This paper presents analytical representations for an optimal insurance contract under distortion risk measure and in the presence of model uncertainty. We incorporate ambiguity aversion and distortion risk measure through the model of Robert and Therond [(2014) ASTIN Bulletin: The Journal of the IAA, 44(2), 277–302.] as per the framework of Klibanoff et al. [(2005) A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.]. Explicit optimal insurance indemnity functions are derived when the decision maker (DM) applies Value-at-Risk as risk measure and is ambiguous about the loss distribution. Our results show that: (1) under model uncertainty, ambiguity aversion results in a distorted probability distribution over the set of possible models with a bias in favor of the model which yields a larger risk; (2) a more ambiguity-averse DM would demand more insurance coverage; (3) for a given budget, uncertainties about the loss distribution result in higher risk level for the DM.

Recursive credibility estimation is discussed from the viewpoint of linear filtering theory. A conjunction of geometric interpretation and the innovation approach leads to general algorithms not developed before. Moreover, covariance characterizations considered by other researchers drop our elegantly as a result of geometric considerations. Examples are presented of Kalman type filters valid for non-Gaussian measurements.

The most important property of a distribution function to be used as a model for the distribution of one claim is of course that it fits the data well enough. If there is no natural truncation point in the data a more formal demand is that all the moments of the distribution function exist. Further, to be of a real value to the statistician, the chosen d.f. ought to be reasonably handy to use. As all the moments of the lognormal d.f. exist the first point to be checked is whether the lognormal d.f. fits the data. The other points on the list below are the qualities that I think are of the greatest value when using a distribution function, i.e. they reflect the handiness of the d.f.

Does the lognormal d.f.

1. Fit the data?

2. Give an unbiased and efficient estimate of the mean? It is important that this estimate is not too difficult to compute.

3. Give a practicable confidence interval of the mean?

4. Give a known distribution function of the estimate of the risk premium?

This paper is an attempt to give an affirmative answer to these questions. As the lognormal distribution function has been treated in the monograph “The lognormal distribution” by J. Aitchison and J. A. C. Brown (Cambridge University Press) the theory of this distribution function will not be dealt with more than necessary for the context.

Adverse weather-related risk is a main source of crop production loss and a big concern for agricultural insurers and reinsurers. In response, weather risk hedging may be valuable, however, due to basis risk it has been largely unsuccessful to date. This research proposes the Lévy subordinated hierarchical Archimedean copula model in modelling the spatial dependence of weather risk to reduce basis risk. The analysis shows that the Lévy subordinated hierarchical Archimedean copula model can improve the hedging performance through more accurate modelling of the dependence structure of weather risks and is more efficient in hedging extreme downside weather risk, compared to the benchmark copula models. Further, the results reveal that more effective hedging may be achieved as the spatial aggregation level increases. This research demonstrates that hedging weather risk is an important risk management method, and the approach outlined in this paper may be useful to insurers and reinsurers in the case of agriculture, as well as for other related risks in the property and casualty sector.

We revisit the “full picture” of the claims development uncertainty in Mack’s (1993) distribution-free stochastic chain ladder model. We derive the uncertainty estimators in a new and easily understandable way, which is much simpler than the derivation found so far in the literature, and compare them with the well known estimators of Mack and of Merz–Wüthrich.

Our uncertainty estimators of the one-year run-off risks are new and different to the Merz–Wüthrich formulas. But if we approximate our estimators by a first order Taylor expansion, we obtain equivalent but simpler formulas. As regards the ultimate run-off risk, we obtain the same formulas as Mack for single accident years and an equivalent but better interpretable formula for the total over all accident years.

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.

We apply Lemaire's algorithm and a non-parametric mixed Poisson fit to a motor insurance portfolio in order to find the true claim frequency and claim amount distributions. The algorithm we develop accounts for the fact that observed distributions are distorted by bonus hunger, when a bonus-malus system is used by the insurer.

In this paper we explore the problem of economic capital allocations in the context of non-negative multivariate (insurance) risks possessing a dependence structure. We derive a general result and illustrate it with a number of useful examples. One such example, for instance, develops explicit expressions for the discussed economic capital decomposition rule when the underlying portfolio consists of dependent compound Poisson risks.

This paper presents a continuous time version of a stochastic investment model originally due to Wilkie. The model is constructed via stochastic differential equations. Explicit distributions are obtained in the case where the SDEs are driven by Brownian motion, which is the continuous time analogue of the time series with white noise residuals considered by Wilkie. In addition, the cases where the driving “noise” are stable processes and Gamma processes are considered.

Credibility theory is concerned with the problem of forecasting the mean performance (claim frequency, total losses, etc.) of an individual risk, selected from a collective of heterogeneous risks, based upon the statistics of the collective, and upon the individual's experience data. The classic results, derived by American actuaries in the 1920's, were further strengthened by Bailey and Mayerson in 1950 and 1965, who showed that these results were exact Bayesian for certain risk distributions. Bühlmann, in 1967, then showed that the credibility formulae were the best least-squares linearized approximation to the exact Bayesian forecast, for general risk distributions.

This paper extends credibility theory to the problem of forecasting the distribution of individual risk, based upon collective statistics and individual experience data. Although the problem is, in principle, solved by finding a Bayesian conditional distribution, this approach requires a detailed knowledge of collective structure. The credible distribution, on the other hand, requires fewer prior statistics, and is also a best least-squares linearized approximation to the exact Bayesian distribution.

Following the theoretical development, detailed computational results are given.

In the wake of the technical development of our era we are increasingly faced with claims of extremely large amounts.

Whereas even in the past century large claims were mostly due to elemental or natural forces such as earthquakes, hurricanes and floods, there are today other causes, conditioned by human factors, that have considerably increased in importance. Another fact to be observed is that the risks that are passed on to the insurance market assume a more serious character year by year. Large building complexes, giant tankers, containers, atomic power stations, dams and jumbo jets pose for the insurers problems which are in no way easy to solve, since the risk covers required touch the limits of market capacity and their rating is subject to great uncertainties. At the same time, the readiness to underwrite such covers is a matter of utmost importance for the private insurance industry.

For the coverage of such large risks, various insurance forms have been developed in the past years within the range of non-proportional methods. Besides the excess of loss and stop loss covers in their usual form, the cumulative loss cover and the largest claim cover have attracted particular attention. The cumulative risk cover has, for example, been dealt with in [11]) and has also been frequently offered in practice. The coverage of the largest claim or, more generally, of the sum of the n largest claims does not appear to have gained a proper foothold in practice in spite of various valuable contributions in this field—considering particularly [2], [3], [5], [8] and [14]. The fact that it was during 1963/64 that various authors analysed these problems is no coincidence, for it was then that in two ASTIN Colloquia these topics were discussed.

Actuaries have always been in search of ways to determine premiums which match the risks insured as closely as possible. They do this by differentiating between them on the basis of observable risk factors. In practice, many examples of such risk factors are being used: age and sex for life insurance; location, type of building etc. for fire insurance. Motor insurance is perhaps the most characteristic branch with respect to this phenomenon: in tariffs we find factors like weight, price or cylinder capacity of the car, age of the driver, area of residence, past claims experience (Bonus/Malus), annual mileage etc.

Outsiders may not always be very positive about such a refined premium differentiation. The basis of insurance, they say, should be solidarity among insureds; premium differentiation is basically opposed to this. Another statement heard in the field is: “Premium differentiation ultimately results in letting every individual pay his own claims, it is the end of insurance”.

Much confusion arises during discussions about this subject, especially between actuaries and non-actuaries. We will therefore first give a mathematical definition of solidarity, (Section 2), followed by a brief description of certain trends in society which might bring insurers to deliberately drop certain risk factors from their tariffs in order to increase solidarity (Section 3). The consequences of doing so are examined and it is shown that increased solvency requirements will in the end prove to be ineffective. A possible solution is a voluntary transfer of premium between companies (Section 4). The situation is illustrated by an example of health insurance in the Netherlands, where proposals to arrive at such transfeis are presently being discussed.

Maximum likelihood estimation in case of a Poisson or Gamma distribution with loglinear parametrization for the mean is quite akin. The asymptotic variance-covariance matrix for the maximum likelihood estimator is derived as well as a linear estimator, which can serve as a starting value for the nonlinear search procedure.

In this paper, it is shown how to approximate theoretical premium calculation principles in order to make them useful in practice. The method relies on stochastic extrema in moment spaces and is illustrated with the aid of the exponential principle.

Enterprise risk management has become a major focus for insurers and reinsurers. Capitalization and pricing decisions are recognized as critical to firm value maximization. Market imperfections including frictional costs of capital such as taxes, agency costs, and financial distress costs are an important motivation for enterprise risk management. Risk management reduces the volatility of financial performance and can have a significant impact on firm value maximization by reducing the impact of frictional costs. Insurers operate in imperfect markets where demand elasticity of policyholders and preferences for financial quality of insurers are important determinants of capitalization and pricing strategies. In this paper, we analyze the optimization of enterprise or firm value in a model with market imperfections. A realistic model of an insurer is developed and calibrated.

Frictional costs, imperfectly competitive demand elasticity, and preferences for financial quality are explicitly modelled and implications for enterprise risk management are quantified.

This paper presents a Bayesian approach using Markov chain Monte Carlo methods and the generalized-t (GT) distribution to predict loss reserves for the insurance companies. Existing models and methods cannot cope with irregular and extreme claims and hence do not offer an accurate prediction of loss reserves. To develop a more robust model for irregular claims, this paper extends the conventional normal error distribution to the GT distribution which nests several heavy-tailed distributions including the Student-t and exponential power distributions. It is shown that the GT distribution can be expressed as a scale mixture of uniforms (SMU) distribution which facilitates model implementation and detection of outliers by using mixing parameters. Different models for the mean function, including the log-ANOVA, log-ANCOVA, state space and threshold models, are adopted to analyze real loss reserves data. Finally, the best model is selected according to the deviance information criterion (DIC).

This paper investigates a high-dimensional vector-autoregressive (VAR) model in mortality modeling and forecasting. We propose an extension of the sparse VAR (SVAR) model fitted on the log-mortality improvements, which we name “spatially penalized smoothed VAR” (SSVAR). By adaptively penalizing the coefficients based on the distances between ages, SSVAR not only allows a flexible data-driven sparsity structure of the coefficient matrix but simultaneously ensures interpretable coefficients including cohort effects. Moreover, by incorporating the smoothness penalties, divergence in forecast mortality rates of neighboring ages is largely reduced, compared with the existing SVAR model. A novel estimation approach that uses the accelerated proximal gradient algorithm is proposed to solve SSVAR efficiently. Similarly, we propose estimating the precision matrix of the residuals using a spatially penalized graphical Lasso to further study the dependency structure of the residuals. Using the UK and France population data, we demonstrate that the SSVAR model consistently outperforms the famous Lee–Carter, Hyndman–Ullah, and two VAR-type models in forecasting accuracy. Finally, we discuss the extension of the SSVAR model to multi-population mortality forecasting with an illustrative example that demonstrates its superiority in forecasting over existing approaches.

We develop quantile functions from regression models in order to derive risk margin and to evaluate capital in non-life insurance applications. By utilizing the entire range of conditional quantile functions, especially higher quantile levels, we detail how quantile regression is capable of providing an accurate estimation of risk margin and an overview of implied capital based on the historical volatility of a general insurers loss portfolio. Two modeling frameworks are considered based around parametric and non-parametric regression models which we develop specifically in this insurance setting. In the parametric framework, quantile functions are derived using several distributions including the flexible generalized beta (GB2) distribution family, asymmetric Laplace (AL) distribution and power-Pareto (PP) distribution. In these parametric model based quantile regressions, we detail two basic formulations. The first involves embedding the quantile regression loss function from the nonparameteric setting into the argument of the kernel of a parametric data likelihood model, this is well known to naturally lead to the AL parametric model case. The second formulation we utilize in the parametric setting adopts an alternative quantile regression formulation in which we assume a structural expression for the regression trend and volatility functions which act to modify a base quantile function in order to produce the conditional data quantile function. This second approach allows a range of flexible parametric models to be considered with different tail behaviors. We demonstrate how to perform estimation of the resulting parametric models under a Bayesian regression framework. To achieve this, we design Markov chain Monte Carlo (MCMC) sampling strategies for the resulting Bayesian posterior quantile regression models. In the non-parametric framework, we construct quantile functions by minimizing an asymmetrically weighted loss function and estimate the parameters under the AL proxy distribution to resemble the minimization process. This quantile regression model is contrasted to the parametric AL mean regression model and both are expressed as a scale mixture of uniform distributions to facilitate efficient implementation. The models are extended to adopt dynamic mean, variance and skewness and applied to analyze two real loss reserve data sets to perform inference and discuss interesting features of quantile regression for risk margin calculations.

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.