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":null,"pages":null},"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}
{"title":"Target benefit pension plan with longevity risk and intergenerational equity – CORRIGENDUM","authors":"X. Rong, Cheng Tao, H Zhao","doi":"10.1017/asb.2023.9","DOIUrl":"https://doi.org/10.1017/asb.2023.9","url":null,"abstract":"","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74569885","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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Abstract We formalize a consumption–investment–insurance problem with the distinction of a state-dependent relative risk aversion. The state dependence refers to the state of the finite state Markov chain that also formalizes insurable risks such as health and lifetime uncertainty. We derive and analyze the implicit solution to the problem, compare it with special cases in the literature, and illustrate the range of results in a disability model where the relative risk aversion is preserved, decreases, or increases upon disability.
{"title":"Optimal consumption, investment, and insurance under state-dependent risk aversion","authors":"Mogens Steffensen, Julie Søe","doi":"10.1017/asb.2022.25","DOIUrl":"https://doi.org/10.1017/asb.2022.25","url":null,"abstract":"Abstract We formalize a consumption–investment–insurance problem with the distinction of a state-dependent relative risk aversion. The state dependence refers to the state of the finite state Markov chain that also formalizes insurable risks such as health and lifetime uncertainty. We derive and analyze the implicit solution to the problem, compare it with special cases in the literature, and illustrate the range of results in a disability model where the relative risk aversion is preserved, decreases, or increases upon disability.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75420299","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 a stochastic model for a target benefit pension plan suffering from rising longevity and falling fertility. Policies for postponing retirement are carried out to hedge the payment difficulties caused by the aging population. The plan members’ contributions are set in advance while the pension payments reflect intergenerational equity by a target payment level and intergenerational risk sharing by an adjustment. The pension fund is invested in both a risk-free asset and a risky asset. Applying the stochastic optimal control methods, we derive analytic solutions for optimal investment and benefit payment strategies which minimize the benefit risk. Besides, an optimal delayed retirement age which can hedge against the aging phenomenon under certain parameters is given. Therefore, it can provide a basis for quantifying the delay of retirement time.
{"title":"Target benefit pension plan with longevity risk and intergenerational equity","authors":"X. Rong, Chengcheng Tao, H Zhao","doi":"10.1017/asb.2022.27","DOIUrl":"https://doi.org/10.1017/asb.2022.27","url":null,"abstract":"Abstract We study a stochastic model for a target benefit pension plan suffering from rising longevity and falling fertility. Policies for postponing retirement are carried out to hedge the payment difficulties caused by the aging population. The plan members’ contributions are set in advance while the pension payments reflect intergenerational equity by a target payment level and intergenerational risk sharing by an adjustment. The pension fund is invested in both a risk-free asset and a risky asset. Applying the stochastic optimal control methods, we derive analytic solutions for optimal investment and benefit payment strategies which minimize the benefit risk. Besides, an optimal delayed retirement age which can hedge against the aging phenomenon under certain parameters is given. Therefore, it can provide a basis for quantifying the delay of retirement time.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75509457","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 an optimal investment problem under a joint limited expected relative loss and portfolio insurance constraint with a general random benchmark. By making use of a static Lagrangian method in a complete market setting, the optimal wealth and investment strategy can be fully determined along with the existence and uniqueness of the Lagrangian multipliers. Our numerical demonstration for various commonly used random benchmarks shows a trade-off between the portfolio outperformance and underperformance relative to the benchmark, which may not be captured by the widely used Omega ratio and its utility-transformed version, reflecting the impact of the benchmarking loss constraint. Furthermore, we develop a new portfolio performance measurement indicator that incorporates the agent’s utility loss aversion relative to the benchmark via solving an equivalent optimal asset allocation problem with a benchmark-reference-based preference. We show that the expected utility performance is well depicted by looking at this new portfolio performance ratio, suggesting a more suitable portfolio performance measurement under a limited loss constraint relative to a possibly random benchmark.
{"title":"Portfolio performance under benchmarking relative loss and portfolio insurance: From omega ratio to loss aversion","authors":"Tak Wa Ng, Thai Q. Nguyen","doi":"10.1017/asb.2022.26","DOIUrl":"https://doi.org/10.1017/asb.2022.26","url":null,"abstract":"Abstract We study an optimal investment problem under a joint limited expected relative loss and portfolio insurance constraint with a general random benchmark. By making use of a static Lagrangian method in a complete market setting, the optimal wealth and investment strategy can be fully determined along with the existence and uniqueness of the Lagrangian multipliers. Our numerical demonstration for various commonly used random benchmarks shows a trade-off between the portfolio outperformance and underperformance relative to the benchmark, which may not be captured by the widely used Omega ratio and its utility-transformed version, reflecting the impact of the benchmarking loss constraint. Furthermore, we develop a new portfolio performance measurement indicator that incorporates the agent’s utility loss aversion relative to the benchmark via solving an equivalent optimal asset allocation problem with a benchmark-reference-based preference. We show that the expected utility performance is well depicted by looking at this new portfolio performance ratio, suggesting a more suitable portfolio performance measurement under a limited loss constraint relative to a possibly random benchmark.","PeriodicalId":8617,"journal":{"name":"ASTIN Bulletin","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76532030","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：