Pub Date : 2021-10-26DOI: 10.1080/03461238.2023.2234916
Thomas Bernhardt, Ge Qu
The stability of income payments in a pooled annuity fund is studied. In those funds, members receive a fluctuating income depending on their experienced mortality in exchange for their pension savings. The focus is on describing the influence of different initial savings on the ability of the fund to provide a stable income in retirement. Because of this, members coincide in their characteristics except for their initial savings. We identify a term, which we dub ``implied number of homogeneous members'', that directly links the initial savings to the size of the income fluctuations. Our main contribution is the analysis of this term and the development of a criterion to answer the question of whether or not a given group of same-aged people should pool their funds together.
{"title":"Wealth heterogeneity in a closed pooled annuity fund","authors":"Thomas Bernhardt, Ge Qu","doi":"10.1080/03461238.2023.2234916","DOIUrl":"https://doi.org/10.1080/03461238.2023.2234916","url":null,"abstract":"The stability of income payments in a pooled annuity fund is studied. In those funds, members receive a fluctuating income depending on their experienced mortality in exchange for their pension savings. The focus is on describing the influence of different initial savings on the ability of the fund to provide a stable income in retirement. Because of this, members coincide in their characteristics except for their initial savings. We identify a term, which we dub ``implied number of homogeneous members'', that directly links the initial savings to the size of the income fluctuations. Our main contribution is the analysis of this term and the development of a criterion to answer the question of whether or not a given group of same-aged people should pool their funds together.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"254 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89059624","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}
Pub Date : 2021-10-11DOI: 10.1080/03461238.2021.1998921
Martin Bladt
In this paper, we demonstrate through the use of matrix calculus a transparent analysis of fractional inhomogeneous Markov models for life insurance where transition matrices commute. The resulting formulae are intuitive matrix generalizations of their single-state counterparts, and the absorption times are matrix versions of well-known scalar distributions. A further advantage of this approach is that it allows extending the analysis to the non-Markovian case where sojourns are Mittag-Leffler distributed, and where the absorption times are fractional phase-type distributed. Considering deterministic time transforms gives rise to fractional inhomogeneous phase-type distributions as absorption times. The latter underlying processes are an example of a regime where not only the present but also the history of a policyholder influences its future evolution. The sub-exponential nature of stable distributions translates into the multi-state insurance model as a random longevity risk at any given state of the chain.
{"title":"Fractional inhomogeneous multi-state models in life insurance","authors":"Martin Bladt","doi":"10.1080/03461238.2021.1998921","DOIUrl":"https://doi.org/10.1080/03461238.2021.1998921","url":null,"abstract":"In this paper, we demonstrate through the use of matrix calculus a transparent analysis of fractional inhomogeneous Markov models for life insurance where transition matrices commute. The resulting formulae are intuitive matrix generalizations of their single-state counterparts, and the absorption times are matrix versions of well-known scalar distributions. A further advantage of this approach is that it allows extending the analysis to the non-Markovian case where sojourns are Mittag-Leffler distributed, and where the absorption times are fractional phase-type distributed. Considering deterministic time transforms gives rise to fractional inhomogeneous phase-type distributions as absorption times. The latter underlying processes are an example of a regime where not only the present but also the history of a policyholder influences its future evolution. The sub-exponential nature of stable distributions translates into the multi-state insurance model as a random longevity risk at any given state of the chain.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"55 1","pages":"510 - 531"},"PeriodicalIF":1.8,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86675551","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}
Pub Date : 2021-10-10DOI: 10.1080/03461238.2021.1978535
W. Neuhaus
Traditional claim estimation in general insurance works with accident year cohorts and development patterns. For the impending International Financial Reporting Standard (IFRS) 17 Insurance Contracts and often for reinsurance purposes, claim estimates must be split by contract year. This paper proposes to add contract year as a cohort classifier and to adjust the development patterns accordingly. To this end, we use the continuous time models of Hesselager and Norberg. Having contract year as an additional cohort classifier, display of claim estimates by contract year and/or accident year becomes a simple matter of summation across the appropriate dimensions. The continuous time model also enables us to derive mutually consistent development patterns for discrete time intervals of different length, such as years and quarters. In addition to delivering consistent development patterns in discrete time, continuous time modelling offers the advantage of requiring only a fixed number of model parameters. Although most of the derivations in this paper are explained in terms of claim numbers, the mechanics can also be applied to claim payments.
{"title":"Consistent development patterns","authors":"W. Neuhaus","doi":"10.1080/03461238.2021.1978535","DOIUrl":"https://doi.org/10.1080/03461238.2021.1978535","url":null,"abstract":"Traditional claim estimation in general insurance works with accident year cohorts and development patterns. For the impending International Financial Reporting Standard (IFRS) 17 Insurance Contracts and often for reinsurance purposes, claim estimates must be split by contract year. This paper proposes to add contract year as a cohort classifier and to adjust the development patterns accordingly. To this end, we use the continuous time models of Hesselager and Norberg. Having contract year as an additional cohort classifier, display of claim estimates by contract year and/or accident year becomes a simple matter of summation across the appropriate dimensions. The continuous time model also enables us to derive mutually consistent development patterns for discrete time intervals of different length, such as years and quarters. In addition to delivering consistent development patterns in discrete time, continuous time modelling offers the advantage of requiring only a fixed number of model parameters. Although most of the derivations in this paper are explained in terms of claim numbers, the mechanics can also be applied to claim payments.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"29 1","pages":"933 - 945"},"PeriodicalIF":1.8,"publicationDate":"2021-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86520574","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}
Pub Date : 2021-10-03DOI: 10.1080/03461238.2021.1980430
Jackie Li, David G. W. Pitt, Han Li
Past mortality experience has shown that the variability in mortality levels is not constant and can be higher than what the usual Poisson assumption implies. This paper proposes two ways to tackle the heterogeneity often present in mortality data. First, an additional dispersion submodel is developed and combined with the mean model to perform a joint modelling of the mean and the dispersion. Moreover, a flexible group of distributions called the Tweedie family is adopted to model the number of deaths. Using Australian and other mortality data, the results of this study show that this Tweedie double modelling framework can generally improve the fitting performance and also leads to a more adequate allowance for longevity risk when valuing pension annuities.
{"title":"Dispersion modelling of mortality for both sexes with Tweedie distributions","authors":"Jackie Li, David G. W. Pitt, Han Li","doi":"10.1080/03461238.2021.1980430","DOIUrl":"https://doi.org/10.1080/03461238.2021.1980430","url":null,"abstract":"Past mortality experience has shown that the variability in mortality levels is not constant and can be higher than what the usual Poisson assumption implies. This paper proposes two ways to tackle the heterogeneity often present in mortality data. First, an additional dispersion submodel is developed and combined with the mean model to perform a joint modelling of the mean and the dispersion. Moreover, a flexible group of distributions called the Tweedie family is adopted to model the number of deaths. Using Australian and other mortality data, the results of this study show that this Tweedie double modelling framework can generally improve the fitting performance and also leads to a more adequate allowance for longevity risk when valuing pension annuities.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"89 1","pages":"356 - 374"},"PeriodicalIF":1.8,"publicationDate":"2021-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90578754","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}
Pub Date : 2021-10-03DOI: 10.1080/03461238.2021.1979639
Tzuling Lin, Cary Chi‐liang Tsai
As world populations age along with speedy internationalization, forecasting mortality for multiple countries or populations with similar socio-economic conditions or close cultural connections has become essential. We apply the hierarchical Bayesian theory to the random walk with drift model governing the dynamics of the logarithm of central death rates for each age and population. Using the mortality data for both genders of three developed countries for an age span 25–84 and a series of fitting age-year windows, and further extending the data set to include both genders of twenty countries, we conclude that the proposed hierarchical Bayesian framework can more accurately capture the mortality trends and overall outperforms the Lee–Carter model and its three extensions in mortality forecasting. Grouping both genders of the twenty countries in three ways, we find that the expected improvement rates per year in the logarithm of central death rate for all twenty countries converge to about 2% except for the US.
{"title":"Hierarchical Bayesian modeling of multi-country mortality rates","authors":"Tzuling Lin, Cary Chi‐liang Tsai","doi":"10.1080/03461238.2021.1979639","DOIUrl":"https://doi.org/10.1080/03461238.2021.1979639","url":null,"abstract":"As world populations age along with speedy internationalization, forecasting mortality for multiple countries or populations with similar socio-economic conditions or close cultural connections has become essential. We apply the hierarchical Bayesian theory to the random walk with drift model governing the dynamics of the logarithm of central death rates for each age and population. Using the mortality data for both genders of three developed countries for an age span 25–84 and a series of fitting age-year windows, and further extending the data set to include both genders of twenty countries, we conclude that the proposed hierarchical Bayesian framework can more accurately capture the mortality trends and overall outperforms the Lee–Carter model and its three extensions in mortality forecasting. Grouping both genders of the twenty countries in three ways, we find that the expected improvement rates per year in the logarithm of central death rate for all twenty countries converge to about 2% except for the US.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"1 1","pages":"375 - 398"},"PeriodicalIF":1.8,"publicationDate":"2021-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82725378","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}
Pub Date : 2021-09-29DOI: 10.1080/03461238.2022.2161411
Josef Anton Strini, S. Thonhauser
We consider the dividend maximization problem including a ruin penalty in a diffusion environment. The additional penalty term is motivated by a constraint on dividend strategies. Intentionally, we use different discount rates for the dividends and the penalty, which causes time-inconsistency. This allows to study different types of constraints. For the diffusion approximation of the classical surplus process we derive an explicit equilibrium dividend strategy and the associated value function. Inspired by duality arguments, we can identify a particular equilibrium strategy such that for a given initial surplus the imposed constraint is fulfilled. Furthermore, we illustrate our findings with a numerical example.
{"title":"Time-inconsistent view on a dividend problem with penalty","authors":"Josef Anton Strini, S. Thonhauser","doi":"10.1080/03461238.2022.2161411","DOIUrl":"https://doi.org/10.1080/03461238.2022.2161411","url":null,"abstract":"We consider the dividend maximization problem including a ruin penalty in a diffusion environment. The additional penalty term is motivated by a constraint on dividend strategies. Intentionally, we use different discount rates for the dividends and the penalty, which causes time-inconsistency. This allows to study different types of constraints. For the diffusion approximation of the classical surplus process we derive an explicit equilibrium dividend strategy and the associated value function. Inspired by duality arguments, we can identify a particular equilibrium strategy such that for a given initial surplus the imposed constraint is fulfilled. Furthermore, we illustrate our findings with a numerical example.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"1 1","pages":"811 - 833"},"PeriodicalIF":1.8,"publicationDate":"2021-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83624208","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}
Pub Date : 2021-09-08DOI: 10.1080/03461238.2021.1971756
Yu Yuan, Zhibin Liang, Xiaoru Han
In this paper, we determine a robust reinsurance contract from joint interests of the insurer and reinsurer under the framework of Stackelberg differential game. More specifically, the reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal proportional reinsurance to purchase. In order to defend the large shocks of wealth process, a loss-dependent premium principle is applied to the insurer. Meanwhile, we incorporate model uncertainty into the reinsurer's controlled surplus due to the asymmetric information. Under the time-consistent mean-variance criterion, we derive the robust reinsurance contract explicitly by solving the coupled extended Hamilton–Jacobi–Bellman systems. It is interesting to prove that the optimal premium control for the reinsurer is determined by a time-adjusted variance principle. In addition, we find that the reinsurer would like to raise the reinsurance price to guard against the model uncertainty, which consequently decreases the insurer's reinsurance demand. Finally, further analyses are provided to show the necessity of considering the model uncertainty; otherwise, the reinsurance company will suffer a great loss of utility.
{"title":"Robust reinsurance contract with asymmetric information in a stochastic Stackelberg differential game","authors":"Yu Yuan, Zhibin Liang, Xiaoru Han","doi":"10.1080/03461238.2021.1971756","DOIUrl":"https://doi.org/10.1080/03461238.2021.1971756","url":null,"abstract":"In this paper, we determine a robust reinsurance contract from joint interests of the insurer and reinsurer under the framework of Stackelberg differential game. More specifically, the reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal proportional reinsurance to purchase. In order to defend the large shocks of wealth process, a loss-dependent premium principle is applied to the insurer. Meanwhile, we incorporate model uncertainty into the reinsurer's controlled surplus due to the asymmetric information. Under the time-consistent mean-variance criterion, we derive the robust reinsurance contract explicitly by solving the coupled extended Hamilton–Jacobi–Bellman systems. It is interesting to prove that the optimal premium control for the reinsurer is determined by a time-adjusted variance principle. In addition, we find that the reinsurer would like to raise the reinsurance price to guard against the model uncertainty, which consequently decreases the insurer's reinsurance demand. Finally, further analyses are provided to show the necessity of considering the model uncertainty; otherwise, the reinsurance company will suffer a great loss of utility.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"70 1","pages":"328 - 355"},"PeriodicalIF":1.8,"publicationDate":"2021-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80288916","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}
Pub Date : 2021-08-30DOI: 10.1080/03461238.2022.2090272
Karim Barigou, Daniël Linders, Fan Yang
ABSTRACT This paper introduces new valuation schemes called actuarial-consistent valuations for insurance liabilities which depend on both financial and actuarial risks, which imposes that all actuarial risks are priced via standard actuarial principles. We propose to extend standard actuarial principles by a new actuarial-consistent procedure, which we call ‘two-step actuarial valuations’. In the case valuations are coherent, we show that actuarial-consistent valuations are equivalent to two-step actuarial valuations. We also discuss the connection with ‘two-step market-consistent valuations’ from Pelsser, A. & Stadje, M. [(2014). Time-consistent and market-consistent evaluations. Mathematical Finance 24(1), 25–65]. In particular, we discuss how the dependence structure between actuarial and financial risks impacts both actuarial-consistent and market-consistent valuations.
摘要本文介绍了一种新的保险负债估值方案，称为精算一致估值，它既依赖于财务风险，也依赖于精算风险，它要求所有精算风险都通过标准精算原则定价。我们建议通过一种新的精算一致性程序来扩展标准精算原则，我们称之为“两步精算估值”。在估值一致的情况下，我们证明了精算一致估值等同于两步精算估值。我们还讨论了与Pelsser, A. & Stadje, M.(2014)的“两步市场一致估值”的联系。时间一致和市场一致的评估。数学金融，24(1)，25-65]。特别地，我们讨论了精算和金融风险之间的依赖结构如何影响精算一致性和市场一致性估值。
{"title":"Actuarial-consistency and two-step actuarial valuations: a new paradigm to insurance valuation","authors":"Karim Barigou, Daniël Linders, Fan Yang","doi":"10.1080/03461238.2022.2090272","DOIUrl":"https://doi.org/10.1080/03461238.2022.2090272","url":null,"abstract":"ABSTRACT This paper introduces new valuation schemes called actuarial-consistent valuations for insurance liabilities which depend on both financial and actuarial risks, which imposes that all actuarial risks are priced via standard actuarial principles. We propose to extend standard actuarial principles by a new actuarial-consistent procedure, which we call ‘two-step actuarial valuations’. In the case valuations are coherent, we show that actuarial-consistent valuations are equivalent to two-step actuarial valuations. We also discuss the connection with ‘two-step market-consistent valuations’ from Pelsser, A. & Stadje, M. [(2014). Time-consistent and market-consistent evaluations. Mathematical Finance 24(1), 25–65]. In particular, we discuss how the dependence structure between actuarial and financial risks impacts both actuarial-consistent and market-consistent valuations.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"14 1","pages":"191 - 217"},"PeriodicalIF":1.8,"publicationDate":"2021-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84351175","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}
Pub Date : 2021-08-19DOI: 10.1080/03461238.2021.1966831
Christoph Hambel, H. Kraft, Claus Munk
We study canonical consumption-savings problems of an individual involving uninsurable biometric risk. These problems are important in many applications from insurance economics and actuarial science. Since biometric risk is uninsurable, closed-form solutions do not exist and thus the problems must be approached by numerical methods. We propose a powerful approach where the solution is obtained by optimizing over a parametrized family of consumption strategies. In settings with mortality risk, critical illness risk, and habit formation, our solution method outperforms the well-established finite-difference approach both in run time and in precision. Our method also delivers a precision measure and closed-form representations of the optimal controls.
{"title":"Solving life-cycle problems with biometric risk by artificial insurance markets","authors":"Christoph Hambel, H. Kraft, Claus Munk","doi":"10.1080/03461238.2021.1966831","DOIUrl":"https://doi.org/10.1080/03461238.2021.1966831","url":null,"abstract":"We study canonical consumption-savings problems of an individual involving uninsurable biometric risk. These problems are important in many applications from insurance economics and actuarial science. Since biometric risk is uninsurable, closed-form solutions do not exist and thus the problems must be approached by numerical methods. We propose a powerful approach where the solution is obtained by optimizing over a parametrized family of consumption strategies. In settings with mortality risk, critical illness risk, and habit formation, our solution method outperforms the well-established finite-difference approach both in run time and in precision. Our method also delivers a precision measure and closed-form representations of the optimal controls.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"20 1","pages":"307 - 327"},"PeriodicalIF":1.8,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85156654","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}
Pub Date : 2021-08-17DOI: 10.1080/03461238.2021.1962962
M. Escobar-Anel, Markus Wahl, R. Zagst
Regulatory risk constraints as in the European Solvency II standard formula for insurance companies may lead to wealth-dependent constraints on the investment strategy. We develop two solution approaches for portfolio optimization problems in continuous time with wealth-dependent constraint sets. In the first approach, we reduce the optimization problem to an associate problem with constraints independent of wealth and a different utility function. The associate problem is then solved using known convex duality results. In the second approach, we use a change of control. We apply these results to Solvency II constraint sets and find that even for an investor with HARA utility who inherently reduces risk in times of distress, the constraints help to prevent the investor from taking too much risk in an optimistic market. Furthermore, we measure significant loss in utility and reduction in risk caused by the constraints, and we also evaluate the trade-off between these two effects.
{"title":"Portfolio optimization with wealth-dependent risk constraints","authors":"M. Escobar-Anel, Markus Wahl, R. Zagst","doi":"10.1080/03461238.2021.1962962","DOIUrl":"https://doi.org/10.1080/03461238.2021.1962962","url":null,"abstract":"Regulatory risk constraints as in the European Solvency II standard formula for insurance companies may lead to wealth-dependent constraints on the investment strategy. We develop two solution approaches for portfolio optimization problems in continuous time with wealth-dependent constraint sets. In the first approach, we reduce the optimization problem to an associate problem with constraints independent of wealth and a different utility function. The associate problem is then solved using known convex duality results. In the second approach, we use a change of control. We apply these results to Solvency II constraint sets and find that even for an investor with HARA utility who inherently reduces risk in times of distress, the constraints help to prevent the investor from taking too much risk in an optimistic market. Furthermore, we measure significant loss in utility and reduction in risk caused by the constraints, and we also evaluate the trade-off between these two effects.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"85 1","pages":"244 - 268"},"PeriodicalIF":1.8,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84039949","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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