In this paper we examine the claims reserving problem using Tweedie's compound Poisson model. We develop the maximum likelihood and Bayesian Markov chain Monte Carlo simulation approaches to fit the model and then compare the estimated models under different scenarios. The key point we demonstrate relates to the comparison of reserving quantities with and without model uncertainty incorporated into the prediction. We consider both the model selection problem and the model averaging solutions for the predicted reserves. As a part of this process we also consider the sub problem of variable selection to obtain a parsimonious representation of the model being fitted.

The cost of adverse selection in the life and critical illness (CI) insurance markets, brought about by restrictions on insurers' use of genetic test information, has been studied for a variety of rare single-gene disorders. Only now do we have a study of a common disorder (breast cancer) that accounts for the risk associated with multiple genes. Such a collection of genes is called a polygene. We take two approaches to modelling the severity of adverse selection which may result from insurers being unable to take account of tests for polygenes as well as major genes. First, we look at several genetic testing scenarios, with a corresponding range of possible insurance-buying behaviours, in a market model for CI insurance. Because a relatively large proportion of the population is exposed to adverse polygenic risk, the costs of adverse selection are potentially much greater than have been associated with rare single genes. Second, we use utility models to map out when adverse selection will appear, and which risk groups will cause it. Levels of risk aversion consistent with some empirical studies do not lead to significant adverse selection in our model, but lower levels of risk aversion could effectively eliminate the market.

This paper investigates the effects of high or low fair-premium demand elasticity in an insurance market where risk classification is restricted. The effects are represented by the equilibrium premium, and the risk-weighted insurance demand or “loss coverage”. High fair-premium demand elasticity leads to a collapse in loss coverage, with an equilibrium premium close to the risk of the higher-risk population. Low fair-premium demand elasticity leads to an equilibrium premium close to the risk of the lower-risk population, and high loss coverage – possibly higher than under more complete risk classification. The demand elasticity parameters which are required to generate a collapse in coverage in the model in this paper appear higher than the values for demand elasticity which have been estimated in several empirical studies of various insurance markets. This offers a possible explanation of why some insurance markets appear to operate reasonably well under community rating, without the collapse in coverage which insurance folklore suggests.

This paper considers the model suggested by Schnieper (1991), which separates the true IBNR claims from the IBNER. Stochastic models are defined, using both recursive and non-recursive procedures, within the framework of the models described in England and Verrall (2002). Approximations to the prediction error of the reserves are derived analytically.

Some extensions to the original Schnieper model are also disussed, together with other possible applications of this type of model.

In this paper a method for determining benchmark rates for the excess of loss reinsurance of a Motor Third Party Liability insurance portfolio will be developed based on observed market rates. The benchmark rates are expressed as a percentage of the expected premium income that is available to cover the whole risk of the portfolio. The rates are assumed to be based on a compound process with a heavy tailed severity, such as Burr or Pareto distributions. In the absence of claim data these assumptions propagate the theoretical benchmark rate component of the regression model.

Given the whole set of excess of loss reinsurance rates in a given market, the unknown parameters are estimated within the framework of quasi-likelihood estimation. This framework makes it possible to select a theoretical benchmark rate model and to choose a parsimonious submodel for describing the observed market rates over a 4-years observation period. This method is applied to the Belgian Motor Third Party Liability excess of loss rates observed during the years 2001 till 2004.

Generalized linear models (GLMs) are gaining popularity as a statistical analysis method for insurance data. For segmented portfolios, as in car insurance, the question of credibility arises naturally; how many observations are needed in a risk class before the GLM estimators can be considered credible? In this paper we study the limited fluctuations credibility of the GLM estimators as well as in the extended case of generalized linear mixed model (GLMMs). We show how credibility depends on the sample size, the distribution of covariates and the link function. This provides a mechanism to obtain confidence intervals for the GLM and GLMM estimators.

The use of generalized linear models (GLM) to estimate claims reserves has become a standard method in insurance. Most frequently, the exponential dispersion family (EDF) is used; see e.g. England, Verrall. We study the so-called Tweedie EDF and test the sensitivity of the claims reserves and their mean square error of predictions (MSEP) over this family. Furthermore, we develop second order Taylor approximations for the claims reserves and the MSEPs for members of the Tweedie family that are difficult to obtain in practice, but are close enough to models for which claims reserves and MSEP estimations are easy to determine. As a result of multiple case studies, we find that claims reserves estimation is relatively insensitive to which distribution is chosen amongst the Tweedie family, in contrast to the MSEP, which varies widely.

The Benktander (1976) and Neuhaus (1992) credibility claims reserving methods are reconsidered in the framework of a credible loss ratio reserving method. As a main contribution we provide a simple and practical optimal credibility weight for combining the chain-ladder or individual loss ratio reserve (grossed up latest claims experience of an origin period) with the Bornhuetter-Ferguson or collective loss ratio reserve (experience based burning cost estimate of the total ultimate claims of an origin period). The obtained simple optimal credibility weights minimize simultaneously the mean squared error and the variance of the claims reserve. We note also that the standard Chain-Ladder, Cape Cod and Bornhuetter-Ferguson methods can be reinterpreted in the credible context and extended to optimal credible standard methods. The new approach is inspired from Mack (2000). Two advantages over the Mack method are worthwhile to be mentioned. First, a pragmatic estimation of the required parameters leads to a straightforward calculation of the optimal credibility weights and mean squared errors of the credible reserves. An advantage of the collective loss ratio claims reserve over the Bornhuetter-Ferguson reserve in Mack (2000) is that different actuaries come always to the same results provided they use the same actuarial premiums.

Recently, Albrecher and his coauthors have published a series of papers on the ruin probability of the Lévy insurance model under the so-called loss-carry-forward taxation, meaning that taxes are paid at a certain fixed rate immediately when the surplus of the company is at a running maximum. In this paper we assume periodic taxation under which the company pays tax at a fixed rate on its net income during each period. We devote ourselves to deriving explicit asymptotic relations for the ruin probability in the most general Lévy insurance model in which the Lévy measure has a subexponential tail, a convolution-equivalent tail, or an exponential-like tail.

This paper is concerned with clarifying the link between risk measurement and capital efficiency. For this purpose we introduce risk measurement as the minimum cost of making a position acceptable by adding an optimal combination of multiple eligible assets. Under certain assumptions, it is shown that these risk measures have properties similar to those of coherent risk measures. The motivation for this paper was the study of a multi-currency setting where it is natural to use simultaneously a domestic and a foreign asset as investment vehicles to inject the capital necessary to make an unacceptable position acceptable. We also study what happens when one changes the unit of account from domestic to foreign currency and are led to the notion of compatibility of risk measures. In addition, we aim to clarify terminology in the field.

One can find in the literature three main sets of estimators for the variance components in the hierarchical credibility model. This paper presents these estimators in a unified notation, studies some of their properties important for numerical evaluation and compares their relative performance by simulation. The paper also demonstrates how function cm of the R package actuar can be used to fit hierarchical models to insurance data.

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.

Traditionally, actuaries have modeled mortality improvement using deterministic reduction factors, with little consideration of the associated uncertainty. As mortality improvement has become an increasingly significant source of financial risk, it has become important to measure the uncertainty in the forecasts. Probabilistic confidence intervals provided by the widely accepted Lee-Carter model are known to be excessively narrow, due primarily to the rigid structure of the model. In this paper, we relax the model structure by considering individual differences (heterogeneity) in each age-period cell. The proposed extension not only provides a better goodness-of-fit based on standard model selection criteria, but also ensures more conservative interval forecasts of central death rates and hence can better reflect the uncertainty entailed. We illustrate the results using US and Canadian mortality data.

Nowadays, the regime switching model has become a popular model in mathematical finance and actuarial science. The market is not complete when the model has regime switching. Thus, pricing the regime switching risk is an important issue. In Naik (1993), a jump diffusion model with two regimes is studied. In this paper, we extend the model of Naik (1993) to a multi-regime case. We present a trinomial tree method to price options in the extended model. Our results show that the trinomial tree method in this paper is an effective method; it is very fast and easy to implement. Compared with the existing methodologies, the proposed method has an obvious advantage when one needs to price exotic options and the number of regime states is large. Various numerical examples are presented to illustrate the ideas and methodologies.

In this paper, we take the point of view of an insurer dealing with life annuities, which aims at building up a (partial) internal model in order to quantify the impact of mortality risks, namely process and longevity risk, in view of taking appropriate risk management actions. We assume that a life table, providing a best-estimate assessment of annuitants' future mortality is available to the insurer; conversely, the insurer has no access to data sets and the methodology underlying the construction of the life table. Nonetheless, the insurer is aware that, in the presence of mortality risks, a stochastic approach is required. The (projected) life table, which provides a deterministic description of future mortality, should then be used as the basic input of a stochastic model.

The model we propose focuses on the annual number of deaths in a given cohort, which we represent allowing for a random mortality rate. To this purpose, we adopt the widely used Poisson model, first assuming a Gamma-distributed random parameter, and second introducing time-dependence in the parameter itself. Further, we define a Bayesian-inferential procedure for updating the parameters to experience in some situations. The setting we define does not demand advanced analytical tools, while allowing for process and longevity risk in a rigorous way.

The model is then implemented for capital allocation purposes. We investigate the amount of the required capital for a given life annuity portfolio, based on solvency targets which could be adopted within internal models. The outcomes of such an investigation are compared with the capital required according to some standard rules, in particular those proposed within the Solvency 2 project.

This paper demonstrates actuarial applications of modern statistical methods that are applied to detailed, micro-level automobile insurance records. We consider 1993-2001 data consisting of policy and claims files from a major Singaporean insurance company. A hierarchical statistical model, developed in prior work (Frees and Valdez (2008)), is fit using the micro-level data. This model allows us to study the accident frequency, loss type and severity jointly and to incorporate individual characteristics such as age, gender and driving history that explain heterogeneity among policyholders.

Based on this hierarchical model, one can analyze the risk profile of either a single policy (micro-level) or a portfolio of business (macro-level). This paper investigates three types of actuarial applications. First, we demonstrate the calculation of the predictive mean of losses for individual risk rating. This allows the actuary to differentiate prices based on policyholder characteristics. The nonlinear effects of coverage modifications such as deductibles, policy limits and coinsurance are quantified. Moreover, our flexible structure allows us to “unbundle” contracts and price more primitive elements of the contract, such as coverage type. The second application concerns the predictive distribution of a portfolio of business. We demonstrate the calculation of various risk measures, including value at risk and conditional tail expectation, that are useful in determining economic capital for insurance companies. Third, we examine the effects of several reinsurance treaties. Specifically, we show the predictive loss distributions for both the insurer and reinsurer under quota share and excess-of-loss reinsurance agreements. In addition, we present an example of portfolio reinsurance, in which the combined effect of reinsurance agreements on the risk characteristics of ceding and reinsuring company are described.

In this paper we compare the current Solvency II standard and a genuine bottom-up approach to risk aggregation. This is understood to be essential for developing a deeper insight into the possible differences between the diversification assumptions between the standard approach and internal models.

In Kim and Hardy (2007) the exact bootstrap was used to estimate certain risk measures including Value at Risk and the Conditional Tail Expectation. In this paper we continue this work by deriving the influence function of the exact-bootstrapped quantile risk measure. We can use the influence function to estimate the variance of the exact-bootstrap risk measure. We then extend the result to the L-estimator class, which includes the conditional tail expectation risk measure. The resulting formula provides an alternative way to estimate the variance of the bootstrapped risk measures, or the whole L-estimator class in an analytic form. A simulation study shows that this new method is comparable to the ordinary resampling-based bootstrap method, with the advantages of an analytic approach.

We present a model for pricing credit risk protection for a limited liability non-life insurance company. The protection is typically provided by a guaranty fund. In the case of continuous monitoring, i.e., where the market values of the company's assets and liabilities are continuously observable, and where the market values of assets and liabilities follow continuous processes, regulators can liquidate the insurance company at the instant the market value of its assets equals the market value of its liabilities, implying that the credit protection is worthless. When jumps are included in the claims process, the protection provided by the guaranty fund has a strictly positive market value. The ability to continuously monitor asset prices with continuous sample paths eliminates economic losses from default. Our analysis suggests that economic losses from default stem from jumps in continuously observed asset prices and/or that asset prices are not continuously observed.

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.