A critical problem in financial and insurance risk analysis is the calculation of risk margins. When there are a number of risks, the total risk margin is often reduced to reflect diversification. How large should the “diversification benefit” be? And how should the benefit be allocated to the individual risks? We propose a simple statistical solution. While providing a theoretical analysis, the final expressions are readily implemented in practice.

This paper evaluates the long-term risk for equity-linked insurance products. We consider a specific type of equity-linked insurance product with guaranteed minimum maturity benefits (GMMBs), and assume that the underlying equity follows the stochastic volatility model which allows the return's latent volatility component to be short- or long-memory. The explicit form of the quantile reserve or the Value at Risk and its confidence intervals are derived for both the long-memory and short-memory stochastic volatility models. To illustrate the effect of long-memory volatility, we use the S&P 500 index as an example of linked equity. Simulation studies are performed to examine the accuracy of the quantile reserve and to demonstrate the consequence of low coverage probability if model misspecification takes place. The empirical results show that the confidence interval of quantile reserve could be severely underestimated if the long-memory effect in equity volatility is ignored.

As a consequence of pointing out an ambiguity in Renshaw (1994), we show that the Overdispersed Poisson model cannot be generated by random independent intensities. Hence Pearson's chi-square-based estimate is normally unsuitable for GLM (Generalized Linear Model) log link claim frequency analysis in insurance. We propose a new dispersion parameter estimate in the GLM Tweedie model for risk premium. This is better than the Pearson estimate, if there are sufficiently many claims in each tariff cell. Simulation results are given showing the differences between it and the Pearson estimate.

This paper considers insurance claims that are available by cause of loss, or peril. Using this multi-peril information, we investigate multivariate frequency and severity models, emphasizing alternative dependency structures. Although dependency models may be used for many risk management strategies, we focus on ratemaking.

Motivation for this research comes from homeowners insurance and so, for the frequency portion, we consider binary response models. Specifically, we examine several multivariate binary regression models that have appeared in the biomedical literature, focusing on a dependence ratio model. For multivariate severity, we use Gaussian copulas to represent dependencies among gamma regressions.

We calibrate competing models based on a representative sample of over 400,000 records and validate them using a held-out sample of over 350,000 records. We find that methods that allow for cross-dependencies among perils provide important economic value in pricing.

In this paper, we consider the dual of the classical Cramér-Lundberg model when gains follow a phase-type distribution. By using the property of phase-type distribution, two pairs of upcrossing and downcrossing barrier probabilities are derived. Explicit formulas for the expected total discounted dividends until ruin and the Laplace transform of the time of ruin under a variety of dividend strategies can then be obtained without the use of Laplace transforms.

We analyze fire exposure rating for three types of risk profiles: policy profiles, top location profiles and location profiles. Location profiles offer more detailed information than top location profiles, which in turn are better than policy profiles. We prove criteria to ensure that a better quality of risk profile leads to a lower price. These criteria are discussed with respect to standard exposure rating and an alternative method called ‘burning cost-adjusted exposure rating’, where the loss ratio is adjusted by means of the burning cost of a low reference layer. Further, we introduce a family of analytic exposure curves, the so-called EP exposure curves. These curves are useful for practical application, since the criteria given in this paper can be checked easily for exposure curves that can be approximated by EP exposure curves. This is of particular interest for discrete exposure curves. Finally, we apply the results to the MBBEFD exposure curves.

We consider a set of workers' compensation insurance claim data where the aggregate number of losses (claims) reported to insurers are classified by year of occurrence of the event causing loss, the US state in which the loss event occurred and the occupation class of the insured workers to which the loss count relates. An exposure measure, equal to the total payroll of observed workers in each three-way classification, is also included in the dataset. Data are analysed across ten different states, 24 different occupation classes and seven separate observation years. A multiple linear regression model, with only predictors for main effects, could be estimated in 223 + 9 + 1 + 1 = 234 ways, theoretically more than 17 billion different possible models! In addition, one might expect that the number of claims recorded in each year in the same state and relating to the same occupation class, are positively correlated. Different modelling assumptions as to the nature of this correlation should also be considered. On the other hand it may reasonably be assumed that the number of losses reported from different states and from different occupation classes are independent. Our data can therefore be modelled using the statistical techniques applicable to panel data and we work with generalised estimating equations (GEE) in the paper. For model selection, Pan (2001) suggested the use of an alternative to the AIC, namely the quasi-likelihood under independence model criterion (QIC), for model comparison. This paper develops and applies a Gibbs sampling algorithm for efficiently locating, out of the more than 17 billion possible models that could be considered for the analysis, that model with the optimal (least) QIC value. The technique is illustrated using both a simulation study and using workers' compensation insurance claim data.

A methodology for pricing of reinsurance contracts in the presence of a catastrophe bond is developed. An important advantage of this approach is that it allows for the pricing of reinsurance contracts consistent with the observed market prices of catastrophe bonds on the same underlying risk process.

Within the proposed methodology, an appropriate financial pricing formula is derived, under a market implied risk neutral probability measure for both a catastrophe bond and an aggregate excess of loss reinsurance contract, using a generalised Fourier transform. Efficient numerical methods for the evaluation of this formula, such as the Fast Fourier transform and Fractional Fast Fourier transform, are considered.

The methodology is illustrated on several examples including Pareto and Gamma claim severities.

An insurance company entering the property and liability insurance market at the high point of the insurance cycle may decide to slash premiums to gain an advantageous market share. Such aggressive intrusion may call forth a concerted industry response, producing a severe decline in the insurance market price. This can ruin some companies, and agrees with the observation that the insurance cycles are correlated with clustered insolvencies. This paper addresses a quantitative analysis of competition-originated cycles; it explores an interplay of rational aggressive and defensive strategies in the multi-period Lundberg-type controlled risk model.

This paper aims to provide accurate approximations for the quantiles of the conditional expected present value of the payments made by the annuity provider, given the future path of the Lee-Carter time index. Conditional cohort and period life expectancies are also considered. The paper also addresses some associated simulation issues, which, hitherto, have been unresolved.

The separation method was introduced by Verbeek (1972) in order to forecast numbers of excess claims and it was developed further by Taylor (1977) to be applicable to the average claim cost. The separation method differs from the chain-ladder in that when the chain-ladder only assumes claim proportionality between the development years, the separation method also separates the claim delay distribution from influences affecting the calendar years, e.g. inflation. Since the inflation contributes to the uncertainty in the estimate of the claims reserve it is important to consider its impact in the context of risk management, too.

In this paper we present a method for assessing the prediction error distribution of the separation method. To this end we introduce a parametric framework within the separation model which enables joint resampling of claim counts and claim amounts. As a result, the variability of Taylor's predicted reserves can be assessed by extending the parametric bootstrap techniques of Björkwall et al. (2009). The performance of the bootstrapped separation method and chain-ladder is compared for a real data set.

In this paper, we aim to evaluate the distribution of the aggregate claims in the collective risk model. The claim count distribution is firstly assumed to belong to a generalised (a, b, 0) family. A matrix form recursive formula is then derived to evaluate the related compound distribution when individual claim amounts follow a discrete distribution on non-negative integers. The corresponding formula is also given for continuous individual claim amounts. Secondly, we pay particular attention to the recursive formula for compound phase-type distributions, since only certain types of discrete phase-type distributions belong to the generalised (a, b, 0) family. Similar recursive formulae are obtained for discrete and continuous individual claim amount distributions. Finally, numerical examples are presented for three counting distributions.

A model is proposed using the run-off triangle of paid claims and also the numbers of reported claims (in a similar triangular array). These data are usually available, and allow the model proposed to be implemented in a large variety of situations. On the basis of these data, the stochastic model is built from detailed assumptions for individual claims, but then approximated using a compound Poisson framework. The model explicitly takes into account the delay from when a claim is incurred and to when it is reported (the IBNR delay) and the delay from when a claim is reported and to when it is fully paid (the RBNS delay). These two separate sources of delay are estimated separately, unlike most other reserving methods. The results are compared with those of the chain ladder technique.

A general ‘multivariate’ decomposition of covariances is formulated and proved, and its role in the context of financial risk measurement and pricing is demonstrated.

We study solvency of insurers in a practical model where in addition to basic insurance claims and premiums, economic factors like inflation, real growth and returns on the investments affect the capital developments of the companies. The objective is to give qualitative descriptions of risks by means of crude estimates for finite time ruin probabilities. In our setup, the economic factors have a dominant role in the estimates. In addition to this theoretical view, we will focus on applied interpretations of the results by means of discussions and examples.

The first version of the paper was presented in ASTIN Colloquium in Helsinki 2009. The colloquium was dedicated to Teivo Pentikäinen.

We consider estimating tail events using exponential families of importance sampling distributions. When the cannonical sufficient statistic for the exponential family mimics the tail behaviour of the underlying cumulative distribution function, we can achieve bounded relative error for estimating tail probabilities. Examples of rare event simulation from various distributions including Tukey's g&h distribution are provided.

This work deals with prediction of IBNR reserve under a different data ordering of the non-cumulative runoff triangle. The rows of the triangle are stacked, resulting in a univariate time series with several missing values. Under this ordering, two approaches entirely based on state space models and the Kalman filter are developed, implemented with two real data sets, and compared with two well-established IBNR estimation methods — the chain ladder and an overdispersed Poisson regression model. The remarks from the empirical results are: (i) computational feasibility and efficiency; (ii) accuracy improvement for IBNR prediction; and (iii) flexibility regarding IBNR modeling possibilities.

The probability of ruin in continuous and finite time is numerically evaluated in a classical risk process where the premium can be updated according to credibility models and therefore change from year to year. A major consideration in the development of this approach is that it should be easily applicable to large portfolios. Our method uses as a first tool the model developed by Afonso et al. (2009), which is quite flexible and allows premiums to change annually. We extend that model by introducing a credibility approach to experience rating.

We consider a portfolio of risks which satisfy the assumptions of the Bühlmann (1967, 1969) or Bühlmann and Straub (1970) credibility models. We compute finite time ruin probabilities for different scenarios and compare with those when a fixed premium is considered.