1. Arthur CHARPENTIER - Modeling and covering catastrophesModeling and covering catastrophes Arthur CharpentierSao Paulo, April 2009 arthur.charpentier@univ-rennes1.fr http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/ 1

2. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks products and modelsGeneral introductionModeling very large claimsNatural catastrophes and accumulation riskInsurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing coversPricing insurance linked securitiesRisk measures, an economic introductionCalculating risk measures for catastrophic risksPricing cat bonds : the Winterthur examplePricing cat bonds : the Mexican Earthquake2 3. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks products and modelsGeneral introductionModeling very large claimsNatural catastrophes and accumulation riskInsurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing coversPricing insurance linked securitiesRisk measures, an economic introductionCalculating risk measures for catastrophic risksPricing cat bonds : the Winterthur examplePricing cat bonds : the Mexican Earthquake3 4. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Swiss Re (2007). 4 5. Arthur CHARPENTIER - Modeling and covering catastrophes Some stylized factsclimatic 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. Arthur CHARPENTIER - Modeling and covering catastrophesSome stylized facts : natural catastrophesIncludes hurricanes, tornados, winterstorms, earthquakes, tsunamis, hail,drought, oods... Date Loss event RegionOverall losses Insured losses Fatalities 25.8.2005Hurricane Katrina USA125,000 61,0001,322 23.8.1992Hurricane AndrewUSA 26,500 17,000 62 17.1.1994 Earthquake NorthridgeUSA 44,000 15,300 61 21.9.2004 Hurricane Ivan USA, Caribbean23,000 13,000125 19.10.2005Hurricane Wilma Mexico, USA20,000 12,400 42 20.9.2005 Hurricane Rita USA 16,000 12,000 10 11.8.2004Hurricane Charley USA, Caribbean18,0008,000 36 26.9.1991 Typhoon MireilleJapan10,0007,000 629.9.2004Hurricane Frances USA, Caribbean12,0006,000 39 26.12.1999Winter storm Lothar Europe 11,5005,900110Tab. 1 The 10 most expensive natural catastrophes, 1950-2005 (Source : MunichRe (2006)).6 7. Arthur CHARPENTIER - Modeling and covering catastrophes Some stylized facts : man-made catastrophesIncludes industry re, oil & gas explosions, aviation crashes, shipping and raildisasters, mining accidents, collapse of building or bridges, terrorism... DateLocation Plant typeEvent type Loss (property)23.10.1989 Texas, USA petrochemical vapor cloud explosion 83904.05.1988 Nevada, USA chemical explosion38305.05.1988 Louisiana, USA refinery vapor cloud explosion 36814.11.1987 Texas, USA petrochemicalvapor cloud explosion 28207.07.1988North sea platform explosion1,08526.08.1992 Gulf of Mexicoplatform explosion93123.08.1991North seaconcrete jacketmechanical damage47424.04.1988Brazilplateformblowout 421Tab. 2 Onshore and oshore 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. Arthur CHARPENTIER - Modeling and covering catastrophesSome 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. Arthur CHARPENTIER - Modeling and covering catastrophes longevity and mortality risks YearAge5e025e022e0260 years old40 years old20 years old5e035e032e035e04 1899 1948 19975e045e050 20 4060 80100 1900 1920 1940 1960 1980 2000 Age AgeFig. 2 Mortality rate surface (function of age and year).9 10. Arthur CHARPENTIER - Modeling and covering catastrophes What is a large claim ?An academic answer ? Teugels (1982) dened 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 sumof claims, or at least 5% of the net premiums,Answer 3 large claims are every claim for which the actuary has to go andsee one of the chief members of the company.Examples Traditional types of catastrophes, natural (hurricanes, typhoons,earthquakes, oods, tornados...), man-made (res, explosions, businessinterruption...) or new risks (terrorist acts, asteroids, power outages...).From large claims to catastrophe, the dierence is that there is a before thecatastrophe, and an after : something has changed !10 11. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Swiss Re (2008). 11 12. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 12 13. Arthur CHARPENTIER - Modeling and covering catastrophesThe 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 hitinsurers, reinsurers and the retrocession market lack of capacity, strong increase in rate Natural Catastrophes - nonUSA : in Asia (earthquakes, typhoons) and Europe(ood, drought, subsidence) sui generis protection programs in some countries 13 14. Arthur CHARPENTIER - Modeling and covering catastrophes The impact of a catastropheStorms - Europe : high speed wind in Europe and US, considered as insurablemain risk for P&C insurersTerrorism, including nuclear, biologic or bacteriologic weaponslack of capacity, strong social pressure : private/public partnershipsLiabilities, third party damagegrowth in indemnities (jurisdictions) yield unsustainable lossesTransportation (maritime and aircrafts), volatile business, and concentrated market14 15. Arthur CHARPENTIER - Modeling and covering catastrophesProbabilistic concepts in risk managementLet X1 , ..., Xn denote some claim size (per policy or per event), the survival probability or exceedance probability isF (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 1V aR(X, u) = F(u) = F(1 u) i.e. P(X > V aR(X, u)) = 1 u, the return period isT (u) = 1/F (x)(u). 15 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 accontaccumulation and global warming extreme value theory for losses time series theory for occurrence credit risk models for contagion or accumulation 16 17. Arthur CHARPENTIER - Modeling and covering catastrophesUpdating actuarial modelsIn classical actuarial models (from Cramer and Lundberg), one usuallyconsider 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, Cramr-Lundbergs 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 nite.In the case of large risks or catastrophes, claim size has heavy tails (e.g. thevariance is usually innite), but the Poisson assumption for occurrence is stillrelevant. 17 18. Arthur CHARPENTIER - Modeling and covering catastrophesUpdating actuarial modelsNExample For business interruption, the total loss is S =Xi where N isi=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=118 19. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks products and modelsGeneral introductionModeling very large claimsNatural catastrophes and accumulation riskInsurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing coversPricing insurance linked securitiesRisk measures, an economic introductionCalculating risk measures for catastrophic risksPricing cat bonds : the Winterthur examplePricing cat bonds : the Mexican Earthquake 19 20. Arthur CHARPENTIER - Modeling and covering catastrophesExample : business interruptionBusiness interruption claims can be very expensive. Zajdenweber (2001)claimed that it is a noninsurable risk since the pure premium is (theoretically)innite.Remark For the 9/11 terrorist attacks, business interruption represented US$ 11billion. 20 21. Arthur CHARPENTIER - Modeling and covering catastrophes Some results from Extreme Value TheoryWhen modeling large claims (industrial re, business interruption,...) : extremevalue theory framework is necessary.The Pareto distribution appears naturally when modeling observations over agiven threshold,b xF (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 innite.The estimation of b is a crucial issue. 21 22. Arthur CHARPENTIER - Modeling and covering catastrophesCumulative distribution function, with confidence interval1.0 lo#!lo# %areto *lot, ,it. /onfiden/e inter3al 0 lo)arit.m of t.e sur5i5al 6ro7a7ilities !10.8 cumulative probabilities !#0.6 !$0.4 !%0.2 !50.0012 34 501# $%5 logarithm of the losses lo)arit.m of t.e lossesFig. 3 Pareto modeling for business interruption claims.22 23. Arthur CHARPENTIER - Modeling and covering catastrophesWhy 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. Arthur CHARPENTIER - Modeling and covering catastrophes Why the Pareto distribution ? historical perspective Lorenz curve of business interruption claims 1.0 0.873% OFProportion of claim sizeTHE LOSSES 0.6 0.4 20% OF 0.2 THE CLAIMS 0.0 0.00.20.40.60.81.0 Proportion of claims number Fig. 5 The 80-20 Pareto principle.24 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() dened 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. 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/ 1+x if = 0, P(X u > x|X > u) H,(u) (x) = (u) exp x if = 0, (u)for some (). 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. Arthur CHARPENTIER - Modeling and covering catastrophesHow to dene large claims ? Use of the k largest claims : Hills estimatorThe intuitive idea is to t 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 {Xnk+1:n , ..., Xn1:n , Xn:n } denote the set of the k largest claims. Recallthat 1/b, and thenn 1= log(Xnk+i:n ) log(Xnk:n ). k i=127 28. Arthur CHARPENTIER - Modeling and covering catastrophes2.5 Hill estimator of the slope Hill estimator of the 95% VaR 102.0 8quantile (95%) slope (!b) 61.5 41.0 20 200400 600800 1000 1200 0 200 400 6008001000 1200Fig. 6 Pareto modeling for business interruption claims : tail index.28 29. Arthur CHARPENTIER - Modeling and covering catastrophesExtreme value distributions...A natural idea is to t 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 and ,xisigma 0.6179447 2.0453168par.ses, with the following standard errorsxisigma 0.1769205 0.4008392 29 30. Arthur CHARPENTIER - Modeling and covering catastrophes5.0 MLE of the tail index, using Generalized Pareto ModelEstimation of VaR and TVaR (95%)5 e!021 e!024.51!F(x) (on log scale) 95 tail index2 e!034.0 995 e!043.51 e!043.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 5102050 100 200 x (on log scale) Fig. 7 Pareto modeling for business interruption claims : VaR and TVaR.30 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 claims140130120110100908070605040302010199319941995 199619971998 1999 2000 2001 Fig. 8 Pricing of a reinsurance layer. 31 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 19931994 1995 1996 1997 1998 1999 2000 frequency 173152 146131158138 120156136Tab. 3 Number of business interruption claims.32 33. Arthur CHARPENTIER - Modeling and covering catastrophes From the statistical model of claims to the pure premiumFor a claim size x, the reinsurers 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 = , = 8.9 (12 for burning cost... based on 1 claim). 33 34. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks modellingGeneral introductionBusiness interruption and very large claimsNatural catastrophes and accumulation riskInsurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements Risk measures, an economic introduction Calculating risk measures for catastrophic risks Diversication and capital allocation 34 35. Arthur CHARPENTIER - Modeling and covering catastrophesIncreased value at riskIn 1950, 30% of the worlds population (2.5 billion people) lived in cities. In 2000,50% of the worlds 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), $1, 902 billion in New York. 35 36. Arthur CHARPENTIER - Modeling and covering catastrophesTwo techniques to model large risks The actuarial-statistical technique : modeling historical series,The actuary models the occurrence process of events, and model the claim size(of the total event).This is simple but relies on stability assumptions. If not, one should modelchanges in the occurrence process, and should take into account ination orincrease in value-at-risk. The meteorological-engineering technique : modeling natural hazard andexposure.This approach needs a lot of data and information so generate scenarios takingall the policies specicities. Not very exible to estimate return periods, andworks as a black box. Very hard to assess any condence levels. 36 37. Arthur CHARPENTIER - Modeling and covering catastrophes The actuarial-statistical approach Modeling event occurrence, the problem of global warming.Global warming has an impact on climate related hazard (droughts, subsidence,hurricanes, winterstorms, tornados, oods, coastal oods) but not geophysical(earthquakes). Modeling claim size, the problem of increase of value at risk and ination.Pielke & Landsea (1998) normalized losses due to hurricanes, using bothpopulation and wealth increases, with this normalization, the trend of increasingdamage amounts in recent decades disappears.37 38. Arthur CHARPENTIER - Modeling and covering catastrophes Impact of global warming on natural hazard!u#$er o) *urricanes, per 2ear 3853!600825Frequency of hurricanes201510501850 1900 1950 2000Year Fig. 9 Number of hurricanes and major hurricanes per year.38 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 LocationAdjusted loss 28.05.1896Saint Louis, IL 2,916 29.09.1927Saint Louis, IL 1,797 18.04.19253 states (MO, IL, IN) 1,392 10.05.1979Wichita Falls, TX 1,141 09.06.1953Worcester, MA 1,140 06.05.1975Omaha, NE 1,127 08.06.1966Topeka, KS1,126 06.05.1936Gainesville, GA 1,111 11.05.1970Lubbock, TX 1,081 28.06.1924Lorain-Sandusky, OH 1,023 03.05.1999Oklahoma City, OK 909 11.05.1953Waco, TX899 27.04.1890Louisville, KY836Tab. 4 Most damaging tornadoes (from Brooks & Doswell (2001)). 39 40. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2006). 40 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 nancial losses based on deductibles, reinsurance (or retrocession) treaties.41 42. Arthur CHARPENTIER - Modeling and covering catastrophesSource : GIEC (2008). 42 43. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 43 44. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 44 45. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 45 46. Arthur CHARPENTIER - Modeling and covering catastrophesSource : AXA (2008). 46 47. Arthur CHARPENTIER - Modeling and covering catastrophesHurricanes 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 years3 years4 years7 years 14 years60 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 years909 years 2, 000 years Tab. 5 Extremal insured losses (from Wharton Risk Center & RMS).Recall that historical default (yearly) probabilities areAAAAA ABBB BB B0.00% 0.01% 0.05% 0.37% 1.45% 6.59%-10, 000 years 2, 000 years 270 years69 years15 years Tab. 6 Return period of default (from S&Ps (1981-2003)).47 48. Arthur CHARPENTIER - Modeling and covering catastrophesModelling contagion in credit risk modelscat insurance credit riskn total number of insuredn number of credit issuers 1 if policy i claims 1 if issuers i defaults Ii = Ii = 0 if not 0 if notMi total sum insured Mi nominal Xi exposure rate1 Xi recovery rate48 49. Arthur CHARPENTIER - Modeling and covering catastrophes Modelling contagion in credit risk modelsIn CreditMetrics, the idea is to generate random scenario to get the Prot &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. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks modellingGeneral introductionModeling very large claimsNatural catastrophes and accumulation riskInsurance covers against catastrophes, traditional versus alternative techniquesRisk measures and capital requirements Risk measures, an economic introduction Calculating risk measures for catastrophic risks Diversication and capital allocation 50 51. Arthur CHARPENTIER - Modeling and covering catastrophesInsurance 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 lenders guarantee to cancel the loan should the shipment be stolen. cf. cat bonds.51 52. Arthur CHARPENTIER - Modeling and covering catastrophesWhy 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. Arthur CHARPENTIER - Modeling and covering catastrophes Which reinsurance treaty is optimal ?In a proportional agreement, the cedent and the reinsurer will agree on acontractually dened 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 dened. 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 arecovered by the treaty a stop-loss, or excess-of-loss ratio : the deductible and the limit og liability areexpressed as annnual aggregate amounts (usually as percentage of annualpremium).53 54. Arthur CHARPENTIER - Modeling and covering catastrophes Risk management solutions ? Equity holding : holding in solvency margin+ easy and basic buer very expensive Reinsurance and retrocession : transfer of the large risks to better diversiedcompanies+ easy to structure, indemnity based business cycle inuences 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. 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 specied triggeris reached+ large capacities, no credit risk, multi year contracts more and more industry/parametric based, structuration costs 55 56. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 56 57. Arthur CHARPENTIER - Modeling and covering catastrophes57 58. Arthur CHARPENTIER - Modeling and covering catastrophes Trigger denition 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 (auditneeded) industry based index trigger : connected to the accumulated loss of theindustry (PCS)+ simple to use, no moral hazard noncorrelation risk58 59. Arthur CHARPENTIER - Modeling and covering catastrophes Trigger denition for peak risk environmental based index trigger : connected to some climate index (rainfall,windspeed, Richter scale...) measured by national authorities andmeteorological oces+ simple to use, no moral hazard noncorrelation risk, related only to physical features (not nancialconsequences) parametric trigger : a loss event is given by a cat-software, using climateinputs, and exposure data+ few risk for the cedant if the model ts well appears as a black-box 59 60. Arthur CHARPENTIER - Modeling and covering catastrophes ReinsuranceThe insurance approach (XL treaty) 35 30 25REINSURERLoss per event 20 15 INSURER 10 INSURED 5 00.0 0.20.4 0.6 0.8 1.0 Event Fig. 10 The XL reinsurance treaty mechanism. 60 61. Arthur CHARPENTIER - Modeling and covering catastrophes Groupnet W.P.net W.P. loss ratio total Shareholders Funds (2005) (2004)(2005)(2004) Munich Re 17.6 20.5 84.66% 24.324.4 Swiss Re (1)16.52085.78% 15.516 Berkshire Hathaway Re7.88.2 91.48% 40.937.8 Hannover Re7.17.8 85.66%2.93.2 GE Insurance Solutions 5.26.3 164.51% 6.46.4 Lloyds5.14.9 103.2% XL Re3.93.2 99.72% Everest Re 33.5 93.97%3.22.8 Reinsurance Group of America Inc.32.6 1.91.7 PartnerRe2.8 386.97%2.42.6 Transatlantic Holdings Inc.2.72.9 84.99%1.9 2 Tokio Marine 2.12.626.923.9 Scor 22.5 74.08%1.51.4 Odyssey Re 1.71.8 90.54%1.21.2 Korean Re1.51.3 69.66%0.50.4 Scottish Re Group Ltd. 1.50.4 0.90.6 Converium1.42.9 75.31%1.21.3 Sompo Japan Insurance Inc. 1.41.625.3% 15.312.1 Transamerica Re (Aegon)1.30.7 5.55.7 Platinum Underwriters Holdings 1.31.2 87.64%1.20.8 Mitsui Sumitomo Insurance1.31.5 63.18% 16.314.1Tab. 7 Top 25 Global Reinsurance Groups in 2005 (from Swiss Re (2006)).61 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 agencies62 63. Arthur CHARPENTIER - Modeling and covering catastrophes ILW - Insurance Loss WarrantyIndustry loss warranties pay a xed 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. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 64 65. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 65 66. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 66 67. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Banks (2005). 67 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. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Banks (2004). 69 70. Arthur CHARPENTIER - Modeling and covering catastrophesInsurance Linked Securitiesindemnity triggerindex triggerparametric trigger70 71. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 71 72. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 72 73. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 73 74. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 74 75. Arthur CHARPENTIER - Modeling and covering catastrophesMortality bondsSource : Guy Carpenter (2008).75 76. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2006). 76 77. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Goldman Sachs (2006). 77 78. Arthur CHARPENTIER - Modeling and covering catastrophes USAAs 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 principalprotected 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 asdened under USAAs portfolio parameters (indemnity trigger)78 79. Arthur CHARPENTIER - Modeling and covering catastrophes class A-1, rated AAA, hurricane bondSource : Banks (2004). 79 80. Arthur CHARPENTIER - Modeling and covering catastrophes class A-2, rated BB, hurricane bondSource : Banks (2004). 80 81. Arthur CHARPENTIER - Modeling and covering catastrophes81 82. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006). 82 83. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006). 83 84. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006). 84 85. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks modellingGeneral introductionBusiness interruption and very large claimsNatural catastrophes and accumulation riskInsurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing coversPricing insurance linked securitiesRisk measures, an economic introductionCalculating risk measures for catastrophic risksPricing cat bonds : the Winterthur examplePricing cat bonds : the Mexican Earthquake 85 86. Arthur CHARPENTIER - Modeling and covering catastrophessurvey of literature on pricing Fundamental asset pricing theorem, in nance, Cox & Ross (JFE, 1976),Harrison & Kreps (JET, 1979), Harrison & Pliska (SPA, 1981, 1983).Recent general survey Dana & Jeanblanc-Picque (1998). Marchs nanciers 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 inInsurance. Springer Verlag. Denuit & Charpentier (2004). Mathmatiques de lassurance non-vie, tome e 1. Economica.86 87. Arthur CHARPENTIER - Modeling and covering catastrophessurvey 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-dependentmodel. Kluwer Academic Publishers. Gollier (2001). The Economics of Risk and Time. MIT Press. 87 88. Arthur CHARPENTIER - Modeling and covering catastrophesfrom 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 satisfysome 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 insurerssolvency, 4. the average cost should be identiable and quantiable, 5. risks could be pooled so that the law of large numbers can be used(independent and identically distributed, i.e. the portfolio should behomogeneous), 6. there should be no moral hazard, and no adverse selection, 7. there must exist an insurance market, in the sense that demand and supplyshould meet, and a price (equilibrium price) should arise.88 89. Arthur CHARPENTIER - Modeling and covering catastrophes risk premium and regulatory capital (points 4 and 5)Within an homogeneous portfolios (Xi identically distributed), suciently largeX1 + ... + Xn(n ), E(X). If the variance is nite, we can also derive ancondence interval (solvency requirement), i.e. if the Xi s are independent, nXi 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. Arthur CHARPENTIER - Modeling and covering catastrophesindependent risks, large portfolio (e.g. car insurance) independent risks, 10,000 insured qq qqFig. 11 A portfolio of n = 10, 000 insured, p = 1/10.90 91. Arthur CHARPENTIER - Modeling and covering catastrophesindependent risks, large portfolio (e.g. car insurance) independent risks, 10,000 insured, p=1/10distribution de la charge totale, N(np, np(1 p) ), q 0.012 cas indpendant, p=1/10, n=10,000 0.010 RISKBASED CAPITALNEED +7% PREMIUM 0.008 0.006 RUIN(1% SCENARIO) 0.004 0.002 0.000 969 q 900 950 100010501100 11501200Fig. 12 A portfolio of n = 10, 000 insured, p = 1/10. 91 92. Arthur CHARPENTIER - Modeling and covering catastrophesindependent risks, large portfolio (e.g. car insurance) independent risks, 10,000 insured, p=1/10distribution de la charge totale, N(np, np(1 p) ), q 0.012 cas indpendant, p=1/10, n=10,000 0.010 RISKBASED CAPITALNEED +7% PREMIUM 0.008 0.006 RUIN(1% SCENARIO) 0.004 0.002 0.000 986 q 900 9501000 10501100 11501200Fig. 13 A portfolio of n = 10, 000 insured, p = 1/10. 92 93. Arthur CHARPENTIER - Modeling and covering catastrophes independent risks, small portfolio (e.g. re insurance)independent risks, 400 insured q q qq Fig. 14 A portfolio of n = 400 insured, p = 1/10. 93 94. Arthur CHARPENTIER - Modeling and covering catastrophes independent risks, small portfolio (e.g. re insurance) independent risks, 400 insured, p=1/10distribution de la charge totale, N(np, np(1 p) ) ,q 0.06cas indpendant, p=1/10, n=400 0.05RUIN 0.04 (1% SCENARIO) 0.03 RISKBASED CAPITAL 0.02NEED +35% PREMIUM 0.01 0.00 q 393040 50 60 70Fig. 15 A portfolio of n = 400 insured, p = 1/10.94 95. Arthur CHARPENTIER - Modeling and covering catastrophes independent risks, small portfolio (e.g. re insurance) independent risks, 400 insured, p=1/10distribution de la charge totale, N(np, np(1 p) ) ,q 0.06cas indpendant, p=1/10, n=400 0.05RUIN 0.04 (1% SCENARIO) 0.03 RISKBASED CAPITAL 0.02NEED +35% PREMIUM 0.01 0.00 q483040 50 60 70Fig. 16 A portfolio of n = 400 insured, p = 1/10.95 96. Arthur CHARPENTIER - Modeling and covering catastrophesnonindependent risks, large portfolio (e.g. earthquake) independent risks, 10,000 insured qq qqFig. 17 A portfolio of n = 10, 000 insured, p = 1/10, nonindependent.96 97. Arthur CHARPENTIER - Modeling and covering catastrophesnonindependent risks, large portfolio (e.g. earthquake) nonindependent risks, 10,000 insured, p=1/10distribution de la charge totale q 0.012 nonindependant case, p=1/10, n=10,000 0.010RUIN (1% SCENARIO) 0.008 0.006 RISKBASED CAPITAL 0.004 NEED +105% PREMIUM 0.002 0.000 897 q 1000 15002000 2500Fig. 18 A portfolio of n = 10, 000 insured, p = 1/10, nonindependent.97 98. Arthur CHARPENTIER - Modeling and covering catastrophesnonindependent risks, large portfolio (e.g. earthquake) nonindependent risks, 10,000 insured, p=1/10distribution de la charge totale q 0.012 nonindependant case, p=1/10, n=10,000 0.010RUIN (1% SCENARIO) 0.008 0.006 RISKBASED CAPITAL 0.004 NEED +105% PREMIUM 0.002 0.0002013 q 1000 15002000 2500Fig. 19 A portfolio of n = 10, 000 insured, p = 1/10, nonindependent.98 99. Arthur CHARPENTIER - Modeling and covering catastrophesthe pure premium as a technical benchmarkPascal, Fermat, Condorcet, Huygens, dAlembert in the XVIIIth centuryproposed to evaluate the produit scalaire des probabilits 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 Qutelet, the expected value was, in the context of insurance, the price thateguarantees a nancial equilibrium.From this idea, we consider in insurance the pure premium as EP (X). As inCournot (1843), lesprance mathmatique est donc le juste prix des chancesee(or the fair price mentioned in Feller (AS, 1953)).Problem : Saint Peterburgs paradox, i.e. innite mean risks (cf. naturalcatastrophes)99 100. Arthur CHARPENTIER - Modeling and covering catastrophesthe pure premium as a technical benchmarkFor a positive random variable X, recall that EP (X) =P(X > x)dx.0Expected value1.0qq0.8qq Probability level, P0.6q q0.4qq0.2qq0.0q0 2 4 6 8 10Loss value, X Fig. 20 Expected value EP (X) =xdFX (x) =P(X > x)dx.100 101. Arthur CHARPENTIER - Modeling and covering catastrophesfrom 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 ex ]/, 1Ru (X) = log EP (eX ) , i.e. the entropic risk measure.See Cramer (1728), Bernoulli (1738), von Neumann & Morgenstern(PUP, 1944), ... etc.101 102. Arthur CHARPENTIER - Modeling and covering catastrophes Distortion of values versus distortion of probabilitiesExpected utility (power utility function) 1.0qq 0.8qqProbability level, P 0.6qq 0.4q q 0.2qq 0.0 q 02 4 6 8 10Loss value, X Fig. 21 Expected utility u(x)dFX (x). 102 103. Arthur CHARPENTIER - Modeling and covering catastrophes Distortion of values versus distortion of probabilitiesExpected utility (power utility function) 1.0qq 0.8qqProbability level, P 0.6qq 0.4q q 0.2qq 0.0 q 02 4 6 8 10Loss value, X Fig. 22 Expected utility u(x)dFX (x). 103 104. 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 ) Rg (X) = V aR(X, ), if g(x) = min{x/(1 ), 1} Rg (X) = T V aR(X, ) (also called expectedshortfall), Rg (X) = EP (X|X > V aR(X, )).See DAlembert (1754), Schmeidler (PAMS, 1986, E, 1989), Yaari (E, 1987),Denneberg (KAP, 1994)... etc.Remark : Rg (X) will be denoted EgP . But it is not an expected value sinceQ = g P is not a probability measure. 104 105. 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 qProbability level, P 0.6 qq 0.4 qq 0.2 q q 0.0 q 0 2 4 6 8 10 Loss value, XFig. 23 Distorted probabilities g(P(X > x))dx.105 106. 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 qProbability level, P 0.6 qq 0.4 qq 0.2 q q 0.0 q 0 2 4 6 8 10 Loss value, XFig. 24 Distorted probabilities g(P(X > x))dx.106 107. 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. such that U ( ) = EP (U ( S)), U (x) = exp(x), 1i.e. = log EP (eX ). Esschers transform (see Esscher (SAJ, 1936), Bhlmann (AB, 1980)), uEP (X eX ) = EQ (X) = , EP (eX )for some > 0, i.e.dQ eX = X ).dP EP (eWangs premium (see Wang (JRI, 2000)), extending the Sharp ratio concept E(X) =F (x)dx and = (1 (F (x)) + )dx 0 0107 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 payo (in allpossible state in the world) have necessarily the same price.Consider a one-period world, S = S u( increase, d > 1)u0risk 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 payo either Cu or Cd at time1 is the same as any asset with the same payo. Let us consider a replicatingportfolio, i.e. (1 + r) + S = C = max {S u K, 0} u u 0 (1 + r) + Sd = Cd = max {S0 d K, 0} 108 109. Arthur CHARPENTIER - Modeling and covering catastrophes pricing options in complete markets : the binomial caseThe only solution of the system isCu Cd 1 Cu Cd= and = Cu S0 u . S0 u S0 d 1+r S0 u S0 dC0 is the price at time 0 of that portfolio. 11+rd C0 = + S0 = (Cu + (1 ) Cd ) where = ( [0, 1]).1+rud C1Hence C0 = EQ where Q is the probability measure (, 1 ), called risk1+rneutral probability measure. 109 110. Arthur CHARPENTIER - Modeling and covering catastrophesnancial versus actuarial pricing, a numerical example risk-free asset risky asset contingent claim 1.05 110 150 probability 75% 1100 ??? 1.05 70 10probability 25% 31Actuarial pricing : pure premium EP (X) = 150 + 10 = 115 (since 44p = 75%). 1Financial pricing : EQ (X) = 126.19 (since = 87.5%).1+rThe payo can be replicated as follows, 223.81 1.05 + 3.5 110 = 150and thus 223.81 1 + 3.5 100 = 126.19. 223.81 1.05 + 3.5 70 = 10 110 111. Arthur CHARPENTIER - Modeling and covering catastrophes nancial versus actuarial pricing, a numerical exampleComparing binomial risks, from insurance to finance 145EXPONENTIALUTILITY ESSCHER TRANSFORM 140 135Prices 130 FINANCIAL PRICE 125(UNDER RISK NEUTRAL MEASURE) 120 WANG DISTORTED PREMIUM ACTUARIAL PURE PREMIUM 115q 0.00 0.010.020.030.04 0.050.06 Alpha or lambda coefficientsFig. 25 Exponential utility, Esscher transform, Wangs transform...etc. 111 112. Arthur CHARPENTIER - Modeling and covering catastrophesrisk neutral measure or deatorsThe idea of deators is to consider state-space securities contingent claim 1 contingent claim 2 1 0 probability 75% ??? ??? 0 1 probability 25% Then it is possible to replicate those contingent claims 1.667 1.05 + 0.025 110 = 1 2.619 1.05 + 0.02 110 = 0 1.667 1.05 + 0.025 70 = 0 2.619 1.05 + 0.02 70 = 1The market prices of the two assets are then 0.8333 and 0.119. Those prices canthen be used to price any contingent claim.E.g. the nal price should be 150 0.8333 + 10 0.119 = 126.19. 112 113. Arthur CHARPENTIER - Modeling and covering catastrophes Cat bonds versus (traditional) reinsurance : the price A regression model (Lane (2000)) A regression model (Major & Kreps (2002))113 114. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006). 114 115. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 115 116. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 116 117. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 117 118. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Guy Carpenter (2008). 118 119. Arthur CHARPENTIER - Modeling and covering catastrophes119 120. Arthur CHARPENTIER - Modeling and covering catastrophes Cat bonds versus (traditional) reinsurance : the price Using distorted premiums (Wang (2000,2002))If F (x) = P(X > x) denotes the losses survival distribution, the pure premium is (X) = E(X) = 0 F (x)dx. The distorted premium is g (X) = g(F (x))dx, 0where g : [0, 1] [0, 1] is increasing, with g(0) = 0 and g(1) = 1.Example The proportional hazards (PH) transform is obtained when g is apower function.Wang (2000) proposed the following transformation, g() = (1 (F ()) + ),where is the N (0, 1) cdf, and is the market price of risk, i.e. the Sharperatio. More generally, consider g() = t (t1 (F ()) + ), where t is the Student tcdf with degrees of freedom. 120 121. 0 2 4 6 8 1012 14 16 Mosaic 2A Mosaic 2B Halyard Re Yield spread (%) Domestic ReConcentric Re Juno Re Residential ReArthur CHARPENTIER - Modeling and covering catastrophesKelvin 1st eventKelvin 2nd event Gold Eagle A Gold Eagle BNamazu ReEmpirical Lane model Atlas Re AWang model Atlas Re B Atlas Re CSeismic LtdProperty Catastrophe Risk Linked Securities, 2001121 122. Arthur CHARPENTIER - Modeling and covering catastrophesWho might buy cat bonds ?In 2004, 40% of the total amount has been bought by mutual funds, 33% of the total amount has been bought by cat funds, 15% of the total amount has been bought by hedge funds.Opportunity to diversify asset management (theoretical low correlation withother asset classes), opportunity to gain Sharpe ratios through cat bonds excessspread. 122 123. Arthur CHARPENTIER - Modeling and covering catastrophesInsure against natural catastrophes and make money ? Return On Equity, US P&C insurers15KATRINARITA WILMA104 hurricanes NORTHRIDGE5ANDREW0 9/11 19901995 20002005Fig. 26 ROE for P&C US insurance companies. 123 124. Arthur CHARPENTIER - Modeling and covering catastrophesReinsure against natural catastrophes and make money ?Combined Ratio Reinsurance vs. P/C Industry162.4160150 9/11 2004/2005140 ANDREWHURRICANES 129130126.5125.8124.6119.2120115.8115.8114.3113.6110.5110.1110.1111108.8108.5107.4106.9110106.7108106.5105.9104.8106105101.9 100.9100.8100.598.3100 901991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Fig. 27 Combined Ratio for P&C US companies versus reinsurance. 124 125. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Morton Lane (2008). 125 126. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Morton Lane (2008). 126 127. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Morton Lane (2008). 127 128. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006). 128 129. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Lane (2006). 129 130. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks modellingGeneral introductionBusiness interruption and very large claimsNatural catastrophes and accumulation riskInsurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing coversPricing insurance linked securitiesRisk measures, an economic introductionCalculating risk measures for catastrophic risksPricing cat bonds : the Winterthur examplePricing cat bonds : the Mexican Earthquake130 131. Arthur CHARPENTIER - Modeling and covering catastrophesDescription of th WinCat Cat Bond structureSource : Crdit Suisse (1997). e 131 132. Arthur CHARPENTIER - Modeling and covering catastrophesDescription of th WinCat Cat Bond structureSource : Crdit Suisse (1997). e 132 133. Arthur CHARPENTIER - Modeling and covering catastrophesPricingSource : Schmock (1999). 133 134. Arthur CHARPENTIER - Modeling and covering catastrophes Claim number of past eventsSource : Schmock (1999). 134 135. Arthur CHARPENTIER - Modeling and covering catastrophesModeling the number of eventConsider major storm of hail storm events, causing more than 1000 claims. LetN denote the number of events per year. Assume that N P() with parameter > 0. The empirical estimate of is = 17 (17 events in 10 years).10Source : Schmock (1999). 135 136. Arthur CHARPENTIER - Modeling and covering catastrophesGoing further on the Poisson modelSeveral models have been considered in Schmock (1999), such as a linear trend of parameter Source : Schmock (1999). 136 137. Arthur CHARPENTIER - Modeling and covering catastrophesGoing further on the Poisson modelSeveral models have been considered in Schmock (1999), such as a log-linear trend of parameter Source : Schmock (1999). 137 138. Arthur CHARPENTIER - Modeling and covering catastrophesGoing further on the Poisson modelSeveral models have been considered in Schmock (1999), such as a (modied) log-linear trend of parameter Source : Schmock (1999). 138 139. Arthur CHARPENTIER - Modeling and covering catastrophesGoing further on the Poisson modelSeveral models have been considered in Schmock (1999), such as a modied linear trend of parameter Source : Schmock (1999). 139 140. Arthur CHARPENTIER - Modeling and covering catastrophesDistribution for the knock-out eventSince events can be extreme, it might be natural to consider a Paretodistribution, 1 , for x ,P(X x) = x 0 , for x < ,A natural estimator for is 1000.Then classical estimators can be considered for . maximum likelihood,Since likelihood can be writen n log L(x1 , , xn , , )= log i=1 x+1in = n log + n log ( + 1) log xi ,i=1 140 141. Arthur CHARPENTIER - Modeling and covering catastrophesthe maximum (in ) is obtained as n 1 1 xi =log n i=1 nSinece E() = , setn1 n1n1 1xi 1 = = log . n n1 i=1 Then 2 E(1 ) = and V ar(1 ) = .n2 method of moments,For the Pareto distribution 2 E(X) = and V ar(X) = 2 ( 2) , pour > 2. 1 ( 1) 141 142. Arthur CHARPENTIER - Modeling and covering catastrophesThusX2 =. X Asymptotical properties are then ( 1)2 E(2 ) et V ar(2 ) . n( 2) OLS regression,If the logarithm of survival probabilities log[1 F (x)] are linear in log x, i.e.log[1 F (x)] = log F (x) = 0 + 1 log x,we obtain a Pareto distribution. In that caseYi = log[1 F (Xi )] = log F (Xi ) = 0 + 1 log Xi + i .The OLS estimator for = (0 , 1 ) is then n n nn i=1 log Xi log F (Xi ) +i=1 log Xi i=1log F (Xi )1 = 3 = n n 2 n2 i=1 [log Xi ][ i=1 log Xi ] 142 143. Arthur CHARPENTIER - Modeling and covering catastrophesSince F is unknown, it should be replaced by an empirical version, n1F (x) =1(Xi > x)ni=1such that F (Xj ) is equal to 1 rank(Xj )/n. empirical quantiles,For instance, if we expect quantiles of order 10% and 90% to be equal, 0.10 = 1 q10% 0.90 = 1 q90%where q10% and q10% are the empirical quantiles q90%Since= 0.9/0.1 = 9, a natural estimator is thenq10%log 9 4 = .log q90% log q10% 143 144. Arthur CHARPENTIER - Modeling and covering catastrophes Ginis index,In the case of a Pareto distribution 1 11/1G=121 (1 u) du = 02 1Thus, if denotes the empirical version of Ginis index 1+5 = .2 144 145. Arthur CHARPENTIER - Modeling and covering catastrophesIn that case, p6000 = P(X > 6000) 61.37 8.57%. It is also possible to derivebounds for this probability, p6000 [4.5%; 16.2%] with 68% chance.Source : Schmock (1999).145 146. Arthur CHARPENTIER - Modeling and covering catastrophesTo go further, it is also possible to use a Generalized Pareto Distribution 1 1+ x, for x 0,P(X x) = 0 , for x < ,Maximum likelihood estimator are here = 1.3806 and = 660.7.Source : Schmock (1999).146 147. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Schmock (1999).Thus, p6000 = P(X > 6000) 7.575%. 147 148. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Schmock (1999). 148 149. Arthur CHARPENTIER - Modeling and covering catastrophes AgendaCatastrophic risks modellingGeneral introductionBusiness interruption and very large claimsNatural catastrophes and accumulation riskInsurance covers against catastrophes, traditional versus alternative techniquesRisk measures and pricing coversPricing insurance linked securitiesRisk measures, an economic introductionCalculating risk measures for catastrophic risksPricing cat bonds : the Winterthur examplePricing cat bonds : the Mexican Earthquake149 150. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Cabrera (2006). 150 151. Arthur CHARPENTIER - Modeling and covering catastrophesNatural Catastrophes in MexicoFor Fonden, a catastrophe is a claim larger than annual average catastrophe,plus the standard deviation,19961997 1998 1999 2000 2001 20022003 2004 amount109 310 330 739 511 120 261 216 32the average is 292 and standard deviation is 48, thus a catastrophe is obtainedwhen the loss exceed 512.The AIR annual expected loss probabilities Annual expected loss probabilities00zone 10000zone 200 00zone 500 0.63% 0.96% 0.30% 151 152. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Cabrera (2006). 152 153. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Cabrera (2006). 153 154. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Cabrera (2006). 154 155. Arthur CHARPENTIER - Modeling and covering catastrophesSource : Cabrera (2006). 155 156. Arthur CHARPENTIER - Modeling and covering catastrophesThe Mexican Cat BondThe cat bond was issued by a SPV Cayman Islands Cat-Mex Ltd, structured bySwiss Reinsurance Company (SRC) and Deutsche Bank.The 160 million cat bond pays a tranche equal to the LIBOR + 235 basis points.The cat bond is a part of a total coverage of 450 million provided by the reinsurerfor three years against earthquakes risk, with a total premium of 26 millions.Payment of losses is conditional upon conrmation by AIR, which modeled theseismic risk. Only 3 zones (out of 9) are insured in the transaction : zones 1, 2and 5, with coverage of 150 million (in each case).The cat bond payment would be triggered if there is an event, i.e. an earthquakehigher (or equal) than 8M w hitting zone 1 or zone 2, or an eathquake higher orequal than 7.5M w hitting zone 5. 156 157. Arthur CHARPENTIER - Modeling and covering catastrophesInsurance market intensity 1Consider an homogeneous Poisson process with intensity 1 . Under thenon-artbitrage framework, the compounded discount actuarial fair insuranceprice at time t = 0, in the reinsurance market is 3 3H = E 450 1 ( < 3) er = 450 ert t f (t)dt = 450 ert t 1 e1 t dt 0 0i.e. the insurance premium is equal to the value of the expected discounted lossfrom earthquake.With constant interest rate, rt = log(1.0541). Thus 326 = 450 e log(1.0541)t 1 e1 t dt, where 1 e1 t is the probability of 0occurence of an event over period [0, t]. Hence, we get an intensity rate from thereinsurance market 1 0.0214.The probability of having (at least) one event in three years is 0.0624, i.e. 2.15events in one hundred years. 157 158. Arthur CHARPENTIER - Modeling and covering catastrophes Capital market intensity 2The cat bond pays to the investors the principal P equal to 160 million atmaturity T = 3 years, and gives coupon C every 3 months during the bonds lifein case of no event. The coupon bonds pays a xed spread rate z = 235 basispoints over LIBOR.Hence the annual interest rate is r = 5.1439%, and thusr+z5.1439% + 2.35%C= P = 160 = 3.1055 44158 159. Arthur CHARPENTIER - Modeling and covering catastrophesLet G be the random variable representing the investors gain, 1 P = E G(1 + r)12 t 11 = EC 1( > ) + P 1( > 3)t=1 4 (1 + r)t/4(1 + r)3121 1 =Ce2 t/4+ P e32t=1(1 + r)t/4 (1 + r)3Substituting the values of the principal P = 160 million and the coupons 12 e2 t/4 e32C = 3.1055 million, i.e. 160 =3.06t/4+ 160 3. From thist=1 (1.0541) (1.0541)expression, the capital market intensity can be obtained 2 0.0241. Theprobability of having (at least) one event in three years is 0.0699, i.e. 2.4 eventsin one hundred years.159 160. Arthur CHARPENTIER - Modeling and covering catastrophesHistorical intensity 3Finaly, the historical intensity rate that describes the eathquake process 3 canbe obtained.Over 104 years, there were 192 earthquakes higher than 6.5M w, but only halfoccured in the insured zones (mainly zone 2).zone frequency (%) 1 3016% 2 4222% 5 189%other102 53%The probability of occurence of the trigger event is p = 3/192.Hence0.005140 == 1.8504 360 160 161. Arthur CHARPENTIER - Modeling and covering catastrophesConsequently the annual historical intensity is 3 = p = 0.0289.161 162. Arthur CHARPENTIER - Modeling and covering catastrophes References (internet)Artemis (Alternative Risk Transfer Internet Portal) http ://www.artemis.bm/Guy Carpenter http ://www.guycarp.com/Lane Financial LLC http ://www.lanefinancialllc.com/Munich Re http ://www.munichre.com/AON Bneeld http ://www.benfieldgroup.com/Swiss Re http ://www.swissre.com/162 163. Arthur CHARPENTIER - Modeling and covering catastrophes ReferencesBanks, E. (2004). Alternative Risk Transfer : Integrated Risk Managementthrough Insurance, Reinsurance, and the Capital Markets. Wiley.Banks, E. (2005). Catastrophic Risk. Wiley.Cabrera, B., B. (2006). Pricing catastrophic bonds for earthquakes in Mexico.Master thesis, Humboldt-Universitt zu Berlin. aCardenas, V. & Mechler, R. (2005). The Mexico cat bonds - a solution forcountry at risk ? Munich Re Foundation Symposium.Cox, S.H. & Pedersen, H.W. (2000). Catastrophe risk bonds. North AmericanActuarial Journal, 4, 4, 56-..Kreps, R. (2005). Riskiness leverage models. CAS Forum.Lane, M. & Mahul, O. (2008). Catastrophe Risk Pricing : an empirical analysis.World Bank, WPS 4765.Mata, A.J. (2004) Catastrophe Excess of Loss. in Encyclopedia of ActuarialSciences.163 164. Arthur CHARPENTIER - Modeling and covering catastrophesMeyers, G. 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