It is rare that the severity loss distribution for a specific line of business can be derived from first principles. One such example is the use of generalised Pareto distribution for losses above a large threshold (or more accurately: asymptotically), which is dictated by extreme value theory. Most popular distributions, such as the lognormal distribution or the Maxwell-Boltzmann-Bose-Einstein-Fermi-Dirac (MBBEFD), are convenient heuristics with no underlying theory to back them. This paper presents a way to derive a severity distribution for property losses based on modelling a property as a weighted graph, that is, a collection of nodes and weighted arcs connecting these nodes. Each node v (to which a value can also be assigned) corresponds to a room or a unit of the property where a fire can occur, while an arc (v, v′; p) between vertices v and v′ signals that the probability of the fire propagating from v to v′ is p. The paper presents two simple models for fire propagation (the random graph approach and the random time approach) and a model for explosion risk that allow one to calculate the loss distribution for a given property from first principles. The MBBEFD model is shown to be a good approximation for the simulated distribution of losses based on property graphs for both the random graph and the random time approach.

We introduce a new modelling framework to explain socio-economic differences in mortality in terms of an affluence index that combines information on individual wealth and income. The model is illustrated using data on older Danish males over the period 1985–2012 reported in the Statistics Denmark national register database. The model fits the historical mortality data well, captures their key features, generates smoothed death rates that allow us to work with a larger number of sub-groups than has previously been considered feasible, and has plausible projection properties.

The aim of this editorial is to increase the acceptance of neural net modeling in the actuarial community. Neural nets may substantially improve classical actuarial models, if appropriately applied. We illustrate this on a toy example but, in fact, this should be understood as a universal concept. Assume we have a classical regression problem where the distribution of a response Y = Y(x) can be described by covariates x. A common actuarial problem is to determine the premium $\mu ({\textit{\textbf{x}}}) = {\mathbb E} [Y({\textit{\textbf{x}}})]$ as a function of the covariates x. Actuaries have developed excellent skills to solve such problems through finding appropriate regression functions x ↦ μ(x). This editorial shows how these skills can further be improved using the toolbox of neural nets.

For insurance companies in Europe, the introduction of Solvency II leads to a tightening of rules for solvency capital provision. In life insurance, this especially affects retirement products that contain a significant portion of longevity risk (e.g., conventional annuities). Insurance companies might react by price increases for those products, and, at the same time, might think of alternatives that shift longevity risk (at least partially) to policyholders. In the extreme case, this leads to so-called tontine products where the insurance company’s role is merely administrative and longevity risk is shared within a pool of policyholders. From the policyholder’s viewpoint, such products are, however, not desirable as they lead to a high uncertainty of retirement income at old ages. In this article, we alternatively suggest a so-called tonuity that combines the appealing features of tontine and conventional annuity. Until some fixed age (the switching time), a tonuity’s payoff is tontine-like, afterwards the policyholder receives a secure payment of a (deferred) annuity. A tonuity is attractive for both the retiree (who benefits from a secure income at old ages) and the insurance company (whose capital requirements are reduced compared to conventional annuities). The tonuity is a possibility to offer tailor-made retirement products: using risk capital charges linked to Solvency II, we show that retirees with very low or very high risk aversion prefer a tontine or conventional annuity, respectively. Retirees with medium risk aversion, however, prefer a tonuity. In a utility-based framework, we therefore determine the optimal tonuity characterized by the critical switching time that maximizes the policyholder’s lifetime utility.

Delong et al. (2018) presented a theory of fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. In this paper, we investigate in detail two practical applications of our theory of fair valuation. In the first example, we consider the fair valuation of a terminal benefit which is contingent on correlated tradeable and non-tradeable financial risks. In the second example, we consider a portfolio of unit-linked contracts contingent on a non-tradeable insurance and a tradeable financial risk. We derive partial differential equations (PDEs) which characterize the continuous-time fair valuation operators in these two examples and we find explicit solutions to these PDEs. The fair values of the liabilities are decomposed into the best estimate of the liability and a risk margin. The arbitrage-free representations of the fair values of the liabilities are derived and the dynamic hedging strategies associated with the continuous-time fair valuation operators are also established. Detailed interpretations of the results, which should be useful both for researchers and practitioners, are provided.

Using risk-reducing properties of conditional expectations with respect to convex order, Denuit and Dhaene [Denuit, M. and Dhaene, J. (2012). Insurance: Mathematics and Economics 51, 265–270] proposed the conditional mean risk sharing rule to allocate the total risk among participants to an insurance pool. This paper relates the conditional mean risk sharing rule to the size-biased transform when pooled risks are independent. A representation formula is first derived for the conditional expectation of an individual risk given the aggregate loss. This formula is then exploited to obtain explicit expressions for the contributions to the pool when losses are modeled by compound Poisson sums, compound Negative Binomial sums, and compound Binomial sums, to which Panjer recursion applies. Simple formulas are obtained when claim severities are homogeneous. A couple of applications are considered: first, to a peer-to-peer insurance scheme where participants share the first layer of their respective risks while the higher layer is ceded to a (re)insurer; second, to survivor credits to be shared among surviving participants in tontine schemes.

In areas of application, including actuarial science and demography, it is increasingly common to consider a time series of curves; an example of this is age-specific mortality rates observed over a period of years. Given that age can be treated as a discrete or continuous variable, a dimension reduction technique, such as principal component analysis (PCA), is often implemented. However, in the presence of moderate-to-strong temporal dependence, static PCA commonly used for analyzing independent and identically distributed data may not be adequate. As an alternative, we consider a dynamic principal component approach to model temporal dependence in a time series of curves. Inspired by Brillinger’s (1974, Time Series: Data Analysis and Theory. New York: Holt, Rinehart and Winston) theory of dynamic principal components, we introduce a dynamic PCA, which is based on eigen decomposition of estimated long-run covariance. Through a series of empirical applications, we demonstrate the potential improvement of 1-year-ahead point and interval forecast accuracies that the dynamic principal component regression entails when compared with the static counterpart.

We consider the problem of pricing modern guarantee concepts in unit-linked life insurance, where the guaranteed amount grows contingent on the performance of an investment fund that acts simultaneously as the underlying security and the replicating portfolio. Using the Martingale Method, this nonstandard pricing problem can be transformed into a fixed-point problem, whose solution requires the evaluation of conditional expectations of highly path-dependent payoffs. By adapting the least-squares Monte Carlo method for American option pricing problems, we develop a new numerical approach to approximate the value of contingent guarantees and prove its convergence. Our valuation procedure can be applied to large-scale pricing problems, for which existing methods are infeasible, and leads to significant improvements in performance.

In this article, we study parameter uncertainty and its actuarial implications in the context of economic scenario generators. To account for this additional source of uncertainty in a consistent manner, we cast Wilkie’s four-factor framework into a Bayesian model. The posterior distribution of the model parameters is estimated using Markov chain Monte Carlo methods and is used to perform Bayesian predictions on the future values of the inflation rate, the dividend yield, the dividend index return and the long-term interest rate. According to the US data, parameter uncertainty has a significant impact on the dispersion of the four economic variables of Wilkie’s framework. The impact of such parameter uncertainty is then assessed for a portfolio of annuities: the right tail of the loss distribution is significantly heavier when parameters are assumed random and when this uncertainty is estimated in a consistent manner. The risk measures on the loss variable computed with parameter uncertainty are at least 12% larger than their deterministic counterparts.

We propose a two-regime Markov switching copula to depict the evolution of mortality dependence. One regime represents periods of high dependence and the other regime represents periods of low dependence. Each regime features a regular vine (R-vine) copula that, built on bivariate copulas, provides great flexibility for modelling complex high-dimensional dependence. Our estimated model indicates that the years of recovery from extreme mortality deterioration and the years of health care reform more likely fall into the low regime, while the years in which extreme mortality deteriorating events break out and the peaceful years without major mortality-impacting events more likely fall into the high regime. We use a case study to illustrate how the regime-switching copula can be applied to assess the effectiveness of longevity risk hedge with different beliefs about future mortality dependence evolution incorporated.

This paper focuses on the estimation and application aspects of the Erlang count logit-weighted reduced mixture of experts model (EC-LRMoE), which is a fully flexible multivariate insurance claim frequency regression model. We first prove the identifiability property of the proposed model to ensure that it is a suitable candidate for statistical inference. An expectation conditional maximization (ECM) algorithm is developed for efficient model calibrations. Three simulation studies are performed to examine the effectiveness of the proposed ECM algorithm and the versatility of the proposed model. The applicability of the EC-LRMoE is shown through fitting an European automobile insurance data set. Since the data set contains several complex features, we find it necessary to adopt such a flexible model. Apart from showing excellent fitting results, we are able to interpret the fitted model in an insurance perspective and to visualize the relationship between policyholders’ information and their risk level. Finally, we demonstrate how the fitted model may be useful for insurance ratemaking.

The aging of population is perhaps the most important problem that developed countries must face in the near future. Dependency can be seen as a consequence of the process of gradual aging. In a health context, this contingency is defined as a lack of autonomy in performing basic activities of daily living that requires the care of another person or significant help. In Europe in general and in Spain in particular, this phenomena represents a problem with economic, political, social and demographic implications. The prevalence of dependency in the population, as well as its intensity and evolution over the course of a person’s life are issues of greatest importance that should be addressed. The aim of this work is the estimation of life expectancy free of dependency (LEFD) based on functional trajectories to enhance the regular estimation of health expectancy. Using information from the Spanish survey EDAD 2008, we estimate the number of years spent free of dependency for disabled people according to gender, dependency degree (moderate, severe, major) and the earlier or later onset of dependency compared to a central trend. The main findings are as follows: first, we show evidence that to estimate LEFD ignoring the information provided by the functional trajectories may lead to non-representative LEFD estimates; second, in general, dependency-free life expectancy is higher for women than for men. However, its intensity is higher in women with later onset on dependency; Third, the loss of autonomy is higher (and more abrupt) in men than in women. Finally, the diversity of patterns observed at later onset of dependency tends to a dependency extreme-pattern in both genders.

A one-parameter version of the generalised Poisson distribution provided by Consul and Jain (1973) is considered in this paper. The distribution is unimodal with a zero vertex and over-dispersed. A generalised linear model related to this distribution is also presented. Its parameters can be estimated by using a Fisher-Scoring algorithm which is equivalent to iteratively reweighted least squares. Due to its flexibility and capacity to describe highly skewed data with an excessive number of zeros, the model is suitable to be applied in insurance settings as an alternative to the negative binomial and zero-inflated model.

We view the retail non-life insurance decision from the perspective of the insured. We formalize different consumption–insurance problems depending on the flexibility of the insurance contract. For exponential utility and power utility we find the optimal flexible insurance decision or insurance contract. For exponential utility we also find the optimal position in standard contracts that are less flexible and therefore, for certain nonlinear pricing rules, lead to a welfare loss for the individual insuree compared to the optimal flexible insurance decision. For the exponential loss distribution, we quantify a significant welfare loss. This calls for product development in the retail insurance business.

In this paper we suggest a modeling of joint life insurance pricing via Extended Marshall–Olkin (EMO) models and related copulas. These models are based on the combination of two approaches: the absolutely continuous copula approach, where the copula is used to capture dependencies due to environmental factors shared by the two lives, and the classical Marshall–Olkin model, where the association is given by accounting for a fatal event causing the simultaneous death of the two lives. New properties of the EMO model are established and applied to a sample of censored residual lifetimes of couples of insureds extracted from a data set of annuities contracts of a large Canadian life insurance company. Finally, some joint life insurance products are analyzed.

In insurance underwriting, misrepresentation represents the type of insurance fraud when an applicant purposely makes a false statement on a risk factor that may lower his or her cost of insurance. Under the insurance ratemaking context, we propose to use the expectation-maximization (EM) algorithm to perform maximum likelihood estimation of the regression effects and the prevalence of misrepresentation for the misrepresentation model proposed by Xia and Gustafson [(2016) The Canadian Journal of Statistics, 44, 198–218]. For applying the EM algorithm, the unobserved status of misrepresentation is treated as a latent variable in the complete-data likelihood function. We derive the iterative formulas for the EM algorithm and obtain the analytical form of the Fisher information matrix for frequentist inference on the parameters of interest for lognormal losses. We implement the algorithm and demonstrate that valid inference can be obtained on the risk effect despite the unobserved status of misrepresentation. Applying the proposed algorithm, we perform a loss severity analysis with the Medical Expenditure Panel Survey data. The analysis reveals not only the potential impact misrepresentation may have on the risk effect but also statistical evidence on the presence of misrepresentation in the self-reported insurance status.

Incurred but not reported (IBNR) loss reserving is of great importance for Property & Casualty (P&C) insurers. However, the temporal dependence exhibited in the claim arrival process is not reflected in many current loss reserving models, which might affect the accuracy of the IBNR reserve predictions. To overcome this shortcoming, we proposed a marked Cox process and showed its many desirable properties in Badescu et al. (2016).

In this paper, we consider the model estimation and applications. We first present an expectation–maximization (EM) algorithm which guarantees the efficiency of the estimators unlike the moment estimation methods widely used in estimating Cox processes. In addition, the proposed fitting algorithm can be implemented at a reasonable computational cost. We examine the performance of the proposed algorithm through simulation studies. The applicability of the proposed model is tested by fitting it to a real insurance claim data set. Through out-of-sample tests, we find that the proposed model can provide realistic predictive distributions.

As a benchmark mortality model in forecasting future mortality rates and hedging longevity risk, the widely employed Lee–Carter model (Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87, 659–671.) suffers from a restrictive constraint on the unobserved mortality index for ensuring model’s identification and a possible inconsistent inference. Recently, a modified Lee–Carter model (Liu, Q., Ling, C. and Peng, L. (2018) Statistical inference for Lee–Carter mortality model and corresponding forecasts. North American Actuarial Journal, to appear.) removes this constraint and a simple least squares estimation is consistent with a normal limit when the mortality index follows from a unit root or near unit root AR(1) model with a nonzero intercept. This paper proposes a bias-corrected estimator for this modified Lee–Carter model, which is consistent and has a normal limit regardless of the mortality index being a stationary or near unit root or unit root AR(1) process with a nonzero intercept. Applications to the US mortality rates and a simulation study are provided as well.

Frequent large losses from recent catastrophes have caused great concerns among insurers/reinsurers, who then turn to seek mitigations of such catastrophe risks by issuing catastrophe (CAT) bonds and thereby transferring the risks to the bond market. Whereas, the pricing of CAT bonds remains a challenging task, mainly due to the facts that the CAT bond market is incomplete and that the pricing usually requires knowledge about the tail of the risks. In this paper, we propose a general pricing framework based on a product pricing measure, which combines a distorted probability measure that prices the catastrophe risks underlying the CAT bond with a risk-neutral probability measure that prices interest rate risk. We also demonstrate the use of the peaks over threshold (POT) method to uncover the tail risk. Finally, we conduct case studies using Mexico and California earthquake data to demonstrate the applicability of our pricing framework.

We study the propagation of uncertainty from a class of priors introduced by Arias-Nicolás et al. [(2016) Bayesian Analysis, 11(4), 1107–1136] to the premiums (both the collective and the Bayesian), for a wide family of premium principles (specifically, those that preserve the likelihood ratio order). The class under study reflects the prior uncertainty using distortion functions and fulfills some desirable requirements: elicitation is easy, the prior uncertainty can be measured by different metrics, and the range of quantities of interest is easily obtained from the extremal members of the class. We illustrate the methodology with several examples based on different claim counts models.