Abstract The least squares Monte Carlo method has become a standard approach in the insurance and financial industries for evaluating a company’s exposure to market risk. However, the non-linear regression of simulated responses on risk factors poses a challenge in this procedure. This article presents a novel approach to address this issue by employing an a-priori segmentation of responses. Using a K-means algorithm, we identify clusters of responses that are then locally regressed on their corresponding risk factors. The global regression function is obtained by combining the local models with logistic regression. We demonstrate the effectiveness of the proposed local least squares Monte Carlo method through two case studies. The first case study investigates butterfly and bull trap options within a Heston stochastic volatility model, while the second case study examines the exposure to risks in a participating life insurance scenario.
{"title":"Risk management with local least squares Monte Carlo","authors":"Donatien Hainaut, Adnane Akbaraly","doi":"10.1017/asb.2023.25","DOIUrl":"https://doi.org/10.1017/asb.2023.25","url":null,"abstract":"Abstract The least squares Monte Carlo method has become a standard approach in the insurance and financial industries for evaluating a company’s exposure to market risk. However, the non-linear regression of simulated responses on risk factors poses a challenge in this procedure. This article presents a novel approach to address this issue by employing an a-priori segmentation of responses. Using a K-means algorithm, we identify clusters of responses that are then locally regressed on their corresponding risk factors. The global regression function is obtained by combining the local models with logistic regression. We demonstrate the effectiveness of the proposed local least squares Monte Carlo method through two case studies. The first case study investigates butterfly and bull trap options within a Heston stochastic volatility model, while the second case study examines the exposure to risks in a participating life insurance scenario.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"39 1","pages":"489 - 514"},"PeriodicalIF":1.9,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87303405","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 This paper studies dynamic reinsurance contracting and competition problems under model ambiguity in a reinsurance market with one primary insurer and n reinsurers, who apply the variance premium principle and who are distinguished by their levels of ambiguity aversion. The insurer negotiates reinsurance policies with all reinsurers simultaneously, which leads to a reinsurance tree structure with full competition among the reinsurers. We model the reinsurance contracting problems between the insurer and reinsurers by Stackelberg differential games and the competition among the reinsurers by a non-cooperative Nash game. We derive equilibrium strategies in semi-closed form for all the companies, whose objective is to maximize their expected surpluses penalized by a squared-error divergence term that measures their ambiguity. We find that, in equilibrium, the insurer purchases a positive amount of proportional reinsurance from each reinsurer. We further show that the insurer always prefers the tree structure to the chain structure, in which the risk of the insurer is shared sequentially among all reinsurers.
{"title":"Reinsurance games with \u0000$boldsymbol{{n}}$\u0000 variance-premium reinsurers: from tree to chain","authors":"Jingyi Cao, Dongchen Li, V. Young, B. Zou","doi":"10.1017/asb.2023.24","DOIUrl":"https://doi.org/10.1017/asb.2023.24","url":null,"abstract":"Abstract This paper studies dynamic reinsurance contracting and competition problems under model ambiguity in a reinsurance market with one primary insurer and n reinsurers, who apply the variance premium principle and who are distinguished by their levels of ambiguity aversion. The insurer negotiates reinsurance policies with all reinsurers simultaneously, which leads to a reinsurance tree structure with full competition among the reinsurers. We model the reinsurance contracting problems between the insurer and reinsurers by Stackelberg differential games and the competition among the reinsurers by a non-cooperative Nash game. We derive equilibrium strategies in semi-closed form for all the companies, whose objective is to maximize their expected surpluses penalized by a squared-error divergence term that measures their ambiguity. We find that, in equilibrium, the insurer purchases a positive amount of proportional reinsurance from each reinsurer. We further show that the insurer always prefers the tree structure to the chain structure, in which the risk of the insurer is shared sequentially among all reinsurers.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"16 1","pages":"706 - 728"},"PeriodicalIF":1.9,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81940006","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 investigate the feasibility of cyber risk transfer through insurance-linked securities (ILS). On the investor side, we elicit the preferred characteristics of cyber ILS and the corresponding return expectations. We then estimate the cost of equity of insurers and compare it to the Rate on Line expected by investors to match demand and supply in the cyber ILS market. Our results show that cyber ILS will work for both cedents and investors if the cyber risk is sufficiently well understood. Thus, challenges related to cyber risk modeling need to be overcome before a meaningful cyber ILS market may emerge.
{"title":"Cyber insurance-linked securities","authors":"Alexander Braun, M. Eling, Christoph Jaenicke","doi":"10.1017/asb.2023.22","DOIUrl":"https://doi.org/10.1017/asb.2023.22","url":null,"abstract":"Abstract We investigate the feasibility of cyber risk transfer through insurance-linked securities (ILS). On the investor side, we elicit the preferred characteristics of cyber ILS and the corresponding return expectations. We then estimate the cost of equity of insurers and compare it to the Rate on Line expected by investors to match demand and supply in the cyber ILS market. Our results show that cyber ILS will work for both cedents and investors if the cyber risk is sufficiently well understood. Thus, challenges related to cyber risk modeling need to be overcome before a meaningful cyber ILS market may emerge.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"16 1","pages":"684 - 705"},"PeriodicalIF":1.9,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86505262","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 The prominence of the Euler allocation rule (EAR) is rooted in the fact that it is the only return on risk-adjusted capital (RORAC) compatible capital allocation rule. When the total regulatory capital is set using the value-at-risk (VaR), the EAR becomes – using a statistical term – the quantile-regression (QR) function. Although the cumulative QR function (i.e., an integral of the QR function) has received considerable attention in the literature, a fully developed statistical inference theory for the QR function itself has been elusive. In the present paper, we develop such a theory based on an empirical QR estimator, for which we establish consistency, asymptotic normality, and standard error estimation. This makes the herein developed results readily applicable in practice, thus facilitating decision making within the RORAC paradigm, conditional mean risk sharing, and current regulatory frameworks.
{"title":"Estimating the VaR-induced Euler allocation rule","authors":"N.V. Gribkova, J. Su, R. Zitikis","doi":"10.1017/asb.2023.17","DOIUrl":"https://doi.org/10.1017/asb.2023.17","url":null,"abstract":"Abstract The prominence of the Euler allocation rule (EAR) is rooted in the fact that it is the only return on risk-adjusted capital (RORAC) compatible capital allocation rule. When the total regulatory capital is set using the value-at-risk (VaR), the EAR becomes – using a statistical term – the quantile-regression (QR) function. Although the cumulative QR function (i.e., an integral of the QR function) has received considerable attention in the literature, a fully developed statistical inference theory for the QR function itself has been elusive. In the present paper, we develop such a theory based on an empirical QR estimator, for which we establish consistency, asymptotic normality, and standard error estimation. This makes the herein developed results readily applicable in practice, thus facilitating decision making within the RORAC paradigm, conditional mean risk sharing, and current regulatory frameworks.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135215460","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 focus on modelling categorical features and improving predictive power of neural networks with mixed categorical and numerical features in supervised learning tasks. The goal of this paper is to challenge the current dominant approach in actuarial data science with a new architecture of a neural network and a new training algorithm. The key proposal is to use a joint embedding for all categorical features, instead of separate entity embeddings, to determine the numerical representation of the categorical features which is fed, together with all other numerical features, into hidden layers of a neural network with a target response. In addition, we postulate that we should initialize the numerical representation of the categorical features and other parameters of the hidden layers of the neural network with parameters trained with (denoising) autoencoders in unsupervised learning tasks, instead of using random initialization of parameters. Since autoencoders for categorical data play an important role in this research, they are investigated in more depth in the paper. We illustrate our ideas with experiments on a real data set with claim numbers, and we demonstrate that we can achieve a higher predictive power of the network.
{"title":"The use of autoencoders for training neural networks with mixed categorical and numerical features","authors":"Łukasz Delong, Anna Kozak","doi":"10.1017/asb.2023.15","DOIUrl":"https://doi.org/10.1017/asb.2023.15","url":null,"abstract":"Abstract We focus on modelling categorical features and improving predictive power of neural networks with mixed categorical and numerical features in supervised learning tasks. The goal of this paper is to challenge the current dominant approach in actuarial data science with a new architecture of a neural network and a new training algorithm. The key proposal is to use a joint embedding for all categorical features, instead of separate entity embeddings, to determine the numerical representation of the categorical features which is fed, together with all other numerical features, into hidden layers of a neural network with a target response. In addition, we postulate that we should initialize the numerical representation of the categorical features and other parameters of the hidden layers of the neural network with parameters trained with (denoising) autoencoders in unsupervised learning tasks, instead of using random initialization of parameters. Since autoencoders for categorical data play an important role in this research, they are investigated in more depth in the paper. We illustrate our ideas with experiments on a real data set with claim numbers, and we demonstrate that we can achieve a higher predictive power of the network.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"15 1","pages":"213 - 232"},"PeriodicalIF":1.9,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84402899","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 Fuelled by the big data explosion, a new methodology to estimate sub-annual death probabilities has recently been proposed, opening new insurance business opportunities. This new approach exploits all the detailed information available from millions of microdata records to develop seasonal-ageing indexes (SAIs) from which sub-annual (quarterly) life tables can be derived from annual tables. In this paper, we explore whether a shortcut could be taken in the estimation of SAIs and (life insurance) sub-annual death rates. We propose three different approximations, in which estimates are attained by using just a small bunch of thousands of data records and assess their impact on several competitive markets defined from an actual portfolio of life insurance policies. Our analyses clearly point to the shortcuts as good practical alternatives that can be used in real-life insurance markets. Noticeably, we see that embracing the new quarterly based approach, even using only an approximation (shortcut), is economically preferable to using the associated annual table, offering a significant competitive advantage to the company adopting this innovation.
{"title":"Shortcuts for the construction of sub-annual life tables","authors":"J. Pavía, Josep Lledó","doi":"10.1017/asb.2023.16","DOIUrl":"https://doi.org/10.1017/asb.2023.16","url":null,"abstract":"Abstract Fuelled by the big data explosion, a new methodology to estimate sub-annual death probabilities has recently been proposed, opening new insurance business opportunities. This new approach exploits all the detailed information available from millions of microdata records to develop seasonal-ageing indexes (SAIs) from which sub-annual (quarterly) life tables can be derived from annual tables. In this paper, we explore whether a shortcut could be taken in the estimation of SAIs and (life insurance) sub-annual death rates. We propose three different approximations, in which estimates are attained by using just a small bunch of thousands of data records and assess their impact on several competitive markets defined from an actual portfolio of life insurance policies. Our analyses clearly point to the shortcuts as good practical alternatives that can be used in real-life insurance markets. Noticeably, we see that embracing the new quarterly based approach, even using only an approximation (shortcut), is economically preferable to using the associated annual table, offering a significant competitive advantage to the company adopting this innovation.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"37 1","pages":"332 - 350"},"PeriodicalIF":1.9,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89591087","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 study, we quantify the relationship between socio-economic status and life expectancy and identify combinations of socio-economic variables that are particularly useful for explaining mortality differences between neighbourhoods in England. We achieve this by examining socio-economic variation in mortality experiences across small areas in England known as lower layer super output areas (LSOAs). We then consider 12 socio-economic variables that are known to have a strong association with mortality. We estimate the relationship between those variables and mortality rates using a random forest algorithm. Based on the resulting estimate, we then create a new socio-economic mortality index – the Longevity Index for England (LIFE). The index is constructed in a way that eliminates the impact of care homes that might artificially increase mortality rates in LSOAs with care homes compared to LSOAs that do not contain a care home. Using mortality data for different age groups, we make the index age-dependent and investigate the impact of specific socio-economic characteristics on the age-specific mortality risk. We compare the explanatory power of the LIFE index to the English Index of Multiple Deprivation (IMD) as predictors of mortality. While we find that the IMD can explain regional mortality differences to some extent, the LIFE index has significantly greater explanatory power for mortality differences between regions. Our empirical results also indicate that income deprivation amongst the elderly and employment deprivation are the most significant socio-economic factors for explaining mortality variation across LSOAs in England.
{"title":"Modelling socio-economic mortality at neighbourhood level","authors":"Jie Wen, A. Cairns, T. Kleinow","doi":"10.1017/asb.2023.12","DOIUrl":"https://doi.org/10.1017/asb.2023.12","url":null,"abstract":"Abstract In this study, we quantify the relationship between socio-economic status and life expectancy and identify combinations of socio-economic variables that are particularly useful for explaining mortality differences between neighbourhoods in England. We achieve this by examining socio-economic variation in mortality experiences across small areas in England known as lower layer super output areas (LSOAs). We then consider 12 socio-economic variables that are known to have a strong association with mortality. We estimate the relationship between those variables and mortality rates using a random forest algorithm. Based on the resulting estimate, we then create a new socio-economic mortality index – the Longevity Index for England (LIFE). The index is constructed in a way that eliminates the impact of care homes that might artificially increase mortality rates in LSOAs with care homes compared to LSOAs that do not contain a care home. Using mortality data for different age groups, we make the index age-dependent and investigate the impact of specific socio-economic characteristics on the age-specific mortality risk. We compare the explanatory power of the LIFE index to the English Index of Multiple Deprivation (IMD) as predictors of mortality. While we find that the IMD can explain regional mortality differences to some extent, the LIFE index has significantly greater explanatory power for mortality differences between regions. Our empirical results also indicate that income deprivation amongst the elderly and employment deprivation are the most significant socio-economic factors for explaining mortality variation across LSOAs in England.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"61 1","pages":"285 - 310"},"PeriodicalIF":1.9,"publicationDate":"2023-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86582660","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 consider a premium control problem in discrete time, formulated in terms of a Markov decision process. In a simplified setting, the optimal premium rule can be derived with dynamic programming methods. However, these classical methods are not feasible in a more realistic setting due to the dimension of the state space and lack of explicit expressions for transition probabilities. We explore reinforcement learning techniques, using function approximation, to solve the premium control problem for realistic stochastic models. We illustrate the appropriateness of the approximate optimal premium rule compared with the true optimal premium rule in a simplified setting and further demonstrate that the approximate optimal premium rule outperforms benchmark rules in more realistic settings where classical approaches fail.
{"title":"Premium control with reinforcement learning","authors":"L. Palmborg, F. Lindskog","doi":"10.1017/asb.2023.13","DOIUrl":"https://doi.org/10.1017/asb.2023.13","url":null,"abstract":"Abstract We consider a premium control problem in discrete time, formulated in terms of a Markov decision process. In a simplified setting, the optimal premium rule can be derived with dynamic programming methods. However, these classical methods are not feasible in a more realistic setting due to the dimension of the state space and lack of explicit expressions for transition probabilities. We explore reinforcement learning techniques, using function approximation, to solve the premium control problem for realistic stochastic models. We illustrate the appropriateness of the approximate optimal premium rule compared with the true optimal premium rule in a simplified setting and further demonstrate that the approximate optimal premium rule outperforms benchmark rules in more realistic settings where classical approaches fail.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"83 1","pages":"233 - 257"},"PeriodicalIF":1.9,"publicationDate":"2023-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90564327","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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