into the tails of risk: an intervention into the process ...€¦ · 8 see paul embrechts, claudia...

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. 

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.”

2

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.

3

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).

4

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.

5

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.

6

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).

7

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.

8

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.

9

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.

10

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

11

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.

12

Figure3.3,400insuranceriskmodels19

Inaddition,wehaveobtainedsummaryoutputfromstandalonenaturalcatastrophe

modelrunsforpropertyinsurance.

Figure4.400naturalcatastrophemodels

19Forfigures3and4,theCRof4,forexample,indicatesavaluebetween3and4.

13

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.

14

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.

15

(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).

16

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

17

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.

18

tailsandcouldbecomeastandardpartofourdiscussionofriskmodels.Withthe

useofametricliketheCR,webelievetheknowledgeandexperienceofnontechnical

managementandboardmemberscanbebroughtintothediscussionsofriskmodel

parameterization.Theendresultofsuchdiscussionswillbothultimatelyimprove

themodelsandincreasethedegreetowhichtheyareactuallyrelieduponfor

informingimportantdecisionswithinarisk‐takingenterprise.

19

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).

20

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

22

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).

23

Withthatsortofinformationinhand,additionalresearchmightbeundertakento

seeifthereisareasonablewaytotakethethreeparameterswemighthave

separatefromariskmodel—mean,standarddeviationandCR—andcreateasimple

distributionofgainsandlosses.Thatmethodmightwellbedifferentfordifferent

levelsofCR.TherangeofexpectedCRvaluesdeterminedindependentlyofamodel

mightalsobeagoodpieceofinformationtodrivetheactualselectionofPDFforthe

riskmodel.

VerypreliminaryviewsofCRforfullenterpriseriskmodelsthatincludeboth

independentandinterdependentriskssuggesttheeffectofdiversificationisa

smallerCR.FurtherresearchcouldbedonetolookathowtheCRperformsfor

combinationsofindependentandinterdependentPDFstoseeifthereisany

predictablereductioninCRfromthecombination.

Ultimately,thisfurtherresearchmightleadtotheconclusionthatthereisabetter

measureoftailrisk.Butitwouldbeagoodresultifsometailriskmeasurethatcan

bewidelyunderstoodiswidelyadopted.