This paper considers linear fair risk sharing rules and the conditional mean risk sharing rule for independent but heterogeneous losses that are gathered in an insurance pool. It studies the ...asymptotic behavior of individual contributions to total losses when the number of participants to the pool tends to infinity. It is shown that (i) insurance at pure premium is obtained for an infinitely large pool and (ii) the difference between the actual contribution and the pure premium becomes ultimately Normally distributed. The linear fair risk sharing rule approximating the conditional mean risk sharing rule is then identified, providing practitioners with a useful simplification applicable within large pools. Also, the approximate number of participants required to keep the volatility of individual contributions within an acceptable range is obtained from the established asymptotic Normality.
This paper offers a systematic treatment of risk‐sharing rules for insurance losses, based on a list of relevant properties. A number of candidate risk‐sharing rules are considered, including the ...conditional mean risk‐sharing rule proposed in Denuit and Dhaene and the newly introduced quantile risk‐sharing rule. Their compliance with the proposed properties is established. Then, methods for building new risk‐sharing rules are discussed. The results derived in this paper are helpful in the development of peer‐to‐peer insurance (or crowdsurance), as well as to manage contingent risk funds where a given budget is distributed among claimants.
Accurate loss reserves are an important item in the financial statement of an insurance company and are mostly evaluated by macrolevel models with aggregate data in run‐off triangles. In recent ...years, a new set of literature has considered individual claims data and proposed parametric reserving models based on claim history profiles. In this paper, we present a nonparametric and flexible approach for estimating outstanding liabilities using all the covariates associated to the policy, its policyholder, and all the information received by the insurance company on the individual claims since its reporting date. We develop a machine learning–based method and explain how to build specific subsets of data for the machine learning algorithms to be trained and assessed on. The choice for a nonparametric model leads to new issues since the target variables (claim occurrence and claim severity) are right‐censored most of the time. The performance of our approach is evaluated by comparing the predictive values of the reserve estimates with their true values on simulated data. We compare our individual approach with the most used aggregate data method, namely, chain ladder, with respect to the bias and the variance of the estimates. We also provide a short real case study based on a Dutch loan insurance portfolio.
This paper studies diversification effects resulting from pooling insurance losses according to the risk allocation rule proposed by Denuit and Dhaene (2012). General comparison results are ...established for conditional expectations given sums of independent random variables. It is shown that these expectations decrease in the number of terms comprised in the conditioning sums. Additional inequalities are obtained under regression dependence in the sum. These general results are used to derive the monotonicity of the respective contributions of the participants with respect to the convex order, showing that increasing the number of participants is always beneficial under conditional mean risk sharing. New convergence results are obtained, showing that the variance of individual contributions tends to zero in many interesting cases. This provides actuaries with conditions ensuring that the risk can be fully eliminated within the pool, at the limit.
This paper considers a peer-to-peer (P2P) insurance scheme where the higher layer is transferred to a (re-)insurer and retained losses are distributed among participants according to the conditional ...mean risk sharing rule proposed by Denuit and Dhaene (2012). The global retention level of the pool of participants grows proportionally with their number. We study the asymptotic behavior of the individual retention levels, as well as individual cash-backs and stop-loss premiums, as the number of participants increases. The probability that the total loss hits the upper layer protected by the stop-loss treaty is also considered. The results depend on the proportional rate of increase of the global retention level with the number of participants, as well as on the existence of the Esscher transform of the losses brought to the pool.
This paper proposes a new risk-sharing procedure, framed into the classical insurance surplus process. Compared to the standard setting where total losses are shared at the end of the period, losses ...are allocated among participants at their occurrence time in the proposed model. The conditional mean risk-sharing rule proposed by Denuit and Dhaene (2012) is applied to this end. The analysis adopts two different points of views: a collective one for the pool and an individual one for sharing losses and adjusting the amounts of contributions among participants. These two views are compatible under the compound Poisson risk process. Guarantees can also be added by partnering with an insurer.
Survivor funds are financial arrangements where participants agree to share the proceeds of a collective investment pool in a predescribed way depending on their survival. This offers investors a way ...to benefit from mortality credits, boosting financial returns. Following Denuit (2019, ASTIN Bulletin, 49, 591–617), participants are assumed to adopt the conditional mean risk sharing rule introduced in Denuit and Dhaene (2012, Insurance: Mathematics and Economics, 51, 265–270) to assess their respective shares in mortality credits. This paper looks at pools of individuals that are heterogeneous in terms of their survival probability and their contributions. Imposing mild conditions, we show that individual risk can be fully diversified if the size of the group tends to infinity. For large groups, we derive simple, hierarchical approximations of the conditional mean risk sharing rule.
This paper supplements the previous contribution by Denuit and Robert (2021). First, the compound Poisson case is revisited and the strong law of large number is rigorously established for the ...conditional expectations defining the conditional mean risk allocation. Then, a weak law of large numbers is proposed, providing the actuary with a criterion ensuring that the variance of individual contributions tends to 0. This is appealing for applications since this behavior is a key success factor for collaborative insurance pools.
Denuit (
2019
,
2020a
) demonstrated that conditional mean risk sharing introduced by Denuit and Dhaene (
2012
) is the appropriate theoretical tool to share losses in collaborative peer-to-peer ...insurance schemes. Denuit and Robert (
2020a
,
2020b
,
2021
) studied this risk sharing mechanism and established several attractive properties including linear approximations when total losses or the number of participants get large. It is also shown there that the conditional expectation defining the conditional mean risk sharing is asymptotically increasing in the total loss (under mild technical assumptions). This ensures that the risk exchange is Pareto-optimal and that all participants have an interest to keep total losses as small as possible. In this article, we design a flexible system where entry prices can be made attractive compared to the premium of a regular, commercial insurance contract and participants are awarded cash-backs in case of favorable experience while being protected by a stop-loss treaty in the opposite case. Members can also be grouped according to some meaningful criteria, resulting in a hierarchical decomposition of the community. The particular case where realized losses are allocated in proportion to the pure premiums is studied.
Abstract Objectives This paper reviews the published data and reports 3 cases of thrombosis involving CoreValve (Medtronic, Minneapolis, Minnesota) and 1 involving Edward Sapien (Edwards ...Lifesciences, Irvine, California) devices. Three of these cases had pathological findings at autopsy. Background Only a limited number of cases of valve dysfunction with rapid increase of transvalvular aortic gradients or aortic insufficiency post-transcatheter aortic valve replacement (TAVR) have been described. This nonstructural valvular dysfunction has been presumed to be because of early pannus formation or thrombosis. Methods Through reviews of the published reports and 4 clinical cases, pathological and clinical findings of early valve thrombosis are examined to elucidate methods for recognition and identifying potential causes and treatments. Results This paper presents 4 cases, 2 of which had increasing gradients post-TAVR. All 3 pathology cases showed presence of a valve thrombosis in at least 2 TAV leaflets on autopsy, but were not visualized by transthoracic echocardiogram or transesophageal echocardiogram. One case was medically treated with oral anti coagulation with normalization of gradients. The consequence of valve thrombosis in all 3 pathology patients either directly or indirectly played a role in their early demise. At least 18 case reports of early valve thrombosis have been published. In 12 of these cases, the early treatment with anticoagulation therapy resolved the thrombus formation and normalized aortic pressures gradients successfully. Conclusions These 4 cases elucidate the occurrence of valve thrombosis post-TAVR. Consideration should be given to treatment with dual antiplatelet therapy and oral anticoagulation in patients post-TAVR with increasing mean pressure gradients and maximum aortic valve velocity. Further research should be conducted to create guidelines for antithrombotic therapy following TAVR procedure.
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