We compare results for 12 multi-population mortality models fitted to 10 distinct socio-economic groups in England, subdivided using the Index of Multiple Deprivation. Using the Bayes Information Criterion to compare models, we find that a special case of the common age effect (CAE) model fits best in a variety of situations, achieving the best balance between goodness of fit and parsimony. We provide a detailed discussion of key models to highlight which features are important. Group-specific period effects are found to be more important than group-specific age effects, and non-parametric age effects deliver significantly better results than parametric (e.g. linear) age effects. We also find that the addition of cohort effects is beneficial in some cases but not all. The preferred CAE model has the additional benefit of being coherent in the sense of Hyndman et al. ((2013) Demography50(1), 261–283); some of the other models considered are not.

Initial insurance losses are often reported with a textual description of the claim. The claims manager must determine the adequate case reserve for each known claim. In this paper, we present a framework for predicting the amount of loss given a textual description of the claim using a large number of words found in the descriptions. Prior work has focused on classifying insurance claims based on keywords selected by a human expert, whereas in this paper the focus is on loss amount prediction with automatic word selection. In order to transform words into numeric vectors, we use word cosine similarities and word embedding matrices. When we consider all unique words found in the training dataset and impose a generalised additive model to the resulting explanatory variables, the resulting design matrix is high dimensional. For this reason, we use a group lasso penalty to reduce the number of coefficients in the model. The scalable, analytical framework proposed provides for a parsimonious and interpretable model. Finally, we discuss the implications of the analysis, including how the framework may be used by an insurance company and how the interpretation of the covariates can lead to significant policy change. The code can be found in the TAGAM R package (github.com/scottmanski/TAGAM).

Introducing common shocks is a popular dependence modelling approach, with some recent applications in loss reserving. The main advantage of this approach is the ability to capture structural dependence coming from known relationships. In addition, it helps with the parsimonious construction of correlation matrices of large dimensions. However, complications arise in the presence of “unbalanced data”, that is, when (expected) magnitude of observations over a single triangle, or between triangles, can vary substantially. Specifically, if a single common shock is applied to all of these cells, it can contribute insignificantly to the larger values and/or swamp the smaller ones, unless careful adjustments are made. This problem is further complicated in applications involving negative claim amounts. In this paper, we address this problem in the loss reserving context using a common shock Tweedie approach for unbalanced data. We show that the solution not only provides a much better balance of the common shock proportions relative to the unbalanced data, but it is also parsimonious. Finally, the common shock Tweedie model also provides distributional tractability.

The Lee–Carter (LC) model is a basic approach to forecasting mortality rates of a single population. Although extensions of the LC model to forecasting rates for multiple populations have recently been proposed, the structure of these extended models is hard to justify and the models are often difficult to calibrate, relying on customised optimisation schemes. Based on the paradigm of representation learning, we extend the LCmodel to multiple populations using neural networks, which automatically select an optimal model structure. We fit this model to mortality rates since 1950 for all countries in the Human Mortality Database and observe that the out-of-sample forecasting performance of the model is highly competitive.

The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Cox process provides flexibility by letting the intensity not only depending on time but also allowing it to be a stochastic process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of the Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes. We provide simulation algorithms for these processes, which would be useful to statistical analysis, further business applications and research. As an application of the compound processes, numerical comparisons of value-at-risk and tail conditional expectation are made.

Spatial analysis ranges from simple univariate descriptive statistics to complex multivariate analyses and is typically used to investigate spatial patterns or to identify spatially linked consumer behaviours in insurance. This paper investigates if the incorporation of publicly available spatially linked demographic census data at population level is useful in modelling customers’ lapse behaviour (i.e. stopping payment of premiums) in life insurance policies, based on data provided by an insurance company in Ireland. From the insurance company’s perspective, identifying and assessing such lapsing risks in advance permit engagement to prevent such incidents, saving money by re-evaluating customer acquisition channels and improving capital reserve calculation and preparation. Incorporating spatial analysis in lapse modelling is expected to improve lapse prediction. Therefore, a hybrid approach to lapse prediction is proposed – spatial clustering using census data is used to reveal the underlying spatial structure of customers of the Irish life insurer, in conjunction with traditional statistical models for lapse prediction based on the company data. The primary contribution of this work is to consider the spatial characteristics of customers for life insurance lapse behaviour, via the integration of reliable government provided census demographics, which has not been considered previously in actuarial literature. Company decision-makers can use the insights gleaned from this analysis to identify customer subsets to target with personalized promotions to reduce lapse rates, and to reduce overall company risk.

The aim of the paper is to derive a simple, implementable machine learning method for general insurance losses. An algorithm for learning a general insurance loss triangle is developed and justified. An argument is made for applying support vector regression (SVR) to this learning task (in order to facilitate transparency of the learning method as compared to more “black-box” methods such as deep neural networks), and SVR methodology derived is specifically applied to this learning task. A further argument for preserving the statistical features of the loss data in the SVR machine is made. A bespoke kernel function that preserves the statistical features of the loss data is derived from first principles and called the exponential dispersion family (EDF) kernel. Features of the EDF kernel are explored, and the kernel is applied to an insurance loss estimation exercise for homogeneous risk of three different insurers. Results of the cumulative losses and ultimate losses predicted by the EDF kernel are compared to losses predicted by the radial basis function kernel and the chain-ladder method. A backtest of the developed method is performed. A discussion of the results and their implications follows.

This paper introduces a new julia package, LRMoE, a statistical software tailor-made for actuarial applications, which allows actuarial researchers and practitioners to model and analyse insurance loss frequencies and severities using the Logit-weighted Reduced Mixture-of-Experts (LRMoE) model. LRMoE offers several new distinctive features which are motivated by various actuarial applications and mostly cannot be achieved using existing packages for mixture models. Key features include a wider coverage on frequency and severity distributions and their zero inflation, the flexibility to vary classes of distributions across components, parameter estimation under data censoring and truncation and a collection of insurance ratemaking and reserving functions. The package also provides several model evaluation and visualisation functions to help users easily analyse the performance of the fitted model and interpret the model in insurance contexts.

The mvClaim package in R provides flexible modelling frameworks for multivariate insurance claim severity modelling. The current version of the package implements a parsimonious mixture of experts (MoE) model family with bivariate gamma distributions, as introduced in Hu et al., and a finite mixture of copula regressions within the MoE framework as in Hu & O’Hagan. This paper presents the modelling approach theory briefly and the usage of the models in the package in detail. This package is hosted on GitHub at https://github.com/senhu/.

The Scenario Weights for Importance Measurement (SWIM) package implements a flexible sensitivity analysis framework, based primarily on results and tools developed by Pesenti et al. (2019). SWIM provides a stressed version of a stochastic model, subject to model components (random variables) fulfilling given probabilistic constraints (stresses). Possible stresses can be applied on moments, probabilities of given events, and risk measures such as Value-At-Risk and Expected Shortfall. SWIM operates upon a single set of simulated scenarios from a stochastic model, returning scenario weights, which encode the required stress and allow monitoring the impact of the stress on all model components. The scenario weights are calculated to minimise the relative entropy with respect to the baseline model, subject to the stress applied. As well as calculating scenario weights, the package provides tools for the analysis of stressed models, including plotting facilities and evaluation of sensitivity measures. SWIM does not require additional evaluations of the simulation model or explicit knowledge of its underlying statistical and functional relations; hence, it is suitable for the analysis of black box models. The capabilities of SWIM are demonstrated through a case study of a credit portfolio model.