Pub Date : 2023-02-21DOI: 10.1080/03461238.2023.2175326
Giovanni Cardillo, P. Giordani, Susanna Levantesi, A. Nigri, A. Spelta
To design appropriate pension or insurance plans it is crucial to understand mortality heterogeneity across demographic features, such as age, gender, and country. To this aim, we propose a coherent mortality forecasting methodology, which leverages the four-way CANDECOMP/PARAFAC and Vector-Error Correction models. We examine how age groups, years, countries, and gender impact target variables, namely log-centered mortality rates and compositional transformation of mortality data using the Human Mortality Database. The CANDECOMP/PARAFAC model synthesizes the behavior of the target variables through a few latent components and highlights the evolution of the temporal patterns. These patterns are employed to forecast future trajectories of mortality with Vector-Error Correction models, which account for the non-stationarity of the series. We carry out Monte Carlo simulations to obtain the distributions of the time component over the forecasted period 2001–2015, and we evaluate the goodness of the prediction by computing the Root Mean Square Error and the Mean Absolute Error. Our analysis underlines that understanding mortality dynamics in a high-dimensional framework is crucial for demographic assessments and could help design appropriate pension plans that mitigate the burden of increased longevity. The paper provides two steps further on methodological developments in the field of mortality analysis and forecasting in a high-dimensional space by (i) offering a comprehensive picture of mortality data through the four-way decomposition and (ii) designing appropriate forecasting of mortality data which relies on the projection of the temporal component through Vector-Error Correction models.
{"title":"Mortality forecasting using the four-way CANDECOMP/PARAFAC decomposition","authors":"Giovanni Cardillo, P. Giordani, Susanna Levantesi, A. Nigri, A. Spelta","doi":"10.1080/03461238.2023.2175326","DOIUrl":"https://doi.org/10.1080/03461238.2023.2175326","url":null,"abstract":"To design appropriate pension or insurance plans it is crucial to understand mortality heterogeneity across demographic features, such as age, gender, and country. To this aim, we propose a coherent mortality forecasting methodology, which leverages the four-way CANDECOMP/PARAFAC and Vector-Error Correction models. We examine how age groups, years, countries, and gender impact target variables, namely log-centered mortality rates and compositional transformation of mortality data using the Human Mortality Database. The CANDECOMP/PARAFAC model synthesizes the behavior of the target variables through a few latent components and highlights the evolution of the temporal patterns. These patterns are employed to forecast future trajectories of mortality with Vector-Error Correction models, which account for the non-stationarity of the series. We carry out Monte Carlo simulations to obtain the distributions of the time component over the forecasted period 2001–2015, and we evaluate the goodness of the prediction by computing the Root Mean Square Error and the Mean Absolute Error. Our analysis underlines that understanding mortality dynamics in a high-dimensional framework is crucial for demographic assessments and could help design appropriate pension plans that mitigate the burden of increased longevity. The paper provides two steps further on methodological developments in the field of mortality analysis and forecasting in a high-dimensional space by (i) offering a comprehensive picture of mortality data through the four-way decomposition and (ii) designing appropriate forecasting of mortality data which relies on the projection of the temporal component through Vector-Error Correction models.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"32 1","pages":"916 - 932"},"PeriodicalIF":1.8,"publicationDate":"2023-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87775303","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 : 2023-02-14DOI: 10.1080/03461238.2023.2176251
M. Lindholm, F. Lindskog, J. Palmquist
We study non-life insurance pricing and present a general procedure for constructing a distribution-free locally unbiased predictor of the risk premium based on any initially suggested predictor. The resulting predictor is piecewise constant, corresponding to a partition of the covariate space, and by construction auto-calibrated. Two key issues are the appropriate partitioning of the covariate space and the handling of randomly varying durations, acknowledging possible early termination of contracts. A basic idea in the present paper is to partition the predictions from the initial predictor, which as a by-product defines a partition of the covariate space. Two different approaches to create partitions are discussed in detail using (i) duration-weighted equal-probability binning, and (ii) binning by duration-weighted regression trees. Given a partitioning procedure, the size of the partition to be used is obtained using cross-validation. In this way we obtain an automatic data-driven tariffication procedure, where the number of tariff cells corresponds to the size of the partition. We illustrate the procedure based on both simulated and real insurance data, using both simple GLMs and GBMs as initial predictors. The resulting tariffs are shown to have a rather small number of tariff cells while maintaining or improving the predictive performance compared to the initial predictors.
{"title":"Local bias adjustment, duration-weighted probabilities, and automatic construction of tariff cells","authors":"M. Lindholm, F. Lindskog, J. Palmquist","doi":"10.1080/03461238.2023.2176251","DOIUrl":"https://doi.org/10.1080/03461238.2023.2176251","url":null,"abstract":"We study non-life insurance pricing and present a general procedure for constructing a distribution-free locally unbiased predictor of the risk premium based on any initially suggested predictor. The resulting predictor is piecewise constant, corresponding to a partition of the covariate space, and by construction auto-calibrated. Two key issues are the appropriate partitioning of the covariate space and the handling of randomly varying durations, acknowledging possible early termination of contracts. A basic idea in the present paper is to partition the predictions from the initial predictor, which as a by-product defines a partition of the covariate space. Two different approaches to create partitions are discussed in detail using (i) duration-weighted equal-probability binning, and (ii) binning by duration-weighted regression trees. Given a partitioning procedure, the size of the partition to be used is obtained using cross-validation. In this way we obtain an automatic data-driven tariffication procedure, where the number of tariff cells corresponds to the size of the partition. We illustrate the procedure based on both simulated and real insurance data, using both simple GLMs and GBMs as initial predictors. The resulting tariffs are shown to have a rather small number of tariff cells while maintaining or improving the predictive performance compared to the initial predictors.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135726920","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 : 2023-01-30DOI: 10.1080/03461238.2023.2169632
Alexander Schimmele, Klaus D. Schmidt
In a recent paper published in this journal, Genest & Kolev (2021) studied bivariate survival functions following a law of uniform seniority in the sense that these bivariate survival functions can be represented by a univariate one. While in that paper it is assumed that the survival functions are continuous and strictly decreasing on their support, we show that these assumptions are redundant in certain places. We also present simplified proofs on some of its results.
{"title":"A note on bivariate survival functions following a law of uniform seniority","authors":"Alexander Schimmele, Klaus D. Schmidt","doi":"10.1080/03461238.2023.2169632","DOIUrl":"https://doi.org/10.1080/03461238.2023.2169632","url":null,"abstract":"In a recent paper published in this journal, Genest & Kolev (2021) studied bivariate survival functions following a law of uniform seniority in the sense that these bivariate survival functions can be represented by a univariate one. While in that paper it is assumed that the survival functions are continuous and strictly decreasing on their support, we show that these assumptions are redundant in certain places. We also present simplified proofs on some of its results.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"23 5 1","pages":"907 - 915"},"PeriodicalIF":1.8,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77903358","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 : 2023-01-03DOI: 10.1080/03461238.2022.2163512
Zijia Wang, Mohamed Amine Lkabous, D. Landriault
ABSTRACT The threshold dividend strategy, under which dividends are paid only when the insurer's surplus exceeds a pre-determined threshold, has received considerable attention in risk theory. However, in practice, it seems rather unlikely that an insurer will immediately pull back the dividend payments as soon as its surplus level drops below the dividend threshold. Hence, in this paper, we propose a refracted Lévy risk model with delayed dividend pullbacks triggered by a certain Poissonian observation scheme. Leveraging the extensive literature on fluctuation identities for spectrally negative Lévy processes, we obtain explicit expressions for two-sided exit identities of the proposed insurance risk process. Also, penalties are incorporated into the analysis of dividend payouts as a mechanism to penalize for the volatility of the dividend policy and account for an investor's typical preference for more stable cash flows. An explicit expression for the expected (discounted) dividend payouts net of penalties is derived. The criterion for the optimal threshold level that maximizes the expected dividend payouts is also discussed. Finally, several numerical examples are considered to assess the impact of dividend delays on ruin-related quantities. We numerically show that dividend strategies with more steady dividend payouts can be preferred (over the well-known threshold dividend strategy) when penalty fee become too onerous.
{"title":"A refracted Lévy process with delayed dividend pullbacks","authors":"Zijia Wang, Mohamed Amine Lkabous, D. Landriault","doi":"10.1080/03461238.2022.2163512","DOIUrl":"https://doi.org/10.1080/03461238.2022.2163512","url":null,"abstract":"ABSTRACT The threshold dividend strategy, under which dividends are paid only when the insurer's surplus exceeds a pre-determined threshold, has received considerable attention in risk theory. However, in practice, it seems rather unlikely that an insurer will immediately pull back the dividend payments as soon as its surplus level drops below the dividend threshold. Hence, in this paper, we propose a refracted Lévy risk model with delayed dividend pullbacks triggered by a certain Poissonian observation scheme. Leveraging the extensive literature on fluctuation identities for spectrally negative Lévy processes, we obtain explicit expressions for two-sided exit identities of the proposed insurance risk process. Also, penalties are incorporated into the analysis of dividend payouts as a mechanism to penalize for the volatility of the dividend policy and account for an investor's typical preference for more stable cash flows. An explicit expression for the expected (discounted) dividend payouts net of penalties is derived. The criterion for the optimal threshold level that maximizes the expected dividend payouts is also discussed. Finally, several numerical examples are considered to assess the impact of dividend delays on ruin-related quantities. We numerically show that dividend strategies with more steady dividend payouts can be preferred (over the well-known threshold dividend strategy) when penalty fee become too onerous.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"45 7 Spec issue 1","pages":"885 - 906"},"PeriodicalIF":1.8,"publicationDate":"2023-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72999770","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 : 2023-01-03DOI: 10.1080/03461238.2022.2161412
Frank Y. Feng, Xudong Zeng, Guanxi Zhu
We develop a dynamic equilibrium model of insurance pricing in a competitive market consisting of heterogeneous insurance companies. The insurers have different beliefs on expected loss rate of an underlying risk process and the belief divergences are stochastic. The insurers select optimal insurance market shares to maximize their individual utilities. The equilibrium insurance price is formulated when the insurance market is cleared. We provide a general equilibrium framework with a continuum of insurers in the market and then solve for the equilibrium insurance price explicitly in the case of N insurers. We find that the stochastic heterogeneity brings extra volatility to insurance price and makes it dynamic. The mean-reverting divergences of insurers may explain cycles of insurance business documented by empirical studies. Compared to the previous literature of optimal insurance, this paper introduces an asset pricing framework of general equilibrium to the research of insurance pricing.
{"title":"Insurance pricing in an equilibrium model","authors":"Frank Y. Feng, Xudong Zeng, Guanxi Zhu","doi":"10.1080/03461238.2022.2161412","DOIUrl":"https://doi.org/10.1080/03461238.2022.2161412","url":null,"abstract":"We develop a dynamic equilibrium model of insurance pricing in a competitive market consisting of heterogeneous insurance companies. The insurers have different beliefs on expected loss rate of an underlying risk process and the belief divergences are stochastic. The insurers select optimal insurance market shares to maximize their individual utilities. The equilibrium insurance price is formulated when the insurance market is cleared. We provide a general equilibrium framework with a continuum of insurers in the market and then solve for the equilibrium insurance price explicitly in the case of N insurers. We find that the stochastic heterogeneity brings extra volatility to insurance price and makes it dynamic. The mean-reverting divergences of insurers may explain cycles of insurance business documented by empirical studies. Compared to the previous literature of optimal insurance, this paper introduces an asset pricing framework of general equilibrium to the research of insurance pricing.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"94 1","pages":"834 - 852"},"PeriodicalIF":1.8,"publicationDate":"2023-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80698470","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 : 2022-12-29DOI: 10.1080/03461238.2023.2208583
Guohui Guan, Zongxia Liang, Yilun Song
This paper investigates a Stackelberg game between an insurer and a reinsurer under the $alpha$-maxmin mean-variance criterion. The insurer can purchase per-loss reinsurance from the reinsurer. With the insurer's feedback reinsurance strategy, the reinsurer optimizes the reinsurance premium in the Stackelberg game. The financial market consists of cash and stock with Heston's stochastic volatility. Both the insurer and reinsurer maximize their respective $alpha$-maxmin mean-variance preferences in the market. The criterion is time-inconsistent and we derive the equilibrium strategies by the extended Hamilton-Jacobi-Bellman equations. Similar to the non-robust case in Li and Young (2022), excess-of-loss reinsurance is the optimal form of reinsurance strategy for the insurer. The equilibrium investment strategy is determined by a system of Riccati differential equations. Besides, the equations determining the equilibrium reinsurance strategy and reinsurance premium rate are given semi-explicitly, which is simplified to an algebraic equation in a specific example. Numerical examples illustrate that the game between the insurer and reinsurer makes the insurance more radical when the agents become more ambiguity aversion or risk aversion. Furthermore, the level of ambiguity, ambiguity attitude, and risk attitude of the insurer (reinsurer) have similar effects on the equilibrium reinsurance strategy, reinsurance premium, and investment strategy.
本文研究了在$alpha$-maxmin均值-方差准则下保险人与再保险人之间的Stackelberg博弈问题。保险人可以向再保险人购买按损失分保。利用保险人的反馈再保险策略，再保险人在Stackelberg博弈中对再保险保费进行优化。金融市场由现金和股票组成，具有赫斯顿随机波动率。保险人和再保险人都在市场上最大化各自的$alpha$-maxmin均值方差偏好。该准则是时间不一致的，并利用扩展的Hamilton-Jacobi-Bellman方程推导出均衡策略。与Li and Young(2022)的非稳健案例类似，超额赔付再保险是保险人最优的再保险策略形式。均衡投资策略是由一个里卡蒂微分方程组决定的。给出了确定均衡再保险策略和再保险费率的半显式方程，并结合具体实例将其简化为代数方程。数值算例表明，保险人和再保险人之间的博弈使保险更具激进性，当代理人变得更加模糊或风险厌恶时。此外，保险人(再保险人)的歧义程度、歧义态度和风险态度对均衡再保险策略、再保险保费和投资策略有相似的影响。
{"title":"A Stackelberg reinsurance-investment game under α-maxmin mean-variance criterion and stochastic volatility","authors":"Guohui Guan, Zongxia Liang, Yilun Song","doi":"10.1080/03461238.2023.2208583","DOIUrl":"https://doi.org/10.1080/03461238.2023.2208583","url":null,"abstract":"This paper investigates a Stackelberg game between an insurer and a reinsurer under the $alpha$-maxmin mean-variance criterion. The insurer can purchase per-loss reinsurance from the reinsurer. With the insurer's feedback reinsurance strategy, the reinsurer optimizes the reinsurance premium in the Stackelberg game. The financial market consists of cash and stock with Heston's stochastic volatility. Both the insurer and reinsurer maximize their respective $alpha$-maxmin mean-variance preferences in the market. The criterion is time-inconsistent and we derive the equilibrium strategies by the extended Hamilton-Jacobi-Bellman equations. Similar to the non-robust case in Li and Young (2022), excess-of-loss reinsurance is the optimal form of reinsurance strategy for the insurer. The equilibrium investment strategy is determined by a system of Riccati differential equations. Besides, the equations determining the equilibrium reinsurance strategy and reinsurance premium rate are given semi-explicitly, which is simplified to an algebraic equation in a specific example. Numerical examples illustrate that the game between the insurer and reinsurer makes the insurance more radical when the agents become more ambiguity aversion or risk aversion. Furthermore, the level of ambiguity, ambiguity attitude, and risk attitude of the insurer (reinsurer) have similar effects on the equilibrium reinsurance strategy, reinsurance premium, and investment strategy.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"30 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74247470","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 : 2022-11-21DOI: 10.1080/03461238.2022.2144432
Wei Zhong, Dan Zhu, Zhimin Zhang
We present an efficient valuation approach for guaranteed minimum benefits embedded in variable annuity contracts, where the log price follows a jump-diffusion model with stochastic volatilities. In particular, we allow separate Cox-Ingersoll-Ross processes for the underlying volatility and the jump intensity, each correlated with the diffusion term of the spot price. To value the contract under such complex stochastic nature, we rely on the recent advances in the frame dual projection methods with the stochastic process approximated by its expectation. As a byproduct of the transparent analytical expression derived, we derive the associated Greeks that provide a practical basis for risk management. Numerical experiments demonstrate the accuracy and efficiency of the proposed method.
{"title":"Valuation of variable annuities under stochastic volatility and stochastic jump intensity","authors":"Wei Zhong, Dan Zhu, Zhimin Zhang","doi":"10.1080/03461238.2022.2144432","DOIUrl":"https://doi.org/10.1080/03461238.2022.2144432","url":null,"abstract":"We present an efficient valuation approach for guaranteed minimum benefits embedded in variable annuity contracts, where the log price follows a jump-diffusion model with stochastic volatilities. In particular, we allow separate Cox-Ingersoll-Ross processes for the underlying volatility and the jump intensity, each correlated with the diffusion term of the spot price. To value the contract under such complex stochastic nature, we rely on the recent advances in the frame dual projection methods with the stochastic process approximated by its expectation. As a byproduct of the transparent analytical expression derived, we derive the associated Greeks that provide a practical basis for risk management. Numerical experiments demonstrate the accuracy and efficiency of the proposed method.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"330 1 1","pages":"708 - 734"},"PeriodicalIF":1.8,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77341292","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 : 2022-11-16DOI: 10.1080/03461238.2022.2145577
P. Goffard
Insurance loss distributions are characterized by a high frequency of small claim amounts and a lower, but not insignificant, occurrence of large claim amounts. Composite models, which link two probability distributions, one for the ‘body’ and the other for the ‘tail’ of the loss distribution, have emerged in the actuarial literature to take this specificity into account. The parameters of these models summarize the distribution of the losses. One of them corresponds to the breaking point between small and large claim amounts. The composite models are usually fitted using maximum likelihood estimation. A Bayesian approach is considered in this work. Sequential Monte Carlo samplers are used to sample from the posterior distribution and compute the posterior model evidence to both fit and compare the competing models. The method is validated via a simulation study and illustrated on an insurance loss dataset.
{"title":"Sequential Monte Carlo samplers to fit and compare insurance loss models","authors":"P. Goffard","doi":"10.1080/03461238.2022.2145577","DOIUrl":"https://doi.org/10.1080/03461238.2022.2145577","url":null,"abstract":"Insurance loss distributions are characterized by a high frequency of small claim amounts and a lower, but not insignificant, occurrence of large claim amounts. Composite models, which link two probability distributions, one for the ‘body’ and the other for the ‘tail’ of the loss distribution, have emerged in the actuarial literature to take this specificity into account. The parameters of these models summarize the distribution of the losses. One of them corresponds to the breaking point between small and large claim amounts. The composite models are usually fitted using maximum likelihood estimation. A Bayesian approach is considered in this work. Sequential Monte Carlo samplers are used to sample from the posterior distribution and compute the posterior model evidence to both fit and compare the competing models. The method is validated via a simulation study and illustrated on an insurance loss dataset.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"71 1","pages":"765 - 787"},"PeriodicalIF":1.8,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80600603","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 : 2022-11-07DOI: 10.1080/03461238.2022.2139631
Xue Dong, X. Rong, H Zhao
ABSTRACT This paper investigates a non-zero-sum stochastic differential game between two competitive CARA insurers, where we adopt the different classes of premium principles (including the expected value premium principle, the variance premium principle and the exponential premium principle) and each insurer aims to maximize the expected exponential utility of his terminal wealth relative to that of his competitor. Moreover, both insurers are allowed to purchase reinsurance treaty to mitigate individual claim risks and can invest in a financial market consisting of a risk-free asset, a risky asset where the instantaneous rate of investment return follows an Ornstein-Uhlenbeck process which can reflect the changes of bull market and bear market. The optimal reinsurance strategy has a non-trivial structure which is distinguished from the conventional proportional and excess-of-loss reinsurance strategies. Furthermore, we derive the optimal reinsurance and investment strategies under the variance premium principle and expected value principle. In addition, we give another model which considers the correlation between risk model and financial market under the expected value principle. Finally, numerical analyses are provided to analyze the effects of model parameters on the optimal strategies under different cases.
{"title":"Non-zero-sum reinsurance and investment game with non-trivial curved strategy structure under Ornstein–Uhlenbeck process","authors":"Xue Dong, X. Rong, H Zhao","doi":"10.1080/03461238.2022.2139631","DOIUrl":"https://doi.org/10.1080/03461238.2022.2139631","url":null,"abstract":"ABSTRACT This paper investigates a non-zero-sum stochastic differential game between two competitive CARA insurers, where we adopt the different classes of premium principles (including the expected value premium principle, the variance premium principle and the exponential premium principle) and each insurer aims to maximize the expected exponential utility of his terminal wealth relative to that of his competitor. Moreover, both insurers are allowed to purchase reinsurance treaty to mitigate individual claim risks and can invest in a financial market consisting of a risk-free asset, a risky asset where the instantaneous rate of investment return follows an Ornstein-Uhlenbeck process which can reflect the changes of bull market and bear market. The optimal reinsurance strategy has a non-trivial structure which is distinguished from the conventional proportional and excess-of-loss reinsurance strategies. Furthermore, we derive the optimal reinsurance and investment strategies under the variance premium principle and expected value principle. In addition, we give another model which considers the correlation between risk model and financial market under the expected value principle. Finally, numerical analyses are provided to analyze the effects of model parameters on the optimal strategies under different cases.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"5 1","pages":"565 - 597"},"PeriodicalIF":1.8,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88100751","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 : 2022-11-07DOI: 10.1080/03461238.2022.2142157
Jingyi Cao, V. Young
The classical Cramér–Lundberg risk process is commonly used to model the surplus of an insurer; it characterizes the claim arrival process and the claim size random variable Y through a compound Poisson process, along with a constant rate of premium income. When , the process can be approximated by a diffusion process, but that requirement eliminates many heavy-tailed claim models, such as the Pareto with . In this paper, we generalize the well known diffusion approximation by assuming that Y lies in the domain of attraction of an α-stable random variable, for . Then, we construct a sequence of classical Cramér–Lundberg risk processes and show that the sequence converges to an α-stable Lévy motion in the Skorokhod -topology. We prove this convergence by proving the pointwise convergence of the corresponding Laplace exponents of our processes, which to our knowledge, is a new result. To apply this convergence result, we show the convergence of a sequence of Gerber–Shiu distributions of exponential Parisian ruin, and we show the convergence of a sequence of payoff functions for barrier dividend strategies. Both of these applications provide new proofs of the stated limits.
{"title":"Approximating the classical risk process by stable Lévy motion","authors":"Jingyi Cao, V. Young","doi":"10.1080/03461238.2022.2142157","DOIUrl":"https://doi.org/10.1080/03461238.2022.2142157","url":null,"abstract":"The classical Cramér–Lundberg risk process is commonly used to model the surplus of an insurer; it characterizes the claim arrival process and the claim size random variable Y through a compound Poisson process, along with a constant rate of premium income. When , the process can be approximated by a diffusion process, but that requirement eliminates many heavy-tailed claim models, such as the Pareto with . In this paper, we generalize the well known diffusion approximation by assuming that Y lies in the domain of attraction of an α-stable random variable, for . Then, we construct a sequence of classical Cramér–Lundberg risk processes and show that the sequence converges to an α-stable Lévy motion in the Skorokhod -topology. We prove this convergence by proving the pointwise convergence of the corresponding Laplace exponents of our processes, which to our knowledge, is a new result. To apply this convergence result, we show the convergence of a sequence of Gerber–Shiu distributions of exponential Parisian ruin, and we show the convergence of a sequence of payoff functions for barrier dividend strategies. Both of these applications provide new proofs of the stated limits.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":"29 1","pages":"679 - 707"},"PeriodicalIF":1.8,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79472001","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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