High-cardinality categorical features are pervasive in actuarial data (e.g., occupation in commercial property insurance). Standard categorical encoding methods like one-hot encoding are inadequate in these settings.

In this work, we present a novel Generalised Linear Mixed Model Neural Network (“GLMMNet”) approach to the modelling of high-cardinality categorical features. The GLMMNet integrates a generalised linear mixed model in a deep learning framework, offering the predictive power of neural networks and the transparency of random effects estimates, the latter of which cannot be obtained from the entity embedding models. Further, its flexibility to deal with any distribution in the exponential dispersion (ED) family makes it widely applicable to many actuarial contexts and beyond. In order to facilitate the application of GLMMNet to large datasets, we use variational inference to estimate its parameters—both traditional mean field and versions utilising textual information underlying the high-cardinality categorical features.

We illustrate and compare the GLMMNet against existing approaches in a range of simulation experiments as well as in a real-life insurance case study. A notable feature for both our simulation experiment and the real-life case study is a comparatively low signal-to-noise ratio, which is a feature common in actuarial applications. We find that the GLMMNet often outperforms or at least performs comparably with an entity-embedded neural network in these settings, while providing the additional benefit of transparency, which is particularly valuable in practical applications.

Importantly, while our model was motivated by actuarial applications, it can have wider applicability. The GLMMNet would suit any applications that involve high-cardinality categorical variables and where the response cannot be sufficiently modelled by a Gaussian distribution, especially where the inherent noisiness of the data is relatively high.

The distribution of the sum of n mutuality independent random variables with a common distribution f(x) plays an important role in many insurance problems. The paper presents alternative methods of deriving the distribution of the sum of n random variables when f(x) is a mixed density and mass function. The methods are illustrated and compared.

For the purpose of Value-at-Risk (VaR) analysis, a model for the return distribution is important because it describes the potential behavior of a financial security in the future. What is primarily, is the behavior in the tail of the distribution since VaR analysis deals with extreme market situations. We analyze the extension of the normal distribution function to allow for fatter tails and for time-varying volatility. Equally important to the distribution function are the associated parameter values. We argue that parameter uncertainty leads to uncertainty in the reported VaR estimates. There is a tradeoff between more complex tail-behavior and this uncertainty. The “best estimate”-VaR should be adjusted to take account of the uncertainty in the VaR. Finally, we consider the VaR forecast for a portfolio of securities. We propose a method to treat the modeling in a univariate, rather than a multivariate, framework. Such a choice allows us to reduce parameter uncertainty and to model directly the relevant variable.

In this paper, we study the optimal asset allocation problem under a discrete regime switching model. Under the short-selling and leveraging constraints, the existence and uniqueness of the optimal trading strategy are obtained. We also obtain some natural properties of the optimal strategy. In particular, we show that if there exists a stochastic dominance order relationship between the random returns at different regimes, then we can order the optimal proportions we should invest in such regimes.

It is likely that in the future applications of actuarial methods to the decision making in non-life companies will more and more relate to the utility concept as was proposed by K. Borch [1] about fifteen years ago. In this connection it will be important to have workable numerical methods. The calculation of the distribution function of the profit is an unavoidable problem from a practical point of view. Even if it is possible to compute this function today accurately with computers by using the ingenious technique developed by H. Bohman [2], integrals become very laborious when applied to the decision making procedure based on utility concepts. This paper intends to show that the NP-technique,—proposed for the first time by L. Kauppi and P. Ojantakanen in actuarial science [3]—, is particularly suitable in integrals needed for utility calculations.

LetF(x) be the distribution function of the total amount of claims and let its mean, standard deviation, skewness and kurtosis be respectively m, σ, γ1 and γ2. The NP-technique uses the system of equations

where Φ(y) is the standardized normal distribution function. If the parameters m, σ, γ1 and γ2 and F(x) are known, y is directly found from the tables of the normal distribution function, and thereafter the second equation directly gives the value of x. If, vice versa, x and the above parameters are known, F(x) is obtained by solving y from the second equation (1), or, more practically, by using the converted NP-expansion instead of (i), i.e. [4]:

where

Sometimes it is sufficient to use short forms of the formulae (1) and (2), obtained by omitting the terms in the brackets. If these rougher approximations are used, the estimation of the kurtosis γ2 remains unnecessary.

This study introduces an equilibrium approach to price mortality-linked securities in a discrete time economy, assuming that the mortality rate has a transformed normal distribution. This pricing method complements current studies on the valuation of mortality-linked securities, which only have discrete trading opportunities and insufficient market trading data. Like the Wang transform, the valuation relationship is still risk-neutral (preference-free) and the mortality-linked security is priced as the expected value of its terminal payoff, discounted by the risk-free rate. This study provides an example of pricing the Swiss Re mortality bond issued in 2003 and obtains an approximated closed-form solution.

Some of the results obtained in an earlier paper entitled “An Invariance property of the Swiss premium calculation principle” by F. De Vylder and M. Goovaerts (1979) are generalized. For that purpose the notion of additivity and iterativity is extended. Some rather general characterization theorems for some premium calculation principles are obtained.

The notion of ordering and danger of claim size distributions is extended to claim frequency distributions.

Episode Treatment Groups (ETGs) classify related services into medically relevant and distinct units describing an episode of care. Proper model selection for those ETG-based costs is essential to adequately price and manage health insurance risks. The optimal claim cost model (or model probabilities) can vary depending on the disease. We compare four potential models (lognormal, gamma, log-skew-t and Lomax) using four different model selection methods (AIC and BIC weights, Random Forest feature classification and Bayesian model averaging) on 320 ETGs. Using the data from a major health insurer, which consists of more than 33 million observations from 9 million claimants, we compare the various methods on both speed and precision, and also examine the wide range of selected models for the different ETGs. Several case studies are provided for illustration. It is found that Random Forest feature selection is computationally efficient and sufficiently accurate, hence being preferred in this large data set. When feasible (on smaller data sets), Bayesian model averaging is preferred because of the posterior model probabilities.

The study of desirable structural properties that define a marketable insurance contract has been a recurring theme in insurance economic theory and practice. In this article, we develop probabilistic and structural characterizations for insurance indemnities that are universally marketable in the sense that they appeal to all policyholders whose risk preferences respect the convex order. We begin with the univariate case where a given policyholder faces a single risk, then extend our results to the case where multiple risks possessing a certain dependence structure coexist. The non-decreasing and 1-Lipschitz condition, in various forms, is shown to be intimately related to the notion of universal marketability. As the highlight of this article, we propose a multivariate mixture model which not only accommodates a host of dependence structures commonly encountered in practice but is also flexible enough to house a rich class of marketable indemnity schedules.

Hans Bühlmann traite dans son ouvrage “Optimale Prämien-stufensysteme” (1964) principalement le problème de trouver l'estimation optimale de la prime de la n-ième période à partir des observations de sinistres au cours des périodes précédentes.

Max Gürtler a traité dans le “Bonus ou Malus” du Astin Bulletin vol. III part I deux systèmes différents pour la détermination de la prime individuelle pour l'assurance automobile, c'est-à-dire

a) système “Absence de sinistre”

b) système “Nombre de sinistre”.

A l'aide de simples modèles (examples page 56) pour le groupement de portefeuilles de risques bons et lourds Gürtler étudie l'effet des deux systèmes et en tire des conclusions générales sur l'applicabilité des systèmes. Gürtler souligne d'abord que le système b) utilise considérablement mieux l'information, parce que tous les sinistres y sont considéréd et non pas, comme dans le système a), seulement le premier sinistre étant registré pour chaque assurance.

Marcel Derron traite dans “A theoretical study of the no-claim bonus problem” du même cahier de la publication la quote-part du ER (Error ratio) qu'il définit comme une quote-part égale au montant absolut de la différence entre la prime payée et la vraie prime divisée par la prime payée. Il calcule la quote-part du ER pour le modële de Gürtler et pour des modèles alternatifs et cherche le système de bonification qui rend la quote-part du ER aussi petite que possible pour le modèle choisi.

The Pareto-optimal design for profit-sharing is derived under general assumptions as to the utility function of both the insured and the insurer. This generalizes the result of Jones and Gerber and explains commonly used dividend formulas in terms of risk aversion.

In various applications of heavy-tail modelling, the assumed Pareto behaviour is tempered ultimately in the range of the largest data. In insurance applications, claim payments are influenced by claim management and claims may, for instance, be subject to a higher level of inspection at highest damage levels leading to weaker tails than apparent from modal claims. Generalizing earlier results of Meerschaert et al. (2012) and Raschke (2020), in this paper we consider tempering of a Pareto-type distribution with a general Weibull distribution in a peaks-over-threshold approach. This requires to modulate the tempering parameters as a function of the chosen threshold. Modelling such a tempering effect is important in order to avoid overestimation of risk measures such as the value-at-risk at high quantiles. We use a pseudo maximum likelihood approach to estimate the model parameters and consider the estimation of extreme quantiles. We derive basic asymptotic results for the estimators, give illustrations with simulation experiments and apply the developed techniques to fire and liability insurance data, providing insight into the relevance of the tempering component in heavy-tail modelling.

In order to fix our ideas an illustration of the theory for (a) a general elementary random process, (b) a compound Poisson process and (c) a Polya process shall be given here below following Ove Lundberg (On Random Processes and Their Application to Accident and Sickness Statistics, Inaug. Diss., Uppsala 1940).

Let the continuous parameter t* be measured on an absolute scale from a given point of zero and consider the random function N* (t*) which takes only non-negative and integer values with N* (o) = o. This function constitutes a general elementary random process for which the conditional probability that N* (t*) = n relative to the hypothesis that shall be denoted , while the absolute probability that N* (t) = n i.e. shall be written If quantities of lower order than dt* are neglected, we may write for the conditional probability that N* (t* + dt*) = n + 1 relative to thehyp othesis that N* (t*) = n, i.e. is the intensity function of the process which is assumed to be a continuous function of t* (the condition of existence for the integral over the given interval of t* for every n > m may be substituted for the condition of continuity). The expectations for an arbitrary but fix value of t* of N* (t*) and p* (t*) will be denoted by the corresponding symbol with a bar so that

If is uniformly bounded for all n in the interval o ≤ t* < T*, where T* is an arbitrary but fix value of t*, we have i.a. that

In preparing the notes on the subjects for discussion at the 4th ASTIN Colloquium at Trieste [1] I used some of the material which formed the basis of a talk given to the Scandinavian Actuarial Societies in September 1962. The papers presented in Trieste have established on a firm mathematical footing the formula for the excess loss premium derived in my note but the discussions also showed that some of the other 1962 material would be of interest.

It will be appreciated that the question originally proposed, i.e. to calculate an excess of loss reinsurance premium when the only information available is the largest claim experienced in each of a succession of periods, was deliberately phrased in this form as being the most troublesome case likely to arise in practice. Clearly if other information is available it would not be rejected in arriving at a premium, but this immediately extends the problem to one of finding the best methods of combining information of different kinds. For example if, say, the largest 5 claims in each period are known, what technique will make the maximum use of the data? Such extensions of the problem are not discussed in this note. Furthermore, I am not unmindful of the valuable comment by Jung, “that there is a natural law which states that you can never get more out of a mincing machine than what you have put into it” [2].

The chain ladder forecast of outstanding losses is known to be unbiased under suitable assumptions. According to these assumptions, claim payments in any cell of a payment triangle are dependent on those from preceding development years of the same accident year. If all cells are assumed stochastically independent, the forecast is no longer unbiased. Section 5 shows that, under rather general assumptions, it is biased upward. This result is linked to earlier work on some stochastic versions of the chain ladder.

Assessing conditional tail risk at very high or low levels is of great interest in numerous applications. Due to data sparsity in high tails, the widely used quantile regression method can suffer from high variability at the tails, especially for heavy-tailed distributions. As an alternative to quantile regression, expectile regression, which relies on the minimization of the asymmetric l2-norm and is more sensitive to the magnitudes of extreme losses than quantile regression, is considered. In this article, we develop a new estimation method for high conditional tail risk by first estimating the intermediate conditional expectiles in regression framework, and then estimating the underlying tail index via weighted combinations of the top order conditional expectiles. The resulting conditional tail index estimators are then used as the basis for extrapolating these intermediate conditional expectiles to high tails based on reasonable assumptions on tail behaviors. Finally, we use these high conditional tail expectiles to estimate alternative risk measures such as the Value at Risk (VaR) and Expected Shortfall (ES), both in high tails. The asymptotic properties of the proposed estimators are investigated. Simulation studies and real data analysis show that the proposed method outperforms alternative approaches.

Japan is one of the most earthquake-prone countries in the world and a considerable volume of earthquake insurance is written by private insurers. Therefore, a report on earthquake insurance in Japan has been prepared for the 13th ASTIN Colloquium.

We will first summarise, from recent results of seismological studies, the matters which would form the basis for a study of earthquake insurance, and then explain how this insurance actually works in Japan.

In several different countries there has lately been a very unfavourable development as far as fire losses are concerned. There have been many and very bad big fires which have especially affected industry and trade. The facts which can be considered to have caused this development are well-known. Bigger and bigger and more concentrated units of production, business enterprises, and storing form an accumulation of values which in itself implies increasing risks. On account of production-technical and business-economical conditions it has not always been possible to divide the buildings into sections desirable from a fire service point of view. The new production methods have sometimes caused new fire risks.

The Northern Central Office for Fire Insurance Statistics from Mutual Companies (CNÖB) has made an investigation in order to see to what extent the international trend towards increasing fire losses has any equivalence in the Northern countries (Denmark, Finland, Norway, and Sweden). I.a. there was made a graphical representation of how the observed rate of net risk premium of the portfolio has developed during the period 1946–1966, the result of which may be seen from fig. 1. No increasing trend can be observed from this. After having been decreasing during the immediate postwar time, the observed rate of net risk premium has been fairly constant from the beginning of the fifties, perhaps even somewhat decreasing except during the later years. The conclusion is that in so far facts as mentioned above have influenced the development of claims of the portfolios or parts of these in a deteriorating way, there must in that case have been circumstances which have worked as neutralizers.

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.