Large-Loss Behavior of Conditional Mean Risk Sharing

Insurance: Mathematics and Economics

2021

Self Cite

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.

Section: Conditional Mean Risk Sharing Rulementioning

confidence: 99%

Section: Relationships Between Sharing Rulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

From risk sharing to pure premium for a large number of heterogeneous losses

Insurance: Mathematics and Economics

2021

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“…It is worth to stress that these functions are not necessarily increasing. Counterexamples are easy to build when the supports of the risks

X_{i}

differ or when some risks dominate the others, as established in Denuit and Robert (2020a). There are also numerous examples where the functions

h_{i, n}^{⋆}

are increasing.…”

Section: Self‐governing or No‐insurance‐company Modelmentioning

confidence: 99%

Risk sharing under the dominant peer‐to‐peer property and casualty insurance business models

Risk Manage Insurance Review

2021

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This paper purposes to formalize the three business models dominating peer‐to‐peer (P2P) property and casualty insurance: the self‐governing model, the broker model, and the carrier model. The former one develops outside the insurance market whereas the latter ones may originate from the insurance industry, by partnering with an existing company or by issuing a new generation of participating insurance policies where part of the risk is shared within a community and higher losses, exceeding the community's risk‐bearing capacity are covered by an insurance or reinsurance company. The present paper proposes an actuarial modeling based on conditional mean risk sharing, to support the development of this new P2P insurance offer under each of the three business models. In addition, several specific questions are also addressed in the self‐governing model. Considering an economic agent who has to select the optimal pool for a risk to be shared with other participants, it is shown that uniform comparison of the Lorenz or concentration curves associated to the respective total losses of the pools under consideration allows the agent to decide which pool is preferable.

“…5 Applications to compound Panjer-Katz sums 5.1 Compound Panjer-Katz losses Denuit (2019Denuit ( , 2020a and Denuit and Robert (2020a) studied size-biasing and conditional mean risk allocation when individual losses are modeled as compound Panjer-Katz sums consisting of compound Binomial, compound Poisson, and compound Negative Binomial sums. This class of distributions is central to actuarial mathematics so that the results derived in this section are of wide applicability in insurance studies.…”

Section: Application To Conditional Mean Risk Allocationmentioning

confidence: 99%

Polynomial Series Expansions and Moment Approximations for Conditional Mean Risk Sharing of Insurance Losses

Methodol Comput Appl Probab

2021

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This paper exploits the representation of the conditional mean risk sharing allocations in terms of size-biased transforms to derive effective approximations within insurance pools of limited size. Precisely, the probability density functions involved in this representation are expanded with respect to the Gamma density and its associated Laguerre orthonormal polynomials, or with respect to the Normal density and its associated Hermite polynomials when the size of the pool gets larger. Depending on the thickness of the tails of the loss distributions, the latter may be replaced with their Esscher transform (or exponential tilting) of negative order. The numerical method then consists in truncating the series expansions to a limited number of terms. This results in an approximation in terms of the first moments of the individual loss distributions. Compound Panjer-Katz sums are considered as an application. The proposed method is compared with the well-established Panjer recursive algorithm. It appears to provide the analyst with reliable approximations that can be used to tune system parameters, before performing exact calculations.