Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Contributions to quantitative risk management in insurance Stéphane Loisel ISFA, Université Lyon 1 Habilitation à diriger les recherches Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 1 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Classical model A recent correlation crisis in Kruger Park Correlation between short-term mortality indicators can suddenly increase. Here correlation and the marginal risks increase at the same time. Another example: a drive with your mother-in-law. Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 4 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Classical model Classical assumptions and our problem Classical assumptions: R ( t ) = u + ct − � N ( t ) i = 1 X i claim amounts ( X i ) i ≥ 1 : sequence of i.i.d. r.v.’s, with finite mean, The ( X i ) i ≥ 1 are independent from ( N ( t )) t ≥ 0 (e.g. renewal process). Problem: Derive asymptotics of finite-time ruin probabilities for large risks ψ ( u , t ) = P ( ∃ τ ∈ [ 0 , t ] , R ( τ ) < 0 | R ( 0 ) = u ) , in more general models with dependence between claim amounts and possible correlation crises. Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 5 / 37
Claims: dependent Pareto • Compound Poisson risk model ( τ ∼ Exp ( λ )) • Claims X ∼ Exp (Θ) , where Θ ∼ Γ( α, β ) Ruin probability is � λ/ c 1 · β α Γ( α ) θ α − 1 e − βθ d θ Ψ( u ) = 0 � ∞ · β α λ λ θ c e − θ u e Γ( α ) θ α − 1 e − βθ c u + d θ λ/ c � �� � � �� � Ψ θ ( u ) f Θ ( θ ) , Θ ∼ Γ( α,β )
Claims: dependent Pareto 1 − Γ( α, βθ 0 ) Ψ( u ) = Γ( α ) Γ( α − 1 , ( β + u ) θ 0 ) β Γ( α )( βθ 0 ) α − 1 e − βθ 0 ( u + β ) − 1 + (( β + u ) θ 0 ) α − 2 e − ( β + u ) θ 0 � �� � → u →∞ 0 � �� � → u →∞ 1 One can see that the probability of ruin decays to a constant u →∞ Ψ( u ) = 1 − Γ( α, βλ c ) lim > 0 Γ( α ) as fast as u − 1 !! Compared to the independent case...
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Classical model Some types of correlation (not developed in this talk) Classical assumptions: R ( t ) = u + ct − � N ( t ) i = 1 X i claim amounts ( X i ) i ≥ 1 : sequence of i.i.d. r.v.’s, with finite mean, The ( X i ) i ≥ 1 are independent from ( N ( t )) t ≥ 0 (e.g. renewal process). Some models with embedded correlations: c is not constant over time and is adjusted to the observed previous claims (with Bühlmann linear credibility premium principle): impact of using credibility theory on the ruin probability (joint work with J. Trufin, 2009) dependence between the claim arrival process and the claim sizes (earthquake risk, flooding and drought risk, ...). Works of Boudreault et al. (2006), Albrecher et al. (2007), joint work with R. Biard, C. Lefèvre and H. Nagaraja (2010). dependence between claim arrivals and the intensity process: shot-noise processes, cycles influenced by large claims (joint work with M. Bargès and X. Venel, 2009). Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 6 / 37
Introduction Simple structure of dependence Dependence on the history of the process Our problems Flooding-type risk process Stéphane Loisel (ISFA, Lyon) A class of path-dependent claim amounts Istanbul 8 / 30
Introduction Simple structure of dependence Dependence on the history of the process Our problems Rewording flooding risk with simple ideas Stéphane Loisel (ISFA, Lyon) A class of path-dependent claim amounts Istanbul 9 / 30
Introduction Simple structure of dependence Dependence on the history of the process Our problems Earthquake-type risk process Stéphane Loisel (ISFA, Lyon) A class of path-dependent claim amounts Istanbul 10 / 30
Introduction Simple structure of dependence Dependence on the history of the process Our problems Rewording earthquake risk with simple ideas Stéphane Loisel (ISFA, Lyon) A class of path-dependent claim amounts Istanbul 11 / 30
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Dependence between claim amounts Impact of dependence between claim amounts In practice, claim amounts are influenced by common factors. This leads to stochastic correlation models. This correlation may change during time, due to endogenous risk or external shocks (joint works with Wayne Fisher and Shaun Wang (2007) and with Pierre Arnal and Romain Durand (2010)), parameter uncertainty (see Meyers (1999)), ... What happens ◮ if dependence between claim amounts is governed by a Markovian environment process? ◮ if claim amounts of different lines of business suddenly become more dependent (in a common shock model)? ◮ if those correlation crises are triggered by some large claims? Some useful references for sums of dependent risks: among others, Barbe et al. (2006), Albrecher et al. (2006), Kortschak and Albrecher (2008), Demarta (2002). Papers of Vladimir Kaishev and coauthors. Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 7 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk Systemic risk: securitization Does securitization really atomize risk? Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 8 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk Systemic risk: securitization Risk is just transferred and recombined, but does not disappear. If a large risk becomes reality, all counterparts may be affected and/or downgraded. Reinsurer’s default can lead to sudden increase in frequency and correlation of claim amounts for the insurer. How to compute the ruin probability in that case ? Work in progress with C. Blanchet, D. Dorobantu and S. Louhichi. Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 9 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk How can independent risks become suddenly strongly correlated? Endogenous uncertainty: Uncertainty is generated/modified by response of individual entities to events Feedback loop: outcomes → forecasts → decisions → outcomes → revised forecasts → revised decisions → . . . (Millennium Bridge) Statistical relationships are endogenous to the model, and may undergo structural shifts (Goodhart’s Law: Any observed statistical regularity will tend to collapse once pressure is placed upon it for control purposes ) Relevant when individual entities are similar in terms of forecasts and likely reactions to events Relevant when outcomes are sensitive to concerted actions Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 10 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk How can independent risks become suddenly strongly correlated? Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 11 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk How can independent risks become suddenly strongly correlated? Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 12 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk How can independent risks become suddenly strongly correlated? Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 13 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk Analogy with LTCM This analogy is drawn from Danielsson and Shin (2003). Similar potential feedback loops for surrender risk in life insurance. Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 14 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Our framework in Biard et al. (2008) Focus on some heavy-tailed distributions Definition (Regular variation class ( R ) ) F belongs to R − α , α ≥ 0 if and only if F ( xy ) F ( x ) = y − α lim x →∞ for any y > 0. Note that if 0 < α < 1, any X with c.d.f. F has infinite mean. Definition (Multivariate Regular variation class ( MR )) A random vector X = ( X 1 , ..., X n ) belongs to MR − α , α > 0 if and only if there exists a θ ∈ S n − 1 , where S n − 1 is the unit sphere with respect to a norm |·| ; such that P ( | X | > tu , X / | X | ∈ · ) v → t − α P S n − 1 ( θ ∈ · ) , P ( | X | > u ) v → denotes vague convergence on S n − 1 . where Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 15 / 37
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