the destructive power of hurricanes, which are among the major causes of insurance losses worldwide.

This paper presents a normative model for the sequential reinsurance and dividend-payment problem of the Insurance Company (I.C.). Optimal strategies are found in closed form for a class of utility functions. In some sense the model studied can be viewed as an adaptation of Hakansson's investment-consumption model of the individual [3] or a generalization of Frisque's model for the dynamic management of an I.C. [2].

In Section 2 the model is formulated as a discrete time dynamic programming problem. The objective of the I.C. is assumed to be maximization of the expected utility of the dividend streams paid to stock/policy-holders (s/p-holders). The initial reserves level is assumed to be known. The premiums to be collected in each period for selling policies are known in advance. The losses due to claims from policy-holders are random variables independent from period to period. In each period the I.C. must decide on the portion of the reserves to be paid as dividends and on the form and level of reinsurance with a reinsurer that quotes prices for any contract.

Optimal strategies in closed form are found in Section 3 when the utility function of the I.C. is given by the discounted sum of one-period utilities of dividends; and when the one-period utilities belong to the linear risk-tolerance class, which is given by: (la) u(x) = (ax + b)c+1/a(c + 1); ax + b > o, ac < o. (Ib) u(x) = log(ax + b); ax + b > o. (II) u(x) = — eγx; γ > o.

Traditional credibility models have treated the process generating the losses as stable over time, perhaps with a deterministic trend imposed. However, there is ample evidence that these processes are not stable over time. What is required is a method that allows for time-varying parameters in the process, yet still provides the shrinkage needed for sound ratemaking. In this paper we use an automobile insurance example to illustrate how this can be accomplished.

For the classical Cramér-Lundberg risk model, a dividend strategy of threshold type has recently been suggested in the literature. This strategy consists of paying out part of the premium income as dividends to shareholders whenever the free surplus is above a given threshold level. In contrast to the well-known horizontal barrier strategy, the threshold strategy can lead to a positive infinite-horizon survival probability, with reduced profit in terms of dividend payments. In this paper we extend several of these results to a Sparre Andersen model with generalized Erlang(n)-distributed interclaim times. Furthermore, we compare the performance of the threshold strategy to a linear dividend barrier model. In particular, (partial) integro-differential equations for the corresponding ruin probabilities and expected discounted dividend payments are provided for both models and explicitly solved for n = 2 and exponentially distributed claim amounts. Finally, the explicit solutions are used to identify parameter sets for which one strategy outperforms the other and vice versa.

The purpose of this paper is to obtain approximations to the transition intensities defined for a multiple state model for Permanent Health Insurance (PHI) which enables us to analyse PHI claims by cause of disability.

The approximations to the transition intensities are obtained using a set of PHI data classified by 18 sickness categories and the graduations of the transition intensities defined for a simpler model proposed in Continuous Mortality Investigation Reports, 12 (1991).

In order to derive the approximations to the recovery and mortality of the sick intensities for our model, we carry out tests of hypotheses based on the distributions of average sickness durations. The approximations to the sickness intensities are obtained by estimating a statistical model for the number of claim inceptions, which can be formulated as a generalized linear model.

This paper investigates equilibrium in an insurance market where risk classification is restricted. Insurance demand is characterised by an iso-elastic function with a single elasticity parameter. We characterise the equilibrium by three quantities: equilibrium premium; level of adverse selection (in the economist's sense); and “loss coverage”, defined as the expected population losses compensated by insurance. We consider both equal elasticities for high and low risk-groups, and then different elasticities. In the equal elasticities case, adverse selection is always higher under pooling than under risk-differentiated premiums, while loss coverage first increases and then decreases with demand elasticity. We argue that loss coverage represents the efficacy of insurance for the whole population; and therefore that if demand elasticity is sufficiently low, adverse selection is not always a bad thing.