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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.

The quality of estimation of multivariate tails depends significantly on the portion of the sample included in the estimation. A simple approach involving sequential statistical testing is proposed in order to select which observations should be used for estimation of the tail and spectral measures. We prove that the estimator is consistent. We test the proposed method on simulated data, and subsequently apply it to analyze CoVaR for stock and index returns.

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.

We study the propagation of uncertainty from a class of priors introduced by Arias-Nicolás et al. [(2016) Bayesian Analysis, 11(4), 1107–1136] to the premiums (both the collective and the Bayesian), for a wide family of premium principles (specifically, those that preserve the likelihood ratio order). The class under study reflects the prior uncertainty using distortion functions and fulfills some desirable requirements: elicitation is easy, the prior uncertainty can be measured by different metrics, and the range of quantities of interest is easily obtained from the extremal members of the class. We illustrate the methodology with several examples based on different claim counts models.

A generalised property exposure rating framework is presented here to address two issues arising in the standard approach to exposure rating, especially in the context of direct insurance and facultative reinsurance (D&F) property pricing:

1. (a) What to do when the main assumption of exposure rating, scalability – that is, that the probability of a given damage ratio does not depend on the maximum possible loss (MPL) but only on the type of property – breaks down.

2. (b) How to take account of the uncertainty around the MPL, that is, the fact that the MPL (unlike the insured value, IV) is an informed estimate rather than a contractual feature.

The first difficulty is addressed by making the exposure rating framework more flexible by introducing exposure curves that are a mixture of scale-dependent and scale-independent losses, and where the weight given to the two components is a function of the MPL. This allows to model the impact on the expected losses of changes in the underlying deductibles, a classic problem in D&F pricing normally solved with the help of deductible impact tables. The second difficulty is addressed by working out the mathematical implications of having a finite probability of exceeding the MPL and extending the exposure curve up to the maximum of MPL and IV. A practical application of this framework – in which the scale-independent and scale-dependent losses are identified with attritional and large losses respectively – is described. A discussion of how this generalised framework can be calibrated based on actual data is included, and implementation code for the framework is made available.

The Sparre Andersen model assumes that the interclaim times and the amounts of claims are independent random variables, the former identically distributed according to a distribution function K(t), t ≥ o, K(o) = o, the latter identically distributed according to a distribution function P(y) — ∞ < y < ∞ As is well known, the Poisson risk process corresponds to the particular case K(t) = 1 — eβt. In the present paper it is pointed out that another particular case, viz. K(t) = є(t — h), corresponding to a fixed (and thus — strictly speaking—nonrandom) mterclaim time, h, has interesting applications. Thus, the ruin problem considered by Giezendanner, Straub and Wettenschwiler in a paper to the 1972 International Congress of Actuaries in Oslo can be formulated by means of this particular case. The same can be said about the earlier model brought forward by Ammeter in his 1948 paper in Skandinavisk Aktuarietidskrift.

About the contents of the paper the following further information may be given. The general Sparre Andersen model is first presented and then the ruin formulas are given for the case with a positive gross risk premium. Thereafter, a modified and more direct method for deriving certain necessary auxiliary functions is illustrated by examples including 1 a the Giezendanner—Straub—Wettenschwiler model. The rest of the paper contains a discussion from the point of view of the Sparre Andersen model of (1) the discrete (equidistant) inspection of a Poisson process for ruin, (11) the Ammeter model and analogous models, and (111) the Giezendanner—Straub—Wettenschwiler model.