Y. Shimizu, Kana Shirai, Yuta Kojima, Daiki Mitsuda, Mahiro Inoue
Abstract The survival energy model (SEM) is a recently introduced novel approach to mortality prediction, which offers a cohort-wise distribution function of the time of death as the first hitting time of a “survival energy” diffusion process to zero. In this study, we propose a novel SEM that can serve as a suitable candidate in the family of prediction models. We also proposed a method to improve the prediction in an earlier work. We further examine the practical advantages of SEM over existing mortality models.
{"title":"Survival energy models for mortality prediction and future prospects","authors":"Y. Shimizu, Kana Shirai, Yuta Kojima, Daiki Mitsuda, Mahiro Inoue","doi":"10.1017/asb.2023.10","DOIUrl":"https://doi.org/10.1017/asb.2023.10","url":null,"abstract":"Abstract The survival energy model (SEM) is a recently introduced novel approach to mortality prediction, which offers a cohort-wise distribution function of the time of death as the first hitting time of a “survival energy” diffusion process to zero. In this study, we propose a novel SEM that can serve as a suitable candidate in the family of prediction models. We also proposed a method to improve the prediction in an earlier work. We further examine the practical advantages of SEM over existing mortality models.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"105 1","pages":"377 - 391"},"PeriodicalIF":1.9,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85690637","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 determine the fair value of a pension buyout contract under the assumption that changes in mortality can have an impact on financial markets. Our proposed model allows for shocks to occur simultaneously in mortality rates and financial markets, so that strong changes in mortality rates can affect interest rates and asset prices. This approach challenges the common but very strong assumption that mortality and market risk drivers are independent. A simulation-based pricing framework is applied to determine the buyout premium for a hypothetical fully funded pension scheme. The results of an extensive sensitivity analysis show how buyout prices are affected by changes in mortality and financial markets. Surprisingly, we find that the impact of shocks is similar whether or not these shocks occur simultaneously or not, although there are some differences in annuity prices and buyout premiums. We clearly see that the intensity and severity of shocks, and asset price volatility play a dominant role for buyout prices.
{"title":"The impact of simultaneous shocks to financial markets and mortality on pension buy-out prices","authors":"Ayşe Arık, Ö. Uğur, T. Kleinow","doi":"10.1017/asb.2023.11","DOIUrl":"https://doi.org/10.1017/asb.2023.11","url":null,"abstract":"Abstract In this paper, we determine the fair value of a pension buyout contract under the assumption that changes in mortality can have an impact on financial markets. Our proposed model allows for shocks to occur simultaneously in mortality rates and financial markets, so that strong changes in mortality rates can affect interest rates and asset prices. This approach challenges the common but very strong assumption that mortality and market risk drivers are independent. A simulation-based pricing framework is applied to determine the buyout premium for a hypothetical fully funded pension scheme. The results of an extensive sensitivity analysis show how buyout prices are affected by changes in mortality and financial markets. Surprisingly, we find that the impact of shocks is similar whether or not these shocks occur simultaneously or not, although there are some differences in annuity prices and buyout premiums. We clearly see that the intensity and severity of shocks, and asset price volatility play a dominant role for buyout prices.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"357 1","pages":"392 - 417"},"PeriodicalIF":1.9,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76498063","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 introduce the 3-step hedge-based valuation for the valuation of hybrid claims. We consider an insurance portfolio which is exposed to traded risks, diversifiable risks and non-traded systematic risks. The class of 3-step hedge-based valuations is equivalent with the class of fair valuations. Closed-form solutions are derived for a portfolio of unit-linked contracts under the assumption of independence between financial and non-financial risks. We also consider the additive 3-step valuation and show that this additive valuation is a member of the more general class of 3-step hedge-based valuations.
{"title":"The 3-step hedge-based valuation: fair valuation in the presence of systematic risks","authors":"Daniël Linders","doi":"10.1017/asb.2023.8","DOIUrl":"https://doi.org/10.1017/asb.2023.8","url":null,"abstract":"Abstract In this paper, we introduce the 3-step hedge-based valuation for the valuation of hybrid claims. We consider an insurance portfolio which is exposed to traded risks, diversifiable risks and non-traded systematic risks. The class of 3-step hedge-based valuations is equivalent with the class of fair valuations. Closed-form solutions are derived for a portfolio of unit-linked contracts under the assumption of independence between financial and non-financial risks. We also consider the additive 3-step valuation and show that this additive valuation is a member of the more general class of 3-step hedge-based valuations.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"11 1","pages":"418 - 442"},"PeriodicalIF":1.9,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75091937","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}
Frédéric Godin, Emmanuel Hamel, Patrice Gaillardetz, Edwin Hon-Man Ng
Abstract This paper introduces a flexible risk decomposition method for life insurance contracts embedding several risk factors. Hedging can be naturally embedded in the framework. Although the method is applied to variable annuities in this work, it is also applicable in general to other insurance or financial contracts. The approach relies on applying an allocation principle to components of a Shapley decomposition of the gain and loss. The implementation of the allocation method requires the use of a stochastic on stochastic algorithm involving nested simulations. Numerical examples studying the relative impact of equity, interest rate and mortality risk for guaranteed minimal maturity benefit (GMMB) policies conclude our analysis.
{"title":"Risk allocation through shapley decompositions, with applications to variable annuities","authors":"Frédéric Godin, Emmanuel Hamel, Patrice Gaillardetz, Edwin Hon-Man Ng","doi":"10.1017/asb.2023.7","DOIUrl":"https://doi.org/10.1017/asb.2023.7","url":null,"abstract":"Abstract This paper introduces a flexible risk decomposition method for life insurance contracts embedding several risk factors. Hedging can be naturally embedded in the framework. Although the method is applied to variable annuities in this work, it is also applicable in general to other insurance or financial contracts. The approach relies on applying an allocation principle to components of a Shapley decomposition of the gain and loss. The implementation of the allocation method requires the use of a stochastic on stochastic algorithm involving nested simulations. Numerical examples studying the relative impact of equity, interest rate and mortality risk for guaranteed minimal maturity benefit (GMMB) policies conclude our analysis.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"6 1","pages":"311 - 331"},"PeriodicalIF":1.9,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87378569","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 model uncertainty issue is pervasive in virtually all applied fields but especially critical in insurance and finance. To hedge against the uncertainty of the underlying probability distribution, which we refer to as ambiguity, the worst case is often considered in quantifying the underlying risk. However, this worst-case treatment often yields results that are overly conservative. We argue that, in most practical situations, a generic risk is realized from multiple scenarios and the risk in some ordinary scenarios may be subject to negligible ambiguity so that it is safe to trust the reference distributions. Hence, we only need to consider the worst case in the other scenarios where ambiguity is significant. We implement this idea in the study of the worst-case moments of a risk in the hope to alleviate the over-conservativeness issue. Note that the ambiguity in our consideration exists in both the scenario indicator and the risk in the corresponding scenario, leading to a two-fold ambiguity issue. We employ the Wasserstein distance to construct an ambiguity ball. Then, we disentangle the ambiguity along the scenario indicator and the risk in the corresponding scenario, so that we convert the two-fold optimization problem into two one-fold problems. Our main result is a closed-form worst-case moment estimate. Our numerical studies illustrate that the consideration of partial ambiguity indeed greatly alleviates the over-conservativeness issue.
{"title":"Worst-case moments under partial ambiguity","authors":"Q. Tang, Yunshen Yang","doi":"10.1017/asb.2023.3","DOIUrl":"https://doi.org/10.1017/asb.2023.3","url":null,"abstract":"Abstract The model uncertainty issue is pervasive in virtually all applied fields but especially critical in insurance and finance. To hedge against the uncertainty of the underlying probability distribution, which we refer to as ambiguity, the worst case is often considered in quantifying the underlying risk. However, this worst-case treatment often yields results that are overly conservative. We argue that, in most practical situations, a generic risk is realized from multiple scenarios and the risk in some ordinary scenarios may be subject to negligible ambiguity so that it is safe to trust the reference distributions. Hence, we only need to consider the worst case in the other scenarios where ambiguity is significant. We implement this idea in the study of the worst-case moments of a risk in the hope to alleviate the over-conservativeness issue. Note that the ambiguity in our consideration exists in both the scenario indicator and the risk in the corresponding scenario, leading to a two-fold ambiguity issue. We employ the Wasserstein distance to construct an ambiguity ball. Then, we disentangle the ambiguity along the scenario indicator and the risk in the corresponding scenario, so that we convert the two-fold optimization problem into two one-fold problems. Our main result is a closed-form worst-case moment estimate. Our numerical studies illustrate that the consideration of partial ambiguity indeed greatly alleviates the over-conservativeness issue.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"25 1","pages":"443 - 465"},"PeriodicalIF":1.9,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88702005","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 tail index is an important parameter that measures how extreme events occur. In many practical cases, this tail index depends on covariates. In this paper,we assume that it takes a finite number of values over a partition of the covariate space. This article proposes a tail index partition-based rules extraction method that is able to construct estimates of the partition subsets and estimates of the tail index values. The method combines two steps: first an additive tree ensemble based on the Gamma deviance is fitted, and second a hierarchical clustering with spatial constraints is used to estimate the subsets of the partition. We also propose a global tree surrogate model to approximate the partition-based rules while providing an explainable model from the initial covariates. Our procedure is illustrated on simulated data. A real case study on wind property damages caused by tornadoes is finally presented.
{"title":"Tail index partition-based rules extraction with application to tornado damage insurance","authors":"Arthur Maillart, C. Robert","doi":"10.1017/asb.2023.1","DOIUrl":"https://doi.org/10.1017/asb.2023.1","url":null,"abstract":"Abstract The tail index is an important parameter that measures how extreme events occur. In many practical cases, this tail index depends on covariates. In this paper,we assume that it takes a finite number of values over a partition of the covariate space. This article proposes a tail index partition-based rules extraction method that is able to construct estimates of the partition subsets and estimates of the tail index values. The method combines two steps: first an additive tree ensemble based on the Gamma deviance is fitted, and second a hierarchical clustering with spatial constraints is used to estimate the subsets of the partition. We also propose a global tree surrogate model to approximate the partition-based rules while providing an explainable model from the initial covariates. Our procedure is illustrated on simulated data. A real case study on wind property damages caused by tornadoes is finally presented.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"16 1","pages":"258 - 284"},"PeriodicalIF":1.9,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82374475","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 article proposes a continuous time mortality model based on calendar years. Mortality rates belong to a mean-reverting random field indexed by time and age. In order to explain the improvement of life expectancies, the reversion level of mortality rates is the product of a deterministic function of age and of a decreasing jump-diffusion process driving the evolution of longevity. We provide a general closed-form expression for survival probabilities and develop it when the mean reversion level of mortality rates is proportional to a Gompertz–Makeham law. We develop an econometric estimation method and validate the model on the Belgian population.
{"title":"A calendar year mortality model in continuous time","authors":"Donatien Hainaut","doi":"10.1017/asb.2023.2","DOIUrl":"https://doi.org/10.1017/asb.2023.2","url":null,"abstract":"Abstract This article proposes a continuous time mortality model based on calendar years. Mortality rates belong to a mean-reverting random field indexed by time and age. In order to explain the improvement of life expectancies, the reversion level of mortality rates is the product of a deterministic function of age and of a decreasing jump-diffusion process driving the evolution of longevity. We provide a general closed-form expression for survival probabilities and develop it when the mean reversion level of mortality rates is proportional to a Gompertz–Makeham law. We develop an econometric estimation method and validate the model on the Belgian population.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"136 1","pages":"351 - 376"},"PeriodicalIF":1.9,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79617173","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}
Xinqiao Xie, Haiyan Liu, Tiantian Mao, Xiao Bai Zhu
Abstract We study a distributionally robust reinsurance problem with the risk measure being an expectile and under expected value premium principle. The mean and variance of the ground-up loss are known, but the loss distribution is otherwise unspecified. A minimax problem is formulated with its inner problem being a maximization problem over all distributions with known mean and variance. We show that the inner problem is equivalent to maximizing the problem over three-point distributions, reducing the infinite-dimensional optimization problem to a finite-dimensional optimization problem. The finite-dimensional optimization problem can be solved numerically. Numerical examples are given to study the impacts of the parameters involved.
{"title":"Distributionally robust reinsurance with expectile","authors":"Xinqiao Xie, Haiyan Liu, Tiantian Mao, Xiao Bai Zhu","doi":"10.1017/asb.2022.28","DOIUrl":"https://doi.org/10.1017/asb.2022.28","url":null,"abstract":"Abstract We study a distributionally robust reinsurance problem with the risk measure being an expectile and under expected value premium principle. The mean and variance of the ground-up loss are known, but the loss distribution is otherwise unspecified. A minimax problem is formulated with its inner problem being a maximization problem over all distributions with known mean and variance. We show that the inner problem is equivalent to maximizing the problem over three-point distributions, reducing the infinite-dimensional optimization problem to a finite-dimensional optimization problem. The finite-dimensional optimization problem can be solved numerically. Numerical examples are given to study the impacts of the parameters involved.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":"395 1","pages":"129 - 148"},"PeriodicalIF":1.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78943190","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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