Coherent extrapolation of mortality rates at old ages applied to long term care
Abstract: In an insurance context, Long-Term Care (LTC) products cover the risk of permanent loss of autonomy, which is defined by the impossibility or difficulty of performing alone all or part of the activities of daily living (ADL). From an actuarial point of view, knowledge of risk depends on knowledge of the underlying biometric laws, including the mortality of autonomous insureds and the mortality of disabled insureds. Due to the relatively short history of LTC products and the age limit imposed at underwriting, i… Show more
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Presentation of EAJ Issues 14/1-2 - October 7th
The seminar will be opened by Stephane Loisel and closed by Hansjoerg Albrecher. The impact of dependencies between climate risks on the asset and liability side of non-life insurers The aim of this paper is to examine the impact of dependencies between climate transition and physical risks on the default probability and profitability of a non-life insurer focusing on the scenario of a delayed and sudden transition. Toward this end, we suggest a simplified modeling approach for scenario analyses for climate risks affecting assets and liabilities, taking into account potential nonlinear dependence structures. Our results show that dependencies on the liability side and between assets and liabilities in the context of physical-transition scenarios can have a significant impact, particularly on the default risk of a non-life insurer. We additionally analyze the mitigating effects of stop loss reinsurance and risk-adjusted pricing, which—if implementable—seem to be an effective risk management measure against physical climate risks in particular. Including individual customer lifetime value and competing risks in tree-based lapse management strategies A retention strategy based on an enlightened lapse model is a powerful profitability lever for a life insurer. Some machine learning models are excellent at predicting lapse, but from the insurer’s perspective, predicting which policyholder is likely to lapse is not enough to design a retention strategy. In our paper, we define a lapse management framework with an appropriate validation metric based on Customer Lifetime Value and profitability. We include the risk of death in the study through competing risks considerations in parametric and tree-based models and show that further individualization of the existing approaches leads to increased performance. We show that survival tree-based models outperform parametric approaches and that the actuarial literature can significantly benefit from them. Then, we compare, on real data, how this framework leads to increased predicted gains for a life insurer and discuss the benefits of our model in terms of commercial and strategic decision-making. A first look back: model performance under Solvency II We consider an empirical backtesting for the Solvency Capital Required (SCR) under Solvency II. Based on empirical facts that the Basic own Funds (BoF) can be assumed to evolve log-normally and have a much lower volatility than the corresponding equity for our test data, we make a proposal based on Earnings at Risk (EaR) that can be used to reduce the biases from overshooting SCR estimates in a prudential way. Stripping the Swiss discount curve using kernel ridge regression We analyze and implement the kernel ridge regression (KR) method developed in Filipovic et al. ([Stripping the discount curve—a robust machine learning approach. Swiss Finance Institute Research Paper No. 22–24, 2022](https://ssrn.com/abstract=4058150)) to estimate the risk-free discount curve for the Swiss government bond market. We show that the insurance industry standard Smith–Wilson method is a special case of the KR framework. We recapitulate the curve estimation methods of the Swiss Solvency Test (SST) and the Swiss National Bank (SNB). In an extensive empirical study covering the years 2010–2022 we compare the KR curves with the SST and SNB curves. The KR method proves to be robust, flexible, transparent, reproducible and easy to implement, and outperforms the benchmarks in- and out-of-sample. We show the limitations of all methods for extrapolating the yield curve and propose possible solutions for the extrapolation problem. We conclude that the KR method is the preferred method for estimating the discount curve. A new approximation of annuity prices for age–period–cohort models This presentation presents a new general formula for estimating annuity prices within a wide range of stochastic mortality models. The formula is constructed using two building blocks: an approximation technique based on the Wentzel–Kramers–Brillouin method for calculating the sum of correlated lognormal random variables, and an approximate expression for the moment generating function of the lognormal distribution. Notably, this formula is applicable to virtually all age–period–cohort models where period effects are represented by vector autoregressive models. This broad assumption encompasses the majority of existing stochastic mortality models in literature. Through a numerical illustration, we also demonstrate the reliability and precision of our new method in determining annuity prices. Longevity trend in Germany In Germany, a trend for decreasing mortality probabilities has been observed in the last 50 years, yielding an increasing life expectancy. The German Actuarial Association DAV offers a standard method for modeling this longevity trend in calculations concerning life insurance by using the life table DAV 2004R. In this note it is investigated, whether or to which extent the longevity function of the DAV 2004R can be used for calculating the expected total number of deaths in Germany. Detection of interacting variables for generalized linear models via neural networks The quality of generalized linear models (GLMs), frequently used by insurance companies, depends on the choice of interacting variables. The search for interactions is time-consuming, especially for data sets with a large number of variables, depends much on expert judgement of actuaries, and often relies on visual performance indicators. Therefore, we present an approach to automating the process of finding interactions that should be added to GLMs to improve their predictive power. Our approach relies on neural networks and a model-specific interaction detection method, which is computationally faster than the traditionally used methods like Friedman’s H-Statistic or SHAP values. In numerical studies, we provide the results of our approach on artificially generated data as well as open-source data. A neural network approach for the mortality analysis of multiple populations: a case study on data of the Italian population A Neural Network (NN) approach for the modelling of mortality rates in a multi-population framework is compared to three classical mortality models. The NN setup contains two instances of Recurrent NNs, including Long Short-Term Memory (LSTM) and Gated Recurrent Units (GRU) networks. The stochastic approaches comprise the Li and Lee model, the Common Age Effect model of Kleinow, and the model of Plat. All models are applied and compared in a large case study on decades of data of the Italian population as divided in counties. In this case study, a new index of multiple deprivation is introduced and used to classify all Italian counties based on socio-economic indicators, sourced from the local office of national statistics (ISTAT). The aforementioned models are then used to model and predict mortality rates of groups of different socio-economic characteristics, sex, and age. Fairness: plurality, causality, and insurability 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. Adversarial AI in insurance: an overview Artificial Intelligence (AI) is by now quite pervasive in the insurance, data-driven and actuarially relevant practice. AI gives relevant, very much welcome advantages to insurance companies. It also poses challenges. Among those, we focus on Adversarial Attacks, which consist of the creation of modified input data to deceive an AI system and produce false outputs, favorauble to the attacker (for instance a competitor or fraudulent customer). We argue on defence methods, considering that Adversarial Attacks are rarely or never seen before events, called few-shot or zero-shot labels. Coherent extrapolation of mortality rates at old ages applied to long term care In an insurance context, Long-Term Care (LTC) products cover the risk of permanent loss of autonomy, which is defined by the impossibility or difficulty of performing alone all or part of the activities of daily living (ADL). From an actuarial point of view, knowledge of risk depends on knowledge of the underlying biometric laws, including the mortality of autonomous insureds and the mortality of disabled insureds. Due to the relatively short history of LTC products and the age limit imposed at underwriting, insurers lack information at advanced ages. This represents a challenge for actuaries, making it difficult to estimate those biometric laws. In this paper, we propose to complete the missing information at advanced ages on the mortality of autonomous and disabled insured populations using information on the global mortality of the portfolio. In fact, the three previous mortality laws are linked since the portfolio is composed only of autonomous and disabled policyholders. We model the two mortality laws (deaths in autonomy and deaths in LTC) in a Poisson Generalized Linear Model framework, additionally using the P-Splines smoothing method. A constraint is then included to link the mortality laws of the two groups and the global mortality of the portfolio. This new method allows for estimating and extrapolating both mortality laws simultaneously in a consistent manner.
Presentation of EAJ Issues 14/1-2 - October 7th
The seminar will be opened by Stephane Loisel and closed by Hansjoerg Albrecher. The impact of dependencies between climate risks on the asset and liability side of non-life insurers The aim of this paper is to examine the impact of dependencies between climate transition and physical risks on the default probability and profitability of a non-life insurer focusing on the scenario of a delayed and sudden transition. Toward this end, we suggest a simplified modeling approach for scenario analyses for climate risks affecting assets and liabilities, taking into account potential nonlinear dependence structures. Our results show that dependencies on the liability side and between assets and liabilities in the context of physical-transition scenarios can have a significant impact, particularly on the default risk of a non-life insurer. We additionally analyze the mitigating effects of stop loss reinsurance and risk-adjusted pricing, which—if implementable—seem to be an effective risk management measure against physical climate risks in particular. Including individual customer lifetime value and competing risks in tree-based lapse management strategies A retention strategy based on an enlightened lapse model is a powerful profitability lever for a life insurer. Some machine learning models are excellent at predicting lapse, but from the insurer’s perspective, predicting which policyholder is likely to lapse is not enough to design a retention strategy. In our paper, we define a lapse management framework with an appropriate validation metric based on Customer Lifetime Value and profitability. We include the risk of death in the study through competing risks considerations in parametric and tree-based models and show that further individualization of the existing approaches leads to increased performance. We show that survival tree-based models outperform parametric approaches and that the actuarial literature can significantly benefit from them. Then, we compare, on real data, how this framework leads to increased predicted gains for a life insurer and discuss the benefits of our model in terms of commercial and strategic decision-making. A first look back: model performance under Solvency II We consider an empirical backtesting for the Solvency Capital Required (SCR) under Solvency II. Based on empirical facts that the Basic own Funds (BoF) can be assumed to evolve log-normally and have a much lower volatility than the corresponding equity for our test data, we make a proposal based on Earnings at Risk (EaR) that can be used to reduce the biases from overshooting SCR estimates in a prudential way. Stripping the Swiss discount curve using kernel ridge regression We analyze and implement the kernel ridge regression (KR) method developed in Filipovic et al. ([Stripping the discount curve—a robust machine learning approach. Swiss Finance Institute Research Paper No. 22–24, 2022](https://ssrn.com/abstract=4058150)) to estimate the risk-free discount curve for the Swiss government bond market. We show that the insurance industry standard Smith–Wilson method is a special case of the KR framework. We recapitulate the curve estimation methods of the Swiss Solvency Test (SST) and the Swiss National Bank (SNB). In an extensive empirical study covering the years 2010–2022 we compare the KR curves with the SST and SNB curves. The KR method proves to be robust, flexible, transparent, reproducible and easy to implement, and outperforms the benchmarks in- and out-of-sample. We show the limitations of all methods for extrapolating the yield curve and propose possible solutions for the extrapolation problem. We conclude that the KR method is the preferred method for estimating the discount curve. A new approximation of annuity prices for age–period–cohort models This presentation presents a new general formula for estimating annuity prices within a wide range of stochastic mortality models. The formula is constructed using two building blocks: an approximation technique based on the Wentzel–Kramers–Brillouin method for calculating the sum of correlated lognormal random variables, and an approximate expression for the moment generating function of the lognormal distribution. Notably, this formula is applicable to virtually all age–period–cohort models where period effects are represented by vector autoregressive models. This broad assumption encompasses the majority of existing stochastic mortality models in literature. Through a numerical illustration, we also demonstrate the reliability and precision of our new method in determining annuity prices. Longevity trend in Germany In Germany, a trend for decreasing mortality probabilities has been observed in the last 50 years, yielding an increasing life expectancy. The German Actuarial Association DAV offers a standard method for modeling this longevity trend in calculations concerning life insurance by using the life table DAV 2004R. In this note it is investigated, whether or to which extent the longevity function of the DAV 2004R can be used for calculating the expected total number of deaths in Germany. Detection of interacting variables for generalized linear models via neural networks The quality of generalized linear models (GLMs), frequently used by insurance companies, depends on the choice of interacting variables. The search for interactions is time-consuming, especially for data sets with a large number of variables, depends much on expert judgement of actuaries, and often relies on visual performance indicators. Therefore, we present an approach to automating the process of finding interactions that should be added to GLMs to improve their predictive power. Our approach relies on neural networks and a model-specific interaction detection method, which is computationally faster than the traditionally used methods like Friedman’s H-Statistic or SHAP values. In numerical studies, we provide the results of our approach on artificially generated data as well as open-source data. A neural network approach for the mortality analysis of multiple populations: a case study on data of the Italian population A Neural Network (NN) approach for the modelling of mortality rates in a multi-population framework is compared to three classical mortality models. The NN setup contains two instances of Recurrent NNs, including Long Short-Term Memory (LSTM) and Gated Recurrent Units (GRU) networks. The stochastic approaches comprise the Li and Lee model, the Common Age Effect model of Kleinow, and the model of Plat. All models are applied and compared in a large case study on decades of data of the Italian population as divided in counties. In this case study, a new index of multiple deprivation is introduced and used to classify all Italian counties based on socio-economic indicators, sourced from the local office of national statistics (ISTAT). The aforementioned models are then used to model and predict mortality rates of groups of different socio-economic characteristics, sex, and age. Fairness: plurality, causality, and insurability 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. Adversarial AI in insurance: an overview Artificial Intelligence (AI) is by now quite pervasive in the insurance, data-driven and actuarially relevant practice. AI gives relevant, very much welcome advantages to insurance companies. It also poses challenges. Among those, we focus on Adversarial Attacks, which consist of the creation of modified input data to deceive an AI system and produce false outputs, favorauble to the attacker (for instance a competitor or fraudulent customer). We argue on defence methods, considering that Adversarial Attacks are rarely or never seen before events, called few-shot or zero-shot labels. Coherent extrapolation of mortality rates at old ages applied to long term care In an insurance context, Long-Term Care (LTC) products cover the risk of permanent loss of autonomy, which is defined by the impossibility or difficulty of performing alone all or part of the activities of daily living (ADL). From an actuarial point of view, knowledge of risk depends on knowledge of the underlying biometric laws, including the mortality of autonomous insureds and the mortality of disabled insureds. Due to the relatively short history of LTC products and the age limit imposed at underwriting, insurers lack information at advanced ages. This represents a challenge for actuaries, making it difficult to estimate those biometric laws. In this paper, we propose to complete the missing information at advanced ages on the mortality of autonomous and disabled insured populations using information on the global mortality of the portfolio. In fact, the three previous mortality laws are linked since the portfolio is composed only of autonomous and disabled policyholders. We model the two mortality laws (deaths in autonomy and deaths in LTC) in a Poisson Generalized Linear Model framework, additionally using the P-Splines smoothing method. A constraint is then included to link the mortality laws of the two groups and the global mortality of the portfolio. This new method allows for estimating and extrapolating both mortality laws simultaneously in a consistent manner.
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