LARGE-LOSS BEHAVIOR OF CONDITIONAL MEAN RISK SHARING

2020 ◽  
Vol 50 (3) ◽  
pp. 1093-1122 ◽  
Author(s):  
Michel Denuit ◽  
Christian Y. Robert
Keyword(s):  
Risk Sharing ◽  
Negative Binomial ◽  
Empirical Finding ◽  
Total Loss ◽  
Risk Allocation ◽  
Numerical Illustration ◽  
Sharing Rule ◽  
Conditional Mean ◽  
Compound Binomial ◽  
Mean Risk

AbstractWe consider the conditional mean risk allocation for an insurance pool, as defined by Denuit and Dhaene (2012). Precisely, we study the asymptotic behavior of the respective relative contributions of the participants as the total loss of the pool tends to infinity. The numerical illustration in Denuit (2019) suggests that the application of the conditional mean risk sharing rule may produce a linear sharing in the tail of the total loss distribution. This paper studies the validity of this empirical finding in the class of compound Panjer–Katz sums consisting of compound Binomial, compound Poisson, and compound Negative Binomial sums with either Gamma or Pareto severities. It is demonstrated that such a behavior does not hold in general since one term may dominate the other ones conditional of large total loss.

scholarly journals SIZE-BIASED TRANSFORM AND CONDITIONAL MEAN RISK SHARING, WITH APPLICATION TO P2P INSURANCE AND TONTINES

2019 ◽  
Vol 49 (03) ◽  
pp. 591-617 ◽  
Author(s):  
Michel Denuit
Keyword(s):  
Risk Sharing ◽  
Negative Binomial ◽  
Representation Formula ◽  
Individual Risk ◽  
Total Risk ◽  
Sharing Rule ◽  
Conditional Mean ◽  
Aggregate Loss ◽  
Binomial Sums ◽  
Mean Risk

AbstractUsing risk-reducing properties of conditional expectations with respect to convex order, Denuit and Dhaene [Denuit, M. and Dhaene, J. (2012). Insurance: Mathematics and Economics 51, 265–270] proposed the conditional mean risk sharing rule to allocate the total risk among participants to an insurance pool. This paper relates the conditional mean risk sharing rule to the size-biased transform when pooled risks are independent. A representation formula is first derived for the conditional expectation of an individual risk given the aggregate loss. This formula is then exploited to obtain explicit expressions for the contributions to the pool when losses are modeled by compound Poisson sums, compound Negative Binomial sums, and compound Binomial sums, to which Panjer recursion applies. Simple formulas are obtained when claim severities are homogeneous. A couple of applications are considered: first, to a peer-to-peer insurance scheme where participants share the first layer of their respective risks while the higher layer is ceded to a (re)insurer; second, to survivor credits to be shared among surviving participants in tontine schemes.

Convex order and comonotonic conditional mean risk sharing

2012 ◽  
Vol 51 (2) ◽  
pp. 265-270 ◽  
Author(s):  
Michel Denuit ◽  
Jan Dhaene
Keyword(s):  
Risk Sharing ◽  
Convex Order ◽  
Conditional Mean ◽  
Mean Risk

Conditional mean risk sharing in the individual model with graphical dependencies

2021 ◽  
pp. 1-27
Author(s):  
Michel Denuit ◽  
Christian Y. Robert
Keyword(s):  
Graphical Models ◽  
Risk Sharing ◽  
Risk Model ◽  
Allocation Rule ◽  
Individual Risk ◽  
Claim Amount ◽  
Conditional Mean ◽  
Operational Risks ◽  
The Individual ◽  
Mean Risk

Abstract Conditional mean risk sharing appears to be effective to distribute total losses amongst participants within an insurance pool. This paper develops analytical results for this allocation rule in the individual risk model with dependence induced by the respective position within a graph. Precisely, losses are modelled by zero-augmented random variables whose joint occurrence distribution and individual claim amount distributions are based on network structures and can be characterised by graphical models. The Ising model is adopted for occurrences and loss amounts obey decomposable graphical models that are specific to each participant. Two graphical structures are thus used: the first one to describe the contagion amongst member units within the insurance pool and the second one to model the spread of losses inside each participating unit. The proposed individual risk model is typically useful for modelling operational risks, catastrophic risks or cybersecurity risks.

scholarly journals PRINSIP PROPORSIONALITAS DAN GOVERNANCE TERHADAP ALOKASI DAN TRANSFER RISIKO DALAM SKEMA KERJASAMA PUBLIC-PRIVATE PARTNERSHIP (PPP)

Yuridika ◽  
2017 ◽  
Vol 32 (3) ◽  
pp. 541
Author(s):  
Yuniarti Yuniarti ◽  
Fifi Junita
Keyword(s):  
Risk Sharing ◽  
Public Private Partnership ◽  
Government Support ◽  
Risk Allocation ◽  
Infrastructure Development ◽  
Transfer Scheme ◽  
Private Partnership ◽  
Force Majeure ◽  
The Government ◽  
High Level

The high level of Foreign Direct Investment (FDI) is also supported by the availability of infrastructure to the remote area where the investment will be implemented. However, with limited funds from both APBN and APBD, infrastructure development can not be fully done by the government. Therefore, the government will cooperate with the investor (private) in the implementation of infrastructure development known as public private partnership. The main problem in implementing PPP is the allocation of risk to PPP projects. The different bargaining positions between the government and the private sector resulted in the fact that most of them impose risks on private parties (private). Implementation of PPP is closely related to the emergence of various risks including and not limited to regulatory risks, force majeure, etc. If there is no risk allocation arrangement proportionally based on governance principles, it weakens the pattern of PPP cooperation in Indonesia. PPP as one form of risk sharing in infrastructure investment should not release the role and government support to private parties / investors. Even in practice, PPP implementation in Indonesia only relies on BOT (Build Operate and Transfer) scheme which is expected to minimize government support in project implementation. This will ultimately lead to project failure.

scholarly journals An identity based on the generalised negative binomial distribution with applications in ruin theory

2018 ◽  
Vol 13 (2) ◽  
pp. 308-319
Author(s):  
David C. M. Dickson
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Probability Function ◽  
Negative Binomial ◽  
Risk Model ◽  
Case Evaluation ◽  
Time Of Ruin ◽  
Compound Binomial ◽  
Identity Based ◽  
The Time Of Ruin

AbstractIn this study, we show how expressions for the probability of ultimate ruin can be obtained from the probability function of the time of ruin in a particular compound binomial risk model, and from the density of the time of ruin in a particular Sparre Andersen risk model. In each case evaluation of generalised binomial series is required, and the argument of each series has a common form. We evaluate these series by creating an identity based on the generalised negative binomial distribution. We also show how the same ideas apply to the probability function of the number of claims in a particular Sparre Andersen model.

scholarly journals On the Stability of Recursive Formulas

1993 ◽  
Vol 23 (2) ◽  
pp. 227-258 ◽  
Author(s):  
Harry H. Panjer ◽  
Shaun Wang
Keyword(s):  
Binomial Distribution ◽  
Negative Binomial ◽  
Recursive Formula ◽  
Absolute Error ◽  
Recurrence Equation ◽  
Simple Method ◽  
Numerical Examples ◽  
Compound Binomial ◽  
Recursive Formulas ◽  
The Stability

AbstractBased on recurrence equation theory and relative error (rather than absolute error) analysis, the concept and criterion for the stability of a recurrence equation are clarified. A family of recursions, called congruent recursions, is proved to be strongly stable in evaluating its non-negative solutions. A type of strongly unstable recursion is identified. The recursive formula discussed by Panjer (1981) is proved to be strongly stable in evaluating the compound Poisson and the compound Negative Binomial (including Geometric) distributions. For the compound Binomial distribution, the recursion is shown to be unstable. A simple method to cope with this instability is proposed. Many other recursions are reviewed. Illustrative numerical examples are given.

scholarly journals A k-Inflated Negative Binomial Mixture Regression Model: Application to Rate–Making Systems

2018 ◽  
Vol 12 (2) ◽  
Author(s):  
Amir T. Payandeh Najafabadi ◽  
Saeed MohammadPour
Keyword(s):  
Regression Model ◽  
Mixture Models ◽  
Mixture Model ◽  
Negative Binomial ◽  
Mixture Distribution ◽  
Third Party ◽  
Numerical Illustration ◽  
Mixture Regression ◽  
Pure Premium ◽  
Binomial Mixture

Abstract This article introduces a k-Inflated Negative Binomial mixture distribution/regression model as a more flexible alternative to zero-inflated Poisson distribution/regression model. An EM algorithm has been employed to estimate the model’s parameters. Then, such new model along with a Pareto mixture model have employed to design an optimal rate–making system. Namely, this article employs number/size of reported claims of Iranian third party insurance dataset. Then, it employs the k-Inflated Negative Binomial mixture distribution/regression model as well as other well developed counting models along with a Pareto mixture model to model frequency/severity of reported claims in Iranian third party insurance dataset. Such numerical illustration shows that: (1) the k-Inflated Negative Binomial mixture models provide more fair rate/pure premiums for policyholders under a rate–making system; and (2) in the situation that number of reported claims uniformly distributed in past experience of a policyholder (for instance $k_1=1$ and $k_2=1$ instead of $k_1=0$ and $k_2=2$). The rate/pure premium under the k-Inflated Negative Binomial mixture models are more appealing and acceptable.

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