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2015 Enterprise Risk Management Symposium
June 11–12, 2015, National Harbor, Maryland
Into the Tails of Risk: An Intervention
into the Process of Risk Evaluation
By David Ingram
Copyright © 2016 by the Society of Actuaries, Casualty Actuarial Society, and the Professional Risk Managers’ International Association. All rights reserved by the Society of Actuaries, Casualty Actuarial Society, and the Professional Risk Managers’ International Association. Permission is granted to make brief excerpts for a published review. Permission is also granted to make limited numbers of copies of items in this monograph for personal, internal, classroom or other instructional use, on condition that the foregoing copyright notice is used so as to give reasonable notice of the Society of Actuaries’, Casualty Actuarial Society’s, and the Professional Risk Managers’ International Association’s copyright. This consent for free limited copying without prior consent of the Society of Actuaries, Casualty Actuarial Society, and the Professional Risk Managers’ International Association and does not extend to making copies for general distribution, for advertising or promotional purposes, for inclusion in new collective works or for resale. The opinions expressed and conclusions reached by the authors are their own and do not represent any official position or opinion of the Society of Actuaries, Casualty Actuarial Society, or the Professional Risk Managers’ International Association or their members. The organizations make no representation or warranty to the accuracy of the information.
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ThankstoDanielBarYaccov,Ph.D.,IanCook,FIA,andNeilBodoff,FCAS,fortheircomments.Theauthorissolelyresponsibleforhowtheirexcellentadvicewasactuallyused.
IntotheTailsofRisk:AnInterventionintotheProcessofRiskEvaluation
DavidIngram
Abstract
Peoplenaturally observe risk as the rangeof experienced gains andlosses represented in statistical terms by standard deviation.Statistical techniquesareused todevelopvalues forextreme tailsofthedistributionofgainsandlosses.Theseprocessesareessentiallyanextrapolation from the “known” risk of volatility near the mean to“unknown”riskofextremelosses.Thispaperwillproposeatailriskmetric (the coefficient of riskiness) that can be used to enhancediscussionbetweenmodelbuildersandmodelusersaboutthefatnessofthetailsinriskmodels.
Riskmodelsallstartwithobservations.Modelerslookattheobservationsandthe
shapeofaplotoftheobservations.Fromthatshape,themodelerschoosea
mathematicalformulatorepresenttheriskdriver(suchasinterestratesorstock
marketreturns)orforthelossseverityitself.Thoseformulasareknownas
probabilitydistributionfunctions(PDF).
Themostfamousandmostcommonlyusedofthesefunctionsisknownasthe
“normal”curve.Mathematicians(sometimescalledquantsorrocketscientists)
particularlyfavoredtheuseofthenormalPDFbecauseitsmathematical
characteristicsmadeitparticularlyeasytomanipulate,makingrapidanalysisofrisk
functionsbaseduponthenormalPDFpossible.1
1Indeed,theuseofthenormalPDFinfinancecanbetracedtotherediscovered1900thesisofLouisBachelier,TheoryofSpeculation,trans.MarkDavisandAlisonEtheridge(Princeton:PrincetonUniversityPress,2006).BacheliersetsastandardfollowedbymanyofpresentingthenormalPDFasthebasisforstatisticalmodelingoffinancialrisk.Bacheliermayhavealsobeenthefirsttocautionthat“Thecalculusofprobabilitiescannodoubtneverapplytomovementsofstockexchangequotations.”
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Inthe2008globalfinancialcrisis,wefoundthatmanyfinancialmarketriskmodels
basedonthenormalPDFdrasticallyunderestimatedthelikelihoodoflosseswhich
weremuch,muchworsethantheaverage.
Unfortunately,formanypeople,expectationsofextremelosseslearnedfrom
businesscourses,mediaand,tosomeextent,riskmodelsaredrawnfromthevery
samecharacteristicsasthenormalPDF.
ThelanguageofthenormalPDFisourbasiclanguageofrisk.ThenormalPDFis
definedcompletelybyjusttwoterms—meanandstandarddeviation.Wetendto
expectthemeanandthestandarddeviationtotellus“allabout”anynewrisk,
withoutrealizingwearetherebyassumingtheriskisnormal.
ThenormalPDFsaysweshouldexpectabouttwo‐thirdsofourobservationstofall
withinonestandarddeviationofthemeanandover90percentoftheobservations
withintwostandarddeviationsofthemean.Italsosaysitisextremelyunlikelyto
haveanyobservationsbeyondthreestandarddeviationsfromthemean.Infact,
observationsshouldfallwithinthreestandarddeviations99.9percentofthetime
forthenormalPDF.2
AndthatishowwewereabletoconfirmthenormalPDFunderestimatedthe
likelihoodoflargedeviationsfromthemean.DavidViniar,GoldmanSachs’chief
financialofficer,famouslyobservedduringthefinancialcrisis,“Weareseeingthings
thatwere25standarddeviationmoves,severaldaysinarow,”3which,underthe
normalPDF,washighlyunlikelytohappenevenonceinthetimesincethelastice
ageended.
2Ofcourse,thenormalPDFactuallysaysthe99.9thpercentileobservationshouldbe3.09standarddeviationsfromthemean.3PeterThalLarsen,“GoldmanPaysthePriceofBeingBig,”FinancialTimes,August13,2007.
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Theideathatriskfitsanormalcurveissodeeplyembeddedthatalmostall
discussionofViniar’s25‐standard‐deviationstatementwasintheformof
discussionofexactlyhowtocalculatethelikelihoodofa25‐standard‐deviation
moveunderthenormalPDF,insteadofchallengingtheveryideathatthenormal
PDFmightnotbeappropriate.4
TwonotedexceptionstothesegeneralizationsareBenoitMandelbrotandNassim
Taleb.Mandelbrot,inhisworkstudyingpricemovementsincottonmarketsinthe
1960s,suggeststherearesevenstatesofrandomness,onlythefirstofwhichis
properlymodeledbyanormalPDF.5Taleb,inhisbooks,actuallydividestheworld
intotworegimes—MediocristanandExtremistan—wherethenormalPDFexplains
thefirstregimeandaParetoPDFexplainsthesecond.6
Ininsurancemodelingbyactuariesandcatastrophemodelers,theuseofanormal
PDFismuchlessdominant.OtherPDFs,especiallytheParetoPDF,allowforquite
extremevalueswithrelativelyhighlikelihood.Infact,withcertaincalibrations,the
ParetoPDFallowsforinfinitevaluesofmetricslikevariance,somethingthatis
possiblyevenmoreunrealisticthanthenormalPDF’slowlikelihoodforextreme
values.Alternately,somemodelerswhoseetheneedforhigherlikelihoodof
extremevalueswithnormalPDF‐likefeaturesotherwisehaveusedcombinationsof
multiplenormalPDFstoachievethedesired“fattails.”7Othermodelsofasingle
categoryofriskexposuresmaycombinetwoormoredifferentPDFs.Forexample,a
modelofapropertyinsurancelineofaninsurermayconsistofseparatemodelsof
naturalcatastrophelosses,lossesfromlargeexposuresandlossesfromsmalland
4Forexample,see“HowUnluckyis25‐Sigma?”byKevinDowd,JohnCotter,ChrisHumphreyandMargaretWoods.5BenoitB.Mandelbrot,“TheVariationofCertainSpeculativePrices,”JournalofBusiness36.no.4(1963).6NassimNicholasTaleb,FooledbyRandomness:TheHiddenRoleofChanceintheMarketsandinLife(NewYork:RandomHouse,2001);TheBlackSwan:TheImpactoftheHighlyImprobable(NewYork:RandomHouse,2007);andAntifragile:ThingsThatGainFromDisorder(NewYork:RandomHouse,2012).7MaryR.Hardy,“ARegime‐SwitchingModelofLong‐TermStockReturns,”NorthAmericanActuarialJournal5,no.2(2001).
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moderate‐sizedexposures.Eachofthesesubmodelsisoftenbaseduponadifferent
PDF.
EachofthealternatePDFshasdifferentcharacteristicsthathavebeengivennames
bystatisticianssuchasskewness(whichquantifiesasymmetry)andkurtosis(which
quantifiesthesharpnessofthedistribution’speak).Theacceptedwisdomamong
modelersisthatforsomeoneto“understand”amodelofrisk,theymustwalkthe
pathofthemodelers:FollowthemathofthePDFsanddefinitelyunderstandthe
nuancesofskewnessandkurtosis.
Extremevaluetheory(EVT)isanexplicitbuthighlytechnicalapproachtobuilding
statisticalmodelsthatarenotfocusedonfittingthemeanortheobservationsnear
tothemean.EVTfocusesonusingspecificPDFsthatareinherentlyfattailed.The
EVTprocessisdesignedtobedrivenbythedataandtheaxiomsofEVTto
analyticallydeterminethetails,especiallythevaluesbeyondtheobservations.8
Thereisastrongpushfortopmanagersandevenboardmemberstobecomeactive
usersoftheoutcomesofriskmodelsandtoactuallyparticipateintheprocessof
validatingtheriskmodel.Forexample,theriskcommitteecharterofonebanksays
thatboardcommitteewilloversee:
ModelRisk,byreviewingallmodel‐relatedpoliciesandassessments
ofthemostsignificantmodels,ineachcaseannually,andreviewing
modeldevelopmentandvalidationactivitiesperiodically.9
ButboththemathematicalapproachtodescribingthePDFsandtheprocess‐based
explanationsthatrequiresimplyfollowingthemodeler’sthinkingfailtoengender
eitherunderstandingorfaithinthemodel.JPMorganChase&Co.,theoriginal8SeePaulEmbrechts,ClaudiaKlüppelberg,andThomasMikosch,ModellingExtremalEventsforInsuranceandFinance(NewYork:Springer,1997).9SantanderConsumerUSAHoldingsInc.,“BoardEnterpriseRiskCommitteeCharter,”effectiveDec.8,2014.
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proponentofthevalue‐at‐riskmodelsusedextensivelyinbanks,experienceda
majorlossinlate2011andearly2012thatwasinpartattributedtoaflawedrisk
modelupdate.10AccordingtoEsadeBusinessSchoolprofessorPabloTriana:
Theperceptionofabank’sriskshouldnotdependonthetechnicalities
ofamathematicalmodelbutratheroncommonsensicalanalysisof
whatshouldandshouldnotbeacceptable.11
Theremainderofthispaperwillpresentanalternateapproachtodiscussingthe
natureofariskmodel’spredictionofthelikelihoodofanextremedeviation.This
approachwillnotrequireextensivemathematicalorstatisticaleducationonthe
partoftheuser,norwillitrequiremuchinthewayofnewvocabulary.Itwillwork
fromwheremostpeoplestandnowintheirunderstandingofthemathofrisk—with
theconceptsofmeanandstandarddeviation.Thisapproachtopresentinga
measureof“fatnessoftails”doesnotreplaceanythingcurrentlyinwideusefor
discussionsofriskmodelswithnontechnicalusersofriskmodels.Itcouldbea
powerfuladditiontothediscussionofriskmodelswiththosenontechnicalusers
andmayleadtoanimportantchangeintherelationshipbetweenthoseusersand
modelersbyprovidingabasisforcommunicationregardingamostimportant
aspectofthemodels.
10ChristopherWhittall,“Value‐at‐RiskModelMaskedJPMorgan$2blnLoss,”Reuters,May11,2012.11TracyAlloway,“JPMorganLossStokesRiskModelFears,”FinancialTimes,May13,2012.
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ExtrapolatingtheTailsoftheRiskModel
Thestatisticalapproachtobuildingamodelofriskinvolvescollectingobservations
andthenusingthedataalongwithageneralunderstandingoftheunderlying
phenomenatochooseaPDF.TheparametersofthatPDFarethenchosentoabest
fitwithboththedataandthegeneralexpectationsabouttherisk.
Thisprocessisoftenexplainedinthoseterms—fittingoneofseveralcommonPDFs
tothedata.Butanalternateviewoftheprocesswouldbetothinkofitasan
extrapolation.Theobservedvaluesgenerallyfallneartothemean.Underthe
normalPDF,wewouldexpecttheobservationstofallwithinonestandarddeviation
ofthemeanabouttwo‐thirdsofthetimeandwithintwostandarddeviationsalmost
98percentofthetime.Whenmodelingannualresults,itisfairlyunlikelywewill
haveevenoneobservationtoguidethe“fit”atthe99thpercentile.
So,inmostcases,wereallyareusingtheshapeofthePDFtoextrapolatetogeta
99thpercentileor99.5thpercentilevalue.Butourmethodofdescribingourmodels
presentsthatfactinafairlyobtusefashion.Sometimesmodeldocumentation
mentionsthePDFweuseforthisextrapolation.Rarelydoesthedocumentation
discusswhythePDFwaschosenand,whenthisisdiscussed,itisalmostnever
mentionedthatitisjudgmentofthemodelerwhichdrivestheexactselectionofthe
parametersthatwilldeterminetheextremevaluesviatheextrapolationprocess.
Afterthe2001dot‐comstockmarketcrash,manymodelersofstockmarketrisk
adoptedaregime‐switchingmodelasatechniquetocreatethe“fattails”thatmany
realizedweremissingfromstockmarketriskmodels.12.
Buthowfatwerethetailsintheseregime‐switchingmodels?Wouldreportingthe
skewnessandkurtosisoftheresultingmodelhelpwithunderstandingofthemodel?
12MaryR.Hardy,“ARegime‐SwitchingModelofLong‐TermStockReturns,”NorthAmericanActuarialJournal5,no.2(2001).
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Oristheregime‐switchingequityriskmodelnowablackboxthatcanonlybe
understoodbyothermodelers?
Almosteverybusinessdecision‐makerisfamiliarwiththemeaningofaverageand
standarddeviationwhenappliedtobusinessstatistics.Weproposethatthose
commonlyusedandalmostuniversallyunderstoodtermsbeusedasthebasisfora
newmetricof“fatnessoftails.”13
Weusetheideaofextrapolationtoconstructforthisnewproposedmeasureof
fatnessoftails.Thecentralideaisthatwewillhaveathree‐pointdescriptionofour
riskmodel,andwiththesethreetermswecandescribethedegreetowhichwecan
expectarisktohavecommonfluctuationsthatwilldrivevariabilityinexpected
earnings(meanandstandarddeviation)aswellasathirdfactorthatindicatesthe
degreetowhichthisriskmightproduceextremelossesofthesortwegenerallyhold
capitalfor.
CoefficientofRiskiness
Wewilladdjustonetermtoourelementaryvocabularyofrisk—thecoefficientof
riskiness(CR).Thisvaluewillbethethirdtermindescribingtheriskmodel.Itisthe
indicatorofthefatnessofthetailoftheriskmodel.
CR=(V.999− )/
Or,inEnglish,thenumberofstandarddeviationsthe99.9thpercentilevalueisfrom
themean.14
13Manyanalystsrelyonthecoefficientofvariation(CV)forcomparingriskinessofdifferentmodels.TheCVisagoodmeasureforlookingatearningsvolatility,butitdoesnotgivestrongindicationofthefatnessofthetails.Itsdefinition,usingonlymeanandstandarddeviation,alsosupportsapresumptionofthenormalPDF.14Thechoiceof99.9thpercentileisdiscussedintheappendixofthispaper.
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Weusedthisconceptabovewhenwesaidobservationsshouldfallwithinthree
standarddeviations99.9percentofthetimeforthenormalPDF.
TheCRcanbequicklyandeasilycalculatedforalmostallriskmodels.Itcanthenbe
usedtocommunicatethewaytheriskmodelpredictsextremelosses,allowingfor
actualdiscussionofextremelossexpectationswithnonmodelers.Weusethemean
andstandarddeviationindefiningtheCRnotbecausetheyarethemathematically
optimalwaytomeasureextremevaluetendency,butbecausetheyarethetworisk‐
modelingtermsalreadywidelyknowntobusinessleaders.
Potentially,theCRcouldbecomeapartoftheprocessfortheinitialconstructionof
riskmodels,takingthepositionofaBayseanprior15inthecommonsituationwhere
therearenoobservationsoftheextremevalues.And,ifCRhasbeenestablishedasa
commonideawithnonmodelers,theycouldhaveavoiceintheprocessof
determininghowthemodelwillapproachthatpartoftherisk‐modelingpuzzle.
TheCRvaluewillnotbeareliableindicatorformodelswherethestandard
deviationisnotreliable.Itisinstructivetoidentifythecharacteristicsofsuch
modelsandtheunderlyingriskssuchmodelsseektocapture.
CoefficientofRiskinessforVariousProbabilityDistributionFunctions
TheCRforthenormalPDFis3.09.Thisistrueforallmodelsthatusethenormal
PDFbecauseallvaluesofanormalPDFareuniquelydeterminedbythemeanand
standarddeviation.16
15ABayseanpriorisanopinionthatactsasaseedtotheriskmodelatthestageoftheprocesswhenthereisinsufficientdatatofullydefineamathematicalmodel.16Forthereaderwhowishestocheckthis,anExceltableofvaluesformean,standarddeviation,99.9thpercentilevalueandCRcaneasilybeconstructed.Meanandstandarddeviationwouldbevalues,99.9thpercentilevaluewouldbeNorminv(.999,mean,stddev)andtheCRwouldbe99.9thpercentilevaluelessthemeandividedbythestandarddeviation.Tryasmanyvaluesforthemeanandstandarddeviationasyouwish.
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AnothercommonlyusedPDFisthelognormal.Thelognormalmodelhastwo
characteristicsthatmakeitpopularforriskmodels—itdoesnotallownegative
outcomesandithasalimitedpositiveskew.17
Figure1.LognormalPDF—CRfor
variousmeans/standarddeviationcombinations
Mean
100% 80% 40% 20% 10%
Stan
dard
Dev
iatio
n
7% 3.4
3.5
3.9
4.9
6.8
10% 3.5
3.7
4.3
5.7
8.3
15% 3.8
4.0
5.0
7.1
9.9
20% 4.0
4.3
5.7
8.3
10.8
25% 4.3
4.6
6.4
9.2
11.1
30% 4.6
5.0
7.1
9.9
11.1
40% 5.1
5.7
8.3
10.8
10.8
50% 5.7
6.4
9.2
11.1
10.2
60% 6.3
7.1
9.9
11.1
9.6
70% 6.8
7.7
10.4
11.0
9.0
80% 7.3
8.3
10.8
10.8
8.4
90% 7.8
8.8
11.0
10.5
7.9
100% 8.3
9.2
11.1
10.2
7.5
120% 9.0
9.9
11.1
9.6
6.7
Asitturnsout,theCRisafunctionoftheratioofstandarddeviationtomean(also
knownasthecoefficientofvariance)forthelognormalPDF.
17ThenormalPDFisexactlysymmetricalandallowsnegativevalues.ThepositiveskewofthelognormalPDFmeansthatitisnotsymmetrical,extendingmuchfurtherontheright(positive)sideofthemeanthanontheleft(towardzero)side.
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Figure2.Lognormal—CRvs.CV
ThePoissonPDFisalsowidelyusedbecauseofitsrelationshiptothebinomial
distribution.SincethePoissonPDFisfullydeterminedbyasingleparameter,theCR
isalwaysapproximately3.5.
TheParetoPDFanditsclosecousin,theexponentialPDF,areusedforavarietyof
typesofrisks.Theserisksallhavethecharacteristicthattheyareusuallyfairly
benignbutinrareinstancestheyproduceextremelyadverseoutcomes.Operational
risksaresometimesmodeledwithaParetoPDF.Risksfromextremewindstorms
andearthquakesarealsomodeledwithParetoPDFs,asispandemicrisk.
In2006,MandelbrotandTalebtogetherproposedtheuseoftheParetoPDFfor
lookingatvulnerabilitytotailrisks:
Thesame“fractal”scalecanbeusedforstockmarketreturnsand
manyothervariables.Indeed,thisfractalapproachcanprovetobean
extremelyrobustmethodtoidentifyaportfolio’svulnerabilityto
severerisks.Traditional“stresstesting”isusuallydonebyselecting
anarbitrarynumberof“worst‐casescenarios”frompastdata.It
0
2
4
6
8
10
12
0 2 4 6 8 10
CoefficientofRiskiness
CoefficientofVariance
CoRforLognormalDistribution
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assumesthatwheneveronehasseeninthepastalargemoveof,say,
10percent,onecanconcludethatafluctuationofthismagnitude
wouldbetheworstonecanexpectforthefuture.Thismethodforgets
thatcrasheshappenwithoutantecedents.Beforethecrashof1987,
stresstestingwouldnothaveallowedfora22percentmove.18
TheParetoPDFmodelscanproduceawiderangeofCRvalues.Standarddeviation,
thenormalPDFconcept,doesnotalwaysworkwellforaParetoPDF.Intheory,the
standarddeviation(aswellasthemean)canactuallybeinfinite.The
recommendationisthatinplaceofthecalculatedCRvalue,themodelerwould
reportthatthemodelindicateswildrandomness(WR)orextremerandomness
(ER).Thesuggestionisexplainedintheappendix.
Extremevalueanalysisdoesnot,bydesign,permitageneralizedlookatastatistic
likeCRbecauseitisfundamentallyanapproachthatdivorcesthetailriskanalysis
fromthedataregardingthemiddleofthedistributionthatmakeupthemeanand
standarddeviation.However,individualriskmodelsthatblendamodelofexpected
variationaroundthemeanwithaspecificmodeloftheextremesbaseduponthe
generalizedextremevaluedistributioncanproducevalueswhichwouldleadtoaCR
calculation.
ExamplesfromInsuranceRiskModels
Theauthorhasobtainedsummaryinformationfromapproximately3,400modelsof
gross(beforereinsurance)propertyandcasualtyinsurancerisksperformed
between2009and2013byactuariesatWillisRe.
18BenoitMandelbrotandNassimTaleb,“AFocusontheExceptionsThatProvetheRule,”FinancialTimes,March23,2006.
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Figure3.3,400insuranceriskmodels19
Inaddition,wehaveobtainedsummaryoutputfromstandalonenaturalcatastrophe
modelrunsforpropertyinsurance.
Figure4.400naturalcatastrophemodels
19Forfigures3and4,theCRof4,forexample,indicatesavaluebetween3and4.
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Itisinterestingtonotethatnoneofthesemodelsshoweda99.9thpercentileresult
thatwas25standarddeviations.But,asyousee,thenaturalcatastrophemodelsdid
produceCRvaluesashighas18.
WhatyoucanseefromtheseexamplesisthatCRdoesseemtobeboundedforthese
actualmodelsintotherangeof3to18andthatexistingprocessesformodeling
insurancerisksdoalreadyproducearangeofCRvalues.
ASimpleBinomialModel
SomeinsighttothedynamicsofCRcanbereachedbylookingatmodelsofsmall
groupsofindependentrisksthathavelowfrequency.
Figure5.CRforsmallbinomialgroups
Ifwestartwithlookingatagroupof200independentriskexposuresthateachhave
alikelihoodoffivein1,000ofhappeningseparately,theexpectationisforoneloss.
Thestandarddeviationwouldbeoneaswell.The99.9thpercentileresultwouldbe
forfivelosses,resultinginaCRof4.ThatisslightlyhigherthantheexpectedCRfor
thePoissonPDFof3.5,andyouseethatasthegroupsizegetslarger,theCRgets
closerto3.5.
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Figure6.CRforalowerincidencerate
Mean StdDev CR EconomicCapital
Interestrate 12.6M 6.0M 4.5 6.9M
Equity 5.5M 10.0M 3.5 22M
Credit 2.5M 1.5M 6.0 3.5M
Underwriting:Property 20.0M 8.0M 12.2 36.8M
Underwriting:Auto 6.0M 2.5M 3.2 0.5M
Underwriting:Health 10.0M 8.0M 3.8 13.2M
Underwriting:Allother 2.0M 0.7M 4.0 0.1M
Reserves 0.0M $12.0M 4.3 37.8M
Operational 0.0M 0.1M 6.0 0.4M
Figure6showsitispossibletoachievesomewhathigherCRwithagroupwitha
lowermean.
Onehypothesisthatcouldexplainthesesimplecalculationsisthatariskwhichhasa
higherCRissusceptibletoanextremelossforalargefractionoftheexposures
whentheexpectedlossisforasmallfraction.Youcouldsaythereisaconcentrated
exposuretotheextremeevent.Duetotheconcentratedexposuretothelargeevent
(hurricaneorearthquake),inthatevent,theirbookofinsurancecontractsactslikea
verysmallgroupofexposures.Sothebinomialviewoftheseverysmallgroupsmay
wellreproducetheexperienceofalargegroupwithconcentration.
CommunicatingExtremeRiskInherentinRiskModels
JustwalkamileinhismoccasinsBeforeyouabuse,criticizeandaccuse.Ifjustforonehour,youcouldfindawayToseethroughhiseyes,insteadofyourownmuse.
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(MaryT.Lathrap,1895)20
Alltoooften,theexplanationforamodelwillbetoidentifythedatausedto
parameterizethemodel.Sometimes,theresultoftheselectionofPDFismentioned,
sometimesnot.RarelyisthereanydiscussionoftheprocessforselectingthePDF
usedortheimplicationsofthatchoice.
Asmentionedabove,nontechnicalmanagersareusuallyfamiliarwiththeideasof
meanandstandarddeviationasthedefiningtermsforstatisticalmodels.The
coefficientofriskinessdescribedhereisproposedasasubstituteforadiscussionof
thecharacteristicsandimplicationsoftheselectionofPDFthat,ingeneral,is
neededbutisnottakingplace.
TheCR,ifadoptedwidely,couldcometobeusedsimilarlytothemoment
magnitudescaleforearthquakesortheSaffir‐SimpsonHurricaneWindScale.Ifyou
werepresentingamodelofhurricanesorearthquakesandmentionedthatyouhad
modeleda2asthemostsevereevent,everyoneintheroomwouldhaveasenseof
whatthatmeant,eveniftheydonotknowanythingaboutthedetailsofthe
modelingapproach.Theywillhaveanopinionaboutwhethera2istheappropriate
valueforthemostseverepossiblehurricaneorearthquake.Theycaneasily
participateinadiscussionoftheassumptionsofthemodelonthatbasis.
TheCRcouldbecomeasimilartoolforbroadcommunicationofmodelseverity.If
youbelievethatViniar’scommentabout25standarddeviationswasactuallybased
uponameasurement(ratherthanaroundnumberexaggerationtomakeapoint),
thenyouwoulddoubtlessrejectthevalidityofthemodelwithaCRof3or4.If
nontechnicalusersofariskmodelgainedanappreciationofwhichofthecompany’s
20ThePoemsandWrittenAddressesofMaryT.LathrapWithaShortSketchofherLife,ed.JuliaR.Parish(Michigan:TheWomenChristianTemperanceUnion,1895).
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riskshaveCRof3andwhichhave12,itwouldbealargeleapofunderstandingofa
veryimportantcharacteristicoftherisks.
So,asanillustrativeexample,anenterpriseriskmodelmightbedescribedas
follows:
Figure7.Enterpriseriskmodel:illustrativevaluesonly
(thesedonotrepresentanyactualmodel)
Enterprise Risk Model
Mean Std Dev CR Economic Capital
Interest rate 12.6M 6.0M 4.5 6.9
Equity 5.5M 10.0M 3.5 22
Credit 2.5M 1.5M 6.0 3.5
Underwriting: Property 20.0M 8.0M 12.2 36.8
Underwriting: Auto 6.0M 2.5M 3.2 0.5
Underwriting: Health 10.0M 8.0M 3.8 13.2
Underwriting: All other 2.0M 0.7M 4.0 0.1
Reserves 0.0M $12.0M 4.3 37.8
Operational 0.0M 0.1M 6.0 0.4
All risk (after diversification)
60.4M 37M 5.0 69.1
Thenthediscussionoftheriskmodelcanfocusonthethreesetsoffacts
presented—theprojectedmean,theprojectedstandarddeviationandthefatnessof
thetail.Thesethreefactsaboutthemodelcanbecomparedtosimilarfactsabout
thepastexperience.Whatwasthemeanexperienceforeachrisk?Whatwasthe
rangeofthatexperienceasstatedbythestandarddeviation?Whatisthehistorical
fatnessofthetail?21Thediscussioncanthenbeallaboutwhythemodeldoesor
doesnotmatchupwithpastexperience.
21Thehistoricalcoefficientofriskinesscanbedefinedasthehistoricalworstcaselessthehistoricalmeandividedbythehistoricalstandarddeviation.Sinceyouwillalmostneverhaveenoughhistorical
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Thehopeisthatbyturningawayfromthetechnical,statisticaldiscussionabout
choiceofPDFandparameterization,thediscussioncanactuallytapintothe
extensiveknowledge,experienceandgutfeelofthenontechnicalmanagementand
boardmembers.PerhapstheCRcanbecomelikethemomentmagnitudescaleof
riskmodels.Fewpeopleunderstandthescienceormathbehindthemoment
magnitudescale,buteveryoneinanearthquakezonecanexperienceashakeand
comeprettyclosetonailingthescoreofthateventwithoutanyfancyequipment.
Andtheyknowhowtopreparefora4ora5ora6quake.Thesamegoesforthe
Saffir‐Simpsonscale.
Conclusion
“Wouldyoutellme,please,whichwayIoughttogofromhere?”“Thatdependsagooddealonwhereyouwanttogetto,”saidtheCat.“Idon'tmuchcarewhere—”saidAlice.“Thenitdoesn'tmatterwhichwayyougo,”saidtheCat.“—solongasIgetSOMEWHERE,”Aliceaddedasanexplanation.“Oh,you'resuretodothat,”saidtheCat,“ifyouonlywalklongenough.”
(LewisCarroll,AliceinWonderland,1865)
Peoplenaturallyobserveriskintheformoftherangeofexperiencedgainsand
losses.Instatisticalterms,thoseobservationsarerepresentedbystandard
deviation.Statisticaltechniquesthathavelongbeenappliedtoinsurancecompany
riskstodevelopcentralestimatesarebeingusedtocalculatevaluesintheextreme
tailsofthedistributionofgainsandlosses.Theseprocessesareessentiallyan
extrapolationfromthe“known”riskofvolatilitynearthemeanto“unknown”riskof
extremelosses.
Todate,thereisnoestablishedlanguagetotalkaboutthenatureofthat
extrapolation.Thecoefficientofriskinessdescribedhereisanattempttobridgethat
gap.TheCRcanbeusedtodifferentiateriskmodelsaccordingtothefatnessofthe
experiencetocalculatea99.9thpercentilefrequency,thisdiscussionwillalwaysbeabouthowmuchworseweeachthinkitcangetintheextreme.
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tailsandcouldbecomeastandardpartofourdiscussionofriskmodels.Withthe
useofametricliketheCR,webelievetheknowledgeandexperienceofnontechnical
managementandboardmemberscanbebroughtintothediscussionsofriskmodel
parameterization.Theendresultofsuchdiscussionswillbothultimatelyimprove
themodelsandincreasethedegreetowhichtheyareactuallyrelieduponfor
informingimportantdecisionswithinarisk‐takingenterprise.
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Appendix
1.TheExponentialRiskModelProblem
ItwasstatedabovethatsomeexponentialriskmodelswillnotfitwiththeCR
calculation.Thatisapossibleproblem.Theproblemarisesbecauseinsomemodels,
thevarianceandperhapsthemeanvalueisinfinite.
Mandelbrotdescribessevenstatesofrandomness
1. Propermildrandomness(thenormaldistribution)
2. Borderlinemildrandomness(theexponentialdistributionwithλ=1)
3. Slowrandomnesswithfiniteanddelocalizedmoments
4. Slowrandomnesswithfiniteandlocalizedmoments(suchasthelognormal
distribution)
5. Prewildrandomness(Paretodistributionwithα=2−3)
6. Wildrandomness:infinitesecondmoment(varianceisinfinite;Pareto
distributionwithα=1−2)
7. Extremerandomness(meanisinfinite;Paretodistributionwithα≤1)22
Tosolvethatproblems,somemodelsusetruncatedexponentialmodels.Truncated
exponentialmodelswillhavefinitevariancebutmightstillhaveunstablesample
valuesatthe99.9thpercentileandthereforeunstableCR.
Suchextremevaluesasthe99.9thpercentilearemainlyusedbyinsurersthatuse
thetailvalueatrisk(TVaR)astheirprimaryriskmetric.But,aftersayingthat,those
firmsmusthavesolvedthisprobleminordertocalculatetheTVaR.
SoweconcludethereisasolutiontothisproblemforanyriskwheretheTVaRcan
becalculated.Butwesuggestextremecautiontoanymodelersdealingwithwild
22BenoitB.Mandelbrot,FractalsandScalinginFinance:Discontinuity,Concentration,Risk(NewYork:Springer,1997).
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randomness(WR)orextremerandomness(ER).TheCRisnotcalculable.Butiftheir
firmisreallyexposedtowildorextremerandomness,theirproblemsaremuch
largerthanthereliabilityoftheirtailriskmeasure.Werecommendthatina
situationwheretheriskmodeldoesindicatewildorextremerandomness,theCRbe
reportedasWRorER.Wealsopresumethatareportwiththoseindicationswill
leadtoveryintensediscussionsoftherisksbeingmodeled.
2.OtherUsesforCR
Riskmodelingisadifficultandtime‐consumingprocess.Ifwedevelopalanguage
aroundatailriskmetricsuchasCR,itwouldbepossibletoestimateriskmodel‐type
tailresultsbyidentifyingthelikelyleveloftheCRforariskandthencombiningthat
withtheobservedmeanandstandarddeviationofactualexperience.Byturningrisk
calculationintoathree‐parameterproblemwhereoneparameterassuresusthat
thetailswillbeappropriately“fat,”thenourriskmodelresultscanbeeasilyand
quicklyestimated.
Thesequickestimatescanbeusedforreadyriskestimatesandalsoformodel
validation.ThevalidationcanbetochecktheCRforeachsubmodelagainstCRfor
othermodelsofsimilarrisks;thequickestimatesdescribedabovecanbean
independentcalculation.Themodelvalidatorcandevelopatolerancefordeviation
ofactualmodelresultsfromthequickestimateasatriggerformorein‐depth
examinationofparticularsubmodels.
Anotherpossibleareaofapplicationisforveryquickestimatesofeconomiccapital
modeloutcomesbaseduponaggregatedhistoricalexperience.IfexperiencewithCR
measurestellus,forexample,thattheCRforacertainlineofbusinessisusuallyin
therangeof3.5to4.5,wecanestimatethe99.9thpercentilevaluefromthe
historicalstandarddeviationofresultsforthatlineinaggregateandthen
interpolatetogeta99thpercentileor99.5thpercentilevalue.Thismightbeuseful
inpublicdataevaluationsofinsurers.
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3.ChoiceofMetric
Thereisnospecificsciencetotheselectionofthe99.9thpercentilevalueatriskas
thebasisfortheCR.WewentbeyondthecommonVaRpointsof99.5percentor99
percentfollowingtheconceptthatsurveyorsusetofindapoint.Theyusuallymake
theirlinetoapointbeyondthespottobedeterminedtoreducethechanceofvery
localerrorsintheregionofthedesiredpoint.Thiscalculationcouldjustaseasilybe
madewith99thpercentileor99.5thpercentilevalues.Somepractitionershave
suggestedthatsuchalternatevaluesmightbemorestablewiththenumberofactual
simulationrunsthataremadeforsomeoftheriskmodels.Thethinkingbehindthe
selectionof99.9percentwasthataonein1,000wasdefinitely“inthetail”and,to
lookatfatnessoftail,itmightbebettertolookfurtheroutthanthemodelvalues
beingused“allofthetime.”Thechoicewasalsoinfluencedbythefactthatthe
normalPDFvaluefor99.9percentproducedaCRofapproximately3,ratherthan
the2.575829304at99.5percentor2.326347874at99percent.Itseemedeasierto
talkaboutametricthatranfrom3to18,thanforonethatwentfrom2.576to
whatever.
Usingarealvalueratherthananindexhasanadvantageaswell.Firstofall,ifwe
thinkoftheone‐in‐1,000eventasaworst‐caseevent,thenwithusingCR,westart
byremindingfolkstheworstcaseisatleastthreetimesthestandarddeviation.This
isimportantbecauseoftenpeoplearelulledintoafalsesenseofsecuritywhensome
timegoesbywithouttheexperienceofanytailevents.Thenwhenwesayariskhas
aCRof6,thatmeanstheworstcaseissimplysixtimesthestandarddeviation
worsethanthemean.Soifweexpectthingstomostlyfallwithinoneortwo
standarddeviations,thentheCRgivesasenseofhowmuchworsetheworstcase
canbe,withoutcomplicatedmultistepcalculations.
Thequestionofunstablevaluescanberesolved.Ifa99.9thpercentileCRbecomesa
standardvalue,thenoccasionallytheriskmodelscanbelefttorunformore
scenariostoproducestablevaluesatthatreturnperiod.Butifthatisnotaviable
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solutionbecausethemodelsaresimplyunstableatthatreturnperiod,thenthatis
probablyalimitationwhichneedstobeunderstoodbytheusers.
Finally,thethinkingwasthatifCRwouldbeusedeventuallytojudgethe
reasonablenessofanotherpointinthetailsuchas99.5percent,thenthatvalidation
wasmorepowerfulifitcouldbestatedthatthemodelresultswerereasonableout
pastthatvalueandthatthe99.5thpercentilevaluewasconsistentwiththe99.9th
percentilevalue.Byfocusingsolelyonthe99.5thpercentilevalue,modelersand
modelusersruntheriskthattheirmodelsarenotevenviableat99.51thpercentile.
Andafocusonjustthatsinglemetricisitselfdangerous.23
However,eveniftheseargumentsfor99.9thpercentilewerecompelling,itishighly
likelythatsomemodelsmightadopttheideabutnotthecalibration.Sowesuggest
thatwhentheCRiscalculatedtoanythingotherthan99.9thpercentile,thatbe
madeclearwithasubscriptwhichstatesthepercentile(i.e.,CR99.5%).
4.FurtherResearch
Thispaperismeanttobetheintroductionofonepossiblemetricforfatnessoftails.
Ifthereissufficientinterestinusingthismetric,thenitshouldbetestedagainst
variousstandardsofrobustnessforriskmetrics,forexample,thecriteriafor
coherentriskmeasures.24
TherecouldalsoberesearchintotherangeofCRvaluesfordifferentmodelsof
similarrisks.HowwideistherangeofCR?Whatarethedriversofhigherorlower
CRvalueswithinaclassofrisks?HowtopredicttheCRwithoutactuallymodelinga
risk?
23DavidIngram,“RiskandLight,”paper,2010,http://ssrn.com/abstract=1594877.24PhilippeArtzner,FreddyDelbaen,Jean‐MarcEber,andDavidHeath.“CoherentMeasuresofRisk,”MathematicalFinance9,no.3(1999).
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Withthatsortofinformationinhand,additionalresearchmightbeundertakento
seeifthereisareasonablewaytotakethethreeparameterswemighthave
separatefromariskmodel—mean,standarddeviationandCR—andcreateasimple
distributionofgainsandlosses.Thatmethodmightwellbedifferentfordifferent
levelsofCR.TherangeofexpectedCRvaluesdeterminedindependentlyofamodel
mightalsobeagoodpieceofinformationtodrivetheactualselectionofPDFforthe
riskmodel.
VerypreliminaryviewsofCRforfullenterpriseriskmodelsthatincludeboth
independentandinterdependentriskssuggesttheeffectofdiversificationisa
smallerCR.FurtherresearchcouldbedonetolookathowtheCRperformsfor
combinationsofindependentandinterdependentPDFstoseeifthereisany
predictablereductioninCRfromthecombination.
Ultimately,thisfurtherresearchmightleadtotheconclusionthatthereisabetter
measureoftailrisk.Butitwouldbeagoodresultifsometailriskmeasurethatcan
bewidelyunderstoodiswidelyadopted.