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.

Recursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.

The fact that a large proportion of insurance policyholders make no claims during a one-year period highlights the importance of zero-inflated count models when analyzing the frequency of insurance claims. There is a vast literature focused on the univariate case of zero-inflated count models, while work in the area of multivariate models is considerably less advanced. Given that insurance companies write multiple lines of insurance business, where the claim counts on these lines of business are often correlated, there is a strong incentive to analyze multivariate claim count models. Motivated by the idea of Liu and Tian (Computational Statistics and Data Analysis, 83, 200–222; 2015), we develop a multivariate zero-inflated hurdle model to describe multivariate count data with extra zeros. This generalization offers more flexibility in modeling the behavior of individual claim counts while also incorporating a correlation structure between claim counts for different lines of insurance business. We develop an application of the expectation–maximization (EM) algorithm to enable the statistical inference necessary to estimate the parameters associated with our model. Our model is then applied to an automobile insurance portfolio from a major insurance company in Spain. We demonstrate that the model performance for the multivariate zero-inflated hurdle model is superior when compared to several alternatives.

This paper studies bilateral risk sharing under no aggregate uncertainty, where one agent has Expected-Utility preferences and the other agent has Rank-dependent utility preferences with a general probability distortion function. We impose exogenous constraints on the risk exposure for both agents, and we allow for any type or level of belief heterogeneity. We show that Pareto-optimal risk-sharing contracts can be obtained via a constrained utility maximization under a participation constraint of the other agent. This allows us to give an explicit characterization of optimal risk-sharing contracts. In particular, we show that an optimal risk-sharing contract contains allocations that are monotone functions of the likelihood ratio, where the latter is obtained from Lebesgue’s Decomposition Theorem.

In the present paper the author gives net premium formulae for a generalized largest claims reinsurance cover. If the claim sizes are mutually independent and identically 3-parametric Pareto distributed and the number of claims has a Poisson, binomial or negative binomial distribution, formulae are given from which numerical values can easily be obtained. The results are based on identities for compounded order statistics.

We study a stochastic model for a target benefit pension plan suffering from rising longevity and falling fertility. Policies for postponing retirement are carried out to hedge the payment difficulties caused by the aging population. The plan members’ contributions are set in advance while the pension payments reflect intergenerational equity by a target payment level and intergenerational risk sharing by an adjustment. The pension fund is invested in both a risk-free asset and a risky asset. Applying the stochastic optimal control methods, we derive analytic solutions for optimal investment and benefit payment strategies which minimize the benefit risk. Besides, an optimal delayed retirement age which can hedge against the aging phenomenon under certain parameters is given. Therefore, it can provide a basis for quantifying the delay of retirement time.

The paper gives details of a case study in the premium rating of a Householders Contents insurance portfolio. The rating is performed by the fitting of bivariate spline functions to a version of operating ratio described in Section 3.

The use of bivariate splines requires a small amount of mathematical equipment, which is developed in Section 4. The fitting of splines, using regression is carried out in Sections 5 and 6, where the numerical results are given, including some assessment of goodness-of-fit.

Contour maps of the spline surfaces are also given, and used for the selection of geographic areas used for premium rating purposes. These are compared with the areas, past and present, actually used by the insurer concerned.

1. There is a general rule applicable to all insurance and reinsurance fields according to which the level of the so-called technical minimum premium should be fixed such that a certain stability criterion is satisfied for the portfolio under consideration. The two bestknown such criteria are

(i) the probability that there is a technical loss in any of the future years should be less than a given percentage

(ii) the probability that the company gets “ruined” i.e. initial reserves plus accumulated premiums minus accumulated claims becomes negative at any time of a given period in the future should be less than a tolerated percentage.

Confining ourselves to criterion (i) in the present paper we may then say that the problem of calculating technical minimum premiums is broadly spoken equivalent with the problem of estimating loss probabilities. Since an exact calculation of such probabilities is only possible for a few very simple and therefore mostly unrealistic risk models and since e.g. Esscher's method is not always very easy to apply in practice it might be worthwhile to describe in the following an alternative approach using results and techniques from Reliability Theory in order to establish bounds for unknown loss probabilities.

It would have been impossible for me to write this paper without having had the opportunity of numerous discussions with the Reliability experts R. Barlow and F. Proschan while I was at Stanford University. In particular I was told the elegant proof of theorem 3 given below by R. Barlow recently.

Innon-life insurance, the payment history can be predictive of the timing of a settlement for individual claims. Ignoring the association between the payment process and the settlement process could bias the prediction of outstanding payments. To address this issue, we introduce into the literature of micro-level loss reserving a joint modeling framework that incorporates longitudinal payments of a claim into the intensity process of claim settlement. We discuss statistical inference and focus on the prediction aspects of the model. We demonstrate applications of the proposed model in the reserving practice with a detailed empirical analysis using data from a property insurance provider. The prediction results from an out-of-sample validation show that the joint model framework outperforms existing reserving models that ignore the payment–settlement association.

The classical evaluation of pure premiums for excess of loss reinsurance with reinstatements requires the knowldege of the claim size distribution of the insurance risk. In the situation of incomplete information, where only a few characteristics of the aggregate claims to an excess of loss layer can be estimated, the method of stop-loss ordered bounds yields a simple analytical distribution-free approximation to pure premiums of excess of loss reinsurance with reinstatements. It is shown that the obtained approximation is enough accurate for practical purposes and improves the analytical approximations obtained using either a gamma, translated gamma, translated inverse Gaussian or a mixture of the last two distributions.

The primary objective of this scholarly work is to develop two estimation procedures – maximum likelihood estimator (MLE) and method of trimmed moments (MTM) – for the mean and variance of lognormal insurance payment severity data sets affected by different loss control mechanism, for example, truncation (due to deductibles), censoring (due to policy limits), and scaling (due to coinsurance proportions), in insurance and financial industries. Maximum likelihood estimating equations for both payment-per-payment and payment-per-loss data sets are derived which can be solved readily by any existing iterative numerical methods. The asymptotic distributions of those estimators are established via Fisher information matrices. Further, with a goal of balancing efficiency and robustness and to remove point masses at certain data points, we develop a dynamic MTM estimation procedures for lognormal claim severity models for the above-mentioned transformed data scenarios. The asymptotic distributional properties and the comparison with the corresponding MLEs of those MTM estimators are established along with extensive simulation studies. Purely for illustrative purpose, numerical examples for 1500 US indemnity losses are provided which illustrate the practical performance of the established results in this paper.

This paper considers the problem of a lifelong health insurance cover where medical inflation is not sufficiently incorporated in the level premium determined at policy issue. We focus on the setting where changes in health benefits, driven by medical inflation, are accounted for by an appropriate update or indexation of the level premium, the policy value, or both premium and policy value, during the term of the contract. Such an updating mechanism is necessary to restore the actuarial equivalence between future health benefits and surrender values on the one hand, and available policy values and future premiums on the other hand. We extend existing literature (Vercruysse et al., 2013; Denuit et al., 2017) by developing updating mechanisms in a discrete-time framework, where medical inflation is only taken into account ex-post as it emerges over time and where surrender values are allowed for. We propose and design two types of surrender values: based on the ageing provision on the one hand and based directly on the premiums paid until surrender on the other hand. We illustrate our updating strategy with numerical examples, using Belgian data, and investigate the sensitivity of our findings with respect to elements from the technical basis (in particular: the lapse rates) used in the actuarial calculations. Our updating mechanism is generic and useful for a wide range of products in life and health insurance, where some elements of the technical basis are guaranteed while others are subject to revision according to policy conditions.

The theory of linear filtering of stochastic processes provides continuous time analogues of finite-dimensional linear Bayes estimators known to actuaries as credibility methods. In the present paper a selfcontained theory is built for processes of bounded variation, which are of particular relevance to insurance. Two methods for constructing the optimal estimator and its mean squared error are deviced. Explicit solutions are obtained in a continuous time variation of Hachemeister's regression model and in a homogeneous doubly stochastic generalized Poisson process. The traditional discrete time set-up is compared to the one with continuous time, and some merits of the latter are pointed out.

We study the linear Markov property, i.e. the possibility of basing the credibility estimator on data of the most recent time period without loss of accuracy. Necessary and sufficient conditions are derived generally. The meaning of the linear Markov property is also discussed in different experience rating and loss reserving models.

The probability of ruin in continuous and finite time is numerically evaluated in a classical risk process where the premium can be updated according to credibility models and therefore change from year to year. A major consideration in the development of this approach is that it should be easily applicable to large portfolios. Our method uses as a first tool the model developed by Afonso et al. (2009), which is quite flexible and allows premiums to change annually. We extend that model by introducing a credibility approach to experience rating.

We consider a portfolio of risks which satisfy the assumptions of the Bühlmann (1967, 1969) or Bühlmann and Straub (1970) credibility models. We compute finite time ruin probabilities for different scenarios and compare with those when a fixed premium is considered.

We present a savings plan for retirement that removes risk by fixing a constraint on a life-long pension so that it has an upper and a lower bound. This corresponds to the ideas of Nobel laureate R.C. Merton whose implementation has never been published. We show with an illustration that our proposed practical algorithm reproduces the theoretical results after a savings period of around 30 years by using daily, monthly, weekly or yearly updates of the investment positions. We calculate the percentiles of the final accumulated wealth distribution for the adjusted implementation. In the simulated illustration, we observe that the adjusted values converge to the theoretical values of the percentiles when the frequency of update increases. We conclude that monthly adjustments result in a practical way to implement theoretical results that were obtained under the hypothesis of a continuous process by Donnelly et al. (2015). This method is easy to use in practice by pension savers and fund managers.

In the present note we consider the Hipp approximation to the convolution of a finite number of probability distributions on the non-negative integers. It is shown that the moment up to and including order r of the rth order approximation are equal to the corresponding moments of the exact distribution. We also give a relation between the exact and approximated (r + 1)th order moments and indicate how similar relations can be obtained for higher order moments.

At the 1990 ASTIN-colloquium, Schmitter posed the problem of finding the extreme values of the ultimate ruin probability ψ(u) in a risk process with initial capital u, fixed safety margin θ, and mean μ and variance σ2 of the individual claims. This note aims to give some more insight into this problem. Schmitter's conjecture that the maximizing individual claims distribution is always diatomic is disproved by a counterexample. It is shown that if one uses the distribution maximizing the upper bound e−Ru to find a ‘large’ ruin probability among risks with range [0, b], incorrect results are found if b is large or u small.

The related problem of finding extreme values of stop-loss premiums for a compound Poisson (λ) distribution with identical restrictions on the individual claims is analyzed by the same methods. The results obtained are very similar.

The projection of outstanding liabilities caused by incurred losses or claims has played a fundamental role in general insurance operations. Loss reserving methods based on individual losses generally perform better than those based on aggregate losses. This study uses a parametric individual information model taking not only individual losses but also individual information such as age, gender, and so on from policies themselves into account. Based on this model, this study proposes a computation procedure for the projection of the outstanding liabilities, discusses the estimation and statistical properties of the unknown parameters, and explores the asymptotic behaviors of the resulting loss reserving as the portfolio size approaching infinity. Most importantly, this study demonstrates the benefits of individual information on loss reserving. Remarkably, the accuracy gained from individual information is much greater than that from considering individual losses. Therefore, it is highly recommended to use individual information in loss reserving in general insurance.

In this paper a new methodology using the conditional specification technique intoduced by Arnold et al. (1999) is used to obtain bonus-malus premiums. A Poisson distribution for which the parameter is a function of the classical structure parameter is used and a new class of prior distribution arises in a natural way. This model contains, as a particular case, the classical compound Poisson model and is found to be much more robust than earlier ones. An example is given to illustrate our ideas.