This year we celebrate the 300th anniversary of Jakob Bernoulli's path-breaking work Ars conjectandi, which appeared in 1713, eight years after his death. In Part IV of his masterpiece, Bernoulli proves the law of large numbers which is one of the fundamental theorems in probability theory, statistics and actuarial science. We review and comment on his original proof.

The paid-incurred chain (PIC) reserving method is a claims reserving method that allows to combine claims payments and incurred losses information in a mathematical consistent way. The main criticism on the original Bayesian log-normal PIC model presented in Merz–Wüthrich [5] is that it does not respect dependence properties within the observed data. In the present paper, we extend the original Bayesian log-normal PIC model so that dependence is modeled in an appropriate way.

Hedging is one of the most important topics in finance. When a financial market is complete, every contingent claim can be hedged perfectly to eliminate any potential future obligations. When the financial market is incomplete, the investor may eliminate his risk exposure by superhedging. In practice, both hedging strategies are not satisfactory due to their high implementation costs, which erode the chance of making any profit. A more practical and desirable strategy is to resort to the partial hedging, which hedges the future obligation only partially. The quantile hedging of Föllmer and Leukert (Finance and Stochastics, vol. 3, 1999, pp. 251–273), which maximizes the probability of a successful hedge for a given budget constraint, is an example of the partial hedging. Inspired by the principle underlying the partial hedging, this paper proposes a general partial hedging model by minimizing any desirable risk measure of the total risk exposure of an investor. By confining to the value-at-risk (VaR) measure, analytic optimal partial hedging strategies are derived. The optimal partial hedging strategy is either a knock-out call strategy or a bull call spread strategy, depending on the admissible classes of hedging strategies. Our proposed VaR-based partial hedging model has the advantage of its simplicity and robustness. The optimal hedging strategy is easy to determine. Furthermore, the structure of the optimal hedging strategy is independent of the assumed market model. This is in contrast to the quantile hedging, which is sensitive to the assumed model as well as the parameter values. Extensive numerical examples are provided to compare and contrast our proposed partial hedging to the quantile hedging.

This paper analyses the impact of using different correlation assumptions between lines of business when estimating the risk-based capital reserve, the solvency capital requirement (SCR), under Solvency II regulations. A case study is presented and the SCR is calculated according to the standard model approach. Alternatively, the requirement is then calculated using an internal model based on a Monte Carlo simulation of the net underwriting result at a one-year horizon, with copulas being used to model the dependence between lines of business. To address the impact of these model assumptions on the SCR, we conduct a sensitivity analysis. We examine changes in the correlation matrix between lines of business and address the choice of copulas. Drawing on aggregate historical data from the Spanish non-life insurance market between 2000 and 2009, we conclude that modifications of the correlation and dependence assumptions have a significant impact on SCR estimation.

Statistical inference for copulas has been addressed in various research papers. Due to the complicated theoretical results, studies have been carried out mainly in the bivariate case, be it properties of estimators or goodness-of-fit tests. However, from a practical point of view, higher dimensions are of interest. This work presents the results of large-scale simulation studies with particular focus on the question to what extent dimensionality influences point and interval estimators.

We use mean–variance hedging in discrete time in order to value an insurance liability. The prediction of the insurance liability is decomposed into claims development results, that is, yearly deteriorations in its conditional expected values until the liability is finally settled. We assume the existence of a tradeable derivative with binary pay-off written on the claims development result and available in each development period. General valuation formulas are stated and, under additional assumptions, these valuation formulas simplify to resemble familiar regulatory cost-of-capital-based formulas. However, adoption of the mean–variance framework improves upon the regulatory approach by allowing for potential calibration to observed market prices, inclusion of other tradeable assets, and consistent extension to multiple periods. Furthermore, it is shown that the hedging strategy can also lead to increased capital efficiency.

Premium settlement and calculation of reserves and capital requirements are typically based on worst- or just bad-case assumptions on interest rates, mortality rates, and other transition rates between states defined according to the insurance benefits. If interest and transition rates are chosen independently from each other, the worst choice, i.e. the combination of interest rates and transition rates that maximizes the reserve, can be found by dynamic programming. Here, we generalize this idea by choosing the interest and transition rates from a set that allows for mutual dependence. In general, finding the worst case is much more complicated in this situation, but we characterize a set of relatively tractable problems and present a series of examples from this set. Our approach with mutual dependence is relevant e.g. for internal models in Solvency II.

The finite-time ruin problem, which implicitly involves the inversion of the Laplace transform of the time to ruin, has been a long-standing research problem in risk theory. Existing results in the Sparre Andersen risk models are mainly based on an exponential assumption either on the interclaim times or on the claim sizes. In this paper, we utilize the multivariate version of Lagrange expansion theorem to obtain a series expansion for the density of the time to ruin under a more general distribution assumption, namely the combination of n exponentials. A remark is further made to emphasize that this technique can also be applied to other areas of applied probability. For instance, the proposed methodology can be used to obtain the distribution of some first passage times for particular stochastic processes. As an illustration, the duration of a busy period in a queueing risk model will be examined.

This study introduces an equilibrium approach to price mortality-linked securities in a discrete time economy, assuming that the mortality rate has a transformed normal distribution. This pricing method complements current studies on the valuation of mortality-linked securities, which only have discrete trading opportunities and insufficient market trading data. Like the Wang transform, the valuation relationship is still risk-neutral (preference-free) and the mortality-linked security is priced as the expected value of its terminal payoff, discounted by the risk-free rate. This study provides an example of pricing the Swiss Re mortality bond issued in 2003 and obtains an approximated closed-form solution.

We revisit the dividend payment problem in the dual model of Avanzi et al. ([2–4]). Using the fluctuation theory of spectrally positive Lévy processes, we give a short exposition in which we show the optimality of barrier strategies for all such Lévy processes. Moreover, we characterize the optimal barrier using the functional inverse of a scale function. We also consider the capital injection problem of [4] and show that its value function has a very similar form to the one in which the horizon is the time of ruin.

In this paper, we present the full characterization of the iterativity condition for the mean-value principle under the cumulative prospect theory. It turns out that the premium principle is iterative for exactly six pairs of probability distortion functions. Some of the corresponding premium principles are the classical mean-value principle, essential infimum or essential supremum of the random loss. Moreover, from the proof of the main theorem of this paper, it follows that the iterativity of the mean-value principle is equivalent to the iterativity of the generalized Choquet integral.

We study mean–variance hedging of a pure endowment, a term insurance and general annuities by trading in a longevity bond with continuous rate payments proportional to the survival probability. In particular, we discuss the introduction of a gratification annuity as an interesting insurance product for the life insurance market. The optimal hedging strategies are determined via their Galtchouk–Kunita–Watanabe decompositions under specific, yet sufficiently general model assumptions. The results are then further illustrated by assuming a general affine structure of the mortality intensity process. The optimal hedging strategies as well as the residual hedging error of a gratification annuity and a simple life annuity are finally investigated with numerical simulations, which illustrate the nice features of the gratification annuity for the insurance industry.

Insurers have been concerned about surrenders for a long time especially in saving business, where huge sums are at stake. The emergence of the European directive Solvency II, which promotes the development of internal risk models (among which a complete unit is dedicated to surrender risk management), strengthens the necessity to deeply study and understand this risk. In this paper, we investigate the topics of segmenting and modelling surrenders in order to better take into account the main risk factors impacting policyholders' decisions. We find that several complex aspects must be specifically dealt with to predict surrenders, in particular the heterogeneity of behaviour as well as the context faced by the insured. Combining them, we develop a new methodology that seems to provide good results on given business lines, and that moreover can be adapted for other products with little effort.

This paper analyzes the framework of German participating payout life annuities (PLAs), which offer guaranteed minimum benefits as well as participation in insurers' surpluses. We show that the process of sharing surpluses between shareholders and policyholders follows transparent and consistent rules. Subsequently, we develop an asset-liability model for a stylized German life insurer that offers PLAs to evaluate benefit variability and insurer stability given stochastic mortality and capital market developments. Our results suggest that guaranteed benefits can be provided with high credibility via PLAs, while, at the same time, annuitants receive attractive money's worth ratios. Moreover, we show that it might be difficult to offer a fixed benefit annuity providing the same lifetime utility as a PLA for the same premium and a comparably low insolvency risk. Overall, PLA schemes may be an efficient way to deal with risk factors that are highly unpredictable and difficult to hedge over the long run, such as systematic longevity and investment risks.

The evaluation of future cash flows and solvency capital recently gained importance in general insurance. To assist in this process, our paper proposes a novel loss reserving model, designed for individual claims developing in discrete time. We model the occurrence of claims, as well as their reporting delay, the time to the first payment, and the cash flows in the development process. Our approach uses development factors similar to those of the well-known chain–ladder method. We suggest the Multivariate Skew Normal distribution as a multivariate distribution suitable for modeling these development factors. Empirical analysis using a real portfolio and out-of-sample prediction tests demonstrate the relevance of the model proposed.

We discuss a class of counting distributions motivated by a problem in discrete surplus analysis, and special cases of which have applications in stop-loss, discrete Tail value at risk (TVaR) and claim count modelling. Explicit formulas are developed, and the mixed Poisson case is considered in some detail. Simplifications occur for some underlying negative binomial and related models, where in some cases compound geometric distributions arise naturally. Applications to claim count and aggregate claims models are then given.

In classical risk theory, the infinite-time ruin probability of a surplus process Ct is calculated as the probability of the process becoming negative at some point in time. In this paper, we consider a relaxation of the ruin concept to the concept of bankruptcy, according to which one has a positive surplus-dependent probability to continue despite temporary negative surplus. We study the resulting bankruptcy probability for the compound Poisson risk model with exponential claim sizes for different bankruptcy rate functions, deriving analytical results, upper and lower bounds as well as an efficient simulation method. Numerical examples are given and the results are compared with the classical ruin probabilities. Finally, it is illustrated how the analysis can be extended to study the discounted penalty function under this relaxed ruin criterion.