The estimation of outstanding claims is one of the important aspects in the management of the insurance business. Various methods have been widely dealt with in the actuarial literature. Exploration of the inaccuracies involved is traditionally based on a post-facto comparison of the estimates against the actual outcomes of the settled claims. However, until recent years it has not been usual to consider the inaccuracies inherent in claims reserving in the context of more comprehensive (risk theoretical) models, the purpose of which is to analyse the insurer as a whole. Important parts of the technique which will be outlined in this paper can be incorporated into over-all risk theory models to introduce the uncertainty involved with technical reserves as one of the components in solvency and other analyses (Pentikäinen et al. (1989)).

The idea in this paper is to describe a procedure by which one can explore how various reserving methods react to fictitious variations, fluctuations, trends, etc. which might influence the claims process, and, what is most important, how they reflect on the variables indicating the financial position of the insurer. For this purpose, a claims process is first postulated and claims are simulated and ordered to correspond to an actual handling of the observed claims of a fictitious insurer. Next, the simulation program will ‘mime’ an actuary who is calculating the claims reserve on the basis of these ‘observed’ claims data. Finally, the simulation is further continued thus generating the settlement of the reserved claims. The difference between reserved amounts and settled amounts gives the reserving (run-off) error in this particular simulated case. By repeating the simulation numerous times (Monte Carlo method) the distribution of the error can be estimated as well as its effect on the total outcome of the insurer.

Models for the risk business of an insurance company are often constructed by weighting pure Poisson models. In this paper it is verified that it is possible to calculate the probability of ruin in such weighted models by weighting ruin probabilities of pure Poisson models.

A recent survey of actuarial practitioners in North America shows that smoothed-market actuarial asset values are commonly used in funding valuations of defined benefit pension plans. Four methods of calculating such values are reported in the actuarial literature but only qualitative descriptions of the methods are given. This paper provides mathematical descriptions of the “average of market”, “weighted average”, “deferred recognition” and “write-up” actuarial values. They are shown to be based on either arithmetic or exponential smoothing. Provided the same form of smoothing is used, the four methods are equivalent.

Second order Bayes estimators, being the main tool in second order optimal statistical theory, provide a natural basis for a new approach to the problem of the prediction of functions of expectation functional for members of an exponential dispersion family. A general formula, providing such prediction up to the term of the order 1/n, is suggested and the application to the problem of the prediction of the tail of distributions is demonstrated. The results are illustrated with normal and gamma claim sizes. The numerical experiment demonstrates the high effectiveness of the approach even for small sample sizes.

The model introduced may be treated as a mixed two-way analysis of variance with fixed company effects and random time effects. Further, the risk volumes are integrated into the model in such a way that the unexplained variance is inversely proportional to the risk volume of each company. The proposed model is used to analyze loss ratio data from the general insurance market in Kuwait. The maximum likelihood estimates of the structural parameters are obtained. These estimates are then used to compute the loss ratios and solvency margins for the four domestic insurance companies.

The estimation of stop loss premiums can be based on some knowledge about the distribution function of the sum of all claims in a year (assuming that the stop loss insurance relates to a period of one calender year). Generally speaking there are two methods to obtain this knowledge about the distribution function.

1. The first method is to construct a distribution function from data concerning:

a. the distribution function of the number of claims per year, taking into account the variability of the parameter(s) of this distribution function.

b. the distribution function of the insured sums.

c. the distribution function of partial claims.

d. the correlation between the insured sum and the probability of occurring of a claim.

e. the probability of contagion.

2. The second method is to derive a distribution function from the year's totals of claims over a long series of years, expressed in e.g: units of the totals of insured sums in that years.

In practice it is often difficult to find a useful basis to apply one of these methods. Data concerning the distribution function of the number of claims per year, of the insured sums, and of partial claims are mostly available, but often nothing is known about the correlation between the insured sum and the probability of occurring of a claim.

The second method is mostly not applicable because, if the year's totals of claims over a long series of, by preference recent, years are available, these data often turn out to be heterogeneous or to be correlated with time. If, in that case, only the data of the most recent years are used, the number of these data is often a too small basis for the construction of a distribution function.

The model uncertainty issue is pervasive in virtually all applied fields but especially critical in insurance and finance. To hedge against the uncertainty of the underlying probability distribution, which we refer to as ambiguity, the worst case is often considered in quantifying the underlying risk. However, this worst-case treatment often yields results that are overly conservative. We argue that, in most practical situations, a generic risk is realized from multiple scenarios and the risk in some ordinary scenarios may be subject to negligible ambiguity so that it is safe to trust the reference distributions. Hence, we only need to consider the worst case in the other scenarios where ambiguity is significant. We implement this idea in the study of the worst-case moments of a risk in the hope to alleviate the over-conservativeness issue. Note that the ambiguity in our consideration exists in both the scenario indicator and the risk in the corresponding scenario, leading to a two-fold ambiguity issue. We employ the Wasserstein distance to construct an ambiguity ball. Then, we disentangle the ambiguity along the scenario indicator and the risk in the corresponding scenario, so that we convert the two-fold optimization problem into two one-fold problems. Our main result is a closed-form worst-case moment estimate. Our numerical studies illustrate that the consideration of partial ambiguity indeed greatly alleviates the over-conservativeness issue.

The variance of statistical estimates of outstanding claim payments for long-tailed general insurance portfolios is examined. The variance's three components are discussed. As there is no accepted technique for measuring this variance three methods are investigated empirically for its measurement—a parametric method, the jackknife method, and the bootstrap method. No method stands out as superior to the others and it is recommended that all three be evaluated and used to gauge the possible errors in the estimation of outstanding claims.

1. The comfound Poisson process in the wide sense is defined as a process for which the probability distribution of the number i of changes in the random function attached to the process, while the parameter passes from o to a fixed value τ of the parameter measured on a suitable scale, is given by the Laplace-Stieltjes integral

where U(ν, τ) for a fixed value of τ defines the distribution of ν. U(ν, τ) is called the risk distribution and is either τ-independent or, dependent on ν, τ.

2. The compound Poisson process in the narrow sense is defined as a process for which the probability distribution of the number of changes can be written in the form of (I) with a τ-independent risk distribution.

In their general form these processes have been analyzed by Ove Lundberg (1940). For such processes the following relation holds for the probability of i changes in the interval ο to τ, P̅i (τ) say

this relation does not hold for processes with τ-dependent risk distribution. Hofmann (1955) has introduced a sub-set of the processes concerned in this section for which the probability for non-occurrence of a change in the interval o to τ is defined as a solution of the differential equation

and ϰ ≥ o; the solutions may be written in the form where η is independent of and of two alternative forms one for ϰ = I and one for other values of ϰ. The probabilities for i changes in the interval o to τ in the processes defined by the solutions of Hofmann's equation are derived by Leibniz's formula, and are designated by and, in this paper, called Hofmann probabilities.

The UK Biobank project is a proposed large-scale investigation of the combined effects of genotype and environmental exposures on the risk of common diseases. It is intended to recruit 500,000 subjects aged 40-69, to obtain medical histories and blood samples at outset, and to follow them up for at least 10 years. This will have a major impact on our knowledge of multifactorial genetic disorders, rather than the rare but severe single-gene disorders that have been studied to date. What use may insurance companies make of this knowledge, particularly if genetic tests can identify persons at different risk? We describe here a simulation study of the UK Biobank project. We specify a simple hypothetical model of genetic and environmental influences on the risk of heart attack. A single simulation of UK Biobank consists of 500,000 life histories over 10 years; we suppose that case-control studies are carried out to estimate age-specific odds ratios, and that an actuary uses these odds ratios to parameterise a model of critical illness insurance. From a large number of such simulations we obtain sampling distributions of premium rates in different strata defined by genotype and environmental exposure. We conclude that the ability of such a study reliably to discriminate between different underwriting classes is limited, and depends on large numbers of cases being analysed.

The sensitivity of the ruin probability depending on the claim size distribution has been the topic of several discussion papers in recent ASTIN Bulletins. This discussion was initiated by a question raised by Schmitter at the ASTIN Colloquium 1990 and attempts to make further contributions to this problem. We find the necessary and sufficient conditions for fitting three given moments by diatomic and diexponential distributions. We consider three examples drawn from fire (large spread), individual life (medium spread) and group life (small spread) insurance data, fit them with diatomics and diexponentials whenever the necessary and sufficient conditions are met, and compute the ruin probabilities using well known formulas for discrete and for combination of exponentials claim amounts. We then compare our approximations with the exact values that appeared in the literature. Finally we propose using diatomic and diexponential claim distributions as tools to study the Schmitter problem.

Several diagonal-based tail dependence indices have been suggested in the literature to quantify tail dependence. They have well-developed statistical inference theories but tend to underestimate tail dependence. For those problems when assessing the maximal strength of dependence is important (e.g., co-movements of financial instruments), the maximal tail dependence index was introduced, but it has so far lacked empirical estimators and statistical inference results, thus hindering its practical use. In the present paper, we suggest an empirical estimator for the index, explore its statistical properties, and illustrate its performance on simulated data.

The present paper introduces a simple aggregated reserving model based on claim count and payment dynamics, which allows for claim closings and re-openings. The modelling starts off from individual Poisson process claim dynamics in discrete time, keeping track of accident year, reporting year and payment delay. This modelling approach is closely related to the one underpinning the so-called double chain-ladder model, and it allows for producing separate reported but not settled and incurred but not reported reserves. Even though the introduction of claim closings and re-openings will produce new types of dependencies, it is possible to use flexible parametrisations in terms of, for example, generalised linear models (GLM) whose parameters can be estimated based on aggregated data using quasi-likelihood theory. Moreover, it is possible to obtain interpretable and explicit moment calculations, as well as having consistency of normalised reserves when the number of contracts tend to infinity. Further, by having access to simple analytic expressions for moments, it is computationally cheap to bootstrap the mean squared error of prediction for reserves. The performance of the model is illustrated using a flexible GLM parametrisation evaluated on non-trivial simulated claims data. This numerical illustration indicates a clear improvement compared with models not taking claim closings and re-openings into account. The results are also seen to be of comparable quality with machine learning models for aggregated data not taking claim openness into account.

We propose an approximation scheme for the computation of the risk measures of guaranteed minimum maturity benefits (GMMBs) and guaranteed minimum death benefits (GMDBs), based on the evaluation of single integrals under conditional moment matching. This procedure is computationally efficient in comparison with standard analytical methods while retaining a high degree of accuracy, and it allows one to deal with the case of additional earnings and the computation of related sensitivities.

We study the optimal investment strategy to minimize the probability of lifetime ruin under a general mortality hazard rate. We explore the error between the minimum probability of lifetime ruin and the achieved probability of lifetime ruin if one follows a simple investment strategy inspired by earlier work in this area. We also include numerical examples to illustrate the estimation. We show that the nearly optimal probability of lifetime ruin under the simplified investment strategy is quite close to the original minimum probability of lifetime ruin under reasonable parameter values.

We consider variation of observed claim frequencies in non-life insurance, modeled by Poisson regression with overdispersion. In order to quantify how much variation between insurance policies that is captured by the rating factors, one may use the coefficient of determination, R2, the estimated proportion of total variation explained by the model. We introduce a novel coefficient of individual determination (CID), which excludes noise variance and is defined as the estimated fraction of total individual variation explained by the model. We argue that CID is a more relevant measure of explained variation than R2 for data with Poisson variation. We also generalize previously used estimates and tests of overdispersion and introduce new coefficients of individual explained and unexplained variance.

Application to a Swedish three year motor TPL data set reveals that only 0.5% of the total variation and 11% of the total individual variation is explained by a model with seven rating factors, including interaction between sex and age. Even though the amount of overdispersion is small (4.4% of the noise variance) it is still highly significant. The coefficient of variation of explained and unexplained individual variation is 29% and 81% respectively.

This study considers a pure exchange economy with insurance against ambiguous loss. Ambiguity preferences are represented by the dual theory of the smooth ambiguity model from Iwaki and Osaki (2014). The economic premium principle of Bühlmann (1980, 1984) is extended to ambiguity. We also perform some comparative statics and present sufficient conditions under which an increase in ambiguity aversion increases insurance demand and insurance premiums. Contrary to the result in Tsanakas and Christofides (2006), the optimal demand for insurance is not always comonotonic, because our model permits an economy comprising both ambiguity averse and ambiguity loving agents.

The distribution-free chain ladder of Mack justified the use of the chain ladder predictor and enabled Mack to derive an estimator of conditional mean squared error of prediction for the chain ladder predictor. Classical insurance loss models, that is of compound Poisson type, are not consistent with Mack’s distribution-free chain ladder. However, for a sequence of compound Poisson loss models indexed by exposure (e.g., number of contracts), we show that the chain ladder predictor and Mack’s estimator of conditional mean squared error of prediction can be derived by considering large exposure asymptotics. Hence, quantifying chain ladder prediction uncertainty can be done with Mack’s estimator without relying on the validity of the model assumptions of the distribution-free chain ladder.