Non-life insurance payouts consist of two factors: claimsizes and claim frequency. When calculating e.g. next years premium, it is vital to correctly model these factors and to estimate the unknown parameters. A standard way is to separately estimate in the claimsize and the claim frequency models.

Often there is a deductible with each single claim, and this deductible can be quite large, particularly in inhomogeneous cases such as industrial fire insurance or marine insurance. Not taking the deductibles into account can lead to serious bias in the estimates and consequent implications when applying the model.

When the deductibles are nonidentical, in a full maximum likelihood estimation all unknown parameters have to be estimated simultaneously. An alternative is to use pseudo-maximum likelihood, i.e. first estimate the claimsize model, taking the deductibles into account, and then use the estimated probability that a claim exceeds the deductible as an offset in the claim frequency estimation. This latter method is less efficient, but due to complexity or time considerations, it may be the preferred option.

In this paper we will provide rather general formulas for the relative efficiency of the pseudo maximum likelihood estimators in the i.i.d. case. Two special cases will be studied in detail, and we conclude the paper by comparing the methods on some marine insurance data.

This paper introduces a new framework for modelling the joint development over time of mortality rates in a pair of related populations with the primary aim of producing consistent mortality forecasts for the two populations. The primary aim is achieved by combining a number of recent and novel developments in stochastic mortality modelling, but these, additionally, provide us with a number of side benefits and insights for stochastic mortality modelling. By way of example, we propose an Age-Period-Cohort model which incorporates a mean-reverting stochastic spread that allows for different trends in mortality improvement rates in the short-run, but parallel improvements in the long run. Second, we fit the model using a Bayesian framework that allows us to combine estimation of the unobservable state variables and the parameters of the stochastic processes driving them into a single procedure. Key benefits of this include dampening down of the impact of Poisson variation in death counts, full allowance for paramater uncertainty, and the flexibility to deal with missing data. The framework is designed for large populations coupled with a small sub-population and is applied to the England & Wales national and Continuous Mortality Investigation assured lives males populations. We compare and contrast results based on the two-population approach with single-population results.

Conditional indexation offers a middle way between defined benefit and defined contribution pension schemes. In this paper, we consider a fully-funded pension scheme with conditional indexation. We show how the pension fund can be managed to reduce the risks associated with promised pension benefits when declared benefits are adjusted regularly during the working life. In particular, we derive an investment strategy that provides protection against underfunding at retirement and which is self-financing on average. Our results are illustrated in an extensive simulation study.

In this paper we propose an additive mixture model, where one component is the Generalized Pareto distribution (GPD) that allows us to estimate extreme quantiles. GPD plays an important role in modeling extreme quantiles for the wide class of distributions belonging to the maximum domain of attraction of an extreme value model. One of the main difficulty with this modeling approach is the choice of the threshold u, such that all observations greater than u enter into the likelihood function of the GPD model. Difficulties are due to the fact that GPD parameter estimators are sensible to the choice of u. In this work we estimate u, and other parameters, using suitable priors in a Bayesian approach. In particular, we propose to model all data, extremes and non-extremes, using a semiparametric model for data below u, and the GPD for the exceedances over u. In contrast to the usual estimation techniques for u, in this setup we account for uncertainty on all GPD parameters, including u, via their posterior distributions. A Monte Carlo study shows that posterior credible intervals also have frequentist coverages. We further illustrate the advantages of our approach on two applications from insurance.

In this paper we develop a full stochastic cash flow model of outstanding liabilities for the model developed in Verrall, Nielsen and Jessen (2010). This model is based on the simple triangular data available in most non-life insurance companies. By using more data, it is expected that the method will have less volatility than the celebrated chain ladder method. Eventually, our method will lead to lower solvency requirements for those insurance companies that decide to collect counts data and replace their conventional chain ladder method.

The chain ladder is considered in relation to certain recursive and non-recursive models of claim observations. The recursive models resemble the (distribution free) Mack model but are augmented with distributional assumptions. The non-recursive models are generalisations of Poisson cross-classified structure for which the chain ladder is known to be maximum likelihood. The error distributions considered are drawn from the exponential dispersion family.

Each of these models is examined with respect to sufficient statistics and completeness (Section 5), minimum variance estimators (Section 6) and maximum likelihood (Section 7). The chain ladder is found to provide maximum likelihood and minimum variance unbiased estimates of loss reserves under a wide range of recursive models. Similar results are obtained for a much more restricted range of non-recursive models.

These results lead to a full classification of this paper's chain ladder models with respect to the estimation properties (bias, minimum variance) of the chain ladder algorithm (Section 8).

This paper intends to develop a feasible framework which incorporates ambiguity aversion into the pricing of insurance products and investigate the implications of ambiguity aversion on the pricing by comparing it with risk aversion. As applications of the framework, we present the closed-form pricing formulae for some insurance products appearing in life insurance and property insurance. Our model confirms that the effects of ambiguity aversion on the pricing of insurance do differ from those of risk aversion. Implications of our model are consistent with some empirical evidences documented in the literature. Our results suggest that taking advantage of natural hedge mechanism can help us control the effects of model uncertainty.

In this contribution, a new measure of comonotonicity for m-dimensional vectors is introduced, with values between zero, representing the independent situation, and one, reflecting a completely comonotonic situation. The main characteristics of this coefficient are examined, and the relations with common dependence measures are analysed. A sample-based version of the comonotonicity coefficient is also derived. Special attention is paid to the explanation of the accuracy of the convex order bound method of Goovaerts, Dhaene et al. in the case of cash flows with Gaussian discounting processes.

In this paper, we investigate the computation of the moments of the compound Poisson sums with discounted claims when introducing dependence between the interclaim time and the subsequent claim size. The dependence structure between the two random variables is defined by a Farlie-Gumbel-Morgenstern copula. Assuming that the claim distribution has finite moments, we give expressions for the first and the second moments and then we obtain a general formula for any mth order moment. The results are illustrated with applications to premium calculation and approximations based on moment matching methods.

Many papers in the literature have adopted the expected utility paradigm to analyze insurance decisions. Insurance companies manage policies by growing, by adding independent risks. Even if adding risks generally ultimately decreases the probability of insolvency, the impact on the insurer's expected utility is less clear. Indeed, it is not true that the risk aversion toward the additional loss generated by a new policy included in an insurance portfolio always decreases with the number of contracts already underwritten. The present paper derives conditions under which zero-utility premium principles are subadditive for independent risks. It is shown that subadditivity is the exception rather than the rule: the zero-utility premium principle generates a superadditive risk premium for most common utility functions. For instance, all completely monotonic utility functions generate superadditive zero-utility premiums. The main message of the present paper is thus that the zero-utility premium for a marginal policy is generally not sufficient to guarantee the formation of insurance portfolios without additional capital.

In this paper the catastrophe bond prices, as determined by the market, are analysed. The limited published work in this area has been carried out mainly by cat bond investors and is based either on intuition, or on simple linear regression on one factor or on comparisons of the process of cat bonds with similar features. In this paper a Generalised Additive Model is fitted to the market data. The statistical significance of different factors which may affect the cat bond prices is examined and the effect of these factors on the prices is measured. A statistical framework and analysis ould provide insight into the cat bond pricing and could have applications among other things in the construction of a cat bond portfolio, cat bond price indices and in understanding changes of the price of risk over time.