Modern Actuarial Risk Theory: Using R

393 Pages·2008·2 MB·English

by  Professor Rob Kaas

Modern Actuarial Risk Theory
Rob Kaas • Marc Goovaerts
Jan Dhaene • Michel Denuit
Modern Actuarial Risk
Theory
Using R
Second Edition
Professor Rob Kaas Professor Marc Goovaerts
UvA / KE UvA / KE
Roetersstraat 11 Roetersstraat 11
1018 WB Amsterdam 1018 WB Amsterdam
The Netherlands The Netherlands
[email protected] K.U. Leuven
University of Amsterdam
Naamsestraat 69
Professor Jan Dhaene
3000 Leuven
AFI (Accountancy, Finance, Insurance)
Belgium
Research Group
[email protected]
Faculteit Economie en Bedrijfswetenschappen
K.U. Leuven
University of Amsterdam Professor Michel Denuit
Naamsestraat 69 Institut de Statistique
3000 Leuven Université Catholique de Louvain
Belgium Voie du Roman Pays 20
[email protected] 1348 Louvain-la-Neuve
Belgium
[email protected]
ISBN: 978-3-540-70992-3 e-ISBN: 978-3-540-70998-5
Library of Congress Control Number: 2008931511
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Foreword to the First Edition
Studythepastifyouwoulddefinethefuture—
Confucius,551BC-479BC
RiskTheoryhasbeenidentifiedandrecognizedasanimportantpartofactuarialed-
ucation;thisisforexampledocumentedbytheSyllabusoftheSocietyofActuaries
andbytherecommendationsoftheGroupeConsultatif.Henceitisdesirabletohave
adiversityoftextbooksinthisarea.
This text in risk theory is original in several respects. In the language of figure
skating or gymnastics, the text has two parts, the compulsory part and the free-
stylepart.ThecompulsorypartincludesChapters1–4,whicharecompatiblewith
officialmaterialoftheSocietyofActuaries.Thisfeaturemakesthetextalsouseful
tostudentswhopreparethemselvesfortheactuarialexams.Otherchaptersaremore
of a free-style nature, for example Chapter 7 (Ordering of Risks, a speciality of
the authors). And I would like to mention Chapters 9 and 11 in particular. To my
knowledge, this is the first text in risk theory with an introduction to Generalized
LinearModels.
Specialpedagogicaleffortshavebeenmadethroughoutthebook.Theclearlan-
guage and the numerous exercises are an example for this. Thus the book can be
highlyrecommendedasatextbook.
Icongratulatetheauthorstotheirtext,andIwouldliketothankthemalsointhe
name of students and teachers that they undertook the effort to translate their text
intoEnglish.Iamsurethatthetextwillbesuccessfullyusedinmanyclassrooms.
Lausanne,2001 HansGerber
v
Preface to the Second Edition
WhenItookoffice,onlyhighenergyphysicistshadeverheardof
whatiscalledtheWorldwideWeb...Nowevenmycathasits
ownpage—BillClinton,1996
Thisbookgivesacomprehensivesurveyofnon-lifeinsurancemathematics.Origi-
nallywrittenforusewiththeactuarialscienceprogramsattheUniversitiesofAm-
sterdam and Leuven, it is now inuse at many other universities, as well as for the
non-academic actuarial education program organized by the Dutch Actuarial So-
ciety. It provides a link to the further theoretical study of actuarial science. The
methodspresentedcannotonlybeusedinnon-lifeinsurance,butarealsoeffective
inotherbranchesofactuarialscience,aswellas,ofcourse,inactuarialpractice.
Apart from the standard theory, this text contains methods directly relevant for
actuarialpractice,forexampletheratingofautomobileinsurancepolicies,premium
principlesandriskmeasures,andIBNRmodels.Also,theimportantactuarialstatis-
ticaltooloftheGeneralizedLinearModelsisstudied.Thesemodelsprovideextra
possibilities beyond ordinary linear models and regression that are the statistical
tools of choice for econometricians. Furthermore, a short introduction is given to
credibilitytheory.Anothertopicwhichalwayshasenjoyedtheattentionofriskthe-
oreticiansisthestudyoforderingofrisks.Thebookreflectsthestateoftheartin
actuarialrisktheory;manyresultspresentedwerepublishedintheactuariallitera-
tureonlyrecently.
Inthissecondeditionofthebook,wehaveaimedtomakethetheoryevenmore
directly applicable by using the software R. It provides an implementation of the
language S, not unlike S-Plus. It is not just a set of statistical routines but a full-
fledgedobjectorientedprogramminglanguage.Othersoftwaremayprovidesimilar
capabilities, but the great advantage of R is that it is open source, hence available
toeveryonefreeofcharge.Thisiswhywefeeljustifiedinimposingitontheusers
ofthisbookasadefactostandard.Ontheinternet,alotofdocumentationaboutR
canbefound.InanAppendix,wegivesomeexamplesofuseofR.Afterageneral
introduction,explaininghowitworks,westudyaproblemfromriskmanagement,
tryingtoforecastthefuturebehaviorofstockpriceswithasimplemodel,basedon
stockpricesofthreerecentyears.Next,weshowhowtouseRtogeneratepseudo-
randomdatasetsthatresemblewhatmightbeencounteredinactuarialpractice.
vii
viii Preface
Modelsandparadigmsstudied
Thetimeaspectisessentialinmanymodelsoflifeinsurance.Betweenpayingpre-
miumsandcollectingtheresultingpension,decadesmayelapse.Thistimeelement
is less prominent in non-life insurance. Here, however, the statistical models are
generally more involved. The topics in the first five chapters of this textbook are
basicfornon-lifeactuarialscience.Theremainingchapterscontainshortintroduc-
tionstoothertopicstraditionallyregardedasnon-lifeactuarialscience.
1.Theexpectedutilitymodel
The very existence of insurers can be explained by the expected utility model. In
thismodel,aninsuredisariskaverseandrationaldecisionmaker,whobyvirtueof
Jensen’sinequalityisreadytopaymorethantheexpectedvalueofhisclaimsjustto
beinasecurefinancialposition.Themechanismthroughwhichdecisionsaretaken
underuncertaintyisnotbydirectcomparisonoftheexpectedpayoffsofdecisions,
butratheroftheexpectedutilitiesassociatedwiththesepayoffs.
2.Theindividualriskmodel
In the individual risk model, as well as in the collective risk model below, the to-
tal claims on a portfolio of insurance contracts is the random variable of interest.
Wewanttocompute,forexample,theprobabilitythatacertaincapitalwillbesuf-
ficient to pay these claims, or the value-at-risk at level 99.5% associated with the
portfolio,beingthe99.5%quantileofitscumulativedistributionfunction(cdf).The
totalclaimsismodeledasthesumofallclaimsonthepolicies,whichareassumed
independent.Suchclaimscannotalwaysbemodeledaspurelydiscreterandomvari-
ables,noraspurelycontinuousones,andweuseanotation,involvingStieltjesinte-
gralsanddifferentials,encompassingboththeseasspecialcases.
Theindividualmodel,thoughthemostrealisticpossible,isnotalwaysverycon-
venient, because the available dataset is not in any way condensed. The obvious
technique to use in this model is convolution, but it is generally quite awkward.
Using transforms like the moment generating function sometimes helps. The Fast
FourierTransform(FFT)techniquegivesafastwaytocomputeadistributionfrom
itscharacteristicfunction.ItcaneasilybeimplementedinR.
Wealsopresentapproximationsbasedonfittingmomentsofthedistribution.The
CentralLimitTheorem,fittingtwomoments,isnotsufficientlyaccurateintheim-
portant right-hand tail of the distribution. So we also look at some methods using
threemoments:thetranslatedgammaandthenormalpowerapproximation.
3.Collectiveriskmodels
Amodelthatisoftenusedtoapproximatetheindividualmodelisthecollectiverisk
model.Inthismodel,aninsuranceportfolioisregardedasaprocessthatproduces
claimsovertime.Thesizesoftheseclaimsaretakentobeindependent,identically
distributedrandomvariables,independentalsoofthenumberofclaimsgenerated.
This makes the total claims the sum of a random number of iid individual claim
amounts.UsuallyoneassumesadditionallythatthenumberofclaimsisaPoisson
variatewiththerightmean,orallowsforsomeoverdispersionbytakinganegative
Preface ix
binomial claim number. For the cdf of the individual claims,one takes an average
ofthecdfsoftheindividualpolicies.Thisleadstoaclosefittingandcomputation-
ally tractable model. Several techniques, including Panjer’s recursion formula, to
computethecdfofthetotalclaimsmodeledthiswayarepresented.
For some purposes it is convenient to replace the observed claim severity dis-
tributionbyaparametriclossdistribution.Familiesthatmaybeconsideredarefor
examplethegammaandthelognormaldistributions.Wepresentanumberofsuch
distributions,andalsodemonstratehowtoestimatetheparametersfromdata.Fur-
ther, we show how to generate pseudo-random samples from these distributions,
beyondthestandardfacilitiesofferedbyR.
4.Theruinmodel
Theruinmodeldescribesthestabilityofaninsurer.Startingfromcapitaluattime
t=0,hiscapitalisassumedtoincreaselinearlyintimebyfixedannualpremiums,
butitdecreaseswithajumpwheneveraclaimoccurs.Ruinoccurswhenthecapital
isnegativeatsomepointintime.Theprobabilitythatthiseverhappens,underthe
assumptionthattheannualpremiumaswellastheclaimgeneratingprocessremain
unchanged, is a good indication of whether the insurer’s assets match his liabili-
ties sufficiently. If not, one may take out more reinsurance, raise the premiums or
increasetheinitialcapital.
Analyticalmethodstocomputeruinprobabilitiesexistonlyforclaimsdistribu-
tions that are mixtures and combinations of exponential distributions. Algorithms
exist for discrete distributions with not too many mass points. Also, tight upper
and lower bounds can be derived. Instead of looking at the ruin probability ψ(u)
with initial capital u, often one just considers an upper bound e−Ru for it (Lund-
berg), where the number R is the so-called adjustment coefficient and depends on
theclaimsizedistributionandthesafetyloadingcontainedinthepremium.
Computing a ruin probability assumes the portfolio to be unchanged eternally.
Moreover, it considers just the insurance risk, not the financial risk. Therefore not
much weight should be attached to its precise value beyond, say, the first relevant
decimal.Thoughsomeclaimthatsurvivalprobabilitiesare‘thegoalofrisktheory’,
manyactuarialpractitionersareoftheopinionthatruintheory,howevertopicalstill
in academic circles, is of no significance to them. Nonetheless, we recommend to
study at least the first three sections of Chapter 4, which contain the description
ofthePoissonprocessaswellassomekeyresults.Asimpleproofisprovidedfor
Lundberg’sexponentialupperbound,aswellasaderivationoftheruinprobability
incaseofexponentialclaimsizes.
5.Premiumprinciplesandriskmeasures
Assumingthatthecdfofariskisknown,oratleastsomecharacteristicsofitlike
meanandvariance,apremiumprincipleassignstotheriskarealnumberusedasa
financialcompensationfortheonewhotakesoverthisrisk.Notethatwestudyonly
riskpremiums,disregardingsurchargesforcostsincurredbytheinsurancecompany.
Bythelawoflargenumbers,toavoideventualruinthetotalpremiumshouldbeat
leastequaltotheexpectedtotalclaims,butadditionally,therehastobealoadingin
x Preface
thepremiumtocompensatetheinsurerformakingavailablehisriskcarryingcapac-
ity.Fromthisloading,theinsurerhastobuildareservoirtodrawuponinadverse
times, so as to avoid getting in ruin. We present a number of premium principles,
together with the most important properties that characterize premium principles.
The choice of a premium principle depends heavily on the importance attached to
suchproperties.Thereisnopremiumprinciplethatisuniformlybest.
Risk measures also attach a real number to some risky situation. Examples are
premiums, infinite ruin probabilities, one-year probabilities of insolvency, the re-
quiredcapitaltobeabletopayallclaimswithaprescribedprobability,theexpected
valueoftheshortfallofclaimsoveravailablecapital,andmore.
6.Bonus-malussystems
Withsometypesofinsurance,notablycarinsurance,chargingapremiumbasedex-
clusivelyonfactorsknownaprioriisinsufficient.Toincorporatetheeffectofrisk
factorsofwhichtheuseasratingfactorsisinappropriate,suchasraceorquiteoften
sexofthepolicyholder,andalsoofnon-observablefactors,suchasstateofhealth,
reflexesandaccidentproneness,manycountriesapplyanexperienceratingsystem.
Suchsystemsontheonehandusepremiumsbasedonapriorifactorssuchastype
of coverage and list-price or weight of a car, on the other hand they adjust these
premiums by using a bonus-malus system, where one gets more discount after a
claim-freeyear,butpaysmoreafterfilingoneormoreclaims.Inthisway,premi-
ums are charged that reflect the exact driving capabilities of the driver better. The
situationcanbemodeledasaMarkovchain.
The quality of a bonus-malus system is determined by the degree in which the
premium paid is in proportion to the risk. The Loimaranta efficiency equals the
elasticity of the mean premium against the expected number of claims. Finding it
involvescomputingeigenvectorsoftheMarkovmatrixoftransitionprobabilities.R
providestoolstodothis.
7.Orderingofrisks
It is the very essence of the actuary’s profession to be able to express preferences
betweenrandomfuturegainsorlosses.Therefore,stochasticorderingisavitalpart
ofhiseducationandofhistoolbox.SometimesithappensthatfortwolossesX and
Y,itisknownthateverysensibledecisionmakerpreferslosingX,becauseY isin
a sense ‘larger’ than X. It may also happen that only the smaller group of all risk
averse decision makers agree about which risk to prefer. In this case, riskY may
belargerthanX,ormerelymore‘spread’,whichalsomakesarisklessattractive.
Whenweinterpret‘morespread’ashavingthickertailsofthecumulativedistribu-
tionfunction,wegetamethodoforderingrisksthathasmanyappealingproperties.
Forexample,thepreferredlossalsooutperformstheotheroneasregardszeroutility
premiums, ruin probabilities, and stop-loss premiums for compound distributions
with these risks as individual terms. It can be shown that the collective model of
Chapter 3 is more spread than the individual model it approximates, hence using
thecollectivemodel,inmostcases,leadstomoreconservativedecisionsregarding
premiums to be asked, reserves to be held, and values-at-risk. Also, we can prove
Preface xi
thatthestop-lossinsurance,demonstratedtobeoptimalasregardsthevarianceof
theretainedriskinChapter1,isalsopreferable,otherthingsbeingequal,intheeyes
ofallriskaversedecisionmakers.
Sometimes,stop-losspremiumshavetobesetunderincompleteinformation.We
give a method to compute the maximal possible stop-loss premium assuming that
themean,thevarianceandanupperboundforariskareknown.
Intheindividualandthecollectivemodel,aswellasinruinmodels,weassume
that the claim sizes are stochastically independent non-negative random variables.
Sometimesthisassumptionisnotfulfilled,forexamplethereisanobviousdepen-
dencebetweenthemortalityrisksofamarriedcouple,betweentheearthquakerisks
ofneighboringhouses,andbetweenconsecutivepaymentsresultingfromalifein-
surancepolicy,notonlyifthepaymentsstoporstartincaseofdeath,butalsoincase
ofarandomforceofinterest.Wegiveashortintroductiontotheriskorderingthat
appliesforthiscase.Itturnsoutthatstop-losspremiumsforasumofrandomvari-
ables with an unknown joint distribution but fixed marginals are maximal if these
variablesareasdependentasthemarginaldistributionsallow,makingitimpossible
thattheoutcomeofoneis‘hedged’byanother.
In finance, frequently one has to determine the distribution of the sum of de-
pendentlognormalrandomvariables.Weapplythetheoryoforderingofrisksand
comonotonicitytogiveboundsforthatdistribution.
Wealsogiveashortintroductioninthetheoryoforderingofmultivariaterisks.
Onemightsaythattworandomsvariablesaremorerelatedthananotherpairwith
the same marginals if their correlation is higher. But a more robust criterion is to
restrict this to the case that their joint cdf is uniformly larger. In that case it can
beprovedthatthesumoftheserandomvariablesislargerinstop-lossorder.There
areboundsforjointscdfsdatingbacktoFre´chetinthe1950’sandHo¨ffdinginthe
1940’s.Forarandompair(X,Y),thecopulaisthejointcdfoftheranksF (X)and
X
F (Y).Usingthesmallestandthelargestcopula,itispossibletoconstructrandom
Y
pairswitharbitraryprescribedmarginalsand(rank)correlations.
8.Credibilitytheory
The claims experience on a policy may vary by two different causes. The first is
thequalityoftherisk,expressedthroughariskparameter.Thisrepresentstheaver-
ageannualclaimsinthehypotheticalsituationthatthepolicyismonitoredwithout
change over a very long period of time. The other is the purely random good and
bad luck of the policyholder that results in yearly deviations from the risk para-
meter. Credibility theory assumes that the risk quality is a drawing from a certain
structuredistribution,andthatconditionallygiventheriskquality,theactualclaims
experienceisasamplefromadistributionhavingtheriskqualityasitsmeanvalue.
Thepredictorfornextyear’sexperiencethatislinearintheclaimsexperienceand
optimalinthesenseofleastsquaresturnsouttobeaweightedaverageoftheclaims
experience of the individual contract and the experience for the whole portfolio.
Theweightfactoristhecredibilityattachedtotheindividualexperience,henceitis
called the credibility factor, and the resulting premiums are called credibility pre-
xii Preface
miums.Asaspecialcase,westudyabonus-malussystemforcarinsurancebased
onaPoisson-gammamixturemodel.
CredibilitytheoryisactuallyaBayesianinferencemethod.Bothcredibilityand
generalizedlinearmodels(seebelow)areinfactspecialcasesofso-calledGeneral-
izedLinearMixedModels(GLMM),andtheRfunctionglmmisabletodealwith
boththerandomandthefixedparametersinthesemodels.
9.Generalizedlinearmodels
Many problems in actuarial statistics are Generalized Linear Models (GLM). In-
steadofassuminganormallydistributederrorterm,othertypesofrandomnessare
allowedaswell,suchasPoisson,gammaandbinomial.Also,theexpectedvaluesof
thedependentvariablesneednotbelinearintheregressors.Theymayalsobesome
functionofalinearformofthecovariates,forexamplethelogarithmleadingtothe
multiplicativemodelsthatareappropriateinmanyinsurancesituations.
Thisway,onecanforexampletackletheproblemofestimatingthereservetobe
keptforIBNRclaims,seebelow.Butonecanalsoeasilyestimatethepremiumsto
bechargedfordriversfromregioniinbonusclass jwithcarweightw.
In credibility models, there are random group effects, but in GLMs the effects
arefixed,thoughunknown.TheglmmfunctioninRcanhandleamultitudeofmod-
els,includingthosewithbothrandomandfixedeffects.
10.IBNRtechniques
Animportantstatisticalproblemforthepracticingactuaryistheforecastingofthe
total of the claims that are Incurred, But Not Reported, hence the acronym IBNR,
ornotfullysettled.Mosttechniquestodetermineestimatesforthistotalarebased
on so-called run-off triangles, in which claim totals are grouped by year of origin
anddevelopmentyear.Manytraditionalactuarialreservingmethodsturnouttobe
maximumlikelihoodestimationsinspecialcasesofGLMs.
Wedescribetheworkingsoftheubiquitouschainladdermethodtopredictfuture
losses,aswellas,briefly,theBornhuetter-Fergusonmethod,whichaimstoincorpo-
rateactuarialknowledgeabouttheportfolio.Wealsoshowhowthesemethodscan
beimplementedinR,usingtheglmfunction.Inthissameframework,manyexten-
sionsandvariantsofthechainladdermethodcaneasilybeintroduced.Englandand
Verrallhaveproposedmethodstodescribethepredictionerrorwiththechainladder
method,bothananalyticalestimateofthevarianceandabootstrappingmethodto
obtainanestimateforthepredictivedistribution.WedescribeanRimplementation
ofthesemethods.
11.MoreonGLMs
For the second edition, we extended the material in virtually all chapters, mostly
involvingtheuseofR,butwealsoaddsomemorematerialonGLMs.Webriefly
recapitulatetheGauss-Markovtheoryofordinarylinearmodelsfoundinmanyother
textsonstatisticsandeconometrics, andexplain howthealgorithmbyNelderand
Wedderburn works, showing how it can be implemented in R. We also study the
stochasticcomponentofaGLM,statingthattheobservationsareindependentran-