Pub Date : 2025-07-30DOI: 10.1016/j.insmatheco.2025.103136
Yaojun Zhang , Lanpeng Ji , Georgios Aivaliotis , Charles C. Taylor
This paper proposes three types of Bayesian CART (or BCART) models for aggregate claim amount, namely, frequency-severity models, sequential models and joint models. We propose a general framework for BCART models applicable to data with multivariate responses, which is particularly useful for the joint BCART models with a bivariate response: the number of claims and the aggregate claim amount. To facilitate frequency-severity modeling, we investigate BCART models for the right-skewed and heavy-tailed claim severity data using various distributions. We discover that the Weibull distribution is superior to gamma and lognormal distributions, due to its ability to capture different tail characteristics in tree models. Additionally, we find that sequential BCART models and joint BCART models, which can incorporate more complex dependence between the number of claims and severity, are beneficial and thus preferable to the frequency-severity BCART models in which independence is commonly assumed. The effectiveness of these models' performance is illustrated by carefully designed simulations and real insurance data.
{"title":"Bayesian CART models for aggregate claim modeling","authors":"Yaojun Zhang , Lanpeng Ji , Georgios Aivaliotis , Charles C. Taylor","doi":"10.1016/j.insmatheco.2025.103136","DOIUrl":"10.1016/j.insmatheco.2025.103136","url":null,"abstract":"<div><div>This paper proposes three types of Bayesian CART (or BCART) models for aggregate claim amount, namely, frequency-severity models, sequential models and joint models. We propose a general framework for BCART models applicable to data with multivariate responses, which is particularly useful for the joint BCART models with a bivariate response: the number of claims and the aggregate claim amount. To facilitate frequency-severity modeling, we investigate BCART models for the right-skewed and heavy-tailed claim severity data using various distributions. We discover that the Weibull distribution is superior to gamma and lognormal distributions, due to its ability to capture different tail characteristics in tree models. Additionally, we find that sequential BCART models and joint BCART models, which can incorporate more complex dependence between the number of claims and severity, are beneficial and thus preferable to the frequency-severity BCART models in which independence is commonly assumed. The effectiveness of these models' performance is illustrated by carefully designed simulations and real insurance data.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"125 ","pages":"Article 103136"},"PeriodicalIF":2.2,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144749841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-30DOI: 10.1016/j.insmatheco.2025.103135
Xi Xin , Giles Hooker , Fei Huang
The adoption of artificial intelligence (AI) across industries has led to the widespread use of complex black-box models and interpretation tools for decision making. This paper proposes an adversarial framework to uncover the vulnerability of permutation-based interpretation methods for machine learning tasks, with a particular focus on partial dependence (PD) plots. This adversarial framework modifies the original black box model to manipulate its predictions for instances in the extrapolation domain. As a result, it produces deceptive PD plots that can conceal discriminatory behaviors while preserving most of the original model's predictions. This framework can produce multiple fooled PD plots via a single model. By using real-world datasets including an auto insurance claims dataset and COMPAS (Correctional Offender Management Profiling for Alternative Sanctions) dataset, our results show that it is possible to intentionally hide the discriminatory behavior of a predictor and make the black-box model appear neutral through interpretation tools like PD plots while retaining almost all the predictions of the original black-box model. Managerial insights for regulators and practitioners are provided based on the findings.
{"title":"Pitfalls in machine learning interpretability: Manipulating partial dependence plots to hide discrimination","authors":"Xi Xin , Giles Hooker , Fei Huang","doi":"10.1016/j.insmatheco.2025.103135","DOIUrl":"10.1016/j.insmatheco.2025.103135","url":null,"abstract":"<div><div>The adoption of artificial intelligence (AI) across industries has led to the widespread use of complex black-box models and interpretation tools for decision making. This paper proposes an adversarial framework to uncover the vulnerability of permutation-based interpretation methods for machine learning tasks, with a particular focus on partial dependence (PD) plots. This adversarial framework modifies the original black box model to manipulate its predictions for instances in the extrapolation domain. As a result, it produces deceptive PD plots that can conceal discriminatory behaviors while preserving most of the original model's predictions. This framework can produce multiple fooled PD plots via a single model. By using real-world datasets including an auto insurance claims dataset and COMPAS (Correctional Offender Management Profiling for Alternative Sanctions) dataset, our results show that it is possible to intentionally hide the discriminatory behavior of a predictor and make the black-box model appear neutral through interpretation tools like PD plots while retaining almost all the predictions of the original black-box model. Managerial insights for regulators and practitioners are provided based on the findings.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"125 ","pages":"Article 103135"},"PeriodicalIF":2.2,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144749694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-07DOI: 10.1016/j.insmatheco.2025.103130
Takaaki Koike , Cathy W.S. Chen , Edward M.H. Lin
Capital allocation is a procedure for quantifying the contribution of each source of risk to aggregated risk. The gradient allocation rule, also known as the Euler principle, is a prevalent rule of capital allocation under which the allocated capital captures the diversification benefit of the marginal risk as a component of the overall risk. This paper concentrates on Expected Shortfall (ES) as a regulatory standard and focuses on the gradient allocations of ES, also called ES contributions (ESCs). We present the comprehensive treatment of backtesting the tuple of ESCs in the framework of the traditional and comparative backtests based on the concepts of joint identifiability and multi-objective elicitability. For robust forecast evaluation against the choice of scoring function, we also extend the Murphy diagram, a graphical tool to check whether one forecast dominates another under a class of scoring functions, to the case of ESCs. Finally, leveraging the recent concept of multi-objective elicitability, we propose a novel semiparametric model for forecasting dynamic ESCs based on a compositional regression model. In an empirical analysis of stock returns we evaluate and compare a variety of models for forecasting dynamic ESCs and demonstrate the solid performance of the proposed model.
{"title":"Forecasting and backtesting gradient allocations of expected shortfall","authors":"Takaaki Koike , Cathy W.S. Chen , Edward M.H. Lin","doi":"10.1016/j.insmatheco.2025.103130","DOIUrl":"10.1016/j.insmatheco.2025.103130","url":null,"abstract":"<div><div>Capital allocation is a procedure for quantifying the contribution of each source of risk to aggregated risk. The gradient allocation rule, also known as the Euler principle, is a prevalent rule of capital allocation under which the allocated capital captures the diversification benefit of the marginal risk as a component of the overall risk. This paper concentrates on Expected Shortfall (ES) as a regulatory standard and focuses on the gradient allocations of ES, also called ES contributions (ESCs). We present the comprehensive treatment of backtesting the tuple of ESCs in the framework of the traditional and comparative backtests based on the concepts of joint identifiability and multi-objective elicitability. For robust forecast evaluation against the choice of scoring function, we also extend the Murphy diagram, a graphical tool to check whether one forecast dominates another under a class of scoring functions, to the case of ESCs. Finally, leveraging the recent concept of multi-objective elicitability, we propose a novel semiparametric model for forecasting dynamic ESCs based on a compositional regression model. In an empirical analysis of stock returns we evaluate and compare a variety of models for forecasting dynamic ESCs and demonstrate the solid performance of the proposed model.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"124 ","pages":"Article 103130"},"PeriodicalIF":1.9,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144588381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-05DOI: 10.1016/j.insmatheco.2025.103131
Yuyu Chen , Paul Embrechts , Ruodu Wang
We study the optimal decisions and equilibria of agents who aim to minimize their risks by allocating their positions over extremely heavy-tailed (i.e., infinite-mean) and possibly dependent losses. The loss distributions of our focus are super-Pareto distributions, which include the class of extremely heavy-tailed Pareto distributions. Using a recent result on stochastic dominance, we show that for a portfolio of super-Pareto losses, non-diversification is preferred by decision makers equipped with well-defined and monotone risk measures. The phenomenon that diversification is not beneficial in the presence of super-Pareto losses is further illustrated by an equilibrium analysis in a risk exchange market. First, agents with super-Pareto losses will not share risks in a market equilibrium. Second, transferring losses from agents bearing super-Pareto losses to external parties without any losses may arrive at an equilibrium which benefits every party involved.
{"title":"Risk exchange under infinite-mean Pareto models","authors":"Yuyu Chen , Paul Embrechts , Ruodu Wang","doi":"10.1016/j.insmatheco.2025.103131","DOIUrl":"10.1016/j.insmatheco.2025.103131","url":null,"abstract":"<div><div>We study the optimal decisions and equilibria of agents who aim to minimize their risks by allocating their positions over extremely heavy-tailed (i.e., infinite-mean) and possibly dependent losses. The loss distributions of our focus are super-Pareto distributions, which include the class of extremely heavy-tailed Pareto distributions. Using a recent result on stochastic dominance, we show that for a portfolio of super-Pareto losses, non-diversification is preferred by decision makers equipped with well-defined and monotone risk measures. The phenomenon that diversification is not beneficial in the presence of super-Pareto losses is further illustrated by an equilibrium analysis in a risk exchange market. First, agents with super-Pareto losses will not share risks in a market equilibrium. Second, transferring losses from agents bearing super-Pareto losses to external parties without any losses may arrive at an equilibrium which benefits every party involved.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"124 ","pages":"Article 103131"},"PeriodicalIF":1.9,"publicationDate":"2025-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-05DOI: 10.1016/j.insmatheco.2025.103132
Hong-Jie Li , Xing-Gang Luo , Zhong-Liang Zhang , Shen-Wei Huang , Wei Jiang
Usage-based insurance (UBI) charges drivers differently through telematics-based driving risk assessments. While current UBI pricing models differentiate driving risks, their overly discriminative prices may expel risky drivers, whose driving behaviors could have been modified, thereby incurring insurers' losses in profits. We propose a new UBI pricing model to address this problem by incorporating customer retention into the conventional UBI framework. Specifically, our model offers targeted discounts based on drivers' price sensitivity to retain those who may terminate the insurance contract, as well as provides concrete suggestions to help them modify unsafe driving behaviors. Using empirical data from a major Chinese auto insurer, we confirm that our model yields higher profits for insurers over the UBI pricing model that does not account for customer retention, and exemplify how suggestions for drivers can be drawn from driving profiles.
{"title":"A usage-based insurance (UBI) pricing model considering customer retention","authors":"Hong-Jie Li , Xing-Gang Luo , Zhong-Liang Zhang , Shen-Wei Huang , Wei Jiang","doi":"10.1016/j.insmatheco.2025.103132","DOIUrl":"10.1016/j.insmatheco.2025.103132","url":null,"abstract":"<div><div>Usage-based insurance (UBI) charges drivers differently through telematics-based driving risk assessments. While current UBI pricing models differentiate driving risks, their overly discriminative prices may expel risky drivers, whose driving behaviors could have been modified, thereby incurring insurers' losses in profits. We propose a new UBI pricing model to address this problem by incorporating customer retention into the conventional UBI framework. Specifically, our model offers targeted discounts based on drivers' price sensitivity to retain those who may terminate the insurance contract, as well as provides concrete suggestions to help them modify unsafe driving behaviors. Using empirical data from a major Chinese auto insurer, we confirm that our model yields higher profits for insurers over the UBI pricing model that does not account for customer retention, and exemplify how suggestions for drivers can be drawn from driving profiles.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"124 ","pages":"Article 103132"},"PeriodicalIF":1.9,"publicationDate":"2025-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-25DOI: 10.1016/j.insmatheco.2025.103128
Melanie Averhoff, Julie Thøgersen
This paper provides a study of how experience rating on both claim frequency and severity impacts the solvency of an insurance business in the continuous-time Cramér Lundberg model. This is done by treating the claim parameters as random outcomes and continuously updating the premiums using Bayesian estimators. In the analysis, the claim sizes conditional on the severity parameter are assumed to be light-tailed. The main contributions are large deviation results where the asymptotic ruin probability is found for a model updating the premium based upon both frequency and severity. This asymptotic ruin probability is lower and decays faster compared to the one of a model which updates the premium solely based on claim frequency. Our findings are illustrated with examples, where the conditional claim size and the severity parameter are parametrised.
{"title":"Experience rating in the Cramér-Lundberg model","authors":"Melanie Averhoff, Julie Thøgersen","doi":"10.1016/j.insmatheco.2025.103128","DOIUrl":"10.1016/j.insmatheco.2025.103128","url":null,"abstract":"<div><div>This paper provides a study of how experience rating on both claim frequency and severity impacts the solvency of an insurance business in the continuous-time Cramér Lundberg model. This is done by treating the claim parameters as random outcomes and continuously updating the premiums using Bayesian estimators. In the analysis, the claim sizes conditional on the severity parameter are assumed to be light-tailed. The main contributions are large deviation results where the asymptotic ruin probability is found for a model updating the premium based upon both frequency and severity. This asymptotic ruin probability is lower and decays faster compared to the one of a model which updates the premium solely based on claim frequency. Our findings are illustrated with examples, where the conditional claim size and the severity parameter are parametrised.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"124 ","pages":"Article 103128"},"PeriodicalIF":1.9,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144518560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-25DOI: 10.1016/j.insmatheco.2025.103127
Ruotian Ti , Ximin Rong , Cheng Tao , Hui Zhao
With the progressive aging of populations, the significance of long-term care (LTC) services in aging societies is growing. In this paper, we integrate LTC services with pensions, studying a stochastic model for a care-dependent target benefit pension (TBP) plan. The plan members' target benefit rates are set according to the care cost for three different health states, i.e., healthy, mildly disabled and severely disabled states. The pension liability evaluation is defined as the potential compensation to all active and retired members, under the assumption of the pension fund default. The objective of minimizing the benefit gap and liability gap is achieved by addressing a stochastic optimal control problem. Then, we derive analytic solutions for optimal investment and benefit payment strategies by employing the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Numerical results show that under a fixed aggregate contribution of the care-dependent TBP, a slight decrease in the target benefit for healthy retirees leads to a significant increase for retirees in both mildly and severely disabled states, thereby improving equity for disabled retirees. Furthermore, we compare the care-dependent TBP with a traditional TBP and a care-dependent tontine in terms of risk sharing, financial stability, and intergenerational equity, highlighting the advantages of the care-dependent TBP.
{"title":"Care-dependent target benefit pension plan with minimum liability gap","authors":"Ruotian Ti , Ximin Rong , Cheng Tao , Hui Zhao","doi":"10.1016/j.insmatheco.2025.103127","DOIUrl":"10.1016/j.insmatheco.2025.103127","url":null,"abstract":"<div><div>With the progressive aging of populations, the significance of long-term care (LTC) services in aging societies is growing. In this paper, we integrate LTC services with pensions, studying a stochastic model for a care-dependent target benefit pension (TBP) plan. The plan members' target benefit rates are set according to the care cost for three different health states, i.e., healthy, mildly disabled and severely disabled states. The pension liability evaluation is defined as the potential compensation to all active and retired members, under the assumption of the pension fund default. The objective of minimizing the benefit gap and liability gap is achieved by addressing a stochastic optimal control problem. Then, we derive analytic solutions for optimal investment and benefit payment strategies by employing the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Numerical results show that under a fixed aggregate contribution of the care-dependent TBP, a slight decrease in the target benefit for healthy retirees leads to a significant increase for retirees in both mildly and severely disabled states, thereby improving equity for disabled retirees. Furthermore, we compare the care-dependent TBP with a traditional TBP and a care-dependent tontine in terms of risk sharing, financial stability, and intergenerational equity, highlighting the advantages of the care-dependent TBP.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"124 ","pages":"Article 103127"},"PeriodicalIF":1.9,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-23DOI: 10.1016/j.insmatheco.2025.103126
Felix Zhu , Yumo Dong , Fei Huang
With the advent of Big Data, machine learning, and artificial intelligence (AI) technologies, actuaries can now develop advanced models in a data-rich environment to achieve better forecasting performance and provide added value in many applications. Traditionally, economic forecasting for actuarial applications is developed using econometric models based on small datasets including only the target variables (usually around 4-6) and their lagged variables. This paper explores the value of economic forecasting using deep learning with a big dataset, Federal Reserve Bank of St Louis (FRED) database, consisting of 121 economic variables and their lagged variables covering periods before, during, and after the global financial crisis (GFC), and during COVID (2019-2021). Four target variables considered in this paper include inflation rate, interest rate, wage rate, and unemployment rate, which are common variables for social security funds forecasting. The proposed model “PCA-Net” combines dimension reduction via principal component analysis (PCA) and Neural Networks (including convolutional neural network (CNN), Long Short-Term Memory (LSTM), and fully-connected layers). PCA-Net generally outperforms the benchmark models based on vector autoregression (VAR) and Wilkie-like models, although the magnitude of its advantage varies across economic variables and forecast horizons. Using conformal prediction, this paper provides prediction intervals to quantify the prediction uncertainty. The model performance is demonstrated using a social security fund forecasting application.
{"title":"Data-rich economic forecasting for actuarial applications","authors":"Felix Zhu , Yumo Dong , Fei Huang","doi":"10.1016/j.insmatheco.2025.103126","DOIUrl":"10.1016/j.insmatheco.2025.103126","url":null,"abstract":"<div><div>With the advent of Big Data, machine learning, and artificial intelligence (AI) technologies, actuaries can now develop advanced models in a data-rich environment to achieve better forecasting performance and provide added value in many applications. Traditionally, economic forecasting for actuarial applications is developed using econometric models based on small datasets including only the target variables (usually around 4-6) and their lagged variables. This paper explores the value of economic forecasting using deep learning with a big dataset, Federal Reserve Bank of St Louis (FRED) database, consisting of 121 economic variables and their lagged variables covering periods before, during, and after the global financial crisis (GFC), and during COVID (2019-2021). Four target variables considered in this paper include inflation rate, interest rate, wage rate, and unemployment rate, which are common variables for social security funds forecasting. The proposed model “PCA-Net” combines dimension reduction via principal component analysis (PCA) and Neural Networks (including convolutional neural network (CNN), Long Short-Term Memory (LSTM), and fully-connected layers). PCA-Net generally outperforms the benchmark models based on vector autoregression (VAR) and Wilkie-like models, although the magnitude of its advantage varies across economic variables and forecast horizons. Using conformal prediction, this paper provides prediction intervals to quantify the prediction uncertainty. The model performance is demonstrated using a social security fund forecasting application.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"124 ","pages":"Article 103126"},"PeriodicalIF":1.9,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144518474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-23DOI: 10.1016/j.insmatheco.2025.103129
Marcus C. Christiansen , Boualem Djehiche
In multistate life insurance, prospective reserves are commonly calculated as expectations conditioned only on the current state of the individual policy, rather than on the full observed past history, which is well motivated in Markov models, but is often done even when the empirical data does not show the Markov property. The resulting as-if-Markov prospective reserves then represent partially portfolio averaged values rather than individual values. This averaging effect is particularly relevant when individual policies are lapsed or modified, where it is common practice to credit the individual reserve to the policyholder, making the cashflow reserve-dependent. Such reserve dependence is normally avoided by applying the Cantelli theorem, but this does not work for as-if-Markov reserves without the Markov property. We show that, under mild technical assumptions, the as-if-Markov prospective reserves are still well defined despite the circularity in their definition, and we explain how they can be computed numerically by fixed-point iteration.
{"title":"As-if-Markov reserves for reserve-dependent payments","authors":"Marcus C. Christiansen , Boualem Djehiche","doi":"10.1016/j.insmatheco.2025.103129","DOIUrl":"10.1016/j.insmatheco.2025.103129","url":null,"abstract":"<div><div>In multistate life insurance, prospective reserves are commonly calculated as expectations conditioned only on the current state of the individual policy, rather than on the full observed past history, which is well motivated in Markov models, but is often done even when the empirical data does not show the Markov property. The resulting as-if-Markov prospective reserves then represent partially portfolio averaged values rather than individual values. This averaging effect is particularly relevant when individual policies are lapsed or modified, where it is common practice to credit the individual reserve to the policyholder, making the cashflow reserve-dependent. Such reserve dependence is normally avoided by applying the Cantelli theorem, but this does not work for as-if-Markov reserves without the Markov property. We show that, under mild technical assumptions, the as-if-Markov prospective reserves are still well defined despite the circularity in their definition, and we explain how they can be computed numerically by fixed-point iteration.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"124 ","pages":"Article 103129"},"PeriodicalIF":1.9,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144470924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-18DOI: 10.1016/j.insmatheco.2025.103125
Ning Wang , Yumo Zhang
This paper investigates an optimal asset-liability management problem involving two strategically interactive managers with ambiguity aversion under a multivariate stochastic covariance model characterized by hybrid stochastic volatility and stochastic interest rates. Two ambiguity-averse managers participate in a financial market comprising a money market account, a market index, a stock, and zero-coupon bonds to enhance profits, where interest rates are determined via an affine model, which includes both the Cox–Ingersoll–Ross model and the Vasicek model as specific instances, while the market index and stock price are driven by a general class of non-Markovian multivariate stochastic covariance models. Moreover, the two competitive managers, subject to idiosyncratic liability commitments and influenced by the random nature of cash inflow or outflow in their investment decision making, have varying risk preferences described by the hyperbolic absolute risk aversion (HARA) utility function, with the power utility function as a special case. Each manager aims to develop a robust investment strategy to outperform their competitors by maximizing the expected terminal utility of the relative surplus in worst-case scenarios. A backward stochastic differential equation method coupled with the martingale optimality principle is used to solve this robust non-Markovian stochastic differential game, resulting in closed-form expressions for robust Nash equilibrium investment strategies, the density generator processes under worst-case probability measures, and the corresponding value functions. Finally, numerical examples are provided to illustrate their financial implications.
{"title":"Robust asset-liability management games in a stochastic market with stochastic cash flows under HARA utility","authors":"Ning Wang , Yumo Zhang","doi":"10.1016/j.insmatheco.2025.103125","DOIUrl":"10.1016/j.insmatheco.2025.103125","url":null,"abstract":"<div><div>This paper investigates an optimal asset-liability management problem involving two strategically interactive managers with ambiguity aversion under a multivariate stochastic covariance model characterized by hybrid stochastic volatility and stochastic interest rates. Two ambiguity-averse managers participate in a financial market comprising a money market account, a market index, a stock, and zero-coupon bonds to enhance profits, where interest rates are determined via an affine model, which includes both the Cox–Ingersoll–Ross model and the Vasicek model as specific instances, while the market index and stock price are driven by a general class of non-Markovian multivariate stochastic covariance models. Moreover, the two competitive managers, subject to idiosyncratic liability commitments and influenced by the random nature of cash inflow or outflow in their investment decision making, have varying risk preferences described by the hyperbolic absolute risk aversion (HARA) utility function, with the power utility function as a special case. Each manager aims to develop a robust investment strategy to outperform their competitors by maximizing the expected terminal utility of the relative surplus in worst-case scenarios. A backward stochastic differential equation method coupled with the martingale optimality principle is used to solve this robust non-Markovian stochastic differential game, resulting in closed-form expressions for robust Nash equilibrium investment strategies, the density generator processes under worst-case probability measures, and the corresponding value functions. Finally, numerical examples are provided to illustrate their financial implications.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"124 ","pages":"Article 103125"},"PeriodicalIF":1.9,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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