This paper gives a method for premium rating by postcode area. The method is based on spatial models in a Bayesian framework and uses the Gibbs sampler for estimation. A summary of the theory of Bayesian spatial methods is given and the data which was analysed by Taylor (1989) is reanalysed. An indication is given of the wide range of models within this class which would be suitable for insurance data. The aim of the paper is to introduce the models and to show how they can be utilised in an insurance setting.

Novel navigation applications provide a driving behavior score for each finished trip to promote safe driving, which is mainly based on experts’ domain knowledge. In this paper, with automobile insurance claims data and associated telematics car driving data, we propose a supervised driving risk scoring neural network model. This one-dimensional convolutional neural network takes time series of individual car driving trips as input and returns a risk score in the unit range of (0,1). By incorporating credibility average risk score of each driver, the classical Poisson generalized linear model for automobile insurance claims frequency prediction can be improved significantly. Hence, compared with non-telematics-based insurers, telematics-based insurers can discover more heterogeneity in their portfolio and attract safer drivers with premiums discounts.

The concept of utility is twofold. One may think of utility:

1) as a tool to describe a “fair game”

2) as a quantity that ought to be maximized.

The first line of thought was initiated by Daniel Bernoulli in connection with the St. Petersburg Paradox. In recent decades, actuaries, economists, operations researchers and statisticians (this order is alphabetical) have been concerned mostly with optimization problems, which belong to the second category. Most of the actuarial models can be found in a paper by Borch [4] as well as in the texts by Beard, Pesonen and Pentikainen [3], Bühlmann [6], Seal [15], and Wolff [17].

We shall adopt the first variant and stipulate the existence of a utility function such that the surplus process of an insurance company is a fair game in terms of utility. This condition is naturally satisfied under the following procedure: a) a utility function is selected, possibly resulting from a compromise between an insurance company and supervising authorities, b) whenever the company makes a decision that affects the surplus, it should not affect the expected utility of the surplus.

Mathematically, this simply means that the utility of the surplus is a martingale. Therefore martingale theory (that was initiated by Doob) is the natural framework in which we shall study the model. We shall utilize one of the most powerful tools provided by this theory, the Martingale Convergence Theorem.

Christofides (1998) studies the proportional hazards (PH) transform of Wang (1995) and shows that for some parametric families, the PH premium principle reduces to the standard deviation (SD) premium principle. Christofides conjectures that for a parametric family of distributions with constant skewness, the PH premium principle reduces to the SD principle. I will show that this conjecture is false in general but that it is true for location-scale families and for certain other families.

Wang's premium principle has been established as a sound measure of risk in Wang (1995, 1996), Wang, Young, and Panjer (1997), and Wang and Young (1998). Determining when the SD premium principle is a special case of Wang's premium principle is important because it will help identify circumstances under which the more easily applied SD premium principle is a reliable measure of risk.

First, recall that a distortion g is a non-decreasing function from [0, 1] onto itself. Wang's premium principle, with a fixed distortion g, associates the following certainty equivalent with a random variable X, (Wang, 1996) and (Denneberg, 1994):

in which Sx is the decumulative distribution function (ddf) of X, Sx(t) = Pr(X > t), t ∈ R. If g is a power distortion, g(p) = pc, then Hg is the proportional hazards (PH) premium principle (Wang, 1995).

Second, recall that a location-scale family of ddfs is , in which Sz is a fixed ddf. Alternatively, if Z has ddf Sz, then {X = μ + σZ: μ∈ R, σ > 0} forms a location-scale family of random variables, and the ddf of . Examples of location-scale families include the normal, Cauchy, logistic, and uniform families (Lehmann, 1991, pp. 20f). In the next proposition, I show that Wang's premium principle reduces to the SD premium principle on a location-scale family. Christofides (1998) observes this phenomenon in several special cases.

The paid-incurred chain (PIC) reserving method is a claims reserving method that allows to combine claims payments and incurred losses information in a mathematical consistent way. The main criticism on the original Bayesian log-normal PIC model presented in Merz–Wüthrich [5] is that it does not respect dependence properties within the observed data. In the present paper, we extend the original Bayesian log-normal PIC model so that dependence is modeled in an appropriate way.

We develop Hachemeister's regression model in credibility theory (without proofs) and indicate how the involved structural parameters can be estimated from the observable variables (with proofs for the simple results and those not yet published).

Large families of unbiased estimators are available. From the practical viewpoint this is rather a handicap because it creates the problem to decide what estimators actually to use. In order to fix optimal estimators, we adopt the small-sample criterion of minimum-variance. But in the research for general solutions three kinds of difficulties arise.

(i) The calculations become too lengthy.

(ii) The optimal estimators depend on some of the parameters to be estimated. (Then we call them pseudo-estimators).

(iii) The optimal estimators depend on new structural parameters defined in terms of fourth-order moments.

Only a compromise allows to cope with this reality. Situation (iii) creates new estimation problems. They can only be avoided at the cost of the introduction of special assumptions or approximations. Then problem (i) is more or less automatically solved. By an obvious method of successive approximations pseudo-estimators can serve as true estimators. Thus (ii) is no real problem.

Frangos and Vrontos (2001) proposed an optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. In this paper, we introduce a generalized form of those obtained previously.

In this paper we study some bivariate counting distributions that are obtained by the trivariate reduction method. We work with Poisson compound distributions and we use their good properties in order to derive recursive algorithms for the bivariate distribution and bivariate aggregate claims distribution. A data set is also fitted.

Outlier observations caused by big claims or by an event producing a series of claims are a special problem in ratemaking and in tariff calculation. The authors believe that combining credibility and robust statistics is the right answer to this problem. The main idea is to robustify the individual claims experience by using a robust estimator Ti instead of the individual mean and to look at the credibility estimator based on the robust statistics {Ti: i = 1, 2, …} . Choosing a particular influence function leads to datatrimming with an observation-dependent trimming point.

In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the finite time ruin probability is well given when the claim amount distribution is a mixed exponential. As its consequence, a well-known result about ultimate ruin probability in the classical risk model is obtained.

In the present paper the author investigates the problem of calculating the net premium for some versions of the largest claims reinsurance cover. A very handy recursive rating method is derived by applying some recursion formulas for the expectations of order statistics.

To evaluate a large portfolio of variable annuity (VA) contracts, many insurance companies rely on Monte Carlo simulation, which is computationally intensive. To address this computational challenge, machine learning techniques have been adopted in recent years to estimate the fair market values (FMVs) of a large number of contracts. It is shown that bootstrapped aggregation (bagging), one of the most popular machine learning algorithms, performs well in valuing VA contracts using related attributes. In this article, we highlight the presence of prediction bias of bagging and use the bias-corrected (BC) bagging approach to reduce the bias and thus improve the predictive performance. Experimental results demonstrate the effectiveness of BC bagging as compared with bagging, boosting, and model points in terms of prediction accuracy.

We investigate a concept of multivariate pricing, which includes claim history for more than one line of business and is a generalization of the Bühlmann-Straub model. The multivariate credibility model is extended to allow for the age of claims to influence the estimation of future claims. The model is applied to data from a portfolio of commercial lines of business.

An insurance company is considered as an intermediary between policyholders and the capital market. By applying the traditional and the generalized version of the capital asset pricing model, a class of premium principles can be derived. This class is fully compatible with Bühlmann's economic premium principle. Moreover, insurance premiums can be directly related to risk premiums on the stock exchange.

The premiums for a bonus-malus system which stays in financial equilibrium over the years are calculated. This is done by minimizing a quadratic function of the difference between the premium for an optimal BMS with an infinite number of classes and the premium for a BMS with a finite number of classes, weighted by the stationary probability of being in a certain class, and by imposing various constraints on the system.

The implementation of hedging strategies for variable annuity products requires the calculation of market risk sensitivities (or “Greeks”). The complex, path-dependent nature of these products means that these sensitivities are typically estimated by Monte Carlo methods. Standard market practice is to use a “bump and revalue” method in which sensitivities are approximated by finite differences. As well as requiring multiple valuations of the product, this approach is often unreliable for higher-order Greeks, such as gamma, and alternative pathwise (PW) and likelihood-ratio estimators should be preferred. This paper considers a stylized guaranteed minimum withdrawal benefit product in which the reference equity index follows a Heston stochastic volatility model in a stochastic interest rate environment. The complete set of first-order sensitivities with respect to index value, volatility and interest rate and the most important second-order sensitivities are calculated using PW, likelihood-ratio and mixed methods. It is observed that the PW method delivers the best estimates of first-order sensitivities while mixed estimation methods deliver considerably more accurate estimates of second-order sensitivities; moreover there are significant computational gains involved in using PW and mixed estimators rather than simple BnR estimators when many Greeks have to be calculated.

Mixed Poisson distributions are widely used in various disciplines including actuarial applications. The family of mixed Poisson distributions contains several members according to the choice of the mixing distribution for the parameter of the Poisson distribution. Very few of them have been studied in depth, mainly because of algebraic intractability. In this paper we will describe an EM type algorithm for maximum likelihood estimation for mixed Poisson distributions. The main achievement is that it reduces the problem of estimation to one of estimation of the mixing distribution which is usually easier. Variants of the algorithm work even when the probability function of the mixed distribution is not known explicitly but we have only an approximation of it. Other discrete distributions are treated as well.

This paper addresses some of the problems a majority of retired individuals face: Why and in what proportion should they invest in a life annuity to maximize the utility of their future consumption or a bequest? The market considered in this work is made up of three assets: a life annuity, a risky asset and a cash account. As this problem doesn’t accept any suitable explicit solution, it is numerically solved by the Markov Chain approximation developed by Kushner and Dupuis. Without a bequest motive, we observe that the optimal planning of consumption is divided into two periods and that optimal asset allocation should include the risky asset. Next, the influence of a bequest on consumption and investment pattern is developed. We demonstrate that even with a bequest motive, pensioners should allocate a part of their wealth to the purchase of life annuities.

The authors have studied the combined data on claims in fire insurance of dwelling houses reported 1958-1969 by Swedish fire insurance companies The claims were cleared of deductibles and adjusted according to a suitable index Only losses above the largest deductible (in real value) applied during the observation period were included.

The material contains four different classes according to the fire resistibillty of the building construction For international comparisons, the pure classes B1 (“stone” dwellings) and B4 (wooden houses) are of interest The distribution of the claims could be well approximated by the log-normal distribution in B1and by the Pareto distribution in B4 An equally good or better fit was obtained by assuming the original loss, reported or not, being distributed according to these distributions and applying the distributions, conditioned by the loss being larger than the deductible in both cases the distribution parameters are functions of the insurance amount in such a way, that the mean value of the loss is described as a power of this amount.

The authors refrain from any theoretical arguments for the general applicability of the distributions used They observe, however, the good approximation by wellknown parametric distributions which facilitates many actuarial taks, such as the determination of first loss premiums, deductible premium factors, excess-of-loss premiums etc The agreement between model and reality make these functions fit for use in the models underlying the general risk theory and in the more comprehensive models of the non-life insurance business.

We consider several one-period reinsurance models and derive a rule which minimizes the ruin probability of the cedent for a fixed reinsurance risk premium. The premium is calculated according to the economic principle, generalized zero-utility principle, Esscher principle or mean-variance principles. It turns out that a truncated stop loss is an optimal treaty in the class of all reinsurance contracts. The result is also valid for models not involving ruin probability. An example is the Arrow model with the Kahneman-Tversky value function.