The normal inverted gamma mixture or generalized Student t and the symmetric double Weibull, as well as their logarithmic counterparts, are proposed for modeling some loss distributions in non-life insurance and daily index return distributions in financial markets. For three specific data sets, the overall goodness-offit from these models, as measured simultaneously by the negative log-likelihood, chi-square and minimum distance statistics, is found to be superior to that of various “good” competitive models including the log-normal, the Burr, and the symmetric α-stable distribution. Furthermore, the study justifies on a statistical basis different important models of financial returns like the model of Black-Scholes (1973), the log-Laplace model of Hürlimann (1995), the normal mixture by Praetz (1972), the symmetric α-stable model by Mandelbrot (1963) and Fama (1965), and the recent double Weibull as limiting geometric-multiplication stable scheme in Mittnik and Rachev (1993). As an application, the prediction of one-year index returns from daily index returns is discussed.

The very title of this paper may cause some surprise, since economic theory so far has found virtually no application in insurance. Insurance is obviously an economic activity, and it is indeed strange that general economic theory should seem inapplicable to insurance.

This apparent paradox may to some extent be explained by the historic development. Actuarial mathematics and the essential scientific basis of insurance were developed into a self-contained and fairly complete theory long before economists could claim the name of science for their subject. Actuaries and other insurance people may from time to time have turned to economic theory for help on their problems. In most cases they must have turned away in disappointment, being convinced that actuarial mathematics was well ahead of general economic theory.

The last point is brought out clearly in a paper on the safety loading of insurance premiums which Tauber (19) presented at the Sixth International Congress of Actuaries in Vienna in 1909. In the introduction to this paper Tauber seems to consider risk bearing as a service and appears to assume that it like all other goods and services must have a price, determined by supply and demand in the market. Tauber did not develop this idea, apparently because the economic theory of his time was utterly unable to analyse the problem. That his approach was sound, and that the problem can be formulated and solved by modern economic theory, has been indicated in a previous paper (6).

During the last thirty years there has been an extremely rapid development in economic theory. The “General Theory” of Keynes (12) which appeared in 1936, is usually considered to have caused a revolution in economic theory. However, this “revolution” has not lead to any developments which seem to have an immediate application in insurance. Post- and pre-Keynesian economics are equally powerless when confronted with the problem which Tauber tried to formulate. It is therefore premature to conclude from the title of this paper that economic theory has caught up with actuarial science, and that the theory of insurance has now become a part of a general theory comprising all economic activities.

A claims reserving method is reviewed which was introduced by Gunnar Benktander in 1976. It is a very intuitive credibility mixture of Bornhuetter/Ferguson and Chain Ladder. In this paper, the mean squared errors of all 3 methods are calculated and compared on the basis of a very simple stochastic model. The Benktander method is found to have almost always a smaller mean squared error than the other two methods and to be almost as precise as an exact Bayesian procedure.

Financial products are priced using risk-neutral expectations justified by hedging portfolios that (as accurate as possible) match the product’s payoff. In insurance, premium calculations are based on a real-world best-estimate value plus a risk premium. The insurance risk premium is typically reduced by pooling of (in the best case) independent contracts. As hybrid life insurance contracts depend on both financial and insurance risks, their valuation requires a hybrid valuation principle that combines the two concepts of financial and actuarial valuation. The aim of this paper is to present a novel three-step projection algorithm to valuate hybrid contracts by decomposing their payoff in three parts: a financial, hedgeable part, a diversifiable actuarial part, and a residual part that is neither hedgeable nor diversifiable. The first two parts of the resulting premium are directly linked to their corresponding hedging and diversification strategies, respectively. The method allows for a separate treatment of unsystematic, diversifiable mortality risk and systematic, aggregate mortality risk related to, for example, epidemics or population-wide improvements in life expectancy. We illustrate our method in the case of CAT bonds and a pure endowment insurance contract with profit and compare the three-step method to alternative valuation operators suggested in the literature.

It has become standard practice in the non-life insurance industry to employ generalized linear models (GLMs) for insurance pricing. However, these GLMs traditionally work only with a priori characteristics of policyholders, while nowadays we increasingly have a posteriori information of individual customers available across multiple product categories. In this paper, we therefore develop a framework to capture this a posteriori information over several product lines using a dynamic claim score. More specifically, we extend the bonus-malus-panel model of Boucher and Inoussa (2014) and Boucher and Pigeon (2018) to include claim scores from other product categories and to allow for nonlinear effects of these scores. The application of the proposed multi-product framework to a Dutch property and casualty insurance portfolio shows that customers’ individual claims experience can have a significant impact on the risk classification. Moreover, it indicates that considerably more profits can be gained by accounting for their multi-product claims experience.

The probability density function of the time of ruin in the classical model with exponential claim sizes is obtained directly by inversion of the associated Laplace transform. This result is then used to obtain explicit closed-form expressions for the moments. The form of the density is examined for various parameter choices.

In an arbitrage-free economy, it is well-known that financial risks can be priced using equivalent martingale measures. We establish in this paper that, for general stochastic processes, the Wang Transform does not lead to a price which is consistent with the arbitrage-free price. Based on these results we must conclude that the Wang Transform cannot be a universal framework for pricing financial and insurance risks.

This paper studies deep learning approaches to find optimal reinsurance and dividend strategies for insurance companies. Due to the randomness of the financial ruin time to terminate the control processes, a Markov chain approximation-based iterative deep learning algorithm is developed to study this type of infinite-horizon optimal control problems. The optimal controls are approximated as deep neural networks in both cases of regular and singular types of dividend strategies. The framework of Markov chain approximation plays a key role in building the iterative equations and initialization of the algorithm. We implement our method to classic dividend and reinsurance problems and compare the learning results with existing analytical solutions. The feasibility of our method for complicated problems has been demonstrated by applying to an optimal dividend, reinsurance and investment problem under a high-dimensional diffusive model with jumps and regime switching.

This paper is concerned with two methods to estimate the parameters of the Poisson-Goncharov distribution introduced recently by Lefevre and Picard (1996). These methods are applied to fit, inter alia, the six observed claims distributions, from automobile insurance third party liability portfolios, studied by Gossiaux and Lemaire (1981) and analysed afterwards by several authors.

This paper deals with the Bonus-Malus system obtained when the claims frequency is submitted to trend. This system is specified in the two particular cases of Poisson-Gamma and Poisson-Inverse Gaussian distributions. The theoretical results are checked on data issued from automobile insurance policies observed during three years.

In this paper, a new methodology based on the use of the inverse of the circular tangent function that allows us to add a scale parameter (say α) to an initial survival function is presented. The latter survival function is determined as limiting case when α tends to zero. By choosing as parent the classical Pareto survival function, the Pareto ArcTan (PAT) distribution is obtained. After providing a comprehensive analysis of its statistical properties, theoretical results with reference to insurance are illustrated. Its performance is compared, by means of the well-known Norwegian fire insurance data, with other existing heavy-tailed distributions in the literature such as Pareto, Stoppa, Shifted Lognormal, Inverse Gamma and Fréchet distributions.

The Cox regression model is a standard tool in survival analysis for studying the dependence of a hazard rate on covariates (parametrically) and time (nonparametrically). This paper is a case study intended to indicate possible applications to non-life insurance, particularly occurrence of claims and rating.

We studied individuals from one Danish county holding policies in auto, property and household insurance simultaneously at some point during the four year period 1988-1991 in one company. The hazard of occurrence of claims of each type was studied as function of calendar time, time since the last claim of each type, age of policy holder, urbanization and detailed type of insurance. Particular emphasis was given to the technical advantages and disadvantages (particularly the complicated censoring patterns) of considering the nonparametrically underlying time as either calendar time or time since last claim. In the former case the theory is settled, but the results are somewhat complicated. The latter choice leads to several issues still under active methodological development. We develop a goodness-of-fit criterion which shows the lack of fit of some models, for which the practical conclusions might otherwise have been useful.

This paper shows how the multivariate finite time ruin probability function, in a phase-type environment, inherits the phase-type structure and can be efficiently approximated with only one Laplace transform inversion.

From a theoretical point of view, we also provide below a generalization of Thorin’s formula (1971) for the double Laplace transform of the finite time ruin probability, by considering also the deficit at ruin; the model is that of a Sparre Andersen (renewal) risk process with phase-type interarrival times.

In the case when the claims distribution is of phase-type as well, we obtain also an alternative formula for the single Laplace transform in time (or “exponentially killed probability’’), in terms of the roots with positive real part of the Lundberg’s equations, which complements Asmussen’s representation (1992) in terms of the roots with negative real part.

This paper examines the validity of some stylized statements that can be found in the actuarial literature about random effects models. Specifically, the actual meaning of the estimated parameters and the nature of the residual heterogeneity are discussed. A numerical illustration performed on a Belgian motor third party liability portfolio supports this discussion.

For the premium calculation the insurer will split up his collectivity of risks into risk groups which are homogeneous with respect to some directly observable risk factors. All risks of such a risk group will be charged the same base premium. But it is clear that by such an a priori classification not all determined factors can be taken into consideration, so that there will still remain accident proneness differentials within a risk group. Since these differentials will be reflected in the course of time by the claim experience of each risk, the insurer can come to a fair tarification by adjusting, each period, the base premium according to the individual claim experience of the risk. Such a system in which earlier neglected risk factors are taken into account a posteriori is an individual experience rating system. Our main interest goes to the following side-effect of experience rating: since an unfavourable claim experience results in a premium increase, an experience rated policyholder is stimulated to self-insure small damages. This phenomenon is well know in connection with bonus-malus systems in motor-car insurance, which explains why it is called “bonus-hunger”.

In the present paper a continuous time model for the bonus-malus system is set up which takes into account this hunger for bonus. An insured causing an accident will decide according to a certain decision rule whether to file a claim with his insurance company. The relevant information that he needs to make this decision is: his current risk class, the number of claims he has already filed during that period and the moment at which the decision is to be made.

In non-life insurance, business sustainability requires accurate and robust predictions of reserves related to unpaid claims. To this aim, two different approaches have historically been developed: aggregated loss triangles and individual claim reserving. The former has reached operational great success in the past decades, whereas the use of the latter still remains limited. Through two illustrative examples and introducing an appropriate tree-based algorithm, we show that individual claim reserving can be really promising, especially in the context of long-term risks.

In this paper, we develop a theoretical basis for drawing up a “Swedish” type actuarial balance sheet for a defined benefit pay-as-you-go (DB PAYG) scheme with retirement and disability benefits. Our model enables us to obtain the system's expected average turnover duration, measure the scheme's solvency and explore the phenomenon identified as “pension reclassification”, a widespread practice that masks the system's real status unless further pension information becomes available. The model is clearly linked to actuarial practice in social security and gives partial support to the practical adaptation of Swedish methodology carried out by OSFI (2012) in applying the concept of the contribution asset to the Canadian Pension Plan (CPP) balance sheet, which includes disability and survivor benefits.

Excess claims lead to an unsatisfactory behavior of standard linear credibility estimators. We suggest in this paper to use robust methods in order to obtain better estimators. Our first proposal is the linear credibility estimator with the claims replaced by a robust M-estimator of scale calculed from the claims. This corresponds to a truncation of the claims with a truncation point depending on the data and different for each contract. We discuss the properties of the robust M-estimator and present several examples. In order to improve the performance for a very small number of years, we propose a second estimator, which incorporates information from other claims into the M-estimator.

In this paper, we present recursive formulae for the ruin probability at or before a certain claim arrival instant for some particular continuous time risk model. The claim number process underlying this risk model is a renewal process with either Erlang or a mixture of exponentials inter-claim times (ICTs). The claim sizes (CSs) are independent and distributed in Erlang's family, i.e., they can have different parameters, which yields a non-homogeneous risk process. We present the corresponding recursive algorithm used to evaluate the above mentioned ruin probability and we illustrate it on several numerical examples in which we vary the model's parameters to assess the impact of the non-homogeneity on the resulting ruin probability.

A new two-parameter family of analytical functions will be introduced for the modelling of loss distributions and exposure curves. The curve family contains the Maxwell-Boltzmann, the Bose-Einstein and the Fermi-Dirac distributions, which are well known in statistical mechanics. The functions can be used for the modelling of loss distributions on the finite interval [0, 1] as well as on the interval [0, ∞]. The functions defined on the interval [0, 1] are discussed in detail and related to several Swiss Re exposure curves used in practice. The curves can be fitted to the first two moments μ and σ of a loss distribution or to the first moment μ and the total loss probability p.