In this article we want to motivate and analyse a wide family of reserving models, called linear stochastic reserving methods (LSRMs). The main idea behind them is the assumption that the (conditionally) expected changes of claim properties during a development period are proportional to exposures which depend linearly on the past. This means the discussion about the choice of reserving methods can be based on heuristic reasons about exposures driving the claims development, which in our opinion is much better than a pure philosophic approach. Moreover, the assumptions of LSRMs do not include the independence of accident periods.

We will see that many common reserving methods, like the Chain-Ladder-Method, the Bornhuetter-Ferguson-Method and the Complementary-Loss-Ratio-Method, can be interpreted in this way. But using the LSRM framework you can do more. For instance you can couple different triangles via exposures. This leads to reserving methods which look at a whole bundle of triangles at once and use the information of all triangles in order to estimate the future development of each of them.

We will present unbiased estimators for the expected ultimate and estimators for the mean squared error of prediction, which may become an integral part of IFRS 4. Moreover, we will look at the one period solvency reserving risk, which already is an important part of Solvency II, and present a corresponding estimator.

Finally we will present two examples that illustrate some features of LSRMs.

In this paper we describe a new approach to modelling the development of claims run-off triangles. This method replaces the usual ad hoc practical process of extrapolating a development pattern to obtain tail factors with an objective procedure. An example is given, illustrating the results in a practical context, and the WinBUGS code is supplied.

By adding the information of reported count data to a classical triangle of reserving data, we derive a suprisingly simple method for forecasting IBNR and RBNS claims. A simple relationship between development factors allows to involve and then estimate the reporting and payment delay. Bootstrap methods provide prediction errors and make possible the inference about IBNR and RBNS claims, separately.

Global warming has more than doubled the likelihood of extreme weather events, e.g. the 2003 European heat wave, the growing intensity of rain and snow in the Northern Hemisphere, and the increasing risk of flooding in the United Kingdom. It has also induced an increasing number of deadly tropical cyclones with a continuing trend. Many individual meteorological dynamic simulations and statistical models are available for forecasting hurricanes but they neither forecast well hurricane intensity nor produce clear-cut consensus. We develop a novel hurricane forecasting model by straddling two seemingly unrelated disciplines — physical science and finance — based on the well known price discovery function of trading in financial markets. Traders of hurricane derivative contracts employ all available forecasting models, public or proprietary, to forecast hurricanes in order to make their pricing and trading decisions. By using transactional price changes of these contracts that continuously clear the market supply and demand as the predictor, and with calibration to extract the embedded hurricane information by developing hurricane futures and futures option pricing models, one can gain a forward-looking market-consensus forecast out of all of the individual forecasting models employed. Our model can forecast when a hurricane will make landfall, how destructive it will be, and how this destructive power will evolve from inception to landing. While the NHC (National Hurricane Center) blends 50 plus individual forecasting results for its consensus model forecasts using a subjective approach, our aggregate is market-based. Believing their proprietary forecasts are sufficiently different from our market-based forecasts, traders could also examine the discrepancy for a potential trading opportunity using hurricane derivatives. We also provide a real case analysis of Hurricane Irene in 2011 using our methodology.

In the paper we introduce a generalization of the mean-value principle under Cumulative Prospect Theory. This new method involves some well-known ways of pricing insurance contracts described in the actuarial literature. Properties of this premium principle, such as translation and scale invariance, additivity for independent risks, risk loading and others are studied.

Actuaries are often faced with the task of estimating tails of loss distributions from just a few observations. Thus estimates of tail probabilities (reinsurance prices) and percentiles (solvency capital requirements) are typically subject to substantial parameter uncertainty. We study the bias and MSE of estimators of tail probabilities and percentiles, with focus on 1-parameter exponential families. Using asymptotic arguments it is shown that tail estimates are subject to significant positive bias. Moreover, the use of bootstrap predictive distributions, which has been proposed in the actuarial literature as a way of addressing parameter uncertainty, is seen to double the estimation bias. A bias corrected estimator is thus proposed. It is then shown that the MSE of the MLE, the parametric bootstrap and the bias corrected estimators only differ in terms of order O(n−2), which provides decision-makers with some flexibility as to which estimator to use. The accuracy of asymptotic methods, even for small samples, is demonstrated exactly for the exponential and related distributions, while other 1-parameter distributions are considered in a simulation study. We argue that the presence of positive bias may be desirable in solvency capital calculations, though not necessarily in pricing problems.

In this paper, we introduce a class of multivariate Erlang mixtures and present its desirable properties. We show that a multivariate Erlang mixture could be an ideal multivariate parametric model for insurance modeling, especially when modeling dependence is a concern. When multivariate losses are governed by a multivariate Erlang mixture, many quantities of interest such as joint density and Laplace transform, moments, and Kendall's tau have a closed form. Further, the class is closed under convolutions and mixtures, which enables us to model aggregate losses in a straightforward way. We also introduce a new concept called quasi-comonotonicity that can be useful to derive an upper bound for individual losses in a multivariate stochastic order and upper bounds for stop-loss premiums of the aggregate loss. Finally, an EM algorithm tailored to multivariate Erlang mixtures is presented and numerical experiments are performed to test the efficiency of the algorithm.

We propose a nonhomogeneous Poisson hidden Markov model for a time series ofclaim counts that accounts for both seasonal variations and random fluctuations in the claims intensity. It assumes that the parameters of the intensity function for the nonhomogeneous Poisson distribution vary according to an (unobserved) underlying Markov chain. This can apply to natural phenomena that evolve in a seasonal environment. For example, hurricanes that are subject to random fluctuations (El Niño-La Niña cycles) affect insurance claims. The Expectation-Maximization (EM) algorithm is used to calculate the maximum likelihood estimators for the parameters of this dynamic Poisson hidden Markov model. Statistical applications of this model to Atlantic hurricanes and tropical storms data are discussed.

We study pricing and hedging for an insurance payment process. We investigate a Black-Scholes financial model with stochastic coefficients and a payment process with death, survival and annuity claims driven by a point process with a stochastic intensity. The dependence of the claims and the intensity on the financial market and on an additional background noise (correlated index, longevity risk) is allowed. We establish a general modeling framework for no-good-deal, local mean-variance and ambiguity risk pricing and hedging. We show that these three valuation approaches are equivalent under appropriate formulations. We characterize the price and the hedging strategy as a solution to a backward stochastic differential equation. The results could be applied to pricing and hedging of variable annuities, surrender options under an irrational lapse behavior and mortality derivatives.

Experience and exposure rating are traditionally considered to be independent but complementary methods for pricing property per risk excess of loss reinsurance. Strengths and limitations of these techniques are well-known. In practice, both methods often lead to quite different prices. In this paper we show that limitations of traditional experience rating can be overcome by taking into account historical profile information by means of exposure curves. For pricing unused or rarely used capacity, we propose to use exposure rating, calibrated on the experience rate of a working layer. We compare the method presented with more traditional methods based on the information which is generally available to the reinsurer.

A prudent assessment of dependence is crucial in many stochastic models for insurance risks. Copulas have become popular to model such dependencies. However, estimation procedures for copulas often lead to large parameter uncertainty when observations are scarce. In this paper, we propose a Bayesian method which combines prior information (e.g. from regulators), observations and expert opinion in order to estimate copula parameters and determine the estimation uncertainty. The combination of different sources of information can significantly reduce the parameter uncertainty compared to the use of only one source. The model can also account for uncertainty in the marginal distributions. Furthermore, we describe the methodology for obtaining expert opinion and explain involved psychological effects and popular fallacies. We exemplify the approach in a case study.

The bootstrap method BICH is given for estimating mean square prediction errors and predictive distributions of non-life claim reserves under weak conditions. The dates of claim occurrence, reporting and finalization and the payment dates and amounts of individual finalized historic claims form a claim set from which samples with replacement are drawn. We assume that all claims are independent and that the historic claims are distributed as the object claims, possibly after inflation adjustment and segmentation on a background variable, whose distribution could have changed over time due to portfolio change. Also we introduce the new reserving function RDC, using all these dates and payments for reserve predictions. We study three reserving functions: chain ladder, the Schnieper (1991) method and RDC. Checks with simulated cases obeying the assumptions of Mack (1999) for chain ladder and Liu and Verrall (2009) for Schnieper's method, respectively, confirm the validity of our method. BICH is used to compare the three reserving functions, of which RDC is found overall best in simulated cases.

In this paper we calculate premiums which are based on the minimization of the Expected Tail Loss or Conditional Tail Expectation (CTE) of absolute loss functions. The methodology generalizes well known premium calculation procedures and gives sensible results in practical applications. The choice of the absolute loss becomes advisable in this context since its CTE is easy to calculate and to understand in intuitive terms. The methodology also can be applied to the calculation of the VaR and CTE of the loss associated with a given premium.

The optimal quantization theory is applied for approximating law-invariant comonotonic coherent risk measures. Simple Lp-norm estimates for the risk measures provide the rate of convergence of that approximation as the number of quantization points goes to infinity.

The claims development result (CDR) is one of the major risk drivers in the profit and loss statement of a general insurance company. Therefore, the CDR has become a central object of interest under new solvency regulation. In current practice, simple methods based on the first two moments of the CDR are implemented to find a proxy for the distribution of the CDR. Such approximations based on the first two moments are rather rough and may fail to appropriately describe the shape of the distribution of the CDR. In this paper we provide an analysis of higher moments of the CDR. Within a Bayes chain ladder framework we consider two different models for which it is possible to derive analytical solutions for the higher moments of the CDR. Based on higher moments we can e.g. calculate the skewness and the excess kurtosis of the distribution of the CDR and obtain refined approximations. Moreover, a case study investigates and answers questions raised in IASB.