Spatial data are a rich source of information for actuarial applications: knowledge of a risk’s location could improve an insurance company’s ratemaking, reserving or risk management processes. Relying on historical geolocated loss data is problematic for areas where it is limited or unavailable. In this paper, we construct spatial embeddings within a complex convolutional neural network representation model using external census data and use them as inputs to a simple predictive model. Compared to spatial interpolation models, our approach leads to smaller predictive bias and reduced variance in most situations. This method also enables us to generate rates in territories with no historical experience.

Telematicsdevices installed in insured vehicles provide actuaries with new risk factors, such as the time of the day, average speeds, and other driving habits. This paper extends the multivariate mixed model describing the joint dynamics of telematics data and claim frequencies proposed by Denuit et al. (2019a) by allowing for signals with various formats, not necessarily integer-valued, and by replacing the estimation procedure with the Expected Conditional Maximization algorithm. A numerical study performed on a database related to Pay-How-You-Drive, or PHYD motor insurance illustrates the relevance of the proposed approach for practice.

We consider the following question: given information on individual policyholder characteristics, how can we ensure that insurance prices do not discriminate with respect to protected characteristics, such as gender? We address the issues of direct and indirect discrimination, the latter resulting from implicit learning of protected characteristics from nonprotected ones. We provide rigorous mathematical definitions for direct and indirect discrimination, and we introduce a simple formula for discrimination-free pricing, that avoids both direct and indirect discrimination. Our formula works in any statistical model. We demonstrate its application on a health insurance example, using a state-of-the-art generalized linear model and a neural network regression model. An important conclusion is that discrimination-free pricing in general requires collection of policyholders’ discriminatory characteristics, posing potential challenges in relation to policyholder’s privacy concerns.

Innon-life insurance, the payment history can be predictive of the timing of a settlement for individual claims. Ignoring the association between the payment process and the settlement process could bias the prediction of outstanding payments. To address this issue, we introduce into the literature of micro-level loss reserving a joint modeling framework that incorporates longitudinal payments of a claim into the intensity process of claim settlement. We discuss statistical inference and focus on the prediction aspects of the model. We demonstrate applications of the proposed model in the reserving practice with a detailed empirical analysis using data from a property insurance provider. The prediction results from an out-of-sample validation show that the joint model framework outperforms existing reserving models that ignore the payment–settlement association.

The present paper introduces a simple aggregated reserving model based on claim count and payment dynamics, which allows for claim closings and re-openings. The modelling starts off from individual Poisson process claim dynamics in discrete time, keeping track of accident year, reporting year and payment delay. This modelling approach is closely related to the one underpinning the so-called double chain-ladder model, and it allows for producing separate reported but not settled and incurred but not reported reserves. Even though the introduction of claim closings and re-openings will produce new types of dependencies, it is possible to use flexible parametrisations in terms of, for example, generalised linear models (GLM) whose parameters can be estimated based on aggregated data using quasi-likelihood theory. Moreover, it is possible to obtain interpretable and explicit moment calculations, as well as having consistency of normalised reserves when the number of contracts tend to infinity. Further, by having access to simple analytic expressions for moments, it is computationally cheap to bootstrap the mean squared error of prediction for reserves. The performance of the model is illustrated using a flexible GLM parametrisation evaluated on non-trivial simulated claims data. This numerical illustration indicates a clear improvement compared with models not taking claim closings and re-openings into account. The results are also seen to be of comparable quality with machine learning models for aggregated data not taking claim openness into account.

In this paper, we present a method for generating a copula by composing two arbitrary n-dimensional copulas via a vector of bivariate functions, where the resulting copula is named as the multivariate composite copula. A necessary and sufficient condition on the vector guaranteeing the composite function to be a copula is given, and a general approach to construct the vector satisfying this necessary and sufficient condition via bivariate copulas is provided. The multivariate composite copula proposes a new framework for the construction of flexible multivariate copula from existing ones, and it also includes some known classes of copulas. It is shown that the multivariate composite copula has a clear probability structure, and it satisfies the characteristic of uniform convergence as well as the reproduction property for its component copulas. Some properties of multivariate composite copulas are discussed. Finally, numerical illustrations and an empirical example on financial data are provided to show the advantages of the multivariate composite copula, especially in capturing the tail dependence.

In this paper, a new three-parameter discrete family of distributions, the $r{\mathcal B}ell$ family, is introduced. The family is based on series expansion of the r-Bell polynomials. The proposed model generalises the classical Poisson and the recently proposed Bell and Bell–Touchard distributions. It exhibits interesting stochastic properties. Its probabilities can be computed by a recursive formula that allows us to calculate the probability function of the amount of aggregate claims in the collective risk model in terms of an integral equation. Univariate and bivariate regression models are presented. The former regression model is used to explain the number of out-of-use claims in an automobile insurance portfolio, by showing a good out-of-sample performance. The latter is used to describe the number of out-of-use and parking claims jointly. This family provides an alternative to other traditionally used distributions to describe count data such as the negative binomial and Poisson-inverse Gaussian models.

Current approaches to fair valuation in insurance often follow a two-step approach, combining quadratic hedging with application of a risk measure on the residual liability, to obtain a cost-of-capital margin. In such approaches, the preferences represented by the regulatory risk measure are not reflected in the hedging process. We address this issue by an alternative two-step hedging procedure, based on generalised regression arguments, which leads to portfolios that are neutral with respect to a risk measure, such as Value-at-Risk or the expectile. First, a portfolio of traded assets aimed at replicating the liability is determined by local quadratic hedging. Second, the residual liability is hedged using an alternative objective function. The risk margin is then defined as the cost of the capital required to hedge the residual liability. In the case quantile regression is used in the second step, yearly solvency constraints are naturally satisfied; furthermore, the portfolio is a risk minimiser among all hedging portfolios that satisfy such constraints. We present a neural network algorithm for the valuation and hedging of insurance liabilities based on a backward iterations scheme. The algorithm is fairly general and easily applicable, as it only requires simulated paths of risk drivers.

Given the rapid reductions in human mortality observed over recent decades and the uncertainty associated with their future evolution, there have been a large number of mortality projection models proposed by actuaries and demographers in recent years. Many of these, however, suffer from being overly complex, thereby producing spurious forecasts, particularly over long horizons and for small, noisy data sets. In this paper, we exploit statistical learning tools, namely group regularisation and cross-validation, to provide a robust framework to construct discrete-time mortality models by automatically selecting the most appropriate functions to best describe and forecast particular data sets. Most importantly, this approach produces bespoke models using a trade-off between complexity (to draw as much insight as possible from limited data sets) and parsimony (to prevent over-fitting to noise), with this trade-off designed to have specific regard to the forecasting horizon of interest. This is illustrated using both empirical data from the Human Mortality Database and simulated data, using code that has been made available within a user-friendly open-source R package StMoMo.

We consider computation of market values of bonus payments in multi-state with-profit life insurance. The bonus scheme consists of additional benefits bought according to a dividend strategy that depends on the past realization of financial risk, the current individual insurance risk, the number of additional benefits currently held, and so-called portfolio-wide means describing the shape of the insurance business. We formulate numerical procedures that efficiently combine simulation of financial risk with classic methods for the outstanding insurance risk. Special attention is given to the case where the number of additional benefits bought only depends on the financial risk. Methods and results are illustrated via a numerical example.

The Lee–Carter model has become a benchmark in stochastic mortality modeling. However, its forecasting performance can be significantly improved upon by modern machine learning techniques. We propose a convolutional neural network (NN) architecture for mortality rate forecasting, empirically compare this model as well as other NN models to the Lee–Carter model and find that lower forecast errors are achievable for many countries in the Human Mortality Database. We provide details on the errors and forecasts of our model to make it more understandable and, thus, more trustworthy. As NN by default only yield point estimates, previous works applying them to mortality modeling have not investigated prediction uncertainty. We address this gap in the literature by implementing a bootstrapping-based technique and demonstrate that it yields highly reliable prediction intervals for our NN model.