Pub Date : 2022-05-29DOI: 10.1080/03461238.2022.2078221
Yuxuan Liu, Zhengjun Jiang, Yiwen Zhang
The paper investigates ultimate ruin probability, the probability that ruin time is finite, for an insurance company whose risk reserves follow a Markov-modulated jump–diffusion risk model. We use both the Banach contraction principle and q-scale functions to prove that ultimate ruin probability is the only fixed point of a contraction mapping and show that an iterative equation can be employed to calculate ultimate ruin probability by an iterative algorithm of approximating the fixed point. Using q-scale functions and the methodology from Gajek and Rudź [(2018). Banach contraction principle and ruin probabilities in regime-switching models. Insurance: Mathematics and Economics, 80, 45–53] applied to the Markov-modulated jump–diffusion risk model, we get a more explicit Lipschitz constant in the Banach contraction principle and conveniently verify some similar results of their appendix in our case.
{"title":"q-scale function, Banach contraction principle, and ultimate ruin probability in a Markov-modulated jump–diffusion risk model","authors":"Yuxuan Liu, Zhengjun Jiang, Yiwen Zhang","doi":"10.1080/03461238.2022.2078221","DOIUrl":"https://doi.org/10.1080/03461238.2022.2078221","url":null,"abstract":"The paper investigates ultimate ruin probability, the probability that ruin time is finite, for an insurance company whose risk reserves follow a Markov-modulated jump–diffusion risk model. We use both the Banach contraction principle and q-scale functions to prove that ultimate ruin probability is the only fixed point of a contraction mapping and show that an iterative equation can be employed to calculate ultimate ruin probability by an iterative algorithm of approximating the fixed point. Using q-scale functions and the methodology from Gajek and Rudź [(2018). Banach contraction principle and ruin probabilities in regime-switching models. Insurance: Mathematics and Economics, 80, 45–53] applied to the Markov-modulated jump–diffusion risk model, we get a more explicit Lipschitz constant in the Banach contraction principle and conveniently verify some similar results of their appendix in our case.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77388621","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 : 2022-05-18DOI: 10.1080/03461238.2022.2075282
A. Y. Golubin, V. Gridin
The problem of designing an optimal insurance strategy in a modification of the risk process with discrete time is investigated. This model introduces stage-by-stage probabilistic constraints (Value-at-Risk (VaR) constraints) on the insurer's capital increments during each stage. Also, the set of admissible insurances is determined by a safety level reflecting a ‘good’ or ‘bad’ capital increment at the previous stage. The mathematical expectation of the insurer's final capital is used as the objective functional. The total loss of the insurer at each stage is modeled by the Gaussian (normal) distribution with parameters depending on a seded loss function (or, in other words, an insurance policy) selected. In contrast to traditional dynamic optimization models for insurance strategies, the proposed approach allows to construct the value functions (and hence the optimal insurance policies) by simply solving a sequence of static insurance optimization problems. It is demonstrated that the optimal seded loss function at each stage depends on the prescribed value of the safety level: it is either a stop-loss insurance or conditional deductible insurance having a discontinuous point. In order to reduce ex post moral hazard, we also investigate the case, where both parties in an insurance contract are obligated to pay more for a larger realization of loss. This leads to that the optimal seeded loss functions are either stop-loss insurances or unconditional deductible insurances.
{"title":"Optimal insurance strategy in a risk process under a safety level imposed on the increments of the process","authors":"A. Y. Golubin, V. Gridin","doi":"10.1080/03461238.2022.2075282","DOIUrl":"https://doi.org/10.1080/03461238.2022.2075282","url":null,"abstract":"The problem of designing an optimal insurance strategy in a modification of the risk process with discrete time is investigated. This model introduces stage-by-stage probabilistic constraints (Value-at-Risk (VaR) constraints) on the insurer's capital increments during each stage. Also, the set of admissible insurances is determined by a safety level reflecting a ‘good’ or ‘bad’ capital increment at the previous stage. The mathematical expectation of the insurer's final capital is used as the objective functional. The total loss of the insurer at each stage is modeled by the Gaussian (normal) distribution with parameters depending on a seded loss function (or, in other words, an insurance policy) selected. In contrast to traditional dynamic optimization models for insurance strategies, the proposed approach allows to construct the value functions (and hence the optimal insurance policies) by simply solving a sequence of static insurance optimization problems. It is demonstrated that the optimal seded loss function at each stage depends on the prescribed value of the safety level: it is either a stop-loss insurance or conditional deductible insurance having a discontinuous point. In order to reduce ex post moral hazard, we also investigate the case, where both parties in an insurance contract are obligated to pay more for a larger realization of loss. This leads to that the optimal seeded loss functions are either stop-loss insurances or unconditional deductible insurances.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76237948","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 : 2022-04-25DOI: 10.1080/03461238.2022.2061868
D. Falden, Anna Kamille Nyegaard
In a set-up of with-profit life insurance including bonus, we study the calculation of the market reserve, where Management Actions such as investment strategies and bonus allocation strategies depend on the reserve itself. Since the amount of future bonus depends on the retrospective savings account, the introduction of Management Actions that depend on the prospective market reserve results in an entanglement of retrospective and prospective reserves. We study the complications that arise due to the interdependence between retrospective and prospective reserves, and characterize the market reserve by a partial differential equation (PDE). We reduce the dimension of the PDE in the case of linearity, and furthermore, we suggest an approximation of the market reserve based on the forward rate. The quality of the approximation is studied in a numerical example.
{"title":"Reserve-dependent Management Actions in life insurance","authors":"D. Falden, Anna Kamille Nyegaard","doi":"10.1080/03461238.2022.2061868","DOIUrl":"https://doi.org/10.1080/03461238.2022.2061868","url":null,"abstract":"In a set-up of with-profit life insurance including bonus, we study the calculation of the market reserve, where Management Actions such as investment strategies and bonus allocation strategies depend on the reserve itself. Since the amount of future bonus depends on the retrospective savings account, the introduction of Management Actions that depend on the prospective market reserve results in an entanglement of retrospective and prospective reserves. We study the complications that arise due to the interdependence between retrospective and prospective reserves, and characterize the market reserve by a partial differential equation (PDE). We reduce the dimension of the PDE in the case of linearity, and furthermore, we suggest an approximation of the market reserve based on the forward rate. The quality of the approximation is studied in a numerical example.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84765592","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 : 2022-03-28DOI: 10.1080/03461238.2022.2049635
Runhuan Feng, Guojun Gan, Ning Zhang
Variable annuity is arguably the most complex individual retirement planning product in the financial market. Its intricacy stems from a variety of product features including investment options, guaranteed benefits, withdrawal options, etc. In many ways, variable annuities can be viewed as traditional life and annuity products at the next level of sophistication with added financial options. Despite a significant amount of publications by practitioners and academics on the subject matter, there have been few research papers that systematically exploit the basic principles underlying the operation of variable annuities. This survey paper aims to fill in the gap in the literature for an overview of state-of-the-art technology and recent trends in the development of variable annuities.
{"title":"Variable annuity pricing, valuation, and risk management: a survey","authors":"Runhuan Feng, Guojun Gan, Ning Zhang","doi":"10.1080/03461238.2022.2049635","DOIUrl":"https://doi.org/10.1080/03461238.2022.2049635","url":null,"abstract":"Variable annuity is arguably the most complex individual retirement planning product in the financial market. Its intricacy stems from a variety of product features including investment options, guaranteed benefits, withdrawal options, etc. In many ways, variable annuities can be viewed as traditional life and annuity products at the next level of sophistication with added financial options. Despite a significant amount of publications by practitioners and academics on the subject matter, there have been few research papers that systematically exploit the basic principles underlying the operation of variable annuities. This survey paper aims to fill in the gap in the literature for an overview of state-of-the-art technology and recent trends in the development of variable annuities.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81599946","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 : 2022-03-10DOI: 10.1080/03461238.2022.2116725
Benjamin Avanzi, Pingfu Chen, L. Henriksen, Bernard Wong
In this paper, we consider a company whose assets and liabilities evolve according to a correlated bivariate geometric Brownian motion, such as in Gerber and Shiu [(2003). Geometric Brownian motion models for assets and liabilities: From pension funding to optimal dividends. North American Actuarial Journal 7(3), 37–56]. We determine what dividend strategy maximises the expected present value of dividends until ruin in two cases: (i) when shareholders won't cover surplus shortfalls and a solvency constraint [as in Paulsen (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance and Stochastics 7(4), 457–473] is consequently imposed and (ii) when shareholders are always to fund any capital deficiency with capital (asset) injections. In the latter case, ruin will never occur and the objective is to maximise the difference between dividends and capital injections. Developing and using appropriate verification lemmas, we show that the optimal dividend strategy is, in both cases, of barrier type. Both value functions are derived in closed form. Furthermore, the barrier is defined on the ratio of assets to liabilities, which mimics some of the dividend strategies that can be observed in practice by insurance companies. The existence and uniqueness of the optimal strategies are shown. Results are illustrated.
{"title":"On the surplus management of funds with assets and liabilities in presence of solvency requirements","authors":"Benjamin Avanzi, Pingfu Chen, L. Henriksen, Bernard Wong","doi":"10.1080/03461238.2022.2116725","DOIUrl":"https://doi.org/10.1080/03461238.2022.2116725","url":null,"abstract":"In this paper, we consider a company whose assets and liabilities evolve according to a correlated bivariate geometric Brownian motion, such as in Gerber and Shiu [(2003). Geometric Brownian motion models for assets and liabilities: From pension funding to optimal dividends. North American Actuarial Journal 7(3), 37–56]. We determine what dividend strategy maximises the expected present value of dividends until ruin in two cases: (i) when shareholders won't cover surplus shortfalls and a solvency constraint [as in Paulsen (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance and Stochastics 7(4), 457–473] is consequently imposed and (ii) when shareholders are always to fund any capital deficiency with capital (asset) injections. In the latter case, ruin will never occur and the objective is to maximise the difference between dividends and capital injections. Developing and using appropriate verification lemmas, we show that the optimal dividend strategy is, in both cases, of barrier type. Both value functions are derived in closed form. Furthermore, the barrier is defined on the ratio of assets to liabilities, which mimics some of the dividend strategies that can be observed in practice by insurance companies. The existence and uniqueness of the optimal strategies are shown. Results are illustrated.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77963737","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 : 2022-03-09DOI: 10.1080/03461238.2022.2034127
Eric R. Ulm
A number of analytic solutions have been found for Variable Annuity Guaranteed Minimum Death Benefit (GMDB) option values under a variety of mortality laws. To date, the solutions are for Risk-Neutral valuation only. Where policyholder decisions are allowed, it is assumed that they act to maximize the risk-neutral value of the GMDB. We examine situations where the asset allocation decisions are made to maximize expected utility rather than option value. We find analytic solutions for both return of premium and ratchet options at small values of bequest motive for a number of mortality laws.
{"title":"Analytic valuation of GMDB options with utility based asset allocation","authors":"Eric R. Ulm","doi":"10.1080/03461238.2022.2034127","DOIUrl":"https://doi.org/10.1080/03461238.2022.2034127","url":null,"abstract":"A number of analytic solutions have been found for Variable Annuity Guaranteed Minimum Death Benefit (GMDB) option values under a variety of mortality laws. To date, the solutions are for Risk-Neutral valuation only. Where policyholder decisions are allowed, it is assumed that they act to maximize the risk-neutral value of the GMDB. We examine situations where the asset allocation decisions are made to maximize expected utility rather than option value. We find analytic solutions for both return of premium and ratchet options at small values of bequest motive for a number of mortality laws.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84676775","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 : 2022-02-24DOI: 10.1080/03461238.2022.2026459
Ailing Gu, Shumin Chen, Zhongfei Li, F. Viens
This paper first studies the optimal reinsurance problems for two competitive insurers and then studies the optimal reinsurance premium pricing problem for their common reinsurer by using the dynamic programming technique. The two insurers are subject to common insurance systematic risk. Each purchases proportional or excess-of-loss reinsurance for risk control. They aim to maximize the expected utilities of their relative terminal wealth. With the insurers' optimal reinsurance strategies, the reinsurer decides the reinsurance premiums for each insurer, also aiming to maximize the expected utility of her terminal wealth. Thus, the optimal reinsurance pricing problem is formulated as a Stackelberg game between two competitive insurers and a reinsurer, where the reinsurer is the leader, and the insurers are followers. Besides, all three players take model ambiguity into account. We characterize the optimal strategies for the insurers and the reinsurer and provide some numerical examples to show the impact of competition and model ambiguity on the pricing of reinsurance contracts.
{"title":"Optimal reinsurance pricing with ambiguity aversion and relative performance concerns in the principal-agent model","authors":"Ailing Gu, Shumin Chen, Zhongfei Li, F. Viens","doi":"10.1080/03461238.2022.2026459","DOIUrl":"https://doi.org/10.1080/03461238.2022.2026459","url":null,"abstract":"This paper first studies the optimal reinsurance problems for two competitive insurers and then studies the optimal reinsurance premium pricing problem for their common reinsurer by using the dynamic programming technique. The two insurers are subject to common insurance systematic risk. Each purchases proportional or excess-of-loss reinsurance for risk control. They aim to maximize the expected utilities of their relative terminal wealth. With the insurers' optimal reinsurance strategies, the reinsurer decides the reinsurance premiums for each insurer, also aiming to maximize the expected utility of her terminal wealth. Thus, the optimal reinsurance pricing problem is formulated as a Stackelberg game between two competitive insurers and a reinsurer, where the reinsurer is the leader, and the insurers are followers. Besides, all three players take model ambiguity into account. We characterize the optimal strategies for the insurers and the reinsurer and provide some numerical examples to show the impact of competition and model ambiguity on the pricing of reinsurance contracts.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74965563","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 : 2022-02-15DOI: 10.1080/03461238.2022.2037016
Donatien Hainaut, J. Trufin, M. Denuit
Thanks to its outstanding performances, boosting has rapidly gained wide acceptance among actuaries. To speed up calculations, boosting is often applied to gradients of the loss function, not to responses (hence the name gradient boosting). When the model is trained by minimizing Poisson deviance, this amounts to apply the least-squares principle to raw residuals. This exposes gradient boosting to the same problems that lead to replace least-squares with Poisson Generalized Linear Models (GLM) to analyze low counts (typically, the number of reported claims at policy level in personal lines). This paper shows that boosting can be conducted directly on the response under Tweedie loss function and log-link, by adapting the weights at each step. Numerical illustrations demonstrate similar or better performances compared to gradient boosting when trees are used as weak learners, with a higher level of transparency since responses are used instead of gradients.
{"title":"Response versus gradient boosting trees, GLMs and neural networks under Tweedie loss and log-link","authors":"Donatien Hainaut, J. Trufin, M. Denuit","doi":"10.1080/03461238.2022.2037016","DOIUrl":"https://doi.org/10.1080/03461238.2022.2037016","url":null,"abstract":"Thanks to its outstanding performances, boosting has rapidly gained wide acceptance among actuaries. To speed up calculations, boosting is often applied to gradients of the loss function, not to responses (hence the name gradient boosting). When the model is trained by minimizing Poisson deviance, this amounts to apply the least-squares principle to raw residuals. This exposes gradient boosting to the same problems that lead to replace least-squares with Poisson Generalized Linear Models (GLM) to analyze low counts (typically, the number of reported claims at policy level in personal lines). This paper shows that boosting can be conducted directly on the response under Tweedie loss function and log-link, by adapting the weights at each step. Numerical illustrations demonstrate similar or better performances compared to gradient boosting when trees are used as weak learners, with a higher level of transparency since responses are used instead of gradients.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74659288","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 : 2022-02-09DOI: 10.1080/03461238.2022.2030398
Duni Hu, Hailong Wang
We investigate the robust reinsurance demand and price under learning and ambiguity aversion. In the reinsurance contract, the insurer is ambiguity neutral and believes that he is perfectly informed, and the reinsurer is a Bayesian learner and is aware that even the filtered model is the best description of the data-generating process, might not forecast the future claims correctly. The ambiguity-averse reinsurer has a preference for reinsurance contract which is robust to model misspecification. Closed-form expressions for the robust reinsurance demand and price are derived. We find that both the reinsurer's one-sided learning and ambiguity aversion influence the structures and levels of the optimal reinsurance demand and price. Moreover, if the ambiguity-averse reinsurer specifies the suboptimal reinsurance contract as an ambiguity-neutral decision-maker, it will result in significant utility loss and the utility loss increases with ambiguity aversion level and Bayesian volatility.
{"title":"Robust reinsurance contract with learning and ambiguity aversion","authors":"Duni Hu, Hailong Wang","doi":"10.1080/03461238.2022.2030398","DOIUrl":"https://doi.org/10.1080/03461238.2022.2030398","url":null,"abstract":"We investigate the robust reinsurance demand and price under learning and ambiguity aversion. In the reinsurance contract, the insurer is ambiguity neutral and believes that he is perfectly informed, and the reinsurer is a Bayesian learner and is aware that even the filtered model is the best description of the data-generating process, might not forecast the future claims correctly. The ambiguity-averse reinsurer has a preference for reinsurance contract which is robust to model misspecification. Closed-form expressions for the robust reinsurance demand and price are derived. We find that both the reinsurer's one-sided learning and ambiguity aversion influence the structures and levels of the optimal reinsurance demand and price. Moreover, if the ambiguity-averse reinsurer specifies the suboptimal reinsurance contract as an ambiguity-neutral decision-maker, it will result in significant utility loss and the utility loss increases with ambiguity aversion level and Bayesian volatility.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76261569","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 : 2022-02-02DOI: 10.1080/03461238.2022.2028185
Wenjun Jiang
This paper studies the design of Pareto-optimal insurance under the heterogeneous beliefs of the insured and insurer. To accommodate a wide range of belief heterogeneity, we allow the likelihood ratio function to be non-monotone. To prevent the ex post moral hazard issue, the incentive compatibility condition is exogenously imposed to restrict the indemnity function. An implicit characterization of the optimal indemnity function is presented first by using the calculus of variations. Based on the point-wise maximizer to the problem, we partition the domain of loss into disjoint pieces and derive the parametric form of the optimal indemnity function over each piece through its implicit characterization. The main result of this paper generalizes those in the literature and provides insights for related problems.
{"title":"Pareto-optimal insurance under heterogeneous beliefs and incentive compatibility","authors":"Wenjun Jiang","doi":"10.1080/03461238.2022.2028185","DOIUrl":"https://doi.org/10.1080/03461238.2022.2028185","url":null,"abstract":"This paper studies the design of Pareto-optimal insurance under the heterogeneous beliefs of the insured and insurer. To accommodate a wide range of belief heterogeneity, we allow the likelihood ratio function to be non-monotone. To prevent the ex post moral hazard issue, the incentive compatibility condition is exogenously imposed to restrict the indemnity function. An implicit characterization of the optimal indemnity function is presented first by using the calculus of variations. Based on the point-wise maximizer to the problem, we partition the domain of loss into disjoint pieces and derive the parametric form of the optimal indemnity function over each piece through its implicit characterization. The main result of this paper generalizes those in the literature and provides insights for related problems.","PeriodicalId":49572,"journal":{"name":"Scandinavian Actuarial Journal","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72877927","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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