Insurance Mathematics & Economics最新文献_第5页

Beyond annual data: Mortality forecasting with mixed frequency data 超越年度数据：混合频率数据的死亡率预测

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-11-10 DOI: 10.1016/j.insmatheco.2025.103172

Runze Li , Rui Zhou , David Pitt

Existing mortality models typically rely on annual data, with forecasts for a given year based on information available up to the end of the previous year. However, technological advances have enabled the collection and production of weekly and monthly death data, offering new opportunities to improve forecasting. Using only annual data overlooks the full range of available information. In this paper, we propose mixed frequency sampling (MIDAS) models to integrate monthly death counts (high frequency) with annual mortality rates (low frequency), enabling improved short-term prediction of annual mortality. Extending economic applications of MIDAS, which typically predict single variables such as GDP growth, our MIDAS framework accounts for age dependence unique to age-specific mortality modeling. We also evaluate different weighting functions, a core element in MIDAS that determines the relative importance of high-frequency data at different lags, and identify suitable weighting functions for mortality forecasting. Using U.S. mortality data, we demonstrate that our approach significantly improves prediction accuracy compared to models relying solely on annual data for short-term forecasting. These findings highlight the potential of MIDAS models as a useful tool for accurate and timely mortality forecasts.

现有的死亡率模型通常依赖年度数据，对某一年的预测是根据截至前一年年底的信息作出的。然而，技术进步使每周和每月死亡数据的收集和产生成为可能，为改进预测提供了新的机会。只使用年度数据忽略了所有可用信息。在本文中，我们提出混合频率采样（MIDAS）模型来整合月死亡计数（高频）和年死亡率（低频），从而改进年死亡率的短期预测。MIDAS通常预测GDP增长等单一变量，我们的MIDAS框架扩展了MIDAS的经济应用，考虑了特定年龄死亡率模型特有的年龄依赖性。我们还评估了不同的权重函数，这是MIDAS的一个核心元素，决定了不同滞后时间高频数据的相对重要性，并确定了适用于死亡率预测的权重函数。使用美国死亡率数据，我们证明，与仅依赖年度数据进行短期预测的模型相比，我们的方法显着提高了预测精度。这些发现突出了MIDAS模型作为准确和及时预测死亡率的有用工具的潜力。

{"title":"Beyond annual data: Mortality forecasting with mixed frequency data","authors":"Runze Li , Rui Zhou , David Pitt","doi":"10.1016/j.insmatheco.2025.103172","DOIUrl":"10.1016/j.insmatheco.2025.103172","url":null,"abstract":"<div><div>Existing mortality models typically rely on annual data, with forecasts for a given year based on information available up to the end of the previous year. However, technological advances have enabled the collection and production of weekly and monthly death data, offering new opportunities to improve forecasting. Using only annual data overlooks the full range of available information. In this paper, we propose mixed frequency sampling (MIDAS) models to integrate monthly death counts (high frequency) with annual mortality rates (low frequency), enabling improved short-term prediction of annual mortality. Extending economic applications of MIDAS, which typically predict single variables such as GDP growth, our MIDAS framework accounts for age dependence unique to age-specific mortality modeling. We also evaluate different weighting functions, a core element in MIDAS that determines the relative importance of high-frequency data at different lags, and identify suitable weighting functions for mortality forecasting. Using U.S. mortality data, we demonstrate that our approach significantly improves prediction accuracy compared to models relying solely on annual data for short-term forecasting. These findings highlight the potential of MIDAS models as a useful tool for accurate and timely mortality forecasts.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"126 ","pages":"Article 103172"},"PeriodicalIF":2.2,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618438","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}

引用次数: 0

Risk measures on Musielak-Orlicz spaces: A state-dependent perspective for insurance Musielak-Orlicz空间的风险度量：保险的状态依赖视角

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-11-01 DOI: 10.1016/j.insmatheco.2025.103174

Francesco Strati

This paper introduces a novel framework for risk measures defined on Musielak-Orlicz spaces, incorporating state-dependent Young functions to address the limitations of traditional uniform risk models in capturing heterogeneous tail behaviors. Extending the foundational work of Cheridito and Li [2009] on Orlicz hearts, the study establishes a robust representation theorem for convex monetary risk measures, demonstrating their expression as a maximum over penalized expectations adjusted by state-specific penalty functions. This result accommodates unbounded risks with spatially or temporally varying profiles, a critical enhancement for applications in insurance and finance where heterogeneity is prevalent. The framework subsumes coherent measures as a special case and provides a characterization of optimal probability measures, ensuring computational feasibility. Practical implications are explored through connections to insurance mathematics, including links to star-shaped risk measures, variable annuities via the Q⊙P measure, and a state-dependent generalization of the Haezendonck-Goovaerts risk measure. Additionally, an aggregation technique for portfolio risk across diverse states is proposed, accompanied by illustrative examples such as the Transformed Norm and Entropic Risk Measures. By integrating theoretical rigor with practical relevance, this study offers a versatile tool for risk assessment under complex, state-varying conditions.

本文引入了一种基于Musielak-Orlicz空间的风险度量框架，结合状态相关的Young函数来解决传统统一风险模型在捕获异质尾部行为方面的局限性。本研究扩展了Cheridito和Li[2009]关于Orlicz心的基础工作，建立了凸性货币风险测度的鲁棒表示定理，证明了它们的表达式是由特定国家的惩罚函数调整的最大过惩罚预期。这一结果适应了具有空间或时间变化概况的无界风险，这是对异质性普遍存在的保险和金融应用的关键增强。该框架将相干测度作为一种特殊情况，并提供了最优概率测度的表征，确保了计算的可行性。通过与保险数学的联系探讨了实际意义，包括与星形风险度量的联系，通过Q⊙P度量的可变年金，以及Haezendonck-Goovaerts风险度量的依赖于状态的概括。此外，提出了一种跨不同状态的投资组合风险的聚合技术，并配有转换规范和熵风险度量等说明性示例。通过将理论严谨性与实际相关性相结合，本研究为复杂、状态变化条件下的风险评估提供了一个通用的工具。

{"title":"Risk measures on Musielak-Orlicz spaces: A state-dependent perspective for insurance","authors":"Francesco Strati","doi":"10.1016/j.insmatheco.2025.103174","DOIUrl":"10.1016/j.insmatheco.2025.103174","url":null,"abstract":"<div><div>This paper introduces a novel framework for risk measures defined on Musielak-Orlicz spaces, incorporating state-dependent Young functions to address the limitations of traditional uniform risk models in capturing heterogeneous tail behaviors. Extending the foundational work of Cheridito and Li [2009] on Orlicz hearts, the study establishes a robust representation theorem for convex monetary risk measures, demonstrating their expression as a maximum over penalized expectations adjusted by state-specific penalty functions. This result accommodates unbounded risks with spatially or temporally varying profiles, a critical enhancement for applications in insurance and finance where heterogeneity is prevalent. The framework subsumes coherent measures as a special case and provides a characterization of optimal probability measures, ensuring computational feasibility. Practical implications are explored through connections to insurance mathematics, including links to star-shaped risk measures, variable annuities via the <em>Q</em>⊙<em>P</em> measure, and a state-dependent generalization of the Haezendonck-Goovaerts risk measure. Additionally, an aggregation technique for portfolio risk across diverse states is proposed, accompanied by illustrative examples such as the Transformed Norm and Entropic Risk Measures. By integrating theoretical rigor with practical relevance, this study offers a versatile tool for risk assessment under complex, state-varying conditions.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"125 ","pages":"Article 103174"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528230","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}

引用次数: 0

Additive tree latent variable models with applications to insurance loss prediction 加性树隐变量模型及其在保险损失预测中的应用

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-11-01 DOI: 10.1016/j.insmatheco.2025.103168

Zhihao Wang , Yanlin Shi , Guangyuan Gao

We consider a specific class of regression models with discrete latent variables, which are commonly used in actuarial science and other fields. When fitting these parametric regression models, regression functions are estimated for both the observed response variable and the latent variable, respectively. Feature engineering, variable selection and model selection become challenging due to the involvement of multiple regression functions and latent variable. To address these challenges, we propose additive tree latent variable models. To calibrate these models, we introduce an iteratively re-weighted gradient boosting (IRGB) algorithm that combines the EM algorithm with the gradient boosting. In each iteration, the IRGB algorithm trains only one weak learner in a stagewise manner. Theoretical analysis demonstrates the monotonic behavior of the likelihood in the IRGB algorithm. We further illustrate the advantages of the proposed nonparametric methods through an empirical example of motor insurance claim counts and a case study on French motor third-party liability insurance pure premiums.

我们考虑了一类具有离散潜变量的特殊回归模型，这些模型通常用于精算科学和其他领域。在拟合这些参数回归模型时，分别对观测到的响应变量和潜在变量估计回归函数。由于多元回归函数和潜在变量的涉及，特征工程、变量选择和模型选择变得具有挑战性。为了解决这些挑战，我们提出了加性树潜变量模型。为了校准这些模型，我们引入了一种迭代重加权梯度增强（IRGB）算法，该算法将EM算法与梯度增强相结合。在每次迭代中，IRGB算法只分阶段训练一个弱学习器。理论分析证明了IRGB算法中似然的单调性。我们通过汽车保险索赔计数的实证例子和法国汽车第三方责任保险纯保费的案例研究进一步说明了所提出的非参数方法的优势。

{"title":"Additive tree latent variable models with applications to insurance loss prediction","authors":"Zhihao Wang , Yanlin Shi , Guangyuan Gao","doi":"10.1016/j.insmatheco.2025.103168","DOIUrl":"10.1016/j.insmatheco.2025.103168","url":null,"abstract":"<div><div>We consider a specific class of regression models with discrete latent variables, which are commonly used in actuarial science and other fields. When fitting these parametric regression models, regression functions are estimated for both the observed response variable and the latent variable, respectively. Feature engineering, variable selection and model selection become challenging due to the involvement of multiple regression functions and latent variable. To address these challenges, we propose additive tree latent variable models. To calibrate these models, we introduce an iteratively re-weighted gradient boosting (IRGB) algorithm that combines the EM algorithm with the gradient boosting. In each iteration, the IRGB algorithm trains only one weak learner in a stagewise manner. Theoretical analysis demonstrates the monotonic behavior of the likelihood in the IRGB algorithm. We further illustrate the advantages of the proposed nonparametric methods through an empirical example of motor insurance claim counts and a case study on French motor third-party liability insurance pure premiums.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"125 ","pages":"Article 103168"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145424574","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}

引用次数: 0

Life care reverse mortgages: Monitoring the net cashflows of a new hybrid insurance product 寿险反向抵押贷款：监测一种新型混合保险产品的净现金流

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-11-01 DOI: 10.1016/j.insmatheco.2025.103170

Giovanna Apicella , Emilia Di Lorenzo , Giulia Magni , Marilena Sibillo

Combining retirement income and long-term disability protection is a well-established concept in the literature. We present a new connection between the lifetime annuity provided by a Reverse Mortgage and long-term care (LTC) insurance, jointly managed in a unique bundled product. Indeed, we define the RM (considered as a source of lifetime income) and LTC as two different regimes of a jointly managed insurance product, which we call the “Life Care Reverse Mortgage” (LCRM). From an actuarial perspective, we design the underlying structure of the LCRM and the inherent framework for the ex-ante estimation of a time-dependent profit/loss function, that provides a measure of the expected annual net cashflows for a life insurer holding a portfolio of LCRMs. We perform a numerical application to illustrate the regime-switching mechanism on which the proposed LCRM insurance contracts are based and to quantify over time the lender’s return for a pool of LCRM contracts through the designed time-dependent profit/loss function. Furthermore, we analyse the sensitivity of the portfolio profit/loss function to two sources of uncertainty: health patterns over time and house price dynamics.

将退休收入与长期残疾保障相结合是文献中一个成熟的概念。我们提出了由反向抵押贷款和长期护理（LTC）保险提供的终身年金之间的新联系，共同管理在一个独特的捆绑产品。事实上，我们将RM（被视为终身收入的来源）和LTC定义为联合管理保险产品的两种不同制度，我们称之为“生命护理反向抵押贷款”（LCRM）。从精算的角度来看，我们设计了LCRM的基础结构和时间相关损益函数的预先估计的内在框架，该框架为持有LCRM投资组合的人寿保险公司提供了预期年度净现金流的度量。我们执行了一个数值应用程序来说明提议的LCRM保险合同所基于的制度转换机制，并通过设计的与时间相关的损益函数来量化贷款人对LCRM合同池的回报。此外，我们分析了投资组合盈亏函数对两个不确定性来源的敏感性：健康模式随时间的变化和房价动态。

{"title":"Life care reverse mortgages: Monitoring the net cashflows of a new hybrid insurance product","authors":"Giovanna Apicella , Emilia Di Lorenzo , Giulia Magni , Marilena Sibillo","doi":"10.1016/j.insmatheco.2025.103170","DOIUrl":"10.1016/j.insmatheco.2025.103170","url":null,"abstract":"<div><div>Combining retirement income and long-term disability protection is a well-established concept in the literature. We present a new connection between the lifetime annuity provided by a Reverse Mortgage and long-term care (LTC) insurance, jointly managed in a unique bundled product. Indeed, we define the RM (considered as a source of lifetime income) and LTC as two different regimes of a jointly managed insurance product, which we call the “Life Care Reverse Mortgage” (LCRM). From an actuarial perspective, we design the underlying structure of the LCRM and the inherent framework for the ex-ante estimation of a time-dependent profit/loss function, that provides a measure of the expected annual net cashflows for a life insurer holding a portfolio of LCRMs. We perform a numerical application to illustrate the regime-switching mechanism on which the proposed LCRM insurance contracts are based and to quantify over time the lender’s return for a pool of LCRM contracts through the designed time-dependent profit/loss function. Furthermore, we analyse the sensitivity of the portfolio profit/loss function to two sources of uncertainty: health patterns over time and house price dynamics.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"125 ","pages":"Article 103170"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528229","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}

引用次数: 0

Optimal risk sharing with correlated insurance businesses in a Stackelberg-Nash differential game Stackelberg-Nash微分对策中相关保险企业的最优风险分担

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-11-01 DOI: 10.1016/j.insmatheco.2025.103171

Mengyu Wu , Zhibin Liang , Qingqing Zhang

In this paper, we investigate the optimal risk sharing problem for two insurers under the framework of Stackelberg-Nash differential game. More specifically, two insurers transfer their businesses to each other for achieving the goal of win-win, where both of them act as the leader for pricing while the follower for choosing their own retention level. Based on the game theoretic equilibrium setting and dynamic programming principle, the explicit optimal strategies are derived. We find that insurers will cooperate more eagerly when there is a stronger negative correlation between the businesses of both parties. In order to explore the advantages of risk sharing, we also investigate the optimal reinsurance problem in a traditional Stackelberg game framework. Risk sharing is found to be more advantageous than reinsurance in many cases, especially when the businesses have significant differences, such as a strong negative correlation or a large/small volatility ratio, which means that one of the two businesses is relatively stable while the other fluctuates greatly. Further analysis is given to show the effects of model parameters and the economics interpretations behind them. It is interesting to find that risk-aversion coefficient plays a key role in this Stackelberg-Nash differential game, and the conclusions confirm an obvious fact, that is, risk-averse individuals tend to be more hesitant and conservative when making a decision.

本文研究了在Stackelberg-Nash微分对策框架下两保险公司的最优风险分担问题。更具体地说，两家保险公司相互转让业务是为了实现双赢的目标，双方都是定价的领导者，而追随者则是选择自己的保留水平。基于博弈论均衡设置和动态规划原理，导出了显式优化策略。我们发现，当双方业务负相关程度越强时，保险公司的合作意愿越强。为了探索风险分担的优势，我们还研究了传统Stackelberg博弈框架下的最优再保险问题。在很多情况下，风险分担比再保险更有利，特别是当两项业务存在显著差异时，如负相关关系强或波动比大/小，即两项业务中一项相对稳定，而另一项波动较大。进一步分析了模型参数的影响及其背后的经济学解释。有趣的是，我们发现风险厌恶系数在这个Stackelberg-Nash微分对策中起着关键作用，结论证实了一个明显的事实，即风险厌恶的个体在做决策时往往更加犹豫和保守。

{"title":"Optimal risk sharing with correlated insurance businesses in a Stackelberg-Nash differential game","authors":"Mengyu Wu , Zhibin Liang , Qingqing Zhang","doi":"10.1016/j.insmatheco.2025.103171","DOIUrl":"10.1016/j.insmatheco.2025.103171","url":null,"abstract":"<div><div>In this paper, we investigate the optimal risk sharing problem for two insurers under the framework of Stackelberg-Nash differential game. More specifically, two insurers transfer their businesses to each other for achieving the goal of win-win, where both of them act as the leader for pricing while the follower for choosing their own retention level. Based on the game theoretic equilibrium setting and dynamic programming principle, the explicit optimal strategies are derived. We find that insurers will cooperate more eagerly when there is a stronger negative correlation between the businesses of both parties. In order to explore the advantages of risk sharing, we also investigate the optimal reinsurance problem in a traditional Stackelberg game framework. Risk sharing is found to be more advantageous than reinsurance in many cases, especially when the businesses have significant differences, such as a strong negative correlation or a large/small volatility ratio, which means that one of the two businesses is relatively stable while the other fluctuates greatly. Further analysis is given to show the effects of model parameters and the economics interpretations behind them. It is interesting to find that risk-aversion coefficient plays a key role in this Stackelberg-Nash differential game, and the conclusions confirm an obvious fact, that is, risk-averse individuals tend to be more hesitant and conservative when making a decision.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"125 ","pages":"Article 103171"},"PeriodicalIF":2.2,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578913","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}

引用次数: 0

Cyber risk taxonomies: statistical analysis of cybersecurity risk classifications 网络风险分类：网络安全风险分类的统计分析

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-10-29 DOI: 10.1016/j.insmatheco.2025.103167

Matteo Malavasi , Gareth W. Peters , Stefan Trück , Pavel V. Shevchenko , Jiwook Jang , Georgy Sofronov

Cyber risk classifications are widely used in the modeling of cyber event distributions, yet their effectiveness in out of sample forecasting performance remains underexplored. In this paper, we analyze the most commonly used classifications and argue in favor of switching the attention from goodness-of-fit and in-sample predictive performance, to focusing on the out-of sample forecasting performance. We use a rolling window analysis, to compare cyber risk distribution forecasts via threshold weighted scoring functions. Our results indicate that business motivated cyber risk classifications appear to be too restrictive and not flexible enough to capture the heterogeneity of cyber risk events. We investigate how dynamic and impact-based cyber risk classifiers seem to be better suited in forecasting future cyber risk losses than the other considered classifications. These findings suggest that cyber risk types provide limited forecasting ability concerning cyber event loss severity distribution, and cyber insurance rate-makers should utilize cyber risk types only when modeling the cyber event frequency distribution. Our study offers valuable insights for decision-makers and policymakers alike, contributing to the advancement of scientific knowledge in the field of cyber risk management.

网络风险分类广泛用于网络事件分布的建模，但其在样本外预测性能方面的有效性仍未得到充分探讨。在本文中，我们分析了最常用的分类，并主张将注意力从拟合优度和样本内预测性能转移到关注样本外预测性能。我们使用滚动窗口分析，通过阈值加权评分函数来比较网络风险分布预测。我们的研究结果表明，商业动机的网络风险分类似乎过于严格，不够灵活，无法捕捉网络风险事件的异质性。我们研究了动态和基于影响的网络风险分类器似乎比其他考虑的分类更适合预测未来的网络风险损失。这些发现表明，网络风险类型对网络事件损失严重程度分布的预测能力有限，网络保险费率制定者应仅在对网络事件频率分布建模时使用网络风险类型。我们的研究为决策者和政策制定者提供了有价值的见解，有助于提高网络风险管理领域的科学知识。

{"title":"Cyber risk taxonomies: statistical analysis of cybersecurity risk classifications","authors":"Matteo Malavasi , Gareth W. Peters , Stefan Trück , Pavel V. Shevchenko , Jiwook Jang , Georgy Sofronov","doi":"10.1016/j.insmatheco.2025.103167","DOIUrl":"10.1016/j.insmatheco.2025.103167","url":null,"abstract":"<div><div>Cyber risk classifications are widely used in the modeling of cyber event distributions, yet their effectiveness in out of sample forecasting performance remains underexplored. In this paper, we analyze the most commonly used classifications and argue in favor of switching the attention from goodness-of-fit and in-sample predictive performance, to focusing on the out-of sample forecasting performance. We use a rolling window analysis, to compare cyber risk distribution forecasts via threshold weighted scoring functions. Our results indicate that business motivated cyber risk classifications appear to be too restrictive and not flexible enough to capture the heterogeneity of cyber risk events. We investigate how dynamic and impact-based cyber risk classifiers seem to be better suited in forecasting future cyber risk losses than the other considered classifications. These findings suggest that cyber risk types provide limited forecasting ability concerning cyber event loss severity distribution, and cyber insurance rate-makers should utilize cyber risk types only when modeling the cyber event frequency distribution. Our study offers valuable insights for decision-makers and policymakers alike, contributing to the advancement of scientific knowledge in the field of cyber risk management.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"126 ","pages":"Article 103167"},"PeriodicalIF":2.2,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145536948","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}

引用次数: 0

Performance-based variable premium scheme and reinsurance design 基于绩效的可变保费方案与再保险设计

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-10-22 DOI: 10.1016/j.insmatheco.2025.103169

Ziyue Shi , David Landriault , Fangda Liu

In the literature, insurance and reinsurance pricing is typically determined by a premium principle, characterized by a risk measure that reflects the policy seller’s risk attitude. Building on the work of Meyers (1980) and Chen et al. (2016), we propose a new performance-based variable premium scheme for reinsurance policies, where the premium depends on both the distribution of the ceded loss and the actual realized loss. Under this scheme, the insurer and the reinsurer face a random premium at the beginning of the policy period. Based on the realized loss, the premium is adjusted into either a “reward” or “penalty” scenario, resulting in a discount or surcharge at the end of the policy period. We characterize the optimal reinsurance policy from the insurer’s perspective under this new variable premium scheme. In addition, we formulate a Bowley optimization problem between the insurer and the monopoly reinsurer. Numerical examples demonstrate that, compared to the expected-value premium principle, the reinsurer prefers the variable premium scheme as it reduces the reinsurer’s total risk exposure.

在文献中，保险和再保险定价通常由保费原则决定，其特征是反映保单卖方风险态度的风险度量。在Meyers（1980）和Chen等人（2016）的研究基础上，我们提出了一种新的基于绩效的再保险保单可变保费方案，其中保费取决于已让与损失的分配和实际已实现损失。在此计划下，保险公司和再保险公司在保单期开始时面临随机保费。根据已实现的损失，保费被调整为“奖励”或“惩罚”方案，从而在保单期结束时产生折扣或附加费。我们从保险人的角度刻画了这种新的可变保费方案下的最优再保险政策。此外，我们还建立了保险公司与垄断再保险公司之间的Bowley优化问题。数值算例表明，与期望值保费原则相比，再保险人更倾向于采用可变保费方案，因为它可以减少再保险人的总风险暴露。

{"title":"Performance-based variable premium scheme and reinsurance design","authors":"Ziyue Shi , David Landriault , Fangda Liu","doi":"10.1016/j.insmatheco.2025.103169","DOIUrl":"10.1016/j.insmatheco.2025.103169","url":null,"abstract":"<div><div>In the literature, insurance and reinsurance pricing is typically determined by a premium principle, characterized by a risk measure that reflects the policy seller’s risk attitude. Building on the work of Meyers (1980) and Chen et al. (2016), we propose a new performance-based variable premium scheme for reinsurance policies, where the premium depends on both the distribution of the ceded loss and the actual realized loss. Under this scheme, the insurer and the reinsurer face a random premium at the beginning of the policy period. Based on the realized loss, the premium is adjusted into either a “reward” or “penalty” scenario, resulting in a discount or surcharge at the end of the policy period. We characterize the optimal reinsurance policy from the insurer’s perspective under this new variable premium scheme. In addition, we formulate a Bowley optimization problem between the insurer and the monopoly reinsurer. Numerical examples demonstrate that, compared to the expected-value premium principle, the reinsurer prefers the variable premium scheme as it reduces the reinsurer’s total risk exposure.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"126 ","pages":"Article 103169"},"PeriodicalIF":2.2,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737416","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}

引用次数: 0

Optimal consumption-leisure-investment and retirement choices with nonconcave aspirational utility 具有非凹期望效用的最优消费-休闲-投资与退休选择

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-10-10 DOI: 10.1016/j.insmatheco.2025.103165

Shuang Li , Hui Meng , Ming Zhou

This paper investigates the optimal strategies for investment, consumption, leisure, and voluntary retirement in a continuous-time framework. Beyond standard consumption, we incorporate aspirational consumption which involves a constant and costly expenditure while generating surges in utility, leading to a two-dimensional nonconcave utility function for consumption and leisure. Using the martingale duality method and concavification principle, we classify individuals into “leisure-seekers”, “aspiration-seekers”, and “centrists” based on the prioritization of aspirational consumption versus maximum leisure, and derive semi-closed solutions for the optimal strategies. Through model calibration, we explore the impact of aspiration on optimal policies, finding that the aspiration postpones retirement, strengthens the incentive to invest, and shrinks both leisure and consumption until the aspiration is fulfilled.

本文在连续时间框架下研究了投资、消费、休闲和自愿退休的最优策略。在标准消费之外，我们纳入了期望消费，这涉及到持续和昂贵的支出，同时产生效用激增，导致消费和休闲的二维非凹效用函数。利用鞅对偶方法和凹化原理，基于期望消费与最大休闲的优先性，将个体划分为“休闲寻求者”、“渴望寻求者”和“中间派”，并推导出最优策略的半封闭解。通过模型校准，我们探索了愿望对最优政策的影响，发现愿望推迟了退休时间，增强了投资激励，收缩了休闲和消费，直到愿望得到满足。

引用次数: 0

Numerical methods for computing risk measures of variable annuities under exponential Lévy models 指数lsamvy模型下可变年金风险测度的数值计算方法

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-10-10 DOI: 10.1016/j.insmatheco.2025.103166

Oleg Kudryavtsev , Xiao Wei

We propose an efficient numerical method to calculate the Value-at-Risk and Conditional Tail Expectation for variable annuities under exponential Lévy models. In the proposed approach, the probability density of the net liabilities, expressed in terms of the final position and integral of the exponential Lévy process, is approximated using frame theory and Riesz bases using 3rd order B-splines. The key element of the numerical method is a new algorithm for calculating the integral of the exponential Lévy process, approximated by a discrete sum whose expectation coincides with the expected value of the desired integral. In the main part of our numerical algorithm, to find Value-at-Risk as a quantile of the loss distribution, we numerically solve the equation for the corresponding cumulative distribution function using Newton's method adapted to the probability density approximation by B-splines. Once the Value-at-Risk is found, we calculate the Conditional Tail Expectation using integration by parts, again taking advantage of the properties of cubic B-splines. Numerical experiments on the application of the developed method for the Black-Scholes and CGMY models clearly demonstrate its high accuracy and speed.

我们提出了一种有效的数值方法来计算指数lsamvy模型下可变年金的风险值和条件尾期望。在提出的方法中，净负债的概率密度，以最终位置和指数lsamvy过程的积分表示，使用框架理论和Riesz基使用三阶b样条近似。数值方法的关键是一种新的计算指数型lsamvy过程积分的算法，该算法近似为一个离散和，其期望与期望积分的期望值一致。在我们的数值算法的主要部分，为了找到损失分布的分位数值，我们使用适用于b样条概率密度近似的牛顿方法对相应的累积分布函数的方程进行了数值求解。一旦发现风险值，我们使用分部积分法计算条件尾期望，再次利用三次b样条的性质。将该方法应用于Black-Scholes和CGMY模型的数值实验表明，该方法具有较高的精度和速度。

{"title":"Numerical methods for computing risk measures of variable annuities under exponential Lévy models","authors":"Oleg Kudryavtsev , Xiao Wei","doi":"10.1016/j.insmatheco.2025.103166","DOIUrl":"10.1016/j.insmatheco.2025.103166","url":null,"abstract":"<div><div>We propose an efficient numerical method to calculate the Value-at-Risk and Conditional Tail Expectation for variable annuities under exponential Lévy models. In the proposed approach, the probability density of the net liabilities, expressed in terms of the final position and integral of the exponential Lévy process, is approximated using frame theory and Riesz bases using 3rd order B-splines. The key element of the numerical method is a new algorithm for calculating the integral of the exponential Lévy process, approximated by a discrete sum whose expectation coincides with the expected value of the desired integral. In the main part of our numerical algorithm, to find Value-at-Risk as a quantile of the loss distribution, we numerically solve the equation for the corresponding cumulative distribution function using Newton's method adapted to the probability density approximation by B-splines. Once the Value-at-Risk is found, we calculate the Conditional Tail Expectation using integration by parts, again taking advantage of the properties of cubic B-splines. Numerical experiments on the application of the developed method for the Black-Scholes and CGMY models clearly demonstrate its high accuracy and speed.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"125 ","pages":"Article 103166"},"PeriodicalIF":2.2,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145332457","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}

引用次数: 0

Robust parameter estimation for the Lee-Carter family: A probabilistic principal component approach 李-卡特族的鲁棒参数估计：一种概率主成分方法

IF 2.2 2区经济学 Q2 ECONOMICS

Insurance Mathematics & Economics

Pub Date : 2025-10-01 DOI: 10.1016/j.insmatheco.2025.103164

Yiping Guo , Johnny Siu-Hang Li

Although the impact of outliers on stochastic mortality modelling has been examined, previous studies on this topic focus on how outliers in the estimated time-varying indexes may be detected and/or modelled, with little attention being paid to the adverse effects of outliers on estimation robustness, particularly that pertaining to age-specific parameters. In this paper, we propose a robust estimation method for the Lee-Carter model, through a reformulation of the model into a probabilistic principal component analysis with multivariate t-distributions and an efficient expectation-maximization algorithm for implementation. The proposed method yields significantly more robust parameter estimates, while preserving the fundamental interpretation for the bilinear term in the model as the first principal component and the flexibility of pairing the estimated time-varying parameters with any appropriate time-series process. We also extend the proposed method for use with multi-population generalizations of the Lee-Carter model, allowing for a wider range of applications such as quantification of population basis risk in index-based longevity hedges. Using a combination of real and pseudo datasets, we demonstrate the superiority of the proposed method relative to conventional estimation approaches such as singular value decomposition and maximum likelihood.

虽然已经研究了异常值对随机死亡率建模的影响，但先前关于这一主题的研究主要集中在如何检测和/或建模估计时变指数中的异常值，很少关注异常值对估计稳健性的不利影响，特别是与特定年龄参数有关的不利影响。在本文中，我们通过将Lee-Carter模型重新表述为具有多元t分布的概率主成分分析和有效的期望最大化算法来提出一种稳健的估计方法。该方法显著提高了参数估计的鲁棒性，同时保留了对模型中双线性项作为第一主成分的基本解释，以及将估计的时变参数与任何适当的时间序列过程配对的灵活性。我们还扩展了所提出的方法，用于Lee-Carter模型的多种群推广，允许更广泛的应用，例如基于指数的长寿对冲中种群基础风险的量化。使用真实和伪数据集的组合，我们证明了该方法相对于传统的估计方法（如奇异值分解和最大似然）的优越性。

{"title":"Robust parameter estimation for the Lee-Carter family: A probabilistic principal component approach","authors":"Yiping Guo , Johnny Siu-Hang Li","doi":"10.1016/j.insmatheco.2025.103164","DOIUrl":"10.1016/j.insmatheco.2025.103164","url":null,"abstract":"<div><div>Although the impact of outliers on stochastic mortality modelling has been examined, previous studies on this topic focus on how outliers in the estimated time-varying indexes may be detected and/or modelled, with little attention being paid to the adverse effects of outliers on estimation robustness, particularly that pertaining to age-specific parameters. In this paper, we propose a robust estimation method for the Lee-Carter model, through a reformulation of the model into a probabilistic principal component analysis with multivariate <em>t</em>-distributions and an efficient expectation-maximization algorithm for implementation. The proposed method yields significantly more robust parameter estimates, while preserving the fundamental interpretation for the bilinear term in the model as the first principal component and the flexibility of pairing the estimated time-varying parameters with any appropriate time-series process. We also extend the proposed method for use with multi-population generalizations of the Lee-Carter model, allowing for a wider range of applications such as quantification of population basis risk in index-based longevity hedges. Using a combination of real and pseudo datasets, we demonstrate the superiority of the proposed method relative to conventional estimation approaches such as singular value decomposition and maximum likelihood.</div></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"125 ","pages":"Article 103164"},"PeriodicalIF":2.2,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265697","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}

引用次数: 0