The paper introduces a tree-based varying coefficient model (VCM) where the varying coefficients are modelled using the cyclic gradient boosting machine (CGBM) from Delong et al. (On cyclic gradient boosting machines, 2023). Modelling the coefficient functions using a CGBM allows for dimension-wise early stopping and feature importance scores. The dimension-wise early stopping not only reduces the risk of dimension-specific overfitting, but also reveals differences in model complexity across dimensions. The use of feature importance scores allows for simple feature selection and easy model interpretation. The model is evaluated on the same simulated and real data examples as those used in Richman and Wüthrich (Scand Actuar J 2023:71–95, 2023), and the results show that it produces results in terms of out of sample loss that are comparable to those of their neural network-based VCM called LocalGLMnet.

In applications of predictive modeling, such as insurance pricing, indirect or proxy discrimination is an issue of major concern. Namely, there exists the possibility that protected policyholder characteristics are implicitly inferred from non-protected ones by predictive models and are thus having an undesirable (and possibly illegal) impact on prices. A technical solution to this problem relies on building a best-estimate model using all policyholder characteristics (including protected ones) and then averaging out the protected characteristics for calculating individual prices. However, such an approach requires full knowledge of policyholders' protected characteristics, which may in itself be problematic. Here, we address this issue by using a multi-task neural network architecture for claim predictions, which can be trained using only partial information on protected characteristics and produces prices that are free from proxy discrimination. We demonstrate the proposed method on both synthetic data and a real-world motor claims dataset, in which proxy discrimination can be observed. In both examples we find that the predictive accuracy of the multi-task network is comparable to a conventional feed-forward neural network, when the protected information is available for at least half of the insurance policies. However, the multi-task network has superior performance in the case when the protected information is known for less than half of the insurance policyholders.

The paper introduces a method for creating a categorical generalized linear model (GLM) based on information extracted from a given black-box predictor. The procedure for creating the guided GLM is as follows: For each covariate, including interactions, a covariate partition is created using partial dependence functions calculated based on the given black-box predictor. In order to enhance the predictive performance, an auto-calibration step is used to determine which parts of each covariate partition should be kept, and which parts should be merged. Given the covariate and interaction partitions, a standard categorical GLM is fitted using a lasso penalty. The performance of the proposed method is illustrated using a number of real insurance data sets where gradient boosting machine (GBM) models are used as black-box reference models. From these examples, it is seen that the predictive performance of the guided GLMs is very close to that of the corresponding reference GBMs. Further, in the examples, the guided GLMs have few parameters, making the resulting models easy to interpret. In the numerical illustrations techniques are used to, e.g., identify important interactions both locally and globally, which is essential when, e.g., constructing a tariff.

This article summarizes the main topics, findings, and avenues for future work from the workshop Fairness with a view towards insurance held August 2023 in Copenhagen, Denmark.

The paper discusses duration effects on the consistency of mean parameter and dispersion parameter estimators in exponential dispersion families (EDFs) that are the standard models used for non-life insurance pricing. Focus is on the standard generalised linear model assumptions where both the mean and variance, conditional on duration, are linear functions in terms of duration. We derive simple convergence results that highlight consequences when the linear conditional moment assumptions are not satisfied. These results illustrate that: (i) the resulting mean estimators always have a relevant asymptotic interpretation in terms of the duration adjusted actuarially fair premium—a premium that only agrees with the standard actuarial premium using a duration equal to one, given that the expected value is linear in the duration; (ii) deviance based estimators of the dispersion parameter in an EDF should be avoided in favour of Pearson estimators; (iii) unless the linear moment assumptions are satisfied, consistency of dispersion and plug-in variance estimators can not be guaranteed and may result in spurious over-dispersion. The results provide explicit conditions on the underlying data generating process that will lead to spurious over-dispersion that can be used for model checking. This is illustrated based on real insurance data, where it is concluded that the linear moment assumptions are violated, which results in non-negligible spurious over-dispersion. 

Discrimination and fairness are major concerns in algorithmic models. Thisis particularly true in insurance, where protected policyholder attributes arenot allowed to be used for insurance pricing. Simply disregarding protectedpolicyholder attributes is not an appropriate solution as this still allows forthe possibility of inferring protected attributes from non-protected covari-ates, leading to the phenomenon of proxy discrimination. Although proxydiscrimination is qualitatively different from the group fairness conceptsdiscussed in the machine learning and actuarial literature, group fairnesscriteria have been proposed to control the impact of protected attributeson the calculation of insurance prices. The purpose of this paper is to discussthe relationship between direct and proxy discrimination in insurance andthe most popular group fairness axioms. We provide a technical definitionof proxy discrimination and derive incompatibility results, showing thatavoiding proxy discrimination does not imply satisfying group fairness andvice versa. This shows that the two concepts are materially different. Fur-thermore, we discuss input data pre-processing and model post-processingmethods that achieve group fairness in the sense of demographic parity.As these methods induce transformations that explicitly depend on poli-cyholders’ protected attributes, it becomes ambiguous whether direct andproxy discrimination is, in fact, avoided.

We study non-life insurance pricing and present a general procedure for constructing a distribution-free locally unbiased predictor of the risk premium based on any initially suggested predictor. The resulting predictor is piecewise constant, corresponding to a partition of the covariate space, and by construction auto-calibrated. Two key issues are the appropriate partitioning of the covariate space and the handling of randomly varying durations, acknowledging possible early termination of contracts. A basic idea in the present paper is to partition the predictions from the initial predictor, which as a by-product defines a partition of the covariate space. Two different approaches to create partitions are discussed in detail using (i) duration-weighted equal-probability binning, and (ii) binning by duration-weighted regression trees. Given a partitioning procedure, the size of the partition to be used is obtained using cross-validation. In this way we obtain an automatic data-driven tariffication procedure, where the number of tariff cells corresponds to the size of the partition. We illustrate the procedure based on both simulated and real insurance data, using both simple GLMs and GBMs as initial predictors. The resulting tariffs are shown to have a rather small number of tariff cells while maintaining or improving the predictive performance compared to the initial predictors.

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.

This paper considers a population being affected by a mortality stress during a limited period of time, for example a pandemic. It has recently been suggested that the mortality stress during a pandemic can be viewed as a temporary shift in apparent age. We instead suggest to use a frailty based model where the baseline mortality rate is being stressed. This approach will in a natural way imply post-pandemic mortality rates at the population level. In particular, analytical results concerning the population mortality rate during and after a pandemic are derived. Under general assumptions it is shown that, compared to a non stressed scenario, the mortality is higher during the pandemic and lower after. These general results are exemplified for the Gompertz-Makeham law where more precise results can be obtained using its proportional frailty representation. The results are illustrated based on COVID-19 data for the Swedish population and we estimate the effect of the pandemic on the expected life time of an individual.

We study the optimal portfolio allocation problem from a Bayesian perspective using value at risk (VaR) and conditional value at risk (CVaR) as risk measures. By applying the posterior predictive distribution for the future portfolio return, we derive relevant quantities needed in the computations of VaR and CVaR, and express the optimal portfolio weights in terms of observed data only. This is in contrast to the conventional method where the optimal solution is based on unobserved quantities which are estimated. We also obtain the expressions for the weights of the global minimum VaR (GMVaR) and global minimum CVaR (GMCVaR) portfolios, and specify conditions for their existence. It is shown that these portfolios may not exist if the level used for the VaR or CVaR computation are too low. By using simulation and real market data, we compare the new Bayesian approach to the conventional plug-in method by studying the accuracy of the GMVaR portfolio and by analysing the estimated efficient frontiers. It is concluded that the Bayesian approach outperforms the conventional one, in particular at predicting the out-of-sample VaR.