Systemic risk (SR) has been shown to play an important role in explaining the financial turmoils in the last several decades and understanding this source of risk has been a particular interest amongst academics, practitioners and regulators. The precise mathematical formulation of SR is still scrutinised, but the main purpose is to evaluate the financial distress of a system as a result of the failure of one component of the financial system in question. Many of the mathematical definitions of SR are based on evaluating expectations in extreme regions and therefore, Extreme Value Theory (EVT) represents the key ingredient in producing valuable estimates of SR and even its decomposition per individual components of the entire system. Without doubt, the prescribed dependence model amongst the system components has a major impact over our asymptotic approximations. Thus, this paper considers various well-known dependence models in the EVT literature that allow us to generate SR estimates. Our findings reveal that SR has a significant impact under asymptotic dependence, while weak tail dependence, known as asymptotic independence, produces an insignificant loss over the regulatory capital.

We propose an approximation scheme for the computation of the risk measures of guaranteed minimum maturity benefits (GMMBs) and guaranteed minimum death benefits (GMDBs), based on the evaluation of single integrals under conditional moment matching. This procedure is computationally efficient in comparison with standard analytical methods while retaining a high degree of accuracy, and it allows one to deal with the case of additional earnings and the computation of related sensitivities.

Unlike chain ladder, the loss ratio method requires volume measures. Typically, these volumes are assumed to be known. In practice, however, accurate volume measures are rarely available. We interpret the available volumes as estimators for the true volume measures and analyze the consequences for the loss ratio method. In particular, we calculate the mean squared error of prediction, including uncertainty of volume measures, and derive approximately optimal weights for the observed incremental loss ratios. We then introduce a generalization of the loss ratio method that is tailored to the situation of uncertain volume measures and calculate the prediction uncertainty of this generalized loss ratio method.

We consider compound Poisson claims reserving models applied to the paid claims and to the number of payments run-off triangles. We extend the standard Poisson-gamma assumption to account for over-dispersion in the payment counts and to account for various mean and variance structures in the individual payments. Two generalized linear models are applied consecutively to predict the unpaid claims. A bootstrap is used to estimate the mean squared error of prediction and to simulate the predictive distribution of the unpaid claims. We show that the extended compound Poisson models make reasonable predictions of the unpaid claims.

In this paper, we first propose a statistical model, called the Coherent Incurred Paid model, to predict future claims, using simultaneously the information contained in incurred and paid claims. This model does not assume log-normality of the levels (or normality of the growth rates) and is semi-parametric since it only specifies the first and the second moments; however, in order to evaluate the impact of the normality assumption, we also propose a benchmark Gaussian version of our model. Correlations between growth rates of incurred and paid claims are allowed and the tail development period is estimated. We also provide methods for computing the Claim Development Results and their Values at Risk in the semi-parametric framework. Moreover, we show how to take into account the updating of the estimation in the computation of the Claim Development Results. An application highlights the practical importance of relaxing the normality assumption and of updating the estimation of the parameters.

We establish a “top-down” approximation scheme to approximate loss distributions of reinsurance products and Insurance-Linked Securities based on three input parameters, namely the Attachment Probability, Expected Loss and Exhaustion Probability. Our method is rigorously derived by utilizing a classical result from Extreme-Value Theory, the Pickands–Balkema–de Haan theorem. The robustness of the scheme is demonstrated by proving sharp error-bounds for the approximated curves with respect to the supremum and L2 norms. The practical implications of our findings are examined by applying it to Industry Loss Warranties: the method performs very accurately for each transaction. Our approach can be used in a variety of applications such as vendor model blending, portfolio optimization and premium calculation.

This paper investigates dynamic hedging strategies for pension and annuity liabilities that are exposed to longevity risk. In particular, we consider a hedger who wishes to minimize the variance of her hedging error using index-based longevity-linked derivatives. To cope with the fact that liquidity of longevity-linked derivatives is still limited, we consider a liquidity constrained case where the hedger can only trade longevity-linked derivatives at a frequency lower than other assets. Time-consistent, closed-form solutions of optimal hedging strategies are obtained under a forward mortality framework. In the numerical illustration, we show that lowering the trading of the longevity-linked derivatives to a 2-year frequency only leads to a slight loss of the hedging performance. Moreover, even when the longevity-linked derivatives are traded at a very low (5-year) frequency, dynamic hedging strategies still significantly outperform the static one.

We develop a novel approach for pricing cyber insurance contracts. The considered cyber threats, such as viruses and worms, diffuse in a structured data network. The spread of the cyber infection is modeled by an interacting Markov chain. Conditional on the underlying infection, the occurrence and size of claims are described by a marked point process. We introduce and analyze a new polynomial approximation of claims together with a mean-field approach that allows to compute aggregate expected losses and prices of cyber insurance. Numerical case studies demonstrate the impact of the network topology and indicate that higher order approximations are indispensable for the analysis of non-linear claims.

We investigate the application of natural hedging strategies for long-term care (LTC) insurers by diversifying both longevity and disability risks affecting LTC annuities. We propose two approaches to natural hedging: one built on a multivariate duration, the other on the Conditional Value-at-Risk minimization of the unexpected loss. Both the approaches are extended to the LTC insurance using a multiple state framework. In order to represent the future evolution of mortality and disability transition probabilities, we use the stochastic model of Cairns et al. (2009) with cohort effect under parameter uncertainty through a semi-parametric bootstrap procedure. We calculate the optimal level of a product mix and measure the effectiveness provided by the interaction of LTC stand alone, deferred annuity and whole-life insurance. We compare the results obtained by the two approaches and find that a natural hedging strategy for LTC insurers is attainable with a product mix of LTC and annuities, but including low proportion of LTC.

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.

Following the EU Gender Directive, that obliges insurance companies to charge the same premium to policyholders of different genders, we address the issue of calculating solvency capital requirements (SCRs) for pure endowments and annuities issued to mixed portfolios. The main theoretical result is that, if the unisex fairness principle is adopted for the unisex premium, the SCR at issuing time of the mixed portfolio calculated with unisex survival probabilities is greater than the sum of the SCRs of the gender-based subportfolios. Numerical results show that for pure endowments the gap between the two is negligible, but for lifetime annuities the gap can be as high as 3–4%. We also analyze some conservative pricing procedures that deviate from the unisex fairness principle, and find that they lead to SCRs that are lower than the sum of the gender-based SCRs because the policyholders are overcharged at issuing time.

It is a common belief for actuaries that the heterogeneity of claim severities in a given insurance portfolio tends to increase its dangerousness, which results in requiring more capital for covering claims. This paper aims to investigate the effects of orderings and heterogeneity among scale parameters on the aggregate claim amount when both claim occurrence probabilities and claim severities are dependent. Under the assumption that the claim occurrence probabilities are left tail weakly stochastic arrangement increasing, the actuaries' belief is examined from two directions, i.e., claim severities are comonotonic or right tail weakly stochastic arrangement increasing. Numerical examples are provided to validate these theoretical findings. An application in assets allocation is addressed as well.

Continuous parametric distributions are useful tools for modeling and pricing insurance risks, measuring income inequality in economics, investigating reliability of engineering systems, and in many other areas of application. In this paper, we propose and develop a new method for estimation of their parameters—the method of Winsorized moments (MWM)—which is conceptually similar to the method of trimmed moments (MTM) and thus is robust and computationally efficient. Both approaches yield explicit formulas of parameter estimators for log-location-scale families and their variants, which are commonly used to model claim severity. Large-sample properties of the new estimators are provided and corroborated through simulations. Their performance is also compared to that of MTM and the maximum likelihood estimators (MLE). In addition, the effect of model choice and parameter estimation method on risk pricing is illustrated using actual data that represent hurricane damages in the United States from 1925 to 1995. In particular, the estimated pure premiums for an insurance layer are computed when the lognormal and log-logistic models are fitted to the data using the MWM, MTM and MLE methods.

In this paper, we present closed-type formulas for some multivariate compound distributions with multivariate Sarmanov counting distribution and independent Erlang distributed claim sizes. Further on, we also consider a type-II Pareto dependency between the claim sizes of a certain type. The resulting densities rely on the special hypergeometric function, which has the advantage of being implemented in the usual software. We numerically illustrate the applicability and efficiency of such formulas by evaluating a bivariate cumulative distribution function, which is also compared with the similar function obtained by the classical recursion-discretization approach.

In this paper, we propose a new family of premium calculation principles based on the use of prior information from different sources. Under this framework and based on the use of Ordered Weighted Averaging operators, we provide alternative collective and Bayes premiums and describe some approaches to efficiently compute them. Several examples are detailed to illustrate the performance of the new methods.

Cash balance pension plans with crediting rates linked to long bond yields are relatively common in the United States, but their liabilities are proving very challenging to hedge. In this paper, we consider dynamic hedge strategies using the one-factor and two-factor Hull White models, based on results for the liability valuation from Hardy et al. (2014). The strategies utilise simple hedge portfolios combining one or two zero-coupon bonds, and a money market account. We assess the effectiveness of the strategies by considering how accurately each one would have hedged a 5-year cash balance liability through the past 20 years, using real-world returns and crediting rates, and assuming parameters calibrated using the information available at the time. We show that there is considerable impact of model and parameter uncertainty, with additional, less severe impact from discrete hedging error and transactions costs. Despite this, the dynamic hedge strategies do manage to stabilize surplus substantially, even through the turbulence of the past two decades.

The four-parameter distribution family, the generalised beta type-II (GB2), also known as the transformed beta distribution, has been proposed for modelling insurance losses. As special cases, this family nests many distributions with light and heavy tails, including the lognormal, gamma, Weibull, Burr and generalised gamma distributions. This paper extends the GB2 family to the contaminated GB2 family, which offers many flexible features, including bimodality and a wide range of skewness and kurtosis. Properties of the contaminated distribution are derived and evaluated in a simulation study and the suitability of the contaminated GB2 distribution for actuarial purposes is demonstrated through two real loss data sets. Analysis of tail quantiles for the data suggests large differences in extreme quantile estimates for different loss distribution assumptions, showing that the selection of appropriate distributions has a significant impact for insurance companies.

The hierarchical credibility model was introduced, and extended, in the 70s and early 80s. It deals with the estimation of parameters that characterize the nodes of a tree structure. That model is limited, however, by the fact that its parameters are assumed fixed over time. This causes the model's parameter estimates to track the parameters poorly when the latter are subject to variation over time. This paper seeks to remove this limitation by assuming the parameters in question to follow a process akin to a random walk over time, producing an evolutionary hierarchical model. The specific form of the model is compatible with the use of the Kalman filter for parameter estimation and forecasting. The application of the Kalman filter is conceptually straightforward, but the tree structure of the model parameters can be extensive, and some effort is required to retain organization of the updating algorithm. This is achieved by suitable manipulation of the graph associated with the tree. The graph matrix then appears in the matrix calculations inherent in the Kalman filter. A numerical example is included to illustrate the application of the filter to the model.

This paper examines the importance of pensions (employment and social security), taxes and government transfers for alternative retirement savings drawdown strategies (DS), compared to the conventional approach in published literature of using a gross income concept obtainable from retirement savings alone. Using a lifetime utility framework, our longitudinal dynamic micro-simulation model incorporates risk aversion, stochastic markets, stochastic mortality and the interactions among sources of retirement income within the complex Canadian tax and social benefit system, enabling us to rank commonly advocated DS and to ask whether incorporating pensions, taxes and transfers alters those rankings. Our findings show the importance of treating the evaluation of alternative DS as a comprehensive and integrated problem by including all sources of income — including pensions, taxes and government transfers. Using restricted income measures can potentially lead to simplistic, and possibly misleading, conclusions.

We develop a Gaussian process (GP) framework for modeling mortality rates and mortality improvement factors. GP regression is a nonparametric, data-driven approach for determining the spatial dependence in mortality rates and jointly smoothing raw rates across dimensions, such as calendar year and age. The GP model quantifies uncertainty associated with smoothed historical experience and generates full stochastic trajectories for out-of-sample forecasts. Our framework is well suited for updating projections when newly available data arrives, and for dealing with “edge” issues where credibility is lower. We present a detailed analysis of GP model performance for US mortality experience based on the CDC (Center for Disease Control) datasets. We investigate the interaction between mean and residual modeling, Bayesian and non-Bayesian GP methodologies, accuracy of in-sample and out-of-sample forecasting, and stability of model parameters. We also document the general decline, along with strong age-dependency, in mortality improvement factors over the past few years, contrasting our findings with the Society of Actuaries (SOA) MP-2014 and -2015 models that do not fully reflect these recent trends.