We present a dynamic pay-how-you-drive pricing scheme for motor insurance using telematics signals. More specifically, our approach allows the insurer to apply penalties to a baseline premium on the occurrence of events such as hard acceleration or braking. In addition, we incorporate a bonus-malus system (BMS) adapted for telematics data, providing a credibility component based on past telematics signals to the claim frequency predictions. We purposefully consider a weekly setting for our ratemaking approach to benefit from the signal’s high-frequency rate and to encourage safe driving via dynamic premium corrections. Moreover, we provide a detailed structure that allows our model to benefit from historical records and detailed telematics data collected weekly through an onboard device. We showcase our results numerically in a case study using data from an insurance company.

We propose an individual claims reserving model based on the conditional Aalen–Johansen estimator, as developed in Bladt and Furrer ((2023a) arXiv:2303.02119.). In our approach, we formulate a multi-state problem, where the underlying variable is the individual claim size, rather than time. The states in this model represent development periods, and we estimate the cumulative density function of individual claim sizes using the conditional Aalen–Johansen method as transition probabilities to an absorbing state. Our methodology reinterprets the concept of multi-state models and offers a strategy for modeling the complete curve of individual claim sizes. To illustrate our approach, we apply our model to both simulated and real datasets. Having access to the entire dataset enables us to support the use of our approach by comparing the predicted total final cost with the actual amount, as well as evaluating it in terms of the continuously ranked probability score.

The analysis of insurance and annuity products issued on multiple lives requires the use of statistical models which account for lifetime dependence. This paper presents a Dirichlet process mixture-based approach that allows to model dependent lifetimes within a group, such as married couples, accounting for individual as well as group-specific covariates. The model is analyzed in a fully Bayesian setting and illustrated to jointly model the lifetime of male–female couples in a portfolio of joint and last survivor annuities of a Canadian life insurer. The inferential approach allows to account for right censoring and left truncation, which are common features of data in survival analysis. The model shows improved in-sample and out-of-sample performance compared to traditional approaches assuming independent lifetimes and offers additional insights into the determinants of the dependence between lifetimes and their impact on joint and last survivor annuity prices.

Yield curve extrapolation to unobservable tenors is a key technique for the market-consistent valuation of actuarial liabilities required by Solvency II and forthcoming similar regulations. Since the regulatory method, the Smith–Wilson method, is inconsistent with observable yield curve dynamics, parsimonious parametric models, the Nelson–Siegel model and its extensions, are often used for yield curve extrapolation in risk management. However, it is difficult for the parsimonious parametric models to extrapolate yield curves without excessive volatility because of their limited ability to represent observed yield curves with a limited number of parameters. To extend the representational capabilities, we propose a novel yield curve extrapolation method using machine learning. Using the long short-term memory architecture, we achieve purely data-driven yield curve extrapolation with better generalization performance, stability, and consistency with observed yield curve dynamics than the previous parsimonious parametric models on US and Japanese yield curve data. In addition, our method has model interpretability using the backpropagation algorithm. The findings of this study prove that neural networks, which have recently received considerable attention in mortality forecasting, are useful for yield curve extrapolation, where they have not been used before.

This study introduces an innovative methodology for mortality forecasting, which integrates signature-based methods within the functional data framework of the Hyndman–Ullah (HU) model. This new approach, termed the Hyndman–Ullah with truncated signatures (HUts) model, aims to enhance the accuracy and robustness of mortality predictions. By utilizing signature regression, the HUts model is able to capture complex, nonlinear dependencies in mortality data which enhances forecasting accuracy across various demographic conditions. The model is applied to mortality data from 12 countries, comparing its forecasting performance against variants of the HU models across multiple forecast horizons. Our findings indicate that overall the HUts model not only provides more precise point forecasts but also shows robustness against data irregularities, such as those observed in countries with historical outliers. The integration of signature-based methods enables the HUts model to capture complex patterns in mortality data, making it a powerful tool for actuaries and demographers. Prediction intervals are also constructed with bootstrapping methods.

Various measures have been introduced in the existing literature to evaluate extreme risk exposure under the effect of an observable factor. Due to the nice properties of the higher-moment (HM) coherent risk measure, we propose a conditional version of the HM (CoHM) risk measure by incorporating the information of an observable factor. We conduct an asymptotic analysis of this measure in the presence of extreme risks under the weak contagion at a high confidence level, which is further applied to the special case of the conditional Haezendonck–Goovaerts risk measure (CoHG). Numerical illustrations are also provided to examine the accuracy of the asymptotic formulas and to analyze the sensitivity of the risk contribution of the CoHG. Based on the asymptotic result in the Fréchet case, we propose an estimator for the CoHM via an extrapolation, supported by a simulation study.

This paper introduces a novel theoretical framework that offers a closed-form expression for the tail variance (TV) for the novel family of generalised hyper-elliptical (GHE) distributions. The GHE family combines an elliptical distribution with the generalised inverse Gaussian (GIG) distribution, resulting in a highly adaptable and powerful model. Expanding upon the findings of Ignatieva and Landsman ((2021) Insurance: Mathematics and Economics, 101, 437–465.) regarding the tail conditional expectation (TCE), this study demonstrates the significance of the TV as an additional risk measure that provides valuable insights into the tail risk and effectively captures the variability within the loss distribution’s tail. To validate the theoretical results, we perform an empirical analysis on two specific cases: the Laplace – GIG and the Student-t – GIG mixtures. By incorporating the TV derived for the GHE family, we are able to quantify correlated risks in a multivariate portfolio more efficiently. This contribution is particularly relevant to the insurance and financial industries, as it offers a reliable method for accurately assessing the risks associated with extreme losses. Overall, this paper presents an innovative and rigorous approach that enhances our understanding of risk assessment within the financial and insurance sectors. The derived expressions for the TV in addition to TCE within the GHE family of distributions provide valuable insights and practical tools for effectively managing risk.

In this short note, we show that every convex, order-bounded above functional on a Fréchet lattice is automatically continuous. This improves a result in Ruszczyński and Shapiro ((2006) Mathematics of Operations Research 31(3), 433–452.) and applies to many deviation and variability measures. We also show that an order-continuous, law-invariant functional on an Orlicz space is strongly consistent everywhere, extending a result in Krätschmer et al. ((2017) Finance and Stochastics 18(2), 271–295.).

We study two continuous-time Stackelberg games between a life insurance buyer and seller over a random time horizon. The buyer invests in a risky asset and purchases life insurance, and she maximizes a mean-variance criterion applied to her wealth at death. The seller chooses the insurance premium rate to maximize its expected wealth at the buyer’s random time of death. We consider two life insurance games: one with term life insurance and the other with whole life insurance—the latter with pre-commitment of the constant investment strategy. In the term life insurance game, the buyer chooses her life insurance death benefit and investment strategy continuously from a time-consistent perspective. We find the buyer’s equilibrium control strategy explicitly, along with her value function, for the term life insurance game by solving the extended Hamilton–Jacobi–Bellman equations. By contrast, in the whole life insurance game, the buyer pre-commits to a constant life insurance death benefit and a constant amount to invest in the risky asset. To solve the whole life insurance problem, we first obtain the buyer’s objective function and then we maximize that objective function over constant controls. Under both models, the seller maximizes its expected wealth at the buyer’s time of death, and we use the resulting optimal life insurance premia to find the Stackelberg equilibria of the two life insurance games. We also analyze the effects of the parameters on the Stackelberg equilibria, and we present some numerical examples to illustrate our results.