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.

Generalized additive models (GAMs) are a leading model class for interpretable machine learning. GAMs were originally defined with smooth shape functions of the predictor variables and trained using smoothing splines. Recently, tree-based GAMs where shape functions are gradient-boosted ensembles of bagged trees were proposed, leaving the door open for the estimation of a broader class of shape functions (e.g. Explainable Boosting Machine (EBM)). In this paper, we introduce a competing three-step GAM learning approach where we combine (i) the knowledge of the way to split the covariates space brought by an additive tree model (ATM), (ii) an ensemble of predictive linear scores derived from generalized linear models (GLMs) using a binning strategy based on the ATM, and (iii) a final GLM to have a prediction model that ensures auto-calibration. Numerical experiments illustrate the competitive performances of our approach on several datasets compared to GAM with splines, EBM, or GLM with binarsity penalization. A case study in trade credit insurance is also provided.

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.

Deep neural networks have become an important tool for use in actuarial tasks, due to the significant gains in accuracy provided by these techniques compared to traditional methods, but also due to the close connection of these models to the generalized linear models (GLMs) currently used in industry. Although constraining GLM parameters relating to insurance risk factors to be smooth or exhibit monotonicity is trivial, methods to incorporate such constraints into deep neural networks have not yet been developed. This is a barrier for the adoption of neural networks in insurance practice since actuaries often impose these constraints for commercial or statistical reasons. In this work, we present a novel method for enforcing constraints within deep neural network models, and we show how these models can be trained. Moreover, we provide example applications using real-world datasets. We call our proposed method ICEnet to emphasize the close link of our proposal to the individual conditional expectation model interpretability technique.

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.