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Arthur CHARPENTIER - Modeling and covering catastrophes      Modeling and covering catastrophes                           ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                                     AgendaCatastro...
Arthur CHARPENTIER - Modeling and covering catastrophes                                                     AgendaCatastro...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Swiss Re (2007).                                          ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                       Some stylized facts“climatic...
Arthur CHARPENTIER - Modeling and covering catastrophes                  Some stylized facts : natural catastrophesInclude...
Arthur CHARPENTIER - Modeling and covering catastrophes                 Some stylized facts : man-made catastrophesInclude...
Arthur CHARPENTIER - Modeling and covering catastrophes                    Some stylized facts : ... mortality risk       ...
Arthur CHARPENTIER - Modeling and covering catastrophes                               longevity and mortality risks       ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                   What is a large claim ?An academ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Swiss Re (2008).                                          ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008).                                               ...
Arthur CHARPENTIER - Modeling and covering catastrophes                              The impact of a catastrophe• Property...
Arthur CHARPENTIER - Modeling and covering catastrophes                                   The impact of a catastrophe•    ...
Arthur CHARPENTIER - Modeling and covering catastrophes                Probabilistic concepts in risk managementLet X1 , ....
Arthur CHARPENTIER - Modeling and covering catastrophes                                   Modeling catastrophes• Man-made ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                Updating actuarial modelsIn classic...
Arthur CHARPENTIER - Modeling and covering catastrophes                                Updating actuarial models          ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                                     AgendaCatastro...
Arthur CHARPENTIER - Modeling and covering catastrophes                          Example : business interruptionBusiness i...
Arthur CHARPENTIER - Modeling and covering catastrophes                 Some results from Extreme Value TheoryWhen modelin...
Arthur CHARPENTIER - Modeling and covering catastrophes                                Cumulative distribution function, w...
Arthur CHARPENTIER - Modeling and covering catastrophes      Why the Pareto distribution ? historical perspectiveVilfredo ...
Arthur CHARPENTIER - Modeling and covering catastrophes                   Why the Pareto distribution ? historical perspec...
Arthur CHARPENTIER - Modeling and covering catastrophes Why the Pareto distribution ? mathematical explanationWe consider ...
Arthur CHARPENTIER - Modeling and covering catastrophes                       Large claims and the Pareto modelThe theorem...
Arthur CHARPENTIER - Modeling and covering catastrophes                              How to define large claims ?• Use of t...
Arthur CHARPENTIER - Modeling and covering catastrophes                2.5             Hill estimator of the slope        ...
Arthur CHARPENTIER - Modeling and covering catastrophes                            Extreme value distributions...A natural...
Arthur CHARPENTIER - Modeling and covering catastrophes                5.0   MLE of the tail index, using Generalized Pare...
Arthur CHARPENTIER - Modeling and covering catastrophes From the statistical model of claims to the pure premiumConsider t...
Arthur CHARPENTIER - Modeling and covering catastrophes From the statistical model of claims to the pure premiumThe averag...
Arthur CHARPENTIER - Modeling and covering catastrophes From the statistical model of claims to the pure premiumFor a clai...
Arthur CHARPENTIER - Modeling and covering catastrophes                                                     AgendaCatastro...
Arthur CHARPENTIER - Modeling and covering catastrophes                                    Increased value at riskIn 1950,...
Arthur CHARPENTIER - Modeling and covering catastrophes                      Two techniques to model large risks• The actu...
Arthur CHARPENTIER - Modeling and covering catastrophes                         The actuarial-statistical approach• Modeli...
Arthur CHARPENTIER - Modeling and covering catastrophes                                   Impact of global warming on natu...
Arthur CHARPENTIER - Modeling and covering catastrophes           More natural hazards with higher value at riskThe most d...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2006).                                               ...
Arthur CHARPENTIER - Modeling and covering catastrophes     Cat models : the meteorological-engineering approachThe basic ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : GIEC (2008).                                              ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008).                                               ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008).                                               ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008).                                               ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008).                                               ...
Arthur CHARPENTIER - Modeling and covering catastrophes        Hurricanes in Florida : Rare and extremal events ?Note that...
Arthur CHARPENTIER - Modeling and covering catastrophes                Modelling contagion in credit risk models          ...
Arthur CHARPENTIER - Modeling and covering catastrophes                 Modelling contagion in credit risk modelsIn Credit...
Arthur CHARPENTIER - Modeling and covering catastrophes                                                     AgendaCatastro...
Arthur CHARPENTIER - Modeling and covering catastrophes        Insurance versus credit, an historical background          ...
Arthur CHARPENTIER - Modeling and covering catastrophes                              Why a reinsurance market ?“reinsuranc...
Arthur CHARPENTIER - Modeling and covering catastrophes                     Which reinsurance treaty is optimal ?In a prop...
Arthur CHARPENTIER - Modeling and covering catastrophes                             Risk management solutions ?• Equity ho...
Arthur CHARPENTIER - Modeling and covering catastrophes                             Risk management solutions ?• Industry ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                                          57
Arthur CHARPENTIER - Modeling and covering catastrophes                           Trigger definition for peak risk• indemni...
Arthur CHARPENTIER - Modeling and covering catastrophes                           Trigger definition for peak risk• environ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                             Reinsurance           ...
Arthur CHARPENTIER - Modeling and covering catastrophes     Group                                  net W.P.      net W.P. ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                                 Side carsA hedge f...
Arthur CHARPENTIER - Modeling and covering catastrophes                           ILW - Insurance Loss WarrantyIndustry lo...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Banks (2005).                                             ...
Arthur CHARPENTIER - Modeling and covering catastrophes                             Cat Bonds and securitizationSecutizati...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Banks (2004).                                             ...
Arthur CHARPENTIER - Modeling and covering catastrophes                              Insurance Linked Securities  indemnit...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                          Mortality bondsSource : G...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2006).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Goldman Sachs (2006).                                     ...
Arthur CHARPENTIER - Modeling and covering catastrophes               USAA’s hurricane bond(s) : Residential ReUSAA, mutua...
Arthur CHARPENTIER - Modeling and covering catastrophes   class A-1, rated AAA, hurricane bondSource : Banks (2004).      ...
Arthur CHARPENTIER - Modeling and covering catastrophes   class A-2, rated BB, hurricane bondSource : Banks (2004).       ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                                          81
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006).                                              ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006).                                              ...
Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006).                                              ...
Arthur CHARPENTIER - Modeling and covering catastrophes                                                     AgendaCatastro...
Arthur CHARPENTIER - Modeling and covering catastrophes                            survey of literature on pricing• Fundam...
Arthur CHARPENTIER - Modeling and covering catastrophes                            survey of literature on pricing• Price ...
Arthur CHARPENTIER - Modeling and covering catastrophes                              from mass risk to large risksinsuranc...
Arthur CHARPENTIER - Modeling and covering catastrophes     risk premium and regulatory capital (points 4 and 5)Within an ...
Arthur CHARPENTIER - Modeling and covering catastrophes    independent risks, large portfolio (e.g. car insurance)        ...
Arthur CHARPENTIER - Modeling and covering catastrophes    independent risks, large portfolio (e.g. car insurance)        ...
Arthur CHARPENTIER - Modeling and covering catastrophes    independent risks, large portfolio (e.g. car insurance)        ...
Arthur CHARPENTIER - Modeling and covering catastrophes   independent risks, small portfolio (e.g. fire insurance)         ...
Arthur CHARPENTIER - Modeling and covering catastrophes   independent risks, small portfolio (e.g. fire insurance)         ...
Arthur CHARPENTIER - Modeling and covering catastrophes   independent risks, small portfolio (e.g. fire insurance)         ...
Arthur CHARPENTIER - Modeling and covering catastrophes  nonindependent risks, large portfolio (e.g. earthquake)          ...
Arthur CHARPENTIER - Modeling and covering catastrophes  nonindependent risks, large portfolio (e.g. earthquake)          ...
Arthur CHARPENTIER - Modeling and covering catastrophes  nonindependent risks, large portfolio (e.g. earthquake)          ...
Arthur CHARPENTIER - Modeling and covering catastrophes              the pure premium as a technical benchmarkPascal, Ferm...
Arthur CHARPENTIER - Modeling and covering catastrophes              the pure premium as a technical benchmark            ...
Arthur CHARPENTIER - Modeling and covering catastrophes          from pure premium to expected utility principle          ...
Arthur CHARPENTIER - Modeling and covering catastrophes     Distortion of values versus distortion of probabilities       ...
Arthur CHARPENTIER - Modeling and covering catastrophes     Distortion of values versus distortion of probabilities       ...
Arthur CHARPENTIER - Modeling and covering catastrophes       from pure premium to distorted premiums (Wang)              ...
Arthur CHARPENTIER - Modeling and covering catastrophes     Distortion of values versus distortion of probabilities       ...
Arthur CHARPENTIER - Modeling and covering catastrophes     Distortion of values versus distortion of probabilities       ...
Arthur CHARPENTIER - Modeling and covering catastrophes               some particular cases a classical premiumsThe expone...
Arthur CHARPENTIER - Modeling and covering catastrophes   pricing options in complete markets : the binomial caseIn comple...
Arthur CHARPENTIER - Modeling and covering catastrophes   pricing options in complete markets : the binomial caseThe only ...
Arthur CHARPENTIER - Modeling and covering catastrophes    financial versus actuarial pricing, a numerical example   risk-f...
Arthur CHARPENTIER - Modeling and covering catastrophes   financial versus actuarial pricing, a numerical example          ...
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  1. 1. Arthur CHARPENTIER - Modeling and covering catastrophes Modeling and covering catastrophes Arthur Charpentier Sao Paulo, April 2009 arthur.charpentier@univ-rennes1.fr http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/ 1
  2. 2. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks products and models• General introduction• Modeling very large claims• Natural catastrophes and accumulation risk• Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing covers• Pricing insurance linked securities• Risk measures, an economic introduction• Calculating risk measures for catastrophic risks• Pricing cat bonds : the Winterthur example• Pricing cat bonds : the Mexican Earthquake 2
  3. 3. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks products and models• General introduction• Modeling very large claims• Natural catastrophes and accumulation risk• Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing covers• Pricing insurance linked securities• Risk measures, an economic introduction• Calculating risk measures for catastrophic risks• Pricing cat bonds : the Winterthur example• Pricing cat bonds : the Mexican Earthquake 3
  4. 4. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Swiss Re (2007). 4
  5. 5. Arthur CHARPENTIER - Modeling and covering catastrophes Some stylized facts“climatic risk in numerous branches of industry is more important than the riskof interest rates or foreign exchange risk” (AXA 2004, quoted in Ceres (2004)). Fig. 1 – Major natural catastrophes (Source : Munich Re (2006)). 5
  6. 6. Arthur CHARPENTIER - Modeling and covering catastrophes Some stylized facts : natural catastrophesIncludes hurricanes, tornados, winterstorms, earthquakes, tsunamis, hail,drought, floods... Date Loss event Region Overall losses Insured losses Fatalities 25.8.2005 Hurricane Katrina USA 125,000 61,000 1,322 23.8.1992 Hurricane Andrew USA 26,500 17,000 62 17.1.1994 Earthquake Northridge USA 44,000 15,300 61 21.9.2004 Hurricane Ivan USA, Caribbean 23,000 13,000 125 19.10.2005 Hurricane Wilma Mexico, USA 20,000 12,400 42 20.9.2005 Hurricane Rita USA 16,000 12,000 10 11.8.2004 Hurricane Charley USA, Caribbean 18,000 8,000 36 26.9.1991 Typhoon Mireille Japan 10,000 7,000 62 9.9.2004 Hurricane Frances USA, Caribbean 12,000 6,000 39 26.12.1999 Winter storm Lothar Europe 11,500 5,900 110Tab. 1 – The 10 most expensive natural catastrophes, 1950-2005 (Source : MunichRe (2006)). 6
  7. 7. Arthur CHARPENTIER - Modeling and covering catastrophes Some stylized facts : man-made catastrophesIncludes industry fire, oil & gas explosions, aviation crashes, shipping and raildisasters, mining accidents, collapse of building or bridges, terrorism... Date Location Plant type Event type Loss (property) 23.10.1989 Texas, USA petrochemical∗ vapor cloud explosion 839 04.05.1988 Nevada, USA chemical explosion 383 05.05.1988 Louisiana, USA refinery vapor cloud explosion 368 14.11.1987 Texas, USA petrochemical vapor cloud explosion 282 07.07.1988 North sea platform∗ explosion 1,085 26.08.1992 Gulf of Mexico platform explosion 931 23.08.1991 North sea concrete jacket mechanical damage 474 24.04.1988 Brazil plateform blowout 421Tab. 2 – Onshore and offshore largest property damage losses (from 1970-1999).The largest claim is now the 9/11 terrorist attack, with a US$ 21, 379 millioninsured loss.∗ evaluated loss US$ 2, 155 million and explosion on platform piper Alpha, US$ 3, 409 million (Swiss Re (2006)). 7
  8. 8. Arthur CHARPENTIER - Modeling and covering catastrophes Some stylized facts : ... mortality risk “there seems to be broad agreement that there exists a market price for systematic mortality risk. Howe- ver, there seems to be no agreement on the structure and level of this price, and how it should be incorpo- rated when valuating insurance products or mortality derivatives” Bauer & Russ (2006). “securitization of longevity risk is not only a good method for risk diversifying, but also provides low beta investment assets to the capital market” Liao, Yang & Huang (2007). 8
  9. 9. Arthur CHARPENTIER - Modeling and covering catastrophes longevity and mortality risks Yea r Age 5e−02 5e−02 2e−02 60 years old 40 years old 20 years old 5e−03 5e−03 2e−03 5e−04 1899 1948 1997 5e−04 5e−05 0 20 40 60 80 100 1900 1920 1940 1960 1980 2000 Age Age Fig. 2 – Mortality rate surface (function of age and year). 9
  10. 10. Arthur CHARPENTIER - Modeling and covering catastrophes What is a large claim ?An academic answer ? Teugels (1982) defined “large claims”, Answer 1 “large claims are the upper 10% largest claims”, Answer 2 “large claims are every claim that consumes at least 5% of the sum of claims, or at least 5% of the net premiums”, Answer 3 “large claims are every claim for which the actuary has to go and see one of the chief members of the company”.Examples Traditional types of catastrophes, natural (hurricanes, typhoons,earthquakes, floods, tornados...), man-made (fires, explosions, businessinterruption...) or new risks (terrorist acts, asteroids, power outages...).From large claims to catastrophe, the difference is that there is a before thecatastrophe, and an after : something has changed ! 10
  11. 11. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Swiss Re (2008). 11
  12. 12. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 12
  13. 13. Arthur CHARPENTIER - Modeling and covering catastrophes The impact of a catastrophe• Property damage : houses, cars and commercial structures,• Human casualties (may not be correlated with economic loss),• Business interruptionExample• Natural Catastrophes - USA : succession of natural events that have hit insurers, reinsurers and the retrocession market• lack of capacity, strong increase in rate• Natural Catastrophes - nonUSA : in Asia (earthquakes, typhoons) and Europe (flood, drought, subsidence)• sui generis protection programs in some countries 13
  14. 14. Arthur CHARPENTIER - Modeling and covering catastrophes The impact of a catastrophe• Storms - Europe : high speed wind in Europe and US, considered as insurable• main risk for P&C insurers• Terrorism, including nuclear, biologic or bacteriologic weapons• lack of capacity, strong social pressure : private/public partnerships• Liabilities, third party damage• growth in indemnities (jurisdictions) yield unsustainable losses• Transportation (maritime and aircrafts), volatile business, and concentrated market 14
  15. 15. Arthur CHARPENTIER - Modeling and covering catastrophes Probabilistic concepts in risk managementLet X1 , ..., Xn denote some claim size (per policy or per event),• the survival probability or exceedance probability is F (x) = P(X > x) = 1 − F (x),• the pure premium or expected value is ∞ ∞ E(X) = xdF (x) = F (x)dx, 0 0• the Value-at-Risk or quantile function is −1 −1 V aR(X, u) = F (u) = F (1 − u) i.e. P(X > V aR(X, u)) = 1 − u,• the return period is T (u) = 1/F (x)(u). 15
  16. 16. Arthur CHARPENTIER - Modeling and covering catastrophes Modeling catastrophes• Man-made catastrophes : modeling very large claims,• extreme value theory (ex : business interruption)• Natural Catastrophes : modeling very large claims taking into accont accumulation and global warming• extreme value theory for losses• time series theory for occurrence• credit risk models for contagion or accumulation 16
  17. 17. Arthur CHARPENTIER - Modeling and covering catastrophes Updating actuarial modelsIn classical actuarial models (from Cramer and Lundberg), one usually ´consider• a model for the claims occurrence, e.g. a Poisson process,• a model for the claim size, e.g. a exponential, Weibull, lognormal...For light tailed risk, Cram´r-Lundberg’s theory gives a bound for the ruin eprobability, assuming that claim size is not to large. Furthermore, additionalcapital to ensure solvency (non-ruin) can be obtained using the central limittheorem (see e.g. RBC approach). But the variance has to be finite.In the case of large risks or catastrophes, claim size has heavy tails (e.g. thevariance is usually infinite), but the Poisson assumption for occurrence is stillrelevant. 17
  18. 18. Arthur CHARPENTIER - Modeling and covering catastrophes Updating actuarial models NExample For business interruption, the total loss is S = Xi where N is i=1Poisson, and the Xi ’s are i.i.d. Pareto.Example In the case of natural catastrophes, claim size is not necessarily huge,but the is an accumulation of claims, and the Poisson distribution is not relevant.But if considering events instead of claims, the Poisson model can be relevant.But the Poisson process is nonhomogeneous. NExample For hurricanes or winterstorms, the total loss is S = Xi where N is i=1 NiPoisson, and Xi = Xi,j , where the Xi,j ’s are i.i.d. j=1 18
  19. 19. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks products and models• General introduction• Modeling very large claims• Natural catastrophes and accumulation risk• Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing covers• Pricing insurance linked securities• Risk measures, an economic introduction• Calculating risk measures for catastrophic risks• Pricing cat bonds : the Winterthur example• Pricing cat bonds : the Mexican Earthquake 19
  20. 20. Arthur CHARPENTIER - Modeling and covering catastrophes Example : business interruptionBusiness interruption claims can be very expensive. Zajdenweber (2001)claimed that it is a noninsurable risk since the pure premium is (theoretically)infinite.Remark For the 9/11 terrorist attacks, business interruption represented US$ 11billion. 20
  21. 21. Arthur CHARPENTIER - Modeling and covering catastrophes Some results from Extreme Value TheoryWhen modeling large claims (industrial fire, business interruption,...) : extremevalue theory framework is necessary.The Pareto distribution appears naturally when modeling observations over agiven threshold, b x F (x) = P(X ≤ x) = 1 − , where x0 = exp(−a/b) x0Then equivalently log(1 − F (x)) ∼ a + b log x, i.e. for all i = 1, ..., n, log(1 − Fn (Xi )) ∼ a + b · log Xi .Remark : if −b ≥ 1, then EP (X) = ∞, the pure premium is infinite.The estimation of b is a crucial issue. 21
  22. 22. Arthur CHARPENTIER - Modeling and covering catastrophes Cumulative distribution function, with confidence interval 1.0 lo#!lo# %areto *lot, ,it. /onfiden/e inter3al 0 lo)arit.m of t.e sur5i5al 6ro7a7ilities !1 0.8 cumulative probabilities !# 0.6 !$ 0.4 !% 0.2 !5 0.0 0 1 2 3 4 5 0 1 # $ % 5 logarithm of the losses lo)arit.m of t.e losses Fig. 3 – Pareto modeling for business interruption claims. 22
  23. 23. Arthur CHARPENTIER - Modeling and covering catastrophes Why the Pareto distribution ? historical perspectiveVilfredo Pareto observed that 20% of the population owns 80% of the wealth. 80% of the claims 20% of the losses 20% of the claims 80% of the losses Fig. 4 – The 80-20 Pareto principle.Example Over the period 1992-2000 in business interruption claims in France,0.1% of the claims represent 10% of the total loss. 20% of the claims represent73% of the losses. 23
  24. 24. Arthur CHARPENTIER - Modeling and covering catastrophes Why the Pareto distribution ? historical perspective Lorenz curve of business interruption claims 1.0 0.8 73% OF Proportion of claim size THE LOSSES 0.6 0.4 20% OF 0.2 THE CLAIMS 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Proportion of claims number Fig. 5 – The 80-20 Pareto principle. 24
  25. 25. Arthur CHARPENTIER - Modeling and covering catastrophes Why the Pareto distribution ? mathematical explanationWe consider here the exceedance distribution, i.e. the distribution of X − u giventhat X > u, with survival distribution G(·) defined as F (x + u) G(x) = P(X − u > x|X > u) = F (u)This is closely related to some regular variation property, and only powerfunction my appear as limit when u → ∞ : G(·) is necessarily a power function. The Pareto model in actuarial literatureSwiss Re highlighted the importance of the Pareto distribution in two technicalbrochures the Pareto model in property reinsurance and estimating propertyexcess of loss risk premium : The Pareto model.Actually, we will see that the Pareto model gives much more than only apremium. 25
  26. 26. Arthur CHARPENTIER - Modeling and covering catastrophes Large claims and the Pareto modelThe theorem of Pickands-Balkema-de Haan states that if the X1 , ..., Xn areindependent and identically distributed, for u large enough, −1/ξ    1+ξ x  if ξ = 0, P(X − u > x|X > u) ∼ Hξ,σ(u) (x) = σ(u)  exp − x   if ξ = 0, σ(u)for some σ(·). It simply means that large claims can always be modeled using the(generalized) Pareto distribution.The practical question which always arises is then “what are large claims”, i.e.how to chose u ? 26
  27. 27. Arthur CHARPENTIER - Modeling and covering catastrophes How to define large claims ?• Use of the k largest claims : Hill’s estimatorThe intuitive idea is to fit a linear straight line since for the largest claimsi = 1, ..., n, log(1 − Fn (Xi )) ∼ a + blog Xi . Let bk denote the estimator based onthe k largest claims.Let {Xn−k+1:n , ..., Xn−1:n , Xn:n } denote the set of the k largest claims. Recallthat ξ ∼ −1/b, and then n 1 ξ= log(Xn−k+i:n ) − log(Xn−k:n ). k i=1 27
  28. 28. Arthur CHARPENTIER - Modeling and covering catastrophes 2.5 Hill estimator of the slope Hill estimator of the 95% VaR 10 2.0 8 quantile (95%) slope (!b) 6 1.5 4 1.0 2 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Fig. 6 – Pareto modeling for business interruption claims : tail index. 28
  29. 29. Arthur CHARPENTIER - Modeling and covering catastrophes Extreme value distributions...A natural idea is to fit a generalized Pareto distribution for claims exceeding u,for some u large enough.threshold [1] 3, we chose u = 3p.less.thresh [1] 0.9271357, i.e. we keep to 8.5% largest claimsn.exceed [1] 87method [1] ‘‘ml’’, we use the maximum likelihood technique,par.ests, we get estimators ξ and σ, xi sigma 0.6179447 2.0453168par.ses, with the following standard errors xi sigma 0.1769205 0.4008392 29
  30. 30. Arthur CHARPENTIER - Modeling and covering catastrophes 5.0 MLE of the tail index, using Generalized Pareto Model Estimation of VaR and TVaR (95%) 5 e!02 1 e!02 4.5 1!F(x) (on log scale) 95 tail index 2 e!03 4.0 99 5 e!04 3.5 1 e!04 3.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 5 10 20 50 100 200 x (on log scale) Fig. 7 – Pareto modeling for business interruption claims : VaR and TVaR. 30
  31. 31. Arthur CHARPENTIER - Modeling and covering catastrophes From the statistical model of claims to the pure premiumConsider the following excess-of-loss treaty, with a priority d = 20, and an upperlimit 70. Historical business interruption claims 140 130 120 110 100 90 80 70 60 50 40 30 20 10 1993 1994 1995 1996 1997 1998 1999 2000 2001 Fig. 8 – Pricing of a reinsurance layer. 31
  32. 32. Arthur CHARPENTIER - Modeling and covering catastrophes From the statistical model of claims to the pure premiumThe average number of claims per year is 145, year 1992 1993 1994 1995 1996 1997 1998 1999 2000 frequency 173 152 146 131 158 138 120 156 136 Tab. 3 – Number of business interruption claims. 32
  33. 33. Arthur CHARPENTIER - Modeling and covering catastrophes From the statistical model of claims to the pure premiumFor a claim size x, the reinsurer’s indemnity is I(x) = min{u, max{0, x − d}}.The average indemnity of the reinsurance can be obtained using the Paretomodel, ∞ u E(I(X)) = I(x)dF (x) = (x − d)dF (x) + u(1 − F (u)), 0 dwhere F is a Pareto distribution.Here E(I(X)) = 0.145. The empirical estimate (burning cost) is 0.14.The pure premium of the reinsurance treaty is 20.6.Example If d = 50 and u = ∞, π = 8.9 (12 for burning cost... based on 1 claim). 33
  34. 34. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks modelling• General introduction• Business interruption and very large claims• Natural catastrophes and accumulation risk• Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements• Risk measures, an economic introduction• Calculating risk measures for catastrophic risks• Diversification and capital allocation 34
  35. 35. Arthur CHARPENTIER - Modeling and covering catastrophes Increased value at riskIn 1950, 30% of the world’s population (2.5 billion people) lived in cities. In 2000,50% of the world’s population (6 billon).In 1950 the only city with more than 10 million inhabitants was New York. Therewere 12 in 1990, and 26 are expected by 2015, including• Tokyo (29 million),• New York (18 million),• Los Angeles (14 million).• Increasing value at risk (for all risks)The total value of insured costal exposure in 2004 was• $1, 937 billion in Florida (18 million),• $1, 902 billion in New York. 35
  36. 36. Arthur CHARPENTIER - Modeling and covering catastrophes Two techniques to model large risks• The actuarial-statistical technique : modeling historical series,The actuary models the occurrence process of events, and model the claim size(of the total event).This is simple but relies on stability assumptions. If not, one should modelchanges in the occurrence process, and should take into account inflation orincrease in value-at-risk.• The meteorological-engineering technique : modeling natural hazard and exposure.This approach needs a lot of data and information so generate scenarios takingall the policies specificities. Not very flexible to estimate return periods, andworks as a black box. Very hard to assess any confidence levels. 36
  37. 37. Arthur CHARPENTIER - Modeling and covering catastrophes The actuarial-statistical approach• Modeling event occurrence, the problem of global warming.Global warming has an impact on climate related hazard (droughts, subsidence,hurricanes, winterstorms, tornados, floods, coastal floods) but not geophysical(earthquakes).• Modeling claim size, the problem of increase of value at risk and inflation.Pielke & Landsea (1998) normalized losses due to hurricanes, using bothpopulation and wealth increases, “with this normalization, the trend of increasingdamage amounts in recent decades disappears”. 37
  38. 38. Arthur CHARPENTIER - Modeling and covering catastrophes Impact of global warming on natural hazard !u#$er o) *urricanes, per 2ear 3853!6008 25 Frequency of hurricanes 20 15 10 5 0 1850 1900 1950 2000 Year Fig. 9 – Number of hurricanes and major hurricanes per year. 38
  39. 39. Arthur CHARPENTIER - Modeling and covering catastrophes More natural hazards with higher value at riskThe most damaging tornadoes in the U.S. (1890-1999), adjusted with wealth, arethe following, Date Location Adjusted loss 28.05.1896 Saint Louis, IL 2,916 29.09.1927 Saint Louis, IL 1,797 18.04.1925 3 states (MO, IL, IN) 1,392 10.05.1979 Wichita Falls, TX 1,141 09.06.1953 Worcester, MA 1,140 06.05.1975 Omaha, NE 1,127 08.06.1966 Topeka, KS 1,126 06.05.1936 Gainesville, GA 1,111 11.05.1970 Lubbock, TX 1,081 28.06.1924 Lorain-Sandusky, OH 1,023 03.05.1999 Oklahoma City, OK 909 11.05.1953 Waco, TX 899 27.04.1890 Louisville, KY 836 Tab. 4 – Most damaging tornadoes (from Brooks & Doswell (2001)). 39
  40. 40. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2006). 40
  41. 41. Arthur CHARPENTIER - Modeling and covering catastrophes Cat models : the meteorological-engineering approachThe basic framework is the following, • the natural hazard model : generate stochastic climate scenarios, and assess perils, • the engineering model : based on the exposure, the values, the building, calculate damage, • the insurance model : quantify financial losses based on deductibles, reinsurance (or retrocession) treaties. 41
  42. 42. Arthur CHARPENTIER - Modeling and covering catastrophesSource : GIEC (2008). 42
  43. 43. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 43
  44. 44. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 44
  45. 45. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 45
  46. 46. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 46
  47. 47. Arthur CHARPENTIER - Modeling and covering catastrophes Hurricanes in Florida : Rare and extremal events ?Note that for the probabilities/return periods of hurricanes related to insuredlosses in Florida are the following (source : Wharton Risk Center & RMS) $ 1 bn $ 2 bn $ 5 bn $ 10 bn $ 20 bn $ 50 bn 42.5% 35.9% 24.5% 15.0% 6.9% 1.7% 2 years 3 years 4 years 7 years 14 years 60 years $ 75 bn $ 100 bn $ 150 bn $ 200 bn $ 250 bn 0.81% 0.41% 0.11% 0.03% 0.005% 123 years 243 years 357 years 909 years 2, 000 years Tab. 5 – Extremal insured losses (from Wharton Risk Center & RMS).Recall that historical default (yearly) probabilities are AAA AA A BBB BB B 0.00% 0.01% 0.05% 0.37% 1.45% 6.59% - 10, 000 years 2, 000 years 270 years 69 years 15 years Tab. 6 – Return period of default (from S&P’s (1981-2003)). 47
  48. 48. Arthur CHARPENTIER - Modeling and covering catastrophes Modelling contagion in credit risk models cat insurance credit risk n total number of insured n number of credit issuers 1 if policy i claims 1 if issuers i defaults Ii = Ii = 0 if not 0 if not Mi total sum insured Mi nominal Xi exposure rate 1 − Xi recovery rate 48
  49. 49. Arthur CHARPENTIER - Modeling and covering catastrophes Modelling contagion in credit risk modelsIn CreditMetrics, the idea is to generate random scenario to get the Profit &Loss distribution of the portfolio.• the recovery rate is modeled using a beta distribution,• the exposure rate is modeled using a MBBEFD distribution (see Bernegger (1999)).To generate joint defaults, CreditMetrics proposed a probit model. 49
  50. 50. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks modelling• General introduction• Modeling very large claims• Natural catastrophes and accumulation risk• Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements• Risk measures, an economic introduction• Calculating risk measures for catastrophic risks• Diversification and capital allocation 50
  51. 51. Arthur CHARPENTIER - Modeling and covering catastrophes Insurance versus credit, an historical background The Babylonians developed a system which was re- corded in the famous Code of Hammurabi (1750 BC) and practiced by early Mediterranean sailing mer- chants. If a merchant received a loan to fund his shipment, he would pay the lender an additional sum in exchange for the lender’s guarantee to cancel the loan should the shipment be stolen. cf. cat bonds. 51
  52. 52. Arthur CHARPENTIER - Modeling and covering catastrophes Why a reinsurance market ?“reinsurance is the transfer of part of the hazards of risks that a direct insurerassumes by way of reinsurance contracts or legal provision on behalf of aninsured, to a second insurancce carrier, the reinsurer, who has no directcontractual relationship with the insured” (Swiss Re, introduction to reinsurance)Reinsurance allwo (primary) insurers to increase the maximum amount they caninsure for a given loss : they can optimize their underwriting capacity withoutburdening their need to cover their solvency margin.The law of large number can be used by insurance companies to assess theirprobable annual loss... but under strong assumptions of identical distribution(hence past event can be used to estimate future one) and independence. 52
  53. 53. Arthur CHARPENTIER - Modeling and covering catastrophes Which reinsurance treaty is optimal ?In a proportional agreement, the cedent and the reinsurer will agree on acontractually defined ratio to share (identically) the premiums and the lossesIn a non-proportional reinsurance treaty, the amount up to which the insurer willkeep (entierely) the loss is defined. The reinsurance company will pay the lossabove the deductible (up to a certain limit).The Excess-of-Loss (XL) trearty, as the basis for non-proportional reinsurance,with• a risk XL : any individual claim can trigger the cover• an event (or cat) XL : only a loss event involving several individual claims are covered by the treaty• a stop-loss, or excess-of-loss ratio : the deductible and the limit og liability are expressed as annnual aggregate amounts (usually as percentage of annual premium). 53
  54. 54. Arthur CHARPENTIER - Modeling and covering catastrophes Risk management solutions ?• Equity holding : holding in solvency margin+ easy and basic buffer− very expensive• Reinsurance and retrocession : transfer of the large risks to better diversified companies+ easy to structure, indemnity based− business cycle influences capacities, default risk• Side cars : dedicated reinsurance vehicules, with quota share covers+ add new capacity, allows for regulatory capital relief− short maturity, possible adverse selection 54
  55. 55. Arthur CHARPENTIER - Modeling and covering catastrophes Risk management solutions ?• Industry loss warranties (ILW) : index based reinsurance triggers+ simple to structure, no credit risk− limited number of capacity providers, noncorrelation risk, shortage of capacity• Cat bonds : bonds with capital and/or interest at risk when a specified trigger is reached+ large capacities, no credit risk, multi year contracts− more and more industry/parametric based, structuration costs 55
  56. 56. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 56
  57. 57. Arthur CHARPENTIER - Modeling and covering catastrophes 57
  58. 58. Arthur CHARPENTIER - Modeling and covering catastrophes Trigger definition for peak risk• indemnity trigger : directly connected to the experienced damage+ no risk for the cedant, only one considered by some regulator (NAIC)− time necessity to estimate actual damage, possible adverse selection (audit needed)• industry based index trigger : connected to the accumulated loss of the industry (PCS)+ simple to use, no moral hazard− noncorrelation risk 58
  59. 59. Arthur CHARPENTIER - Modeling and covering catastrophes Trigger definition for peak risk• environmental based index trigger : connected to some climate index (rainfall, windspeed, Richter scale...) measured by national authorities and meteorological offices+ simple to use, no moral hazard− noncorrelation risk, related only to physical features (not financial consequences)• parametric trigger : a loss event is given by a cat-software, using climate inputs, and exposure data+ few risk for the cedant if the model fits well− appears as a black-box 59
  60. 60. Arthur CHARPENTIER - Modeling and covering catastrophes Reinsurance The insurance approach (XL treaty) 35 30 25 REINSURER Loss per event 20 15 INSURER 10 INSURED 5 0 0.0 0.2 0.4 0.6 0.8 1.0 Event Fig. 10 – The XL reinsurance treaty mechanism. 60
  61. 61. Arthur CHARPENTIER - Modeling and covering catastrophes Group net W.P. net W.P. loss ratio total Shareholders’ Funds (2005) (2004) (2005) (2004) Munich Re 17.6 20.5 84.66% 24.3 24.4 Swiss Re (1) 16.5 20 85.78% 15.5 16 Berkshire Hathaway Re 7.8 8.2 91.48% 40.9 37.8 Hannover Re 7.1 7.8 85.66% 2.9 3.2 GE Insurance Solutions 5.2 6.3 164.51% 6.4 6.4 Lloyd’s 5.1 4.9 103.2% XL Re 3.9 3.2 99.72% Everest Re 3 3.5 93.97% 3.2 2.8 Reinsurance Group of America Inc. 3 2.6 1.9 1.7 PartnerRe 2.8 3 86.97% 2.4 2.6 Transatlantic Holdings Inc. 2.7 2.9 84.99% 1.9 2 Tokio Marine 2.1 2.6 26.9 23.9 Scor 2 2.5 74.08% 1.5 1.4 Odyssey Re 1.7 1.8 90.54% 1.2 1.2 Korean Re 1.5 1.3 69.66% 0.5 0.4 Scottish Re Group Ltd. 1.5 0.4 0.9 0.6 Converium 1.4 2.9 75.31% 1.2 1.3 Sompo Japan Insurance Inc. 1.4 1.6 25.3% 15.3 12.1 Transamerica Re (Aegon) 1.3 0.7 5.5 5.7 Platinum Underwriters Holdings 1.3 1.2 87.64% 1.2 0.8 Mitsui Sumitomo Insurance 1.3 1.5 63.18% 16.3 14.1Tab. 7 – Top 25 Global Reinsurance Groups in 2005 (from Swiss Re (2006)). 61
  62. 62. Arthur CHARPENTIER - Modeling and covering catastrophes Side carsA hedge fund that wishes to get into the reinsurance business will start a specialpurpose vehicle with a reinsurer.The hedge fund is able to get into reinsurance without hiring underwriters,buying models, nor getting rated by the rating agencies 62
  63. 63. Arthur CHARPENTIER - Modeling and covering catastrophes ILW - Insurance Loss WarrantyIndustry loss warranties pay a fixed amount based of the amount of industry loss(PCS or SIGMA).Example For example, a $30 million ILW with a $5 billion trigger. Cat bonds and securitizationBonds issued to cover catastrophe risk were developed subsequent to HurricaneAndrewThese bonds are structured so that the investor has a good return if there are noqualifying events and a poor return if a loss occurs. Losses can be triggered on anindustry index or on an indemnity basis. 63
  64. 64. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 64
  65. 65. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 65
  66. 66. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 66
  67. 67. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Banks (2005). 67
  68. 68. Arthur CHARPENTIER - Modeling and covering catastrophes Cat Bonds and securitizationSecutizations in capital markets were intiated with mortgage-backed securities (MBS) collaterized mortgage obligations (CMO) asset-backed securities (ABS) collaterized loan obligations (CLO) collaterized bond obligations (CBO) collaterized debt obligations (CDO) 68
  69. 69. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Banks (2004). 69
  70. 70. Arthur CHARPENTIER - Modeling and covering catastrophes Insurance Linked Securities indemnity trigger index trigger parametric trigger 70
  71. 71. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 71
  72. 72. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 72
  73. 73. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 73
  74. 74. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 74
  75. 75. Arthur CHARPENTIER - Modeling and covering catastrophes Mortality bondsSource : Guy Carpenter (2008). 75
  76. 76. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2006). 76
  77. 77. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Goldman Sachs (2006). 77
  78. 78. Arthur CHARPENTIER - Modeling and covering catastrophes USAA’s hurricane bond(s) : Residential ReUSAA, mutually owned insurance company (auto, householders, dwelling,personal libability for US military personal, and family).Hurricane Andrew (1992) : USD 620 millionEarly 1996, work with AIR and Merrill Lynch (and later Goldman Sachs andLehman Brothers) to transfer a part of their portfolioBond structured to give the insurer cover of the Excess-of-Loss layer above USD1 billon, to a maximum of USD 500 million, at an 80% rate (i.e. 20% coinsured),provided by an insurance vehicule Residential Re, established as a Cayman SPR.The SPR issued notes to investors, in 2 classes of 3 tranches, class A-1, rated AAA, featuring a USD 77 million tranche of principal protected notes, and USD 87 million of principal variable notes, class A-2, rated BB, featuring a USD 313 million of principal variable notes,Trigger is the single occurrence of a class 3-5 hurricane, with ultimate net loss asdefined under USAA’s portfolio parameters (indemnity trigger) 78
  79. 79. Arthur CHARPENTIER - Modeling and covering catastrophes class A-1, rated AAA, hurricane bondSource : Banks (2004). 79
  80. 80. Arthur CHARPENTIER - Modeling and covering catastrophes class A-2, rated BB, hurricane bondSource : Banks (2004). 80
  81. 81. Arthur CHARPENTIER - Modeling and covering catastrophes 81
  82. 82. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006). 82
  83. 83. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006). 83
  84. 84. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006). 84
  85. 85. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks modelling• General introduction• Business interruption and very large claims• Natural catastrophes and accumulation risk• Insurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing covers• Pricing insurance linked securities• Risk measures, an economic introduction• Calculating risk measures for catastrophic risks• Pricing cat bonds : the Winterthur example• Pricing cat bonds : the Mexican Earthquake 85
  86. 86. Arthur CHARPENTIER - Modeling and covering catastrophes survey of literature on pricing• Fundamental asset pricing theorem, in finance, Cox & Ross (JFE, 1976), Harrison & Kreps (JET, 1979), Harrison & Pliska (SPA, 1981, 1983). Recent general survey– Dana & Jeanblanc-Picque (1998). March´s financiers en temps continu : ´ e ´ valorisation et ´quilibre. Economica. e– Duffie (2001). Dynamic Asset Pricing Theory. Princeton University Press.– Bingham & Kiesel (2004). Risk neutral valuation. Springer Verlag• Premium calculation, in insurance.– Buhlmann (1970) Mathematical Methods in Risk Theory. Springer Verlag. ¨– Goovaerts, de Vylder & Haezendonck (1984). Premium Calculation in Insurance. Springer Verlag.– Denuit & Charpentier (2004). Math´matiques de l’assurance non-vie, tome e ´ 1. Economica. 86
  87. 87. Arthur CHARPENTIER - Modeling and covering catastrophes survey of literature on pricing• Price of uncertain quantities, in economics of uncertainty, von Neumann & Morgenstern (1944), Yaari (E, 1987). Recent general survey– Quiggin (1993). Generalized expected utility theory : the rank-dependent model. Kluwer Academic Publishers.– Gollier (2001). The Economics of Risk and Time. MIT Press. 87
  88. 88. Arthur CHARPENTIER - Modeling and covering catastrophes from mass risk to large risksinsurance is “the contribution of the many to the misfortune of the few”. 1. judicially, an insurance contract can be valid only if claim occurrence satisfy some randomness property, 2. the “game rule” (using the expression from Berliner (Prentice-Hall, 1982), i.e. legal framework) should remain stable in time, 3. the possible maximum loss should not be huge, with respect to the insurer’s solvency, 4. the average cost should be identifiable and quantifiable, 5. risks could be pooled so that the law of large numbers can be used (independent and identically distributed, i.e. the portfolio should be homogeneous), 6. there should be no moral hazard, and no adverse selection, 7. there must exist an insurance market, in the sense that demand and supply should meet, and a price (equilibrium price) should arise. 88
  89. 89. Arthur CHARPENTIER - Modeling and covering catastrophes risk premium and regulatory capital (points 4 and 5)Within an homogeneous portfolios (Xi identically distributed), sufficiently large X1 + ... + Xn(n → ∞), → E(X). If the variance is finite, we can also derive a nconfidence interval (solvency requirement), i.e. if the Xi ’s are independent,   n √ Xi ∈ nE(X) ± 1.96 nVar(X)  with probability 95%.   i=1 risk based capital needHigh variance, small portfolio, or nonindependence implies more volatility, andtherefore more capital requirement. 89
  90. 90. Arthur CHARPENTIER - Modeling and covering catastrophes independent risks, large portfolio (e.g. car insurance) independent risks, 10,000 insured q q q q Fig. 11 – A portfolio of n = 10, 000 insured, p = 1/10. 90
  91. 91. Arthur CHARPENTIER - Modeling and covering catastrophes independent risks, large portfolio (e.g. car insurance) independent risks, 10,000 insured, p=1/10 distribution de la charge totale, N(np, np(1 − p) ) , q 0.012 cas indépendant, p=1/10, n=10,000 0.010 RISK−BASED CAPITAL NEED +7% PREMIUM 0.008 0.006 RUIN (1% SCENARIO) 0.004 0.002 0.000 969 q 900 950 1000 1050 1100 1150 1200 Fig. 12 – A portfolio of n = 10, 000 insured, p = 1/10. 91
  92. 92. Arthur CHARPENTIER - Modeling and covering catastrophes independent risks, large portfolio (e.g. car insurance) independent risks, 10,000 insured, p=1/10 distribution de la charge totale, N(np, np(1 − p) ) , q 0.012 cas indépendant, p=1/10, n=10,000 0.010 RISK−BASED CAPITAL NEED +7% PREMIUM 0.008 0.006 RUIN (1% SCENARIO) 0.004 0.002 0.000 986 q 900 950 1000 1050 1100 1150 1200 Fig. 13 – A portfolio of n = 10, 000 insured, p = 1/10. 92
  93. 93. Arthur CHARPENTIER - Modeling and covering catastrophes independent risks, small portfolio (e.g. fire insurance) independent risks, 400 insured q q q q Fig. 14 – A portfolio of n = 400 insured, p = 1/10. 93
  94. 94. Arthur CHARPENTIER - Modeling and covering catastrophes independent risks, small portfolio (e.g. fire insurance) independent risks, 400 insured, p=1/10 distribution de la charge totale, N(np, np(1 − p) ) , q 0.06 cas indépendant, p=1/10, n=400 0.05 RUIN 0.04 (1% SCENARIO) 0.03 RISK−BASED CAPITAL 0.02 NEED +35% PREMIUM 0.01 0.00 q 39 30 40 50 60 70 Fig. 15 – A portfolio of n = 400 insured, p = 1/10. 94
  95. 95. Arthur CHARPENTIER - Modeling and covering catastrophes independent risks, small portfolio (e.g. fire insurance) independent risks, 400 insured, p=1/10 distribution de la charge totale, N(np, np(1 − p) ) , q 0.06 cas indépendant, p=1/10, n=400 0.05 RUIN 0.04 (1% SCENARIO) 0.03 RISK−BASED CAPITAL 0.02 NEED +35% PREMIUM 0.01 0.00 q 48 30 40 50 60 70 Fig. 16 – A portfolio of n = 400 insured, p = 1/10. 95
  96. 96. Arthur CHARPENTIER - Modeling and covering catastrophes nonindependent risks, large portfolio (e.g. earthquake) independent risks, 10,000 insured q q q q Fig. 17 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent. 96
  97. 97. Arthur CHARPENTIER - Modeling and covering catastrophes nonindependent risks, large portfolio (e.g. earthquake) non−independent risks, 10,000 insured, p=1/10 distribution de la charge totale q 0.012 nonindependant case, p=1/10, n=10,000 0.010 RUIN (1% SCENARIO) 0.008 0.006 RISK−BASED CAPITAL 0.004 NEED +105% PREMIUM 0.002 0.000 897 q 1000 1500 2000 2500 Fig. 18 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent. 97
  98. 98. Arthur CHARPENTIER - Modeling and covering catastrophes nonindependent risks, large portfolio (e.g. earthquake) non−independent risks, 10,000 insured, p=1/10 distribution de la charge totale q 0.012 nonindependant case, p=1/10, n=10,000 0.010 RUIN (1% SCENARIO) 0.008 0.006 RISK−BASED CAPITAL 0.004 NEED +105% PREMIUM 0.002 0.000 2013 q 1000 1500 2000 2500 Fig. 19 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent. 98
  99. 99. Arthur CHARPENTIER - Modeling and covering catastrophes the pure premium as a technical benchmarkPascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth centuryproposed to evaluate the “produit scalaire des probabilit´s et des gains”, e n n < p, x >= pi xi = P(X = xi ) · xi = EP (X), i=1 i=1based on the “r`gle des parties”. eFor Qu´telet, the expected value was, in the context of insurance, the price that eguarantees a financial equilibrium.From this idea, we consider in insurance the pure premium as EP (X). As inCournot (1843), “l’esp´rance math´matique est donc le juste prix des chances” e e(or the “fair price” mentioned in Feller (AS, 1953)).Problem : Saint Peterburg’s paradox, i.e. infinite mean risks (cf. naturalcatastrophes) 99
  100. 100. Arthur CHARPENTIER - Modeling and covering catastrophes the pure premium as a technical benchmark ∞For a positive random variable X, recall that EP (X) = P(X > x)dx. 0 Expected value 1.0 q q 0.8 q q Probability level, P 0.6 q q 0.4 q q 0.2 q q 0.0 q 0 2 4 6 8 10 Loss value, X Fig. 20 – Expected value EP (X) = xdFX (x) = P(X > x)dx. 100
  101. 101. Arthur CHARPENTIER - Modeling and covering catastrophes from pure premium to expected utility principle Ru (X) = u(x)dP = P(u(X) > x))dxwhere u : [0, ∞) → [0, ∞) is a utility function.Example with an exponential utility, u(x) = [1 − e−αx ]/α, 1 Ru (X) = log EP (eαX ) , αi.e. the entropic risk measure.See Cramer (1728), Bernoulli (1738), von Neumann & Morgenstern(PUP, 1944), ... etc. 101
  102. 102. Arthur CHARPENTIER - Modeling and covering catastrophes Distortion of values versus distortion of probabilities Expected utility (power utility function) 1.0 q q 0.8 q q Probability level, P 0.6 q q 0.4 q q 0.2 q q 0.0 q 0 2 4 6 8 10 Loss value, X Fig. 21 – Expected utility u(x)dFX (x). 102
  103. 103. Arthur CHARPENTIER - Modeling and covering catastrophes Distortion of values versus distortion of probabilities Expected utility (power utility function) 1.0 q q 0.8 q q Probability level, P 0.6 q q 0.4 q q 0.2 q q 0.0 q 0 2 4 6 8 10 Loss value, X Fig. 22 – Expected utility u(x)dFX (x). 103
  104. 104. Arthur CHARPENTIER - Modeling and covering catastrophes from pure premium to distorted premiums (Wang) Rg (X) = xdg ◦ P = g(P(X > x))dxwhere g : [0, 1] → [0, 1] is a distorted function.Example• if g(x) = I(X ≥ 1 − α) Rg (X) = V aR(X, α),• if g(x) = min{x/(1 − α), 1} Rg (X) = T V aR(X, α) (also called expected shortfall), Rg (X) = EP (X|X > V aR(X, α)).See D’Alembert (1754), Schmeidler (PAMS, 1986, E, 1989), Yaari (E, 1987),Denneberg (KAP, 1994)... etc.Remark : Rg (X) will be denoted Eg◦P . But it is not an expected value sinceQ = g ◦ P is not a probability measure. 104
  105. 105. Arthur CHARPENTIER - Modeling and covering catastrophes Distortion of values versus distortion of probabilities Distorted premium beta distortion function) 1.0 q q 0.8 q q Probability level, P 0.6 q q 0.4 q q 0.2 q q 0.0 q 0 2 4 6 8 10 Loss value, X Fig. 23 – Distorted probabilities g(P(X > x))dx. 105
  106. 106. Arthur CHARPENTIER - Modeling and covering catastrophes Distortion of values versus distortion of probabilities Distorted premium beta distortion function) 1.0 q q 0.8 q q Probability level, P 0.6 q q 0.4 q q 0.2 q q 0.0 q 0 2 4 6 8 10 Loss value, X Fig. 24 – Distorted probabilities g(P(X > x))dx. 106
  107. 107. Arthur CHARPENTIER - Modeling and covering catastrophes some particular cases a classical premiumsThe exponential premium or entropy measure : obtained when the agentas an exponential utility function, i.e. π such that U (ω − π) = EP (U (ω − S)), U (x) = − exp(−αx), 1i.e. π = log EP (eαX ). αEsscher’s transform (see Esscher (SAJ, 1936), B¨hlmann (AB, 1980)), u EP (X · eαX ) π = EQ (X) = , EP (eαX )for some α > 0, i.e. dQ eαX = αX ) . dP EP (eWang’s premium (see Wang (JRI, 2000)), extending the Sharp ratio concept ∞ ∞ E(X) = F (x)dx and π = Φ(Φ−1 (F (x)) + λ)dx 0 0 107
  108. 108. Arthur CHARPENTIER - Modeling and covering catastrophes pricing options in complete markets : the binomial caseIn complete and arbitrage free markets, the price of an option is derived usingthe portfolio replication principle : two assets with the same payoff (in allpossible state in the world) have necessarily the same price.Consider a one-period world,   S = S u( increase, d > 1) u 0risk free asset 1 → (1+r), and risky asset S0 → S1 =  Sd = S0 d( decrease, u < 1)The price C0 of a contingent asset, at time 0, with payoff either Cu or Cd at time1 is the same as any asset with the same payoff. Let us consider a replicatingportfolio, i.e.   α (1 + r) + βS = C = max {S u − K, 0} u u 0  α (1 + r) + βSd = Cd = max {S0 d − K, 0} 108
  109. 109. Arthur CHARPENTIER - Modeling and covering catastrophes pricing options in complete markets : the binomial caseThe only solution of the system is Cu − Cd 1 Cu − Cd β= and α = Cu − S0 u . S0 u − S0 d 1+r S0 u − S0 dC0 is the price at time 0 of that portfolio. 1 1+r−d C0 = α + βS0 = (πCu + (1 − π) Cd ) where π = (∈ [0, 1]). 1+r u−d C1Hence C0 = EQ where Q is the probability measure (π, 1 − π), called risk 1+rneutral probability measure. 109
  110. 110. Arthur CHARPENTIER - Modeling and covering catastrophes financial versus actuarial pricing, a numerical example risk-free asset risky asset contingent claim      1.05  110  150 probability 75%  1→ 100 → ??? →  1.05  70  10 probability 25%  3 1Actuarial pricing : pure premium EP (X) = × 150 + × 10 = 115 (since 4 4p = 75%). 1Financial pricing : EQ (X) = 126.19 (since π = 87.5%). 1+rThe payoff can be replicated as follows,   −223.81 · 1.05 + 3.5 · 110 = 150 and thus −223.81 · 1 + 3.5 · 100 = 126.19.  −223.81 · 1.05 + 3.5 · 70 = 10 110
  111. 111. Arthur CHARPENTIER - Modeling and covering catastrophes financial versus actuarial pricing, a numerical example Comparing binomial risks, from insurance to finance 145 EXPONENTIAL UTILITY ESSCHER TRANSFORM 140 135 Prices 130 FINANCIAL PRICE 125 (UNDER RISK NEUTRAL MEASURE) 120 WANG DISTORTED PREMIUM ACTUARIAL PURE PREMIUM 115 q 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Alpha or lambda coefficients Fig. 25 – Exponential utility, Esscher transform, Wang’s transform...etc. 111
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