Arianna Vallarino, Giovanni Rabitti, Amir Khorrami Chokami
Abstract The ratemaking process is a key issue in insurance pricing. It consists in pooling together policyholders with similar risk profiles into rating classes and assigning the same premium for policyholders in the same class. In actuarial practice, rating systems are typically not based on all risk factors but rather only some of factors are selected to construct the rating classes. The objective of this study is to investigate the selection of risk factors in order to construct rating classes that exhibit maximum internal homogeneity. For this selection, we adopt the Shapley effects from global sensitivity analysis. While these sensitivity indices are used for model interpretability, we apply them to construct rating classes. We provide a new strategy to estimate them, and we connect them to the intra-class variability and heterogeneity of the rating classes. To verify the appropriateness of our procedure, we introduce a measure of heterogeneity specifically designed to compare rating systems with a different number of classes. Using a well-known car insurance dataset, we show that the rating system constructed with the Shapley effects is the one minimizing this heterogeneity measure.
{"title":"Construction of rating systems using global sensitivity analysis: A numerical investigation","authors":"Arianna Vallarino, Giovanni Rabitti, Amir Khorrami Chokami","doi":"10.1017/asb.2023.34","DOIUrl":"https://doi.org/10.1017/asb.2023.34","url":null,"abstract":"Abstract The ratemaking process is a key issue in insurance pricing. It consists in pooling together policyholders with similar risk profiles into rating classes and assigning the same premium for policyholders in the same class. In actuarial practice, rating systems are typically not based on all risk factors but rather only some of factors are selected to construct the rating classes. The objective of this study is to investigate the selection of risk factors in order to construct rating classes that exhibit maximum internal homogeneity. For this selection, we adopt the Shapley effects from global sensitivity analysis. While these sensitivity indices are used for model interpretability, we apply them to construct rating classes. We provide a new strategy to estimate them, and we connect them to the intra-class variability and heterogeneity of the rating classes. To verify the appropriateness of our procedure, we introduce a measure of heterogeneity specifically designed to compare rating systems with a different number of classes. Using a well-known car insurance dataset, we show that the rating system constructed with the Shapley effects is the one minimizing this heterogeneity measure.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we study the optimal VIX-linked target benefit (TB) pension design. By applying the dynamic programming approach, we show the optimal risk-sharing structure for the benefit payment exhibits a linear form that consists of three components: (1) a model-robust performance adjustment, (2) a counter-cyclical volatility adjustment that depends on the VIX index, and (3) a TB level that is partially indexed to the cost-of-living adjustment. Differences between our results and the previous literature are highlighted via both theoretical derivations and numerical illustrations.
{"title":"Optimal VIX-linked structure for the target benefit pension plan","authors":"Lv Chen, Danping Li, Yumin Wang, Xiaobai Zhu","doi":"10.1017/asb.2023.33","DOIUrl":"https://doi.org/10.1017/asb.2023.33","url":null,"abstract":"Abstract In this paper, we study the optimal VIX-linked target benefit (TB) pension design. By applying the dynamic programming approach, we show the optimal risk-sharing structure for the benefit payment exhibits a linear form that consists of three components: (1) a model-robust performance adjustment, (2) a counter-cyclical volatility adjustment that depends on the VIX index, and (3) a TB level that is partially indexed to the cost-of-living adjustment. Differences between our results and the previous literature are highlighted via both theoretical derivations and numerical illustrations.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135883626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the optimal investment-reinsurance problem in the context of equity-linked insurance products. Such products often have a capital guarantee, which can motivate insurers to purchase reinsurance. Since a reinsurance contract implies an interaction between the insurer and the reinsurer, we model the optimization problem as a Stackelberg game. The reinsurer is the leader in the game and maximizes its expected utility by selecting its optimal investment strategy and a safety loading in the reinsurance contract it offers to the insurer. The reinsurer can assess how the insurer will rationally react on each action of the reinsurer. The insurance company is the follower and maximizes its expected utility by choosing its investment strategy and the amount of reinsurance the company purchases at the price offered by the reinsurer. In this game, we derive the Stackelberg equilibrium for general utility functions. For power utility functions, we calculate the equilibrium explicitly and find that the reinsurer selects the largest reinsurance premium such that the insurer may still buy the maximal amount of reinsurance. Since in the equilibrium the insurer is indifferent in the amount of reinsurance, in practice, the reinsurer should consider charging a smaller reinsurance premium than the equilibrium one. Therefore, we propose several criteria for choosing such a discount rate and investigate its wealth-equivalent impact on the expected utility of each party.
{"title":"Risk sharing in equity-linked insurance products: Stackelberg equilibrium between an insurer and a reinsurer","authors":"Yevhen Havrylenko, Maria Hinken, Rudi Zagst","doi":"10.1017/asb.2023.32","DOIUrl":"https://doi.org/10.1017/asb.2023.32","url":null,"abstract":"Abstract We study the optimal investment-reinsurance problem in the context of equity-linked insurance products. Such products often have a capital guarantee, which can motivate insurers to purchase reinsurance. Since a reinsurance contract implies an interaction between the insurer and the reinsurer, we model the optimization problem as a Stackelberg game. The reinsurer is the leader in the game and maximizes its expected utility by selecting its optimal investment strategy and a safety loading in the reinsurance contract it offers to the insurer. The reinsurer can assess how the insurer will rationally react on each action of the reinsurer. The insurance company is the follower and maximizes its expected utility by choosing its investment strategy and the amount of reinsurance the company purchases at the price offered by the reinsurer. In this game, we derive the Stackelberg equilibrium for general utility functions. For power utility functions, we calculate the equilibrium explicitly and find that the reinsurer selects the largest reinsurance premium such that the insurer may still buy the maximal amount of reinsurance. Since in the equilibrium the insurer is indifferent in the amount of reinsurance, in practice, the reinsurer should consider charging a smaller reinsurance premium than the equilibrium one. Therefore, we propose several criteria for choosing such a discount rate and investigate its wealth-equivalent impact on the expected utility of each party.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135885032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"ASB volume 53 issue 3 Cover and Front matter","authors":"","doi":"10.1017/asb.2023.30","DOIUrl":"https://doi.org/10.1017/asb.2023.30","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135304915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A target benefit plan (TBP) is a collective defined contribution (DC) plan that is growing in popularity in Canada. Similar to DC plans, TBPs have fixed contribution rates, but they also implement pooling of longevity and investment risk. In this paper, we formulate a multi-period model that incorporates two sources of risk – asset risk and labor income risk for active members. We present an optimal investment and retirement benefits schedule for TBP members with a fixed contribution rate. Using Australian data from 1965 to 2018, we evaluate the performance of the optimal TBP scheme and compare it to the optimal DC scheme. By adopting the benefit–investment strategy derived in this paper, we demonstrate the stability of benefit distribution over time for a TBP scheme in this stochastic formulation. To outperform the DC scheme’s benefit payment, careful consideration shall be given to the benefit target in the TBP scheme. A high target may not be achievable, while a low target can impede the accumulation momentum of the fund’s wealth in its early stages. Moreover, a TBP fund’s investment strategy is primarily influenced by the wealth target, with more aggressive investments in risky assets as the wealth target increases. This analysis may shed light on the possible improvements to retirement planning in Australia. Although the results are sensitive to the choice of model parameters, overall, the proposed TBP promotes system stability in various scenarios.
{"title":"Target benefit versus defined contribution scheme: a multi-period framework","authors":"Ping Chen, Haixiang Yao, Hailiang Yang, Dan Zhu","doi":"10.1017/asb.2023.27","DOIUrl":"https://doi.org/10.1017/asb.2023.27","url":null,"abstract":"Abstract A target benefit plan (TBP) is a collective defined contribution (DC) plan that is growing in popularity in Canada. Similar to DC plans, TBPs have fixed contribution rates, but they also implement pooling of longevity and investment risk. In this paper, we formulate a multi-period model that incorporates two sources of risk – asset risk and labor income risk for active members. We present an optimal investment and retirement benefits schedule for TBP members with a fixed contribution rate. Using Australian data from 1965 to 2018, we evaluate the performance of the optimal TBP scheme and compare it to the optimal DC scheme. By adopting the benefit–investment strategy derived in this paper, we demonstrate the stability of benefit distribution over time for a TBP scheme in this stochastic formulation. To outperform the DC scheme’s benefit payment, careful consideration shall be given to the benefit target in the TBP scheme. A high target may not be achievable, while a low target can impede the accumulation momentum of the fund’s wealth in its early stages. Moreover, a TBP fund’s investment strategy is primarily influenced by the wealth target, with more aggressive investments in risky assets as the wealth target increases. This analysis may shed light on the possible improvements to retirement planning in Australia. Although the results are sensitive to the choice of model parameters, overall, the proposed TBP promotes system stability in various scenarios.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"54 1","pages":"545 - 579"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82714576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"ASB volume 53 issue 3 Cover and Back matter","authors":"","doi":"10.1017/asb.2023.31","DOIUrl":"https://doi.org/10.1017/asb.2023.31","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"197 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135304923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Modelling mortality co-movements for multiple populations has significant implications for mortality/longevity risk management. This paper assumes that multiple populations are heterogeneous sub-populations randomly drawn from a hypothetical super-population. Those heterogeneous sub-populations may exhibit various patterns of mortality dynamics across different age groups. We propose a hierarchical structure of these age patterns to ensure the model stability and use a Vector Error Correction Model (VECM) to fit the co-movements over time. Especially, a structural analysis based on the VECM is implemented to investigate potential interdependence among mortality dynamics of the examined populations. An efficient Bayesian Markov Chain Monte-Carlo method is also developed to estimate the unknown parameters to address the computational complexity. Our empirical application to the mortality data collected for the Group of Seven nations demonstrates the efficacy of our approach.
{"title":"Multi-population mortality modelling: a Bayesian hierarchical approach","authors":"Jianjie Shi, Yanlin Shi, Pengjie Wang, Dan Zhu","doi":"10.1017/asb.2023.29","DOIUrl":"https://doi.org/10.1017/asb.2023.29","url":null,"abstract":"\u0000 Modelling mortality co-movements for multiple populations has significant implications for mortality/longevity risk management. This paper assumes that multiple populations are heterogeneous sub-populations randomly drawn from a hypothetical super-population. Those heterogeneous sub-populations may exhibit various patterns of mortality dynamics across different age groups. We propose a hierarchical structure of these age patterns to ensure the model stability and use a Vector Error Correction Model (VECM) to fit the co-movements over time. Especially, a structural analysis based on the VECM is implemented to investigate potential interdependence among mortality dynamics of the examined populations. An efficient Bayesian Markov Chain Monte-Carlo method is also developed to estimate the unknown parameters to address the computational complexity. Our empirical application to the mortality data collected for the Group of Seven nations demonstrates the efficacy of our approach.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"128 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75702455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We propose a family of range-based risk measures to generalize the role of value at risk (VaR) in the formulation of range value at risk (RVaR) considering other risk measures induced by a tail level. We discuss this type of measure in detail and its theoretical properties and representations. Moreover, we present a score function to evaluate the forecasts of these measures. In order to present the proposed concepts in an applied way, we performed illustrations using Monte Carlo simulations and real financial data.
{"title":"Range-based risk measures and their applications","authors":"M. Righi, F. Müller","doi":"10.1017/asb.2023.28","DOIUrl":"https://doi.org/10.1017/asb.2023.28","url":null,"abstract":"Abstract We propose a family of range-based risk measures to generalize the role of value at risk (VaR) in the formulation of range value at risk (RVaR) considering other risk measures induced by a tail level. We discuss this type of measure in detail and its theoretical properties and representations. Moreover, we present a score function to evaluate the forecasts of these measures. In order to present the proposed concepts in an applied way, we performed illustrations using Monte Carlo simulations and real financial data.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"215 1","pages":"636 - 657"},"PeriodicalIF":1.9,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72663331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A variable annuity is a modern life insurance product that offers its policyholders participation in investment with various guarantees. To address the computational challenge of valuing large portfolios of variable annuity contracts, several data mining frameworks based on statistical learning have been proposed in the past decade. Existing methods utilize regression modeling to predict the market value of most contracts. Despite the efficiency of those methods, a regression model fitted to a small amount of data produces substantial prediction errors, and thus, it is challenging to rely on existing frameworks when highly accurate valuation results are desired or required. In this paper, we propose a novel hybrid framework that effectively chooses and assesses easy-to-predict contracts using the random forest model while leaving hard-to-predict contracts for the Monte Carlo simulation. The effectiveness of the hybrid approach is illustrated with an experimental study.
{"title":"A hybrid data mining framework for variable annuity portfolio valuation","authors":"Hyukjun Gweon, Shu Li","doi":"10.1017/asb.2023.26","DOIUrl":"https://doi.org/10.1017/asb.2023.26","url":null,"abstract":"Abstract A variable annuity is a modern life insurance product that offers its policyholders participation in investment with various guarantees. To address the computational challenge of valuing large portfolios of variable annuity contracts, several data mining frameworks based on statistical learning have been proposed in the past decade. Existing methods utilize regression modeling to predict the market value of most contracts. Despite the efficiency of those methods, a regression model fitted to a small amount of data produces substantial prediction errors, and thus, it is challenging to rely on existing frameworks when highly accurate valuation results are desired or required. In this paper, we propose a novel hybrid framework that effectively chooses and assesses easy-to-predict contracts using the random forest model while leaving hard-to-predict contracts for the Monte Carlo simulation. The effectiveness of the hybrid approach is illustrated with an experimental study.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"1 1","pages":"580 - 595"},"PeriodicalIF":1.9,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89260785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In pricing insurance contracts based on the individual policyholder’s aggregate losses for non-life insurers, the literature has mainly focused on using detailed information from policies and closed claims. However, the information on open claims can reflect shifts in the distribution of the expected claim payments better than closed claims. Such shifts may be needed to be reflected in the ratemaking process earlier rather than later, especially when insurers are experiencing environmental changes. In practice, actuaries use ad hoc techniques to adjust data to current levels to determine premiums. This paper presents an intuitive ratemaking model, employing a marked Poisson process framework, which ensures that the multivariate risk analysis is done more routinely using all reported claims and makes an adjustment for Incurred But Not Reported claims. Utilizing data from the Wisconsin Local Government Property Insurance Fund, we find that by determining rates based on current data, the proposed ratemaking model leads to better alignment of premiums and provides insurers with a more financially sound portfolio.
{"title":"Ratemaking in a changing environment","authors":"A. Nii-Armah, Nii-Armah Okine","doi":"10.1017/asb.2023.23","DOIUrl":"https://doi.org/10.1017/asb.2023.23","url":null,"abstract":"Abstract In pricing insurance contracts based on the individual policyholder’s aggregate losses for non-life insurers, the literature has mainly focused on using detailed information from policies and closed claims. However, the information on open claims can reflect shifts in the distribution of the expected claim payments better than closed claims. Such shifts may be needed to be reflected in the ratemaking process earlier rather than later, especially when insurers are experiencing environmental changes. In practice, actuaries use ad hoc techniques to adjust data to current levels to determine premiums. This paper presents an intuitive ratemaking model, employing a marked Poisson process framework, which ensures that the multivariate risk analysis is done more routinely using all reported claims and makes an adjustment for Incurred But Not Reported claims. Utilizing data from the Wisconsin Local Government Property Insurance Fund, we find that by determining rates based on current data, the proposed ratemaking model leads to better alignment of premiums and provides insurers with a more financially sound portfolio.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"44 1","pages":"596 - 618"},"PeriodicalIF":1.9,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84241657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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