Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses

2021 ◽  
Author(s):  
Michel Denuit ◽  
Christian Y. Robert
Keyword(s):  
Risk Sharing ◽  
Conditional Mean ◽  
Conditional Tail Expectation ◽  
Mean Risk

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.

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 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.

Culture and Risk Sharing Networks

2013 ◽  
Author(s):  
Joanne Laban
Keyword(s):  
Risk Sharing

Protecting All: Risk Sharing for a Diverse and Diversifying World of Work

2019 ◽  
Author(s):  
Truman Packard ◽  
Ugo Gentilini ◽  
Margaret Grosh ◽  
Philip O’Keefe ◽  
Robert Palacios ◽  
...  
Keyword(s):  
Risk Sharing

Lending to the Borrower from Hell

2017 ◽  
Author(s):  
Mauricio Drelichman ◽  
Hans-Joachim Voth
Keyword(s):  
United States ◽  
Sixteenth Century ◽  
Sovereign Debt ◽  
Risk Sharing ◽  
The United States ◽  
Philip Ii ◽  
Loan Contracts ◽  
The Right ◽  
Intelligent Model ◽  
New Evidence

Why do lenders time and again loan money to sovereign borrowers who promptly go bankrupt? When can this type of lending work? As the United States and many European nations struggle with mountains of debt, historical precedents can offer valuable insights. This book looks at one famous case—the debts and defaults of Philip II of Spain. Ruling over one of the largest and most powerful empires in history, King Philip defaulted four times. Yet he never lost access to capital markets and could borrow again within a year or two of each default. Exploring the shrewd reasoning of the lenders who continued to offer money, the book analyzes the lessons from this historical example. Using detailed new evidence collected from sixteenth-century archives, the book examines the incentives and returns of lenders. It provides powerful evidence that in the right situations, lenders not only survive despite defaults—they thrive. It also demonstrates that debt markets cope well, despite massive fluctuations in expenditure and revenue, when lending functions like insurance. The book unearths unique sixteenth-century loan contracts that offered highly effective risk sharing between the king and his lenders, with payment obligations reduced in bad times. A fascinating story of finance and empire, this book offers an intelligent model for keeping economies safe in times of sovereign debt crises and defaults.

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