Consider the problem of determining the Bayesian credibility mean $E(X_{n+1}|X_1,\cdots, X_n),$ whenever the random claims $X_1,\cdots, X_n,$ given parameter vector $\boldsymbol{\Psi},$ are sampled from the K-component mixture family of distributions, whose members are the union of different families of distributions. This article begins by deriving a recursive formula for such a Bayesian credibility mean. Moreover, under the assumption that using additional information $Z_{i,1},\cdots,Z_{i,m},$ one may probabilistically determine a random claim $X_i$ belongs to a given population (or a distribution), the above recursive formula simplifies to an exact Bayesian credibility mean whenever all components of the mixture distribution belong to the exponential families of distributions. For a situation where a 2-component mixture family of distributions is an appropriate choice for data modelling, using the logistic regression model, it shows that: how one may employ such additional information to derive the Bayesian credibility model, say Logistic Regression Credibility model, for a finite mixture of distributions. A comparison between the Logistic Regression Credibility (LRC) model and its competitor, the Regression Tree Credibility (RTC) model, has been given. More precisely, it shows that under the squared error loss function, it shows the LRC’s risk function dominates the RTC’s risk function at least in an interval which about $0.5.$ Several examples have been given to illustrate the practical application of our findings.

With the recent outbreak of COVID-19, evaluating the epidemic risk appears to be a pressing issue of global concern and one of the major challenges recently. In the fight against pandemics, the ability to understand, model, and forecast the transmission dynamics of infectious diseases plays a crucial role. This paper provides an overview of foundational compartment models and introduces the Susceptible-Exposed-Infected-Containing-3-Substates-Recovered-Dead model to study the dynamics of COVID-19. A meticulous data calibration procedure is employed to study the evolution trend of an actual pandemic using real-world data from Victoria, Australia. Additionally, the paper discusses innovative applications of epidemic models to the insurance industry, which are currently under investigation. Through the use of the newly developed analytically tractable model, insurance companies are able to determine fair premium levels during an outbreak. Moreover, the paper provides practical guidance for insurance companies by examining the variation in reserve levels over time.

In this paper, we discuss the estimation of conditional quantiles of aggregate claim amounts for non-life insurance embedding the problem in a quantile regression framework using the neural network approach. As the first step, we consider the quantile regression neural networks (QRNN) procedure to compute quantiles for the insurance ratemaking framework. As the second step, we propose a new quantile regression combined actuarial neural network (Quantile-CANN) combining the traditional quantile regression approach with a QRNN. In both cases, we adopt a two-part model scheme where we fit a logistic regression to estimate the probability of positive claims and the QRNN model or the Quantile-CANN for the positive outcomes. Through a case study based on a health insurance dataset, we highlight the overall better performances of the proposed models with respect to the classical quantile regression one. We then use the estimated quantiles to calculate a loaded premium following the quantile premium principle, showing that the proposed models provide a better risk differentiation.

This paper provides a review of cyber risk research accomplished in different disciplines, with a primary goal to aid researchers in the field of insurance and actuarial science in identifying potential research gaps as well as leveraging useful models and techniques that have been considered in the literature. We highlight the recent advancements in cyber risk prediction, modeling, management, and insurance achieved in different domains including computer engineering, actuarial science, and business studies. The surveyed works are classified according to their respective modeling approaches, allowing readers to more easily compare the technical aspects of the surveyed works and spot out research gaps based on the research tools of their liking. We conclude this paper with a summary of possible research directions that are identified from the review.

We investigate jointly modelling age–year-specific rates of various causes of death in a multinational setting. We apply multi-output Gaussian processes (MOGPs), a spatial machine learning method, to smooth and extrapolate multiple cause-of-death mortality rates across several countries and both genders. To maintain flexibility and scalability, we investigate MOGPs with Kronecker-structured kernels and latent factors. In particular, we develop a custom multi-level MOGP that leverages the gridded structure of mortality tables to efficiently capture heterogeneity and dependence across different factor inputs. Results are illustrated with datasets from the Human Cause-of-Death Database (HCD). We discuss a case study involving cancer variations in three European nations and a US-based study that considers eight top-level causes and includes comparison to all-cause analysis. Our models provide insights into the commonality of cause-specific mortality trends and demonstrate the opportunities for respective data fusion.

A closed-form solution for zero-coupon bonds is obtained for a version of the discrete-time arbitrage-free Nelson-Siegel model. An estimation procedure relying on a Kalman filter is provided. The model is shown to produce adequate fit when applied to historical Canadian spot rate data and to improve distributional predictive performance over benchmarks. An adaptation of the mixed fund return model from Augustyniak et al. ((2021). ASTIN Bulletin: The Journal of the IAA, 51(1), 131–159.) is also provided to include the discrete-time arbitrage-free Nelson-Siegel model as one of its building blocks.

Insurers and pension funds face the challenges of historically low-interest rates and high volatility in equity markets, that have been accentuated due to the COVID-19 pandemic. Recent advances in equity portfolio management with a target volatility have been shown to deliver improved on average risk-adjusted return, after transaction costs. This paper studies these targeted volatility portfolios in applications to equity, balanced, and target-date funds with varying constraints on leverage. Conservative leverage constraints are particularly relevant to pension funds and insurance companies, with more aggressive leverage levels appropriate for alternative investments. We show substantial improvements in fund performance for differing leverage levels, and of most interest to insurers and pension funds, we show that the highest Sharpe ratios and smallest drawdowns are in targeted volatility-balanced portfolios with equity and bond allocations. Furthermore, we demonstrate the outperformance of targeted volatility portfolios during major stock market crashes, including the crash from the COVID-19 pandemic.

This paper introduces gemact, a Python package for actuarial modeling based on the collective risk model. The library supports applications to risk costing and risk transfer, loss aggregation, and loss reserving. We add new probability distributions to those available in scipy, including the (a, b, 0) and (a, b, 1) discrete distributions, copulas of the Archimedean family, the Gaussian, the Student t and the Fundamental copulas. We provide an implementation of the AEP algorithm for calculating the cumulative distribution function of the sum of dependent, nonnegative random variables, given their dependency structure specified with a copula. The theoretical framework is introduced at the beginning of each section to give the reader with a sufficient understanding of the underlying actuarial models.

This paper introduces DivFolio, a multiperiod portfolio selection and analytic software application that incorporates automated and user-determined divestment practices accommodating Environmental Social Governance (ESG) and portfolio carbon footprint considerations. This freely available portfolio analytics software tool is written in R with a GUI interface developed as an R Shiny application for ease of user experience. Users can utilize this software to dynamically assess the performance of asset selections from global equity, exchange-traded funds, exchange-traded notes, and depositary receipts markets over multiple time periods. This assessment is based on the impact of ESG investment and fossil-fuel divestment practices on portfolio behavior in terms of risk, return, stability, diversification, and climate mitigation credentials of associated investment decisions. We highlight two applications of DivFolio. The first revolves around using sector scanning to divest from a specialized portfolio featuring constituents of the FTSE 100. The second, rooted in actuarial considerations, focuses on divestment strategies informed by environmental risk assessments for mixed pension portfolios in the US and UK.

Traditional techniques for calculating outstanding claim liabilities such as the chain-ladder are notoriously at risk of being distorted by outliers in past claims data. Unfortunately, the literature in robust methods of reserving is scant, with notable exceptions such as Verdonck & Debruyne (2011, Insurance: Mathematics and Economics, 48, 85–98) and Verdonck & Van Wouwe (2011, Insurance: Mathematics and Economics, 49, 188–193). In this paper, we put forward two alternative robust bivariate chain-ladder techniques to extend the approach of Verdonck & Van Wouwe (2011, Insurance: Mathematics and Economics, 49, 188–193). The first technique is based on Adjusted Outlyingness (Hubert & Van der Veeken, 2008. Journal of Chemometrics, 22, 235–246) and explicitly incorporates skewness into the analysis while providing a unique measure of outlyingness for each observation. The second technique is based on bagdistance (Hubert et al., 2016. Statistics: Methodology, 1–23) which is derived from the bagplot; however; it is able to provide a unique measure of outlyingness and a means to adjust outlying observations based on this measure.

Furthermore, we extend our robust bivariate chain-ladder approach to an N-dimensional framework. The implementation of the methods, especially beyond bivariate, is not trivial. This is illustrated on a trivariate data set from Australian general insurers and results under the different outlier detection and treatment mechanisms are compared.

The calculation of life and health insurance liabilities is based on assumptions about mortality and disability rates, and insurance companies face systematic insurance risks if assumptions about these rates change. In this paper, we study how to manage systematic insurance risks in a multi-state setup by considering securities linked to the transition intensities of the model. We assume there exists a market for trading two securities linked to, for instance, mortality and disability rates, the de-risking option and the de-risking swap, and we describe the optimization problem to find the de-risking strategy that minimizes systematic insurance risks in a multi-state setup. We develop a numerical example based on the disability model, and the results imply that systematic insurance risks significantly decrease when implementing de-risking strategies.

This paper focuses on modeling surrender time for policyholders in the context of life insurance. In this setup, a large lapse rate at the first months of a contract is often observed, with a decrease in this rate after some months. The modeling of the time to cancelation must account for this specific behavior. Another stylized fact is that policies which are not canceled in the study period are considered censored. To account for both censoring and heterogeneous lapse rates, this work assumes a Bayesian survival model with a mixture of regressions. The inference is based on data augmentation allowing for fast computations even for datasets of over millions of clients. Moreover, frequentist point estimation based on Expectation–Maximization algorithm is also presented. An illustrative example emulates a typical behavior for life insurance contracts, and a simulated study investigates the properties of the proposed model. A case study is considered and illustrates the flexibility of our proposed model allowing different specifications of mixture components. In particular, the observed censoring in the insurance context might be up to $50\%$ of the data, which is very unusual for survival models in other fields such as epidemiology. This aspect is exploited in our simulated study.

Guaranteed minimum accumulation benefits (GMABs) are retirement savings vehicles that protect the policyholder against downside market risk. This article proposes a valuation method for these contracts based on physics-inspired neural networks (PINNs), in the presence of multiple financial and biometric risk factors. A PINN integrates principles from physics into its learning process to enhance its efficiency in solving complex problems. In this article, the driving principle is the Feynman–Kac (FK) equation, which is a partial differential equation (PDE) governing the GMAB price in an arbitrage-free market. In our context, the FK PDE depends on multiple variables and is difficult to solve using classical finite difference approximations. In comparison, PINNs constitute an efficient alternative that can evaluate GMABs with various specifications without the need for retraining. To illustrate this, we consider a market with four risk factors. We first derive a closed-form expression for the GMAB that serves as a benchmark for the PINN. Next, we propose a scaled version of the FK equation that we solve using a PINN. Pricing errors are analyzed in a numerical illustration.

The coronavirus pandemic has created a new awareness of epidemics, and insurance companies have been reminded to consider the risk related to infectious diseases. This paper extends the traditional multi-state models to include epidemic effects. The main idea is to specify the transition intensities in a Markov model such that the impact of contagion is explicitly present in the same way as in epidemiological models. Since we can study the Markov model with contagious effects at an individual level, we consider individual risk and reserves relating to insurance products, conforming with the standard multi-state approach in life insurance mathematics. We compare our notions with other but related notions in the literature and perform numerical illustrations.

This paper will outline the functionality available in the CovRegpy package which was written for actuarial practitioners, wealth managers, fund managers, and portfolio analysts in the language of Python 3.11. The objective is to develop a new class of covariance regression factor models for covariance forecasting, along with a library of portfolio allocation tools that integrate with this new covariance forecasting framework. The novelty is in two stages: the type of covariance regression model and factor extractions used to construct the covariates used in the covariance regression, along with a powerful portfolio allocation framework for dynamic multi-period asset investment management.

The major contributions of package CovRegpy can be found on the GitHub repository for this library in the scripts: CovRegpy.py, CovRegpy_DCC.py, CovRegpy_RPP.py, CovRegpy_SSA.py, CovRegpy_SSD.py, and CovRegpy_X11.py. These six scripts contain implementations of software features including multivariate covariance time series models based on the regularized covariance regression (RCR) framework, dynamic conditional correlation (DCC) framework, risk premia parity (RPP) weighting functions, singular spectrum analysis (SSA), singular spectrum decomposition (SSD), and X11 decomposition framework, respectively.

These techniques can be used sequentially or independently with other techniques to extract implicit factors to use them as covariates in the RCR framework to forecast covariance and correlation structures and finally apply portfolio weighting strategies based on the portfolio risk measures based on forecasted covariance assumptions. Explicit financial factors can be used in the covariance regression framework, implicit factors can be used in the traditional explicit market factor setting, and RPP techniques with long/short equity weighting strategies can be used in traditional covariance assumption frameworks.

We examine, herein, two real-world case studies for actuarial practitioners. The first of these is a modification (demonstrating the regularization of covariance regression) of the original example from Hoff & Niu ((2012). Statistica Sinica, 22(2), 729–753) which modeled the covariance and correlative relationship that exists between forced expiratory volume (FEV) and age and FEV and height. We examine this within the context of making probabilistic predictions about mortality rates in patients with chronic obstructive pulmonary disease.

The second case study is a more complete example using this package wherein we present a funded and unfunded UK pension example. The decomposition algorithm isolates high-, mid-, and low-frequency structures from FTSE 100 constituents over 20 years. These are used to forecast the forthcoming quarter’s covariance structure to weight the portfolio based on the RPP strategy. These fully funded pensions are compared against the performance of a fully unfunded pension using the FTSE 100 index performance as a proxy.

Calculation of loss scenarios is a fundamental requirement of simulation-based capital models and these are commonly approximated. Within a life insurance setting, a loss scenario may involve an asset-liability optimization. When cashflows and asset values are dependent on only a small number of risk factor components, low-dimensional approximations may be used as inputs into the optimization and resulting in loss approximation. By considering these loss approximations as perturbations of linear optimization problems, approximation errors in loss scenarios can be bounded to first order and attributed to specific proxies. This attribution creates a mechanism for approximation improvements and for the eventual elimination of approximation errors in capital estimates through targeted exact computation. The results are demonstrated through a stylized worked example and corresponding numerical study. Advances in error analysis of proxy models enhance confidence in capital estimates. Beyond error analysis, the presented methods can be applied to general sensitivity analysis and the calculation of risk.

For some time now, Solvency II requires that insurance companies calculate minimum capital requirements to face the risk of insolvency, either in accordance with the Standard Formula or using a full or partial Internal Model. An Internal Model must be based on a market-consistent valuation of assets and liabilities at a 1-year time span, where a real-world probabilistic structure is used for the first year of projection. In this paper, we describe the major risks of a non-life insurance company, i.e. the non-life underwriting risk and market risk, and their interactions, focusing on the non-life premium risk, equity risk, and interest rate risk. This analysis is made using some well-known stochastic models in the financial-actuarial literature and practical insurance business, i.e. the Collective Risk Model for non-life premium risk, the Geometric Brownian Motion for equity risk, and a real-world version of the G2++ Model for interest rate risk, where parameters are calibrated on current and real market data. Finally, we illustrate a case study on a single-line and a multi-line insurance company in order to see how the risk drivers behave in both a stand-alone and an aggregate framework.

Based on the recent papers, two distributions for the total claims amount (loss cost) are considered: compound Poisson-gamma and Tweedie. Each is used as an underlying distribution in the Bonus-Malus Scale (BMS) model. The BMS model links the premium of an insurance contract to a function of the insurance experience of the related policy. In other words, the idea is to model the increase and the decrease in premiums for insureds who do or do not file claims. We applied our approach to a sample of data from a major insurance company in Canada. Data fit and predictability were analyzed. We showed that the studied models are exciting alternatives to consider from a practical point of view, and that predictive ratemaking models can address some important practical considerations.