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.

A premium calculation principle π is called positively homogeneous if π(cX) = cπ(X) for all c > 0 and all random variables X. For all known principles it is shown that this condition is fulfilled if it is satisfied for two specific values of c only, say c = 2 and c = 3, and for only all two point random variables X. In the case of the Esscher principle one value of c suffices. In short this means that local homogeneity implies global homogeneity. From this it follows that in the case of the zero utility principle or Swiss premium calculation principle, the underlying utility function is of a very specific type.

A very general theorem on premium calculation principles which satisfy a weak continuity condition, is added. Among others the proof uses Kroneckers Theorem on Diophantine Approximations.

We revisit the relative retention problem originally introduced by de Finetti using concepts recently developed in risk theory and quantitative risk management. Instead of using the Variance as a risk measure we consider the Expected Shortfall (Tail-Value-at-Risk) and include capital costs and take constraints on risk capital into account. Starting from a risk-based capital allocation, the paper presents an optimization scheme for sharing risk in a multi-risk class environment. Risk sharing takes place between two portfolios and the pricing of risktransfer reflects both portfolio structures. This allows us to shed more light on the question of how optimal risk sharing is characterized in a situation where risk transfer takes place between parties employing similar risk and performance measures. Recent developments in the regulatory domain (‘risk-based supervision’) pushing for common, insurance industry-wide risk measures underline the importance of this question. The paper includes a simple non-life insurance example illustrating optimal risk transfer in terms of retentions of common reinsurance structures.

Kaas, Dannenburg & Goovaerts (1997) generalized Jewell’s theorem on exact credibility, from the classical Bühlmann model to the (weighted) Bühlmann-Straub model. We extend this result further to the “Bühlmann-Straub model with a priori differences” (Bühlmann & Gisler, 2005). It turns out that exact credibility holds for a class of Tweedie models, including the Poisson, gamma and compound Poisson distribution – the most important distributions for insurance applications of generalized linear models (GLMs). Our results can also be viewed as an alternative to the HGLM approach for combining credibility and GLMs, see Nelder and Verrall (1997).

We introduce a class of Bayesian infinite mixture models first introduced by Lo (1984) to determine the credibility premium for a non-homogeneous insurance portfolio. The Bayesian infinite mixture models provide us with much flexibility in the specification of the claim distribution. We employ the sampling scheme based on a weighted Chinese restaurant process introduced in Lo et al. (1996) to estimate a Bayesian infinite mixture model from the claim data. The Bayesian sampling scheme also provides a systematic way to cluster the claim data. This can provide some insights into the risk characteristics of the policyholders. The estimated credibility premium from the Bayesian infinite mixture model can be written as a linear combination of the prior estimate and the sample mean of the claim data. Estimation results for the Bayesian mixture credibility premiums will be presented.

Estimation risk is known to have a huge impact on mean/variance optimized portfolios, which is one of the primary reasons to make standard Markowitz optimization unfeasible in practice. This issue has attracted new interest in the last years, and several approaches to incorporate estimation risk into portfolio selection have been developed only recently. In this article, we review these approaches as well as some older ones and compare them in an empirical out-of-sample study. The approaches can be classified along two criteria. First, we can differentiate heuristic approaches (restricting portfolio weights and employing simulation techniques) and those based on Bayesian statistics (shrinking the portfolios towards a pre-determined target). Second, the assumptions about the return-generating process differ, either assuming returns to be IID distributed or to be partly predictable. The central result of our empirical study is that all of the IID approaches, whether they account for estimation risk or not, are not superior to simple investment strategies like holding the market portfolio. A risk-adjusted outperformance is possible only if sample means are substituted with conditional expected return estimates. Furthermore, the Bayesian approaches reduce turnover and stabilize portfolio weights.

Telematicsdevices installed in insured vehicles provide actuaries with new risk factors, such as the time of the day, average speeds, and other driving habits. This paper extends the multivariate mixed model describing the joint dynamics of telematics data and claim frequencies proposed by Denuit et al. (2019a) by allowing for signals with various formats, not necessarily integer-valued, and by replacing the estimation procedure with the Expected Conditional Maximization algorithm. A numerical study performed on a database related to Pay-How-You-Drive, or PHYD motor insurance illustrates the relevance of the proposed approach for practice.

We consider the class of concave distortion risk measures to study how choice is influenced by the decision-maker's attitude to risk and provide comparative statics results. We also assume ambiguity about the probability distribution of the risk and consider a framework à la Klibanoff, Marinacci and Mukerji (2005; A smooth model of decision making under ambiguity. Econometrica, 73, 1849–1892) to study the value of information that resolves ambiguity. We show that this value increases with greater ambiguity, with greater ambiguity aversion, and in some cases with greater risk aversion. Finally, we examine whether a more risk-averse and a more ambiguity-averse individual will invest in more effort to shift his initial risk distribution to a better target distribution.

The paper incorporates liquid reserves, interest and dividends in the compound Poisson surplus model. When an insurer's surplus is below a certain level, it is kept as liquid reserves. As the surplus attains the level, the excess of the surplus above the level will earn interest at a constant interest rate. If the surplus continues to surpass a higher level, the excess of the surplus above this higher level will be paid out as dividends to the insurer's shareholders at a constant dividend rate or by the threshold strategy. The lower and higher levels are called the liquid reserve level and the threshold level, respectively.

This paper is to discuss the interactions of the liquid reserve level, the interest rate, the threshold level, and the dividend rate in the proposed risk model by studying the expected discounted penalty function and the expected present value of dividends paid up to the time of ruin. We derive expressions for the solutions to both quantities via the approach of integro-differential equation systems. We show that the dividend-penalty identity (Gerber et al. 2006, ASTIN Bulletin) still holds for the threshold strategy with liquid reserves and interest. We illustrate these results by deriving explicit solutions to the probability of ultimate ruin under the threshold strategy when claim sizes are exponentially distributed. In the end, we also discuss the impact of the liquid reserve level, the interest rate, the threshold level, and the dividend rate on the ruin probability by numerical examples.

In this paper we present a method for the numerical evaluation of the ruin probability in continuous and finite time for a classical risk process where the premium can change from year to year. A major consideration in the development of this methodology is that it should be easily applicable to large portfolios. Our method is based on the simulation of the annual aggregate claims and then on the calculation of the ruin probability for a given surplus at the start and at the end of each year. We calculate the within-year ruin probability assuming a translated gamma distribution approximation for aggregate claim amounts.

We illustrate our method by studying the case where the premium at the start of each year is a function of the surplus level at that time or at an earlier time.

The Esscher premium principle has recently had some exposure, namely, with the works of Bühlmann (1980) and Gerber (1980).

Bühlmann (1980) devised the principle and coined the name for it within the framework of utility theory and risk exchange. Geruber (1980), on the other hand, gives further insight into the principle by studying it within the realm of forecasting in much the same spirit as credibility theorists forecast premiums. However, there is an important distinction: the choice of loss function.

The present note sets out to criticize this relatively embryonic principle using decision theoretic arguments and indicates that the Esscher premium is essentially a small perturbation of the well established linearized credibility premium Bühlmann (1970).

Let H denote the Esscher premium principle with loading h > o. That is, if X is an observable random variable and Y is a parameter (a risk or a random quantity) to be forecasted then the Esscher premium is given by

That is, H(Y ∣ X) is the Bayes decision rule for estimating Y given the data X relative to the loss function

where a is the estimate of Y, and of course the loading h is greater than zero.

Now, for the clincher. This loss function is nonsensical from the point of view of estimation. It indicates a loss (or error) to the forecaster that is essentially the antithesis of relative loss.

Assurance companies usually divide their portfolio into a series of tariff groups. This division is necessary for a comparison between the claims and the available premiums in the different tariff groups. By comparing these figures the company is able to verify if the tariff is appropriate or if a modification in line with the empirical data is to be recommended. Apart from this technical argument companies are sometimes compelled to make such a division because of their constitution (e.g. mutual companies) or because it is prescribed by the supervisory authorities. Companies underwriting in various countries are under the obligation, as a rule, to keep special accounts for different countries and currencies. All these reasons lead to a division of the whole business into more or less independent subdivisions which quite obviously have not as much capacity to equalize the risk fluctuations as the whole portfolio.

In every tariff group an exceptional fluctuation may occur either from a large individual claim or from claims under number of policies. This leads to a substantial loss in the relevant profit and loss statement at the end of the year. If this debit balance is carried forward to the new account every year, the account of such a tariff group may show a loss over a number of years. It is obvious that a company will endeavour to avoid such an awkward situation.

This sort of problem is usually dealt with by means of reinsurance and, to some extent this will be the right solution. However if every tariff group should be able to meet its own liabilities, a rather extended reinsurance is necessary. The capacity of the whole portfolio then would be of no importance, which suggests that a solution through reinsurance alone is not entirely satisfactory.

Under certain conditions, a Bonus-Malus system can be interpreted as a Markov chain whose n-step transition probabilities converge to a limit probability distribution. In this paper, the rate of the convergence is studied by means of the eigenvalues of the transition probability matrix of the Markov chain.

In this paper, we consider the problem of Pareto optimal allocation in a general framework, involving preference functionals defined on a general real vector space. The optimization problem is equivalent to a modified sup-convolution of the different agents’ preference functionals. The results are then applied to a multi-period setting and some further characterization of Pareto optimality for an allocation is obtained for expected utility for processes.

In non-life insurance, territory-based risk classification is useful for various insurance operations including marketing, underwriting, ratemaking, etc. This paper proposes a spatially dependent frequency-severity modeling framework to produce territorial risk scores. The framework applies to the aggregated insurance claims where the frequency and severity components examine the occurrence rate and average size of insurance claims in each geographic unit, respectively. We employ the bivariate conditional autoregressive models to accommodate the spatial dependency in the frequency and severity components, as well as the cross-sectional association between the two components. Using a town-level claims data of automobile insurance in Massachusetts, we demonstrate applications of the model output–territorial risk scores–in ratemaking and market segmentation.

This paper analyses the impact of using different correlation assumptions between lines of business when estimating the risk-based capital reserve, the solvency capital requirement (SCR), under Solvency II regulations. A case study is presented and the SCR is calculated according to the standard model approach. Alternatively, the requirement is then calculated using an internal model based on a Monte Carlo simulation of the net underwriting result at a one-year horizon, with copulas being used to model the dependence between lines of business. To address the impact of these model assumptions on the SCR, we conduct a sensitivity analysis. We examine changes in the correlation matrix between lines of business and address the choice of copulas. Drawing on aggregate historical data from the Spanish non-life insurance market between 2000 and 2009, we conclude that modifications of the correlation and dependence assumptions have a significant impact on SCR estimation.

We are dealing with the ruin probability and the expected ruin time in a two state Markov model where the premium is the reciprocal of an integer and the initial surplus is a multiple of the premium.

A conventional practice in standard risk theory considerations has been to assume that claims are paid immediately as they have incurred (see BPP, item 3.1c, BPP is used as an abbreviation for the book “Risk Theory”, 1984 edition, by Beard, Pentikäinen, Pesonen). The delay of the claims settlement has been, of course, a central aspect in reserve calculation theory and practices, and numerous valuable works have been published on this topic in recent years. However, its regard in general model building and in risk theory considerations has gained little attention until recent years. The purpose of this paper is to contribute to this research work by discussing how the “run-off” risk, i.e., the variability due to the delay of the claims payment, could be incorporated into the standard risk theory models as a separate entry (see BPP, item 10.2e) and to find some evaluation of the order of magnitude of the “extra” (if any) fluctuation so rendered. We expect that the proposed technique can also be utilized in testing different reserve calculation methods and in comparing their effectiveness. The main ideas follow very much along the lines given by Rantala in his doctoral thesis (1984).

One should appreciate the fact that any risk theory model can never be more than an idealization of real-life processes. An intricate problem for practitioners is to evaluate the uncertainties ensuing from the fact that the model, more or less, ignores or only approximates the factors affecting the real events, and that the practical applications are often based on and their necessary parameters estimated from observed data that are subject to random fluctuations and to many other kinds of uncertainties. The problem complex of the run-off risk, when understood in a broad sense, is so wide that it requires a series of studies, and this paper should be regarded as a first step only. The posing of the problem follows the conventional risk theory approach by using the mixed compound Poisson process further allowing for long-term variations of risk exposure (“cycles”), and now extending the model to cope with the delayed settlement of the claims. At this stage of the on-going researchwork the impact of the parameter estimation is excluded from consideration. Therefore, our results and the numerical examples, as given in what follows, do not describe the total uncertainty of the claims or the reserves.

This paper considers the portfolio selection and capital injection problem for a diffusion risk model within the classical Black–Scholes financial market. It is assumed that the original surplus process of an insurance portfolio is described by a drifted Brownian motion, and that the surplus can be invested in a risky asset and a risk-free asset. When the surplus hits zero, the company can inject capital to keep the surplus positive. In addition, it is assumed that both fixed and proportional costs are incurred upon each capital injection. Our objective is to minimize the expected value of the discounted capital injection costs by controlling the investment policy and the capital injection policy. We first prove the continuity of the value function and a verification theorem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation. We then show that the optimal investment policy is a solution to a terminal value problem of an ordinary differential equation. In particular, explicit solutions are derived in some special cases and a series solution is obtained for the general case. Also, we propose a numerical method to solve the optimal investment and capital injection policies. Finally, a numerical study is carried out to illustrate the effect of the model parameters on the optimal policies.

We consider the multivariate Value-at-Risk (VaR) and Conditional-Tail-Expectation (CTE) risk measures introduced in Cousin and Di Bernardino (Cousin, A. and Di Bernardino, E. (2013) Journal of Multivariate Analysis, 119, 32–46; Cousin, A. and Di Bernardino, E. (2014) Insurance: Mathematics and Economics, 55(C), 272–282). For absolutely continuous Archimedean copulas, we derive integral formulas for the multivariate VaR and CTE Archimedean risk measures. We show that each component of the multivariate VaR and CTE functional vectors is an integral transform of the corresponding univariate VaR measures. For the class of Archimedean copulas, the marginal components of the CTE vector satisfy the following properties: positive homogeneity (PH), translation invariance (TI), monotonicity (MO), safety loading (SL) and VaR inequality (VIA). In case marginal risks satisfy the subadditivity (MSA) property, the marginal CTE components are also sub-additive and hitherto coherent risk measures in the usual sense. Moreover, the increasing risk (IR) or stop-loss order preserving property of the marginal CTE components holds for the class of bivariate Archimedean copulas. A counterexample to the (IR) property for the trivariate Clayton copula is included.