T2777tk

TRANSACTIONSOFSOCIETYOFACTUARIES1978VOL.
30ASSETSHAREMATHEMATICSPEYTONJ.
HUFFMANABSTRACTThispaperpresentsanewinsightintothemathematicalstructureofassetshare-typecalculations.
Assetsharesareinterpretedasaccumula-tionsofinsurancecashflows,which,alongwiththeirrelatedinvestmentincome,aretreatedasStieltjesintegrals.
Theresultisalogical,system-atic,andgeneralmethodofapproachingtheinsuranceandinvestmentcash-flowelements.
Thetechniquesaredevelopedwithinthecontextofindividuallifeinsuranceassetshares,butmaybeappliedtoamuchbroaderrangeofsituations.
I.
INTRODUCTIONTHEconceptofanassetshareisfamiliartoactuaries.
Traditionally,anassetshareisdefinedas"theestimatedamountattributabletoanindividual[unitofcoverage]iftheaccumulatednetfundsofaclassofalargenumberofidenticalpolicies.
.
,isdividedatsometimetamongallthe[remainingunitsofcoverageonaproratabasis].
''1Theparticularpolicycharacteristicsandex'perienceassumptionsusedreflectthosefactorsthattheactuaryfeelsarerelevanttothepurposeofthecalculation.
Thebasicassetsharecalculationiscomparativelysimpleinboththeoryandpractice,requiringonlyasetofexperienceassumptions,arudimentaryknowledgeofalgebra,acalculator,andafourteen-columnworksheet.
Assetshareshavebeenusedsincethenineteenthcenturyforawidevarietyofpurposes,includingcalculatingandtestingpremiumrates,settingnonforfeiturevalues,establishingdividendscales,testingsolvency,andmakingprojectionsofcountlesstypes.
Earlyactuaries,withlimitedresourcesforcomplexcalculations,madeeithersubjectiveorapproximateadjustmentswhenintroducinganunusualfactorintoanassetshare.
Theincreasedavailabilityandutiliza-tionofhigh-speedcomputershaveledtoavastnumberofmathematicalrefinementstotheassetsharecalcuiation,asrecentstudentsoftheSocietyofActuariesFellowshipexaminationscanconfirmreadily.
How-tCharlesH.
Page,"AssetSharesandModelOmces,"SocietyofActuaries,StudyNote89-22-70,p.
1.
277278ASSETSHAREMATHEMATICSever,therehasbeenlittlereexaminationofthebasicmathematical"natureoftheassetshare.
Theassetshareitselfremainsanexerciseinalgebra(albeitanincreasinglycomplicatedone),anditdeservestobeexaminedfromamoresophisticatedviewpoint.
II.
ANALYSISOFASI~(PLEASSETSHAREFOR.
X{ULAConsiderthefollowingelementaryassetshareformulaforanannualpremiumpolicywithafaceamountof$1,000:1A,=~t[A,_,+GP(1--EP,)--E~I(1+i)(1)-,,_,Cl,00o(1+t,wheret=At=GP=CV,=E~=El=i=qw~_1Policyyear;Assetshareper$1,000unitofcoverageinforceattheendofpolicyyeart;Grosspremium;Cashvalueavailableattheendofpolicyyeart;Percent-of-premiumexpenserateinpolicyyeart;Dollars-per-unitexpenseinpolicyyeart;Interestrate;Probabilityofentranttopolicyyeartterminatingbecauseofdeathduringpolicyyeart;Probabilityofentranttopolicyyeartterminatingbecauseofwithdrawalduringpolicyyeart;1--q~_,--qT-*Probabilityofentranttopolicyyeartenteringpolicyyeart+1inforce.
Also,let/0=Numberofunitsofthepolicyinitiallyissuedattimet--O;1,=l,-1--d~-i--d~-t--Numberofunitssurvivingtopolicydurationt;d~_.
l=l,-lq~-i=Numberofunitsterminatingduringpolicyyeartbecauseofdeath;=Numberofunitsterminatingduringpolicy),eartbecauseofwithdrawal.
ASSETSHAREMATHEMATICS279ConceptofAssetFundFormula(1)givestheassetshareperunitinforce.
Forreasonsthatwillbediscussedsoon,itispreferabletodealintermsofFt,theassetfundper/0initiallyissuedunits,accumulatedatinteresttodurationt.
Thus,F,=ltAt(2)=(F,-I+t,_,[aP(1-FA)-E*,]}(1+i)(3)++)-Aftertheassetfundhasbeencomputedatdurationt,conversiontothetraditionalassetshareperunitinforcemaybeeffected,ofcourse,bydividingtheassetfundbytheunitssurvivingatthesameduration(10.
Conceptually,theassetfundrepresentsashiftfromthepolicyholder'spointofviewtotheinsurer'spointofview.
Theassetshareproratesfundsamongthepoliciessothateachgetsitsshare;theassetfunddoesnot,therebymeasuringtheaccumulatedfundsheldbytheinsurer.
Theprorationcanproducemisleadingresults.
Forexample,considertwoplanswithpremiumssetsothattheassetsharesexceedtherespectivematurityvaluesby$10.
Ifbothmatureatthesamepolicydurationandprojectedmortalityandlapseexperienceisidenticalexceptinthefirstyear,whenonehasa10percentlapseratewhiletheotherhasa40per-centlapserate,thenthefirstpolicyisexpectedtoadd50percentmoretocompanysurplusthanthesecond,althoughtheirassetsharesareidenticalatmaturity.
Thedifferencebetweenthesepolicieswouldbeconspicuoususingassetfunds.
Itisnotsuggestedthattheassetfundissuperiortotheassetshare.
'Rather,theassetfundandassetsharearecomplementary,alternativewaysofviewingthedevelopmentoffundsheldbyaninsurer.
Eachhasadvantages.
Oneadvantageoftheassetfundisthatitdoesnotrequireanormalizationprocessattheendofeachpolicyyear.
Asaresult,theassetfundmaybeviewedasanaccumulationofcashflows.
Theimpactofacashflowontheassetfundisindependentofsubsequentpersistency;itaffectstheassetfundbyitsdollaramountplusaccumulatedinterest.
Thisistheadvantagethatwillbeexploredinthispaper.
TreatmentofInterestOneaspectofassetsharemethodologythathasbeenaccordedlittleattentionisthetreatmentofinterest.
Intheformulaunderconsideration,theassetfundattheendofthepreviousyearplusthepremiumafterexpensesearnafullyear'sinterest.
Thebenefitcashflowsarecharged280ASSETSHAREM.
ATI~fEMATICSinterestdependinguponwhentheyarepaidduringtheyear.
Onlythedeathbenefitisassumedtobepaidatanonintegralpolicyduration.
Forthedeathbenefitinterestelement,-{i(l,000d~_1)isatypicalformulationandgenerallyhasbeenconsideredareasonableandconvenientapproxi-mation.
Amorepreciseexpressionforthisinterestcomponent,basedontheassumptionthatdeathsaredistributeduniformlyovertheyear,is-f[(1+0~-o-111,000~_lds=--11,000dr_,.
oAlternatively,amidyeardeathassumptionresultsin[(1+i)"~1----]l,000dt_l.
Thesimpler-i(1,000da_l)approximatesbothwellandalsohappenstobeconservative.
Whencash-flowelementsotherthandeathbenefitsneedtobeintro-ducedatnonintegralpolicydurations,theactuarygenerallyhastouseimprovisedmethodsfortreatinginterest.
Analternativethatsimplifiesthehandlingofinterestisbasedonthefollowingassumption:INTERESTASSUMPTION:Cashflowswithinapolicyyearearnsimpleinterestfromtheirrespectivedatesofincidencetotheendofthepolicyyear.
Theaccumulatedamountattheendofapolicyyearearnsinterestthereafterattheregularcompoundedannualrate.
Thatis,acashflowcatmomentswithinpolicyyeartaccumulatesto[1+(1--s)i]cattheendofpolicyyeart.
Thecorrespondingaccumula-tionunderthestandardcompoundinterestformulationis(1+i)l-'c.
Reconsideringtheassumptionofauniformdistributionofdeaths,theinterestcomponentsimplifiesto1--f(1--s)i(1,0OO~_,)ds=--i(1,000~_0.
0Infact,themidveardeathassumptionalsosimplifiestobecomeexactlythe"approximation.
"Returningnowtotheassetfundformula(3),thetermscanberear-rangedtoprovideadditionalinsight:F,=Ft_l(4a)+iFt_l(4b)+t,_,[aPO-g~,)-E*,]-1,ooo~_1-d,~_lCV,(4c)+i{l,_,[GP(1--E~)--EsL]--(1,000~4d)ASSETSHAREMATHEMATICS281Theend-of-yearassetfundconsistsofthepreviousyear-end'sassetfund(4a),plusinterestonthatfund(4b),plustheyear'snetinsurancecashflows(4c),plusinterestontheyear'snetinsurancecashflows,withdueregardfortheirincidence(4d).
Theinterestelementassociatedwiththecurrentyear'snetinsurancecashflows(4d)maybeinterpretedasanapplicationoftheInterestAssumptiontoeachoftheinsurancecash-flowcategories(premium,expenses,deathbenefits,andwithdrawalbenefits).
Ill.
ASTIELTJ'ESINTEGRALINTERPRETATIONTheassetfundformulagiveninexpressions(4a)-(4d)demonstratesthatiftheinsurancecashflows(4c)ineachyearareknown,alongwiththeircontributiontoinvestmentincome(4d),thentheassetfundisknown.
Eachpolicyyear'sinsurancecashflowsandrelatedinterestcanbecalculatedinamuchmoregeneralway.
Thefirststepistopartitiontheinsurancecashflowsintoafinitenumberofdistinctcategories.
Inthecaseofthesimpleassetshareconsiderfivesuchcategories,asfollows:1.
Premiumincome=GPlt_l;2.
Percent-of-premiumexpense=--E~GPI,_~;3.
Dollars-per-unitexpense=-Ell,_1;4.
Deathbenefits----1,000da~_1;5.
Withdrawalbenefits=--CVLd~-vThispartitionisamatteroftasteandconvenience;items1-3couldhavebeencombinedintoasinglecategorytogivean"effectivepremium.
"Therequisitecharacteristicisthatthecashflowsofagivencategorybegeneratedbythesameevents.
Ineachcategory,thecashflowisaproductofanamountandarateofpayment.
Forexample,premiumincomeistheproductofGPand1,-1;deathbenefitsaretheproductof--1,000andd~-l.
Ifitisassumedthatdeathsaredistributeduniformlyoverthepolicyyear,theproduct--1,000d~_~isactuallythesumofmomentarycashflowsoverthepolicyyear,thatis,If-1,000d(_,s)=-0Consideringpremiumincomeoverthepolicyyear,notethatannualpremiumsarepaidonlyatthebeginningofthepolicyyear.
Definingastepfunctiong(s)=O,s0,282ASSETSHAREMATHEMATICSpremiumincomealsomaybeexpressedinanintegralform,xfGPdg(s)--GPI,_I.
0Theothercash-flowcategoriesmaybeexpressedsimilarly.
Foranygivencategoryofcashflow,wearelookingataStieltjesintegral.
Givenfunctionsf(s)andg(s)definedontheunitinterval,theStieltjesintegralof"foverg"ontheunitintervalisdenotedby1fr(s)dg(s),0andisdefinedasthefollowinglimit(ifitexists):nlira~f(si)[g(ti)--g(ti-t)],IlalI-~o-.
whereA=(to,q,isanarbitrarypartitionoftheunitinterval,sisanarbitrarilychosenpointin[t~-x,ti],and[IAi]isthelargestintervalill-x,t~].
2Generally,theStieltjesintegralexistsiffiscontinuousandgisofboundedvariation,orviceversa.
Inthecasesconsideredinthispaper,existenceisclearlysatisfied.
Inverbalterms,f(s)isan"amountfunction,"whiledg(s)isan"incidencefunction.
"Thatis,f(s)describestheamountofexpectedvalueofcashflowiftheeventuponwhichitiscontingentoccurs,whiledg(s)describestheexpectedincidenceofeventsgivingrisetocashflows.
Let{Ck}beanarbitrarypartitionofthecashflowsintoafinitenumberofcategoriessuchthateachcategorymaybedescribedasaStieltjesintegraloverthepolicyyear(unitinterval),1C~=ff~(s)dgk(s).
0Thetotalofinsurancecashflowsduringthepolicyyearissimplythesumoftheindividualintegrals:1c,:,rs,(,)d,,(s).
k=lk=i0Finally,considerI~.
,theinvestmentincomeassociatedwithcashflowCk.
Acashflowfk(s)dgk(s)atmomentsearnsi(1--s)f,(s)dg~(s)duringtAngusE.
Taylor,GeneralTheoryofFunctionsandIntegralion(NewYork:BlaisdellPublishingCo.
,1965).
ASSETSHAREMATHEMATICS283thebalanceoftheyear.
Thus,categorykgenerates1I~=fi(1--s)f~(s)dg~(s)01=itCk--fsf,(s)dgk(s)].
0BylettingTkdenotethe"firstnormalizedmomentoffkovergk,"wehave1fsfk(s)dg,(s)T~=oCk#OC,=0,Ck--'0;thenIk=iCk(1--Tk).
Conceptually,Tkistheweightedaveragedurationofincidenceofcash-flowcategoryk.
Thetotalinvestmentincomegeneratedbythepolicyyear'sinsurancecashflowsisIk=iY'~Ck(1--T~).
kmlk-1Restatingformulas(4a)-(4d)ingeneralterms,theassetfundformulabecomesF,=F,_t+iFt_l+Ck-~iECk(1.
--Tk),(5)k~lk~lwhereCkandTkarecomputedforpolicyyeart.
Thisgeneralizedformulacanbeusedforanytypeofinsuranceorannuity.
IV.
ANEXAMPLETodemonstratetheStieltjestechnique,considertheplanofcoverageunderlyingthesimpleassetshare,modifiedsothatmpremiums(GPC=)/m)arepayableeachpolicyyear.
Assumethatthed~_lwithdrawalsoccurringinpolicyyeartaredistributedequallyovereligiblewithdrawaldates(off-premiumduedatewithdrawalsignored),andthatwith-drawalswithinthepolicy3'earreceivetheinterpolatedcashv~lue.
Also,assumethatdeathsaredistributeduniformlyoverthepolicy),ear.
Theamountfunctions,incidencefunctions,cashflows,andaveragedurationsofincidenceofthefivecash-flowcategoriesnowmaybedeveloped.
Forconvenience,dgk(s)willbegivenratherthangk(s).
284ASSETSHAREMATHEMATICS1.
Premiumincome:dD(s)=lt-t+~,=-0,fl(s)=GP(")/m,s=O/m,1/m,.
.
.
,(m--1)/motherwise.
Notethatl,-t+,=l,-1-sda~-i-u(s)d~_lprovidedthatu(s)=(k--1)/m,wherekisthesmmllestintegersuchthatscomeisreceived:Hencef2(s)=--E~GP(~)/m,dg,(s)=dg~(s).
C2=--E~Cl,T2=Ta.
ASSETSHAREMATHEMATICS2853.
Dollars-per-unitexpense,assumedtobeincurredentirelyatthebeginningoftheyear:E~,dg3(s)=l,-i,s'=0=0,otherwise.
ThenCa=--EStlt-1,T3=0.
4.
Deathbenefits,assumedtobeunaffectedbywithdrawalswithinthepolicyyear:f4(s)=-1,000,dg4(s)=d(~_ts).
HenceC4=-1,000~_1,T4={.
5.
Withdrawalbenefits:f~(s)=--[CV,_x+s(CV,-CV,_x)],Thendgs(s)1d'~-----0~1Cn=.
ffs(s)dgs(s)o12mS9n'otherwise.
m+1V,=-[cv,_,+--2~-~(cv,-c1C5T5=OFsfs(s)dgs(s)o=---~CV,_,+I(CV,-CV,_~)--d.
"m,Fm+1(m+1)(2m+1)(CV,-CV,=--L2mCV~_I+6m~Fm+1(m+t)(2m+1)(CV,-CV,_,)]T6=b~CVt-1+6m2m+l-'xIcy,_,+~(cv,-cv,_,)](ifCVt-1=CV,=O,T5=0).
286ASSETSHAREMATHEMATICSThesecash-flowcategoriesarenotintendedtorepresentacontempo-rarysetofassetshareelements.
However,thesetechniqueseasilycanbeusedtoreflectmanymorefactors.
Thepurposeofthiscash-flowelementanalysisistodemonstratetheStieltjestechniques.
Eveninthissimpleexample,manualcalculationofthepremiumincomeandwithdrawalbenefitpieceswouldbeunreasonable.
Thesamecalculations,however,canbeperformedverysimplyonceprogrammedonacomputer.
Theassumptionofauniformdistributionofwithdrawalsovereligiblewithdrawaldatescanbemodifiedtoanarbitrarydisti'ibutionwithoutcomplicatingmattersgreatly.
LetH(s)betheportionofwithdrawalsduringpolicyyeartoccurringbydurations.
Retainingtheprohibitionagainstwithdrawalsatotherthanpremiumduedates,H(s)isamono-tonicallynondecreasingstepfunctionsuchthatitis0fors_1,andisdiscontinuousonlyats=k/mfork--1,.
.
.
,m.
Also,leth(s)=dH(s).
Notethath(s)iszeroeverywhereexceptfors=1/m,2/m,.
.
.
,m/m,andthatf~h(s)ds=2.
Thus,lL-l+,=l,-1--sd~_l--H(s)d~-l.
Usingthismoregeneralrepresentationof/,-1+,,thepremiumincomeandwithdrawalbenefitcashflowsmaybereevaluated.
I*.
Premiumincome:1C,=.
fft(s)dgt(s)0m-].
=__IGp(=>~l,-~+,l=Z*~k.
O"-'[]1ap(,~~]l,_,~_,a(kl,,,)~_,m-1="-t2mSimilarly,[zm6m2"-XH(k/ra)d~t_t],,,-i(,,,-11(2m-I)al,_,-~Tt=~l,-1--X[lt-15*.
Withdrawalbenefits:,.
-1!
]-idH(k/m)g_,.
2mk.
Odgs(s)=d[H(s)d'Tt_l.
ASSETSHAREMATHEMATICSHence,1C5=ffs(s)dgs(s)02871=-,['[CV,_t+s(CV,-CV,_t)ld[tt(s)d,~_t]o---k.
l'/~k=Iandmk2Xh(k/m)CVt_l+__kh(k/m)(CV,--CV,_t.
.
7t~Notethat1"and5*infactreduceto1and5,respectively,whenwith-drawalsareassumedtooccurequallyateacheligiblewithdrawaldate.
Asamplecalculationofanassetshareusingtheassetfundtechniquesandthefivecash-flowcategorieswhoseintegralshavebeenevaluatedhereisshowninTable2oftheAppendix.
V.
CALENDAR-YEARASSETSFIARESVerylittlehasbeenpublishedaboutassetsharesmeasuredoverotherthanpolicy-yearintervals.
Therearemanygoodreasonsforthis.
Forexample,incalculatingassetsharesforrate-makingitistheindividualplan-agecellthatisunderconsideration;hence,itisappropriatetomeasuretheassetsharefrompolicyanniversarytopolicyanniversary.
Inaddition,thereiscomparativelylittledifferencebetween,say,theassetsharemeasuredatthetwentiethdurationandanintermediateassetsharemeasuredbetweenthenineteenthandtwentiethpolicydurations.
Evenwhentheuseofacalendar-yearassetshareisdearlyappropriate(inmodelingapplications,for.
example),actuarieshavecontinuedtouseaggregationsofpolicy-yearassetshares.
Practicalproblemsalsoarise:Whatisacalendar-yearassetshareIsitanassetshareforapolicywhoseissuedateisJune30orJuly1Ifso,whencalculatingtheassetshareatDecember31,haveoneortwosemiannualpremiumsbeen288ASSETSI:IAREMATHEMATICSreceivedCashflowswithinapolicyyeartendtobeskewedsothatpositivecashflowsoccurnearthebeginningoftheyearandnegativecashflowsoccurneartheend;hence,aninterpolatedcashflowcannotbeusedarbitrarily.
Fortunately,Stieltjesintegrationtechniquesprovideananswer.
Howisthecalendaryear-endtobeinterpretedwithrespecttoanassetshareTheJune30/July1issue-dateassumptionproducesalessthansatisfactoryresult.
InFigurel,thefirstyearofapolicyissuedinyearyisdesignatedasAB.
Assumethatpoliciesareissueduniformlyovercalendaryeary,thatis,lodzunitsareissuedatmomentzwithincalendaryeary.
Then,inFigure2thefirstpolicyyearoftheissuesofcalendaryearybecomestheareaABCD.
NotethatlineBDistheendofcalendaryeary,whileBCistheendofthefirstpolicyyear.
Ifthecash-flowamountswithinABDandtheiraveragedurationsofincidencewithinABDaredetermined,thepolicyyear'scashflowswillhavebeensplitintopiecesthatcanbeusedtogeneratecalendar-yearassetshares.
a~01/t/yIA12/31/yCalendarTimeFIo.
10ADl/l/y12/31/yCalendarTimeFro.
2ASSETSHAREMATHEMATICS289Apolicyissuedatmomentzwithincalendaryearywillremainincalendaryearyuntilpolicyduration1--z.
Consideringanarbitrarycash-flowcategory,thecashflowgeneratedinareaABDis11-sC'=fff(s)dg(s)dz.
o0Reversingtheorderofintegrationandsolving,11-aC'=ff(s)fdzdg(s)oo1=f(1--s)f(s)dg(s)0=c(1-T).
Inotherwords,theportionofthecashflowsoccurringincalendaryearyis1--T,whereTisintuitivelytheaveragedurationoftheirincidencewithinthepolicy),ear.
Forexample,anannualpremiumpolicywillhaveT=0forpremiumincome,so1-T=1,indicatingthatallpremiumincomeoccursinyeary.
Next,whatistheinvestmentincomegeneratedbycashflowswithincalcndaryearyUsingthesamearbitrarycash-flowcategory,wehaveI1--sI'=J"J"i(1--z--s)f(s)dg(s)dzo011--8=ill(s)f(1--z--s)dzdg(s)00where1(1--s)2dg(s)=ioff(~)2=iC[(1--2T+M)],11ufs,i(s)dg(S)o/fI(s)g(s)isthe"secondnormalizedmomentoffoverg.
"AswithT,Mismeasuredoverthepolicyyear.
Usingagaintheexampleofthepremiumincomeassociatedwithanannualpremiumpolicy,weobtainM=0.
Asex-pected,I'=iC.
Usingeitherintegrationoralgebra,wefindthatthecashflowandinterestincalendaryeary+1areC"--CTandIt'=/C[{(2T--M)].
290ASSETSHAREMATHEMATICSLet~#tdenotetheassetfundmeasuredatthecalendaryear-endoccurringduringpolicyyeart.
Thenthegeneralizedaccumulationformulacorrespondingtoformula(5)isP,=P,_,+iP,_~+~-1~liCk(1--,Tk)+,_,Ck,-lTk]k=l-(~-12+,-IC~2t_tTk--t-tMk)2(6)Calendar-yearcalculationsareofgreatestutilityformodelingapplica-tions.
Inaddition,acalendar-yearassetsharecanbecomputed.
InordertoconvertF,toanassetshareAtperunitinforce,L,themeannumberofunitsofcoverageinforceatcalendaryear-end,mustbecomputedfrom1l,=fl,_t+.
ds.
oThenfi,tiscalculatedas~,=P,/L.
(7)Calendar-yearreservesperunitinforcealsomustgothroughanormal-izingprocess.
Forexample,ifforthepolicyinquestionV~m{andV~")aretheconsecutiveterminalreservesandp~,oisthenetpremium,thereserveper/oinitialunitsislV*=f{l,_,+,[X(V,_l+V,+P("')]-lt_a+,w(s)P("*)}ds,(8)owherew(s)=(m--k)/mgiventhatkisthelargestintegersuchthat(k-1)/mcomesV,=V*/i,.
(9)Tosummarize,calendar-yearassetfundscanbecalculateddirectlyfromthepolicy-yearcash-flowamountsandfirstandsecondnormalizedmoments.
ThisserendipitousresultfollowsdirectlyfromtheStieltjesintegrationinterpretationofcashflows.
Assetsharescanbecomputedinturnbydividingbythemeannumberofunitsinforceatcalendaryear-end.
Similarly,calendar-yearreservesarecalculatedfromanintegral.
ASSETSHAREMATHEMATICS291VI.
ANEXAMPLEREVISITEDTOextendthecash-flowformulascitedpreviouslytothecalendar-yearcase,itisnecessaryonlytoadd"thirdmoment"functions.
Themoregeneralwithdrawaldistributionwillbeused.
1.
Premiumincome:lC,MI=fs2f(s)dg(s)0"-'(m)'_l_J_aPcomplexityoftheformulasproducedbvStieltjestechniquesmaylendanauraofspuriousprecisiontotheresultingassetshares.
AnimportantresultofapplyingStieltjestechniquestoratemakingisthedifferentiabilitybetweentheassetsharesofvariousplans.
Theassetsharesofsimilarcoveragesshouldvaryinalogicalmanner.
Themorepreciselytheamountandincidencefunctionsaredefined,themorerefinedbecomesthedifferentiationinimpacttoboththepolicyholderandthecompany.
ThedegreeofrefinementsoughtwillvaryfromASSETSHAREMATHEMATICS293applicationtoapplication.
Theconstraintswithinwhichtheactuarymustworkalsowilldiffer,dependingonthepurposeforwhichthecalculationsarebeingmade.
TheStieltjesmethodologyisequallyapplicableforprospectiveandretrospectivecalculationsandcanbeutilizedforotherpurposes,suchasthecalculationofGAAPunitreservefactors.
Stieltjestechniquesalsocanbeadaptedtocalendar-yearissueassumptionsotherthanuniformandtonew-moneyinvestmentmethods.
Inanycase,thetechniquesthathavebeenpresentedwillenableactuariestomaketheircalculationsinaneasierandmoresystematicmanner.
Inaddition,themathematicalstructureunderlyingtheassetsharehopefullyhasbeenmadeclearer.
APPENDIXTodemonstratetheStieltjestechniques,awholelifepolicyissuedatage35onthesemiannualmodeisused.
Thesemiannualmodehasbeenchosensothatmanualverificationoftheresultsisfeasible.
Thepremium,claim,expense,andinterestassumptionsarenotintendedtoreflectcurrentexperience.
Assumptions1.
Grosspremium:$16.
00perunit,payableintwosemiannualinstallments.
2.
Percent-of-premiumexpenses:102percentinpolicyyear1,9.
5percentinpolicyyears2-10,and4.
5percentinpolicyyears11--65,incurredonthepremiumduedate.
3.
Dollars-per-unitexpenses:$12.
00inpolicyyear1and$0.
50inpolicyyears2-65,incurredatthebeginningofeachpolicyyear.
4.
Deathbenefit:$1,000;claimsdistributeduniformlyoverthepolicyyear.
5.
Cashvalues:Minimumcashvaluesbasedon3~percentinterestandcurtatefunctions;interpolatedvaluepayableatmidyear.
6.
Statutoryreserves:Netlevelfractionalpremiumreservesbasedon3~xpercentinterestandimmediatepaymentofdeathclaims;P~)=15.
48563.
7.
Mortality:Modificationof1965-70intercompanyexperience.
8.
Withdrawals:LintonBrates;withdrawalsdistributeduniformlyovereligibleterminationdateswithineachpolicyyearexceptthefirst,whentwo-thirdsareassumedtooccuratmidyear.
9.
Interest:5.
5percentTable1displaystheplandataandexperienceassumptionsinamorecompleteandgraphicform.
Table2presentsthedevelopmentofpolicy-yearassetfundsandassetshares.
Ftiscalculatedbyformula(5)andconvertedtoAtbydividingbyI,Table3developscalendar-yearassetfunds,assetshares,andreserves.
Ftiscomputedbyformula(6)andconvertedtoAtbydividingbyit.
V~'iscom-putedbyformula(8)andconvertedtoVtbydividingbyit.
TABLE1PLANANDEXPERIENCEDATA0.
.
.
.
1.
.
.
.
.
2.
.
.
.
.
3.
.
.
.
.
4.
.
.
.
'.
5.
.
.
.
.
16.
.
17.
.
18.
.
19.
.
~0.
.
/t-I,000.
000799.
160702.
454631.
387574.
979528.
119489.
196456.
928429.
416405.
159383.
746364.
087345.
988329.
305313.
893299.
642285.
807272.
420259.
480246.
965dtodddwGP(')E~E$iDeathqt-iqt-Jt-~1--~tBenefitCVtV~~J0.
000840.
2000.
840200.
00016.
00102.
0012.
005.
501,000013.
360.
001010.
1200.
80795.
89916.
009.
500.
505.
501,000027.
100.
001170.
1000.
82270.
24516.
009.
500.
505.
501,0001141.
190.
001340.
0880.
84655.
56216.
009.
500.
505.
501,0002555.
620.
001500.
0800.
86245.
99816.
009.
500.
505.
501,0004070.
370.
001700.
0720.
89838.
02516.
009.
500.
505.
501,0005585.
420.
001960.
0640.
95931.
30916.
009.
500.
505.
501,00071100.
770.
002210.
0581.
01026.
50216.
009.
500.
505.
501,00087116.
410.
002490.
0541.
06923.
18816.
009.
500.
505.
501,000103132.
350.
002850.
0501.
15520.
25816.
009.
500.
505.
501,000119148.
580.
003230.
0481.
23918.
42016.
004.
500.
505.
501,000136165.
100.
003710.
0461.
35116.
74816.
004.
500.
505.
501,000153181.
870.
004220.
0441.
46015.
22316.
004.
500.
505.
501,000170198.
900.
004800.
0421.
58l13.
83116.
004.
500.
505.
501,000187216.
160.
005400.
0401.
69512.
55616.
004.
500.
505.
501,000205233.
630.
006170.
0401.
84911.
98616.
004.
500.
505.
501,000223251.
290.
006840.
0401.
95511.
43216.
004.
500.
505.
501,000241269.
130.
007500.
0402.
04310.
89716.
004.
500.
505.
501,000259287.
130.
008230.
0402.
13610.
37916.
004.
500.
505.
501,000277305.
270.
009050.
0402.
235I9.
87916.
004.
500.
505.
501,000296323.
55TABLE2POLICY-YEARASSETFUNDSANDASSETSHARES2.
.
L.
~.
.
5.
.
3.
.
).
.
[0.
11.
[2.
13.
t4.
[5.
L6.
CtC~C*C.
C,TtTtT,T,TtFt.
At14,930--15,229--12,000--84000.
232080.
232080.
00.
5GO000.
00000-13,835'--17.
3112,400--1,178--400--80700.
242200.
242200.
00.
5C0000.
00000--4,157--5.
9210,955-1,041--351--822--5800.
243510.
243510.
00.
5(0000.
833334,1416.
569,877,--938--316--846--1,1950.
244290.
244290.
00.
500000.
7907011,26819.
609,012!
--856--287--862--1,6670.
244800.
244800.
00.
500000.
7758617,50633.
158,294t--788--264--898--1,9490.
245310.
245310.
00.
500000.
7682923,11147.
247,698--731--245--959--2,0980.
245810.
245810.
00.
500000.
7649328,26961.
877,201--684--228--1,010--2,2000.
246180.
246180.
00.
500000.
7620533,10477.
096,774--643--215--1,069--2,2960.
246420.
246420.
00.
500000.
7601037,65892.
956,397--608--203--1,155-2,330~0.
246650.
246650.
00.
500000.
7587041,996109.
446,061--273--192--1,239--2,427I0.
246760.
246760.
00.
500000.
7580646,399127.
445,753--259--182--1,351--2,4910.
246850.
246850.
010.
500000.
7571450,568146.
165,469--246--173--1,460--2,5230.
246950.
246950.
00.
500000.
7564154,549165.
655,207--234--165--1,581--2,5280.
247040.
247040.
010.
500000.
7558158,368185.
954,965--223--157--1,695--2,5170.
247130.
247130.
0~0.
500000.
7556162,059207.
114,739--213-150--1,849--2,6190.
247080.
247080.
00.
500000.
7551565,473229.
08[7.
.
.
.
.
.
.
4,519-203-143--1,955--2,7040.
247040.
247040.
00.
5000010.
7547668,669252.
07[8.
.
ii4,307194i136--2,043--2,7730.
247000.
247000.
00.
50000:0.
7544271,676276.
23t94,102--185--130--2,136--2,8280.
246950.
246950,00.
500000.
7541374,499301.
66!
0.
i3,903--176!
--123--2,235--2,8770.
246900.
246900.
00.
500000.
7540877,136328.
45TABLE3CALENDAR-YEARASSETFUNDSANDASSETSHARES1.
.
.
.
.
.
2.
.
.
.
.
.
3.
.
.
.
.
.
4.
.
.
.
.
.
5.
.
.
.
.
.
6.
7.
8.
9.
1011.
.
.
.
.
12.
.
.
.
.
13.
.
.
.
.
14.
.
.
.
.
15.
.
.
.
.
16.
.
.
:.
17.
.
.
.
.
18.
.
.
.
.
19.
.
.
.
.
20.
.
.
.
.
[MtMt.
.
.
0.
116040.
11604.
.
.
0.
121100.
12110.
.
.
0.
121760.
12176.
.
.
0.
122140.
12214.
.
.
0.
122400.
12240.
.
.
0.
122650.
12265.
.
.
0.
122900.
12290.
.
.
0.
123090.
12309.
.
.
0.
123210.
12321.
.
.
0.
123330.
12333.
.
.
0.
123380.
123380.
123430.
12343.
.
.
0.
123470.
12347.
.
.
0.
123520.
12352.
.
.
0.
123560.
12356.
.
.
0.
123540.
12354.
.
.
0.
123520.
12352.
.
.
0.
123500.
12350.
.
.
0.
123470.
12347.
.
.
0.
123450.
12345Ms0.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
00.
0M~0.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
333330.
33333Mb0.
000000.
00000O.
75000O.
686050.
66379O.
65244O.
64739O.
643070.
64015O.
638040.
637100.
635710.
634620.
633720.
633420.
632720.
632140.
631630.
631190.
63112Pt-12,993--6,3372,52210,15216,76022,58927,87432,78437,40741,77746,17550,40654,42358,26361,96065,48768,76471,83574,71777,415v*9,58518,58025,95332,20437,60442,33146,59950,55854,23857,64460,81463,69366,31368,70370,88272,83274,45075,75476,75177,451932.
913774.
782684.
482617.
074563.
049518.
164480.
889449.
797423.
084399.
517378.
521359.
224341.
452325.
057309.
906295.
721281.
971268.
674255.
817243.
378/it--13.
93--8.
183.
6816.
4529.
7743.
5957.
9672.
8988.
42104.
57121.
99140.
32159.
39179.
24199.
93221.
45243.
87267.
37292.
07318.
09Pt10.
2723.
9837.
9252.
1966.
7981.
6996.
90112.
40128.
20144.
28160.
66177.
31194.
212tl.
36228.
72246.
26264.
03281.
96300.
02318.
23DISCUSSIONOFPRECEDINGPAPERPIERREC.
CHOUINARD:Ireadthispaperwithgreatinterest.
Itpresentsarevolutionaryap-proachtothecalculationofassetshares--revolutionaryinthesensethat,oncethecashflows,tCK,andtheiraveragedurations,tTK,havebeendetermined,evaluationoftheassetfundturnsouttobeaproblemofinterestonly;~TKand,TKtakecareofthecontingencies.
Thisdiscussionhasthefollowingtwopurposes:first,tointroducesome"distributionofissues"assumptionsinthemodel,and,second,topresenttheresultsIhaveobtainedincalculatingcalendar-yearassetsharesinadifferentmanner.
DistributionofIssuesAllformulasofSectionsVandVIwerederivedassumingauniformdis-tributionofissuesduringacalendaryear.
AsstatedinMr.
Huffman'sconclusion,themodelneednotbeconfinedtothishypothesis.
Ageneralizeddistributionofissuescouldbeincorporatedinthemodelinthefollowingway:1l-IC'=fOFf(s)r(z)dg(s)dz.
,o01I--SI'=iff(as)~(z)ag(s)az,oowhereC'isanarbitrarycash-flowcategorygeneratedinareaABDofFigure2ofthepaper,I'istheinvestmentincomegeneratedbythiscashflowinthesameareaABD,andr(z)istheannualizedproportionoftotalpoliciesissuedatmomentzduringtheyearofissue.
Inactuarialterms,r(z)isnothingmorethantheforceofissueattimez.
NowletR(z)bethecumulativeportionofpoliciesissuedbytimez,suchthatR(0)=0andR(1)=1.
ThusthedifferentialofR(z),dR(z),representstheinfinitesimalportionoftotalissuessoldatmomentz,thatis,r(z)dz.
Inthecaseofauniformdistributionofissues,R(z)=zandr(z)dz=dz.
However,otherdistributionassumptionsalsocouldbeused.
Forexample,supposethatsalesincreasecontinuouslyatanannualizedrate&Theissuefunctionsr(z)andR(z)thencanbeshowntotakethefollowing297298ASSETSHAREMATIEEMATICSforms:R(z)=~_I'r(z)=~_1"Ifwereplacee~by(1-bj),wherejistheeffectiverateofsalesincreaseduringtheyear,weobtainR(z)=(1q-j)'--1r(z)=8(1+j)"j'jC'andI'thenbecome1'1-*"C'=:f(s):~(1-Fj)'dzdg(s)=-of,~:(s)ag(s)J$=_NC,1where11-j--:,~:_Jand~istheusualcontinuousforborneannuity-certain,atratej,andI'=i=i1I--Sff(s)f(1--z--s)_~(l+jl'dzdg(s)00]"::(,>[(,s>~];ds(,>=i(s)[)ag(s)--,c(~-~+~).
1Onecouldverifythat~-(I-s)liraj-~Oj(1-s)22Anotherdistributionofissuesthatcouldbeassumedisonebasedonthefactthatsomecompaniesexperienceanincreaseinsalesduringpar-ticularmonths.
Letussupposethattwo-thirdsofthetotalannualsalesDISCUSSION299"ofacompanyoccuruniformlybetweenJuly1andDecember31.
Further-more,letusalsoassumeauniformdistributionofsalesduringthefirstsixmonths.
Inthiscase,r(z)wouldbeatwo-stepconstantfunctionr(z)=5,o3-1,c,t'-iil[,.
CVt_lJf-.
~-~m-~tRANKC.
~ETZ:MycomplimentstoMr.
Huffmanforpresentingsuchageneralandpowerfulapproachtoassetsharemathematics.
HispaperisanextremelyvaluableandwelcomeadditiontoactuarialliteratureandmostcertainlyshouldbeincludedinthecourseofreadingforPart8.
TheultimateutilityofMr.
Huffman'sapproachislimitedonlybytheingenuityoftheactuaryapplyingitandbythereasonablenessoftheassumptionsem-ployed.
Severalareasdevelopedinthepaperimpressedmeasbeingpar-ticularlyinterestingandvaluable.
Theabilitytoallowforskewnessofwithdrawalsandtheabilitytocalculatecalendar-yearassetsharesarenoteworthyattributesofMr.
Huffman'sequations.
Thelatterattributeshouldbeveryusefulinmodelingapplications.
Theseparationofin-surancecashflowandinvestmentelementsforeachyearisanovelandvaluablefeaturethatallowsfortheisolationofnew-moneycellsindevelopinginvestment-yearassetshares.
Inordertoillustratehowthegeneralconceptspresentedinthepapercanbeappliedunderadifferentsetofassumptions,IhaveattemptedtorecasttheequationsinSectionIVofthepaperusingratesratherthan306ASSETSHAREMATHEMATICSprobabilitiesandhaveallowedforskewnessofthedeathsaswellasofthewithdrawals.
ThismodificationofMr.
Huffman'sequationsisnotanattempttobemoreprecisebutismerelyanillustrationofhowapar-ticularactuarymightapplythelogicandmethodologypresentedinthepapertoadifferentsetofcircumstances.
Therecastequationsareprefacedbythesetofdefinitionsthatfollows.
Theuseofrates(denotedbyJandH)ratherthanprobabilitiesisforthesakeofsimplicity.
q'~a_l---Deathrateforpolicyyeart;q~Withdrawalrateforpo!
icyyeart;lo=NumberofUnitsofpolicyinitiallyissuedattimet--O;l,=-l,_x(1--q]~a)(1--q'~x)Numberofunitssurvivingtopolicydurationt;J(s)=Proportionofdeathratethathastakeneffectbytimeswithinapolicyyear,0come:A(s)=dgt(s)=GP°")/m;l,_1+,,s=O/m,.
.
.
,(m--1)/m,0otherwise;ouj~,,=,,c!
-rs~'Cgtrs~Gpo.
),.
-t"==~Elt-l+klm;C1mk~O/,-x+k/=-/t_x{1k/,,,.
,a,~,--Sog(r)qt_t[1--H(r)q,_x]drklratwtd--Soh(r)q,_,[1--Y(r)q,_,ldr}=/,_1[1--](k/m)~l~_x--H(k/ra)~_xklm.
td/to+So3(r)q,_lH(r)q,_ldrklm/wtd+soh(,)q,_~J(,')q,_~dr]/t-I[1=--J(k/m)q~n_aH(k/m)q~DISCUSSION307.
k-11)/=,a,,oIt(,/m)S~/,~dJ(r)q,-tq,_li-Oklirat~pd+~.
,J(~/m)S(,_,,/.
.
att(rlq,_,q,_,]i=lifwithdrawalsareassumedtooccurattheendofmthlyintervals,thatis,H(s)=H((k-1)/m),(k-1)/m~l,_l+k/==C1_Gp(m)lt_t[1m--1m--1,,~2mq't~-t2mqi-x+(m-1)(2m-1),a,w'l6m2qt-lqt-ld;Gp(1kClT,=S]sf,(s)dg,(s)l,_,+~/.
,(,.
--1)(qf_,+qT-,)(m-1)2.
,,,.
,d3m--TI-[2~1(m--1)(2m--1),a(ra--1)(2m--1),~6m2qt-i--6ra2q,-1(m--1)2,a,,~-IF.
+"4m~"q'-'q'-'JL'm--1,m--1,w2mq~a-12mq,-i(ra--1)(2m--1),a,,.
'1-l6m2q,-lq,-132.
Percent-of-premiumexpense,assumedtobeincurredatthetimepremiumisreceived:eGPcm)f2(s)-E,;ag2(s)--ag,(s).
mHenceC2=--E~CI,T2=T1.
308ASSETSHAREMATHEMATICS3.
Dollars-per-unitexpense,assumedtobeincurredentirelyatthebe-ginningoftheyear:f~(s)-p.
~;dg3(s)=It-l,s=0=0otherwise.
Then$C3=~Etlt-t,Ta=0.
4.
Deathbenefits:f4(s)=-1,ooo;dg4(s)-l,_tj(s)q~_a[1"~;--tt(s)qt_x]dsH(s)=O,0_O,1f(1+i)~-'f(s)dge~n[(1+a/2)C--aD],(11b)011C=,ff(s)dg,D=fsf(s)dg.
(12)0oThenotationhereisslightlydifferentfromthatpresentedinMr.
Huffman'spaperinthatthesecondintegralinformula(12)wasdefinedthereasCT.
ThishadtheslightadvantagethatCcouldbefactoredfromtheresultingformulasbutproducesthedistinctdisadvantageoftryingtodefineTwhenC--0.
SettingT--0whenC--0hastheerroneousim-plicationthatcashflowsthatnettozeroalwayswillhavesimpleinterestaccumulationsthatalsonettozero.
ThesametypeofproblemoccurswiththeintegralthathedefinesasCM,whichforalaterusewillbedefinedasE,where1E=fs~f(s)dg.
0314ASSETSHAREMATHEMATICSCombining(7a),(7b),(lla),and(llb),weobtainGT'"_Combining(14a),(14b),and(6),wehaveF=t"com-binationsofthesevalues,thatis,F'~~"=[a(aC)++b(aD)+]--[c(AC)'~+d(AO)~-],(16a)F~'"=[c(AC)++d(AD)+1--[a(AC)'~+b(AD)~],(16b)wherea=1+i,b=--i,c=e~n(1+8/2),d=--Be~n,(AC)+=~C~,(1+i)"-',etc.
Applying(16a)and(16b)toMr.
Huffman'snumericalexample,theboundsproducedare77,032_h()itisclearthat,nears=,~V(s)>h(s)foranyc.
Theerrorsatzeroand1,however,canbepositiveornegative.
Inanyevent,theabsolutevalueofthelargerofthesetwoextremeerrorswillbeminimizediftheerrorsareequal.
Thisproducesc=iand8(0)=8(1)=1+i/2--i/&Forthischoiceofc,themaximum"interior"errorinabsolutevaluewilloccurat.
~=1-(1/6)In(i/6)(standardcalculustechnique)andwillbeequalto8(~)=-i/2+(i/6)In(i/6).
Asomewhatlengthycalculationshowsthat8(~)_Combiningequations(6)and(20)producesF~v,whichforMr.
Huffman'sexampleandn=20producesP2~=77,118.
2.
QUADRATICAPPROXIMATIONSByintroducingquadraticapproximationstoh(s),theboundsinformula(17)canbesharpenedconsiderably.
Sinceitisreasonabletodemandthattheapproximatingquadraticsh2(s)willintersecth(s)ats=0ands=1,webeginwiththegeneralformof~(s),whichish2(s)=bs(s--1)--is+1+i.
(21)Byconsideringthederivativesofh(s)andh2(s)ats=0ands=1,itissimpletoshowthat,ifi--~h'(O)andh~(l)>h'(1),whichim-pliesthath~(s)startsoutfroms=0aboveh(s)andendsupats--1frombelowh(s).
Toseethis,letf(s)=h2(s)--h(s).
Then,sinceif(s)iscontinuous,f~(0)>0impliesthatif(s)>0onandif(l)>0impliesthat316ASSETSHAREMATHEMATICSif(s)>0on(1--e,1+e),forsomee>0.
Hence,since/(0)=0andand/(1)=0,wehavef(s)=ff'(t)dt>0on(0,e),01f(s)=--ff'(t)dth(s)on(0,1)istoexpandh~a*(s)--h(s)asaTaylorseriesin~about~=0,keepingsfixed.
Thecoefficientsofthepowersof~willbepolynomialsinsthatarestrictlypositiveforsE(0,1).
Thesametechniqueappliedtoh(s)-h~"(s)willshowthath~i"(s)~es--i,thenclearly~(s)h'~*(s).
Hence,thequadraticsdefinedinequa-tions(23)havetheminimalpropertiesstated.
Otherpropertiesincludethefollowing:a)h~"*(s)istangenttoh(s)ats=1;b)h~"(s)istangenttoh(s)ats=0;c)If11s,=f(hW-h)(~)d~,s,=f(h-h'C'")(s)ds,00then188(1)8,=~+O(a'),i=1,2,(2)82>8x,and1~4(3)S~--St=~+000DISCUSSION317Thetwoinequalities(11)nowcanbewrittenintermsofC,D,andEusing(23a)and(23b),thatis,1f(1+i)l-°f(s)dg(1+i)C-~,aD+(~e*--i)E.
(24b)0From(24a)and(24b),G~~andG~incanbedefinedasinthelinearcasein(14a)and(14b),whereEt+andE7willhavetheobviousmeaning.
DevelopingF~a*andFrainasin(16a)and(16b)andapplyingtheseformulastoMr.
Huffman'sexample,thefollowingboundsaredeveloped:77,116.
85comes77,117.
57.
ItisinterestingtonotethatthevalueofF~[obtainedbyusingthelinearaveragingapproximationwas77,117.
89.
Theaccuracyofthisap-proximationisduelargelytothefactthatthemaximumerrorinh]~(s)is62/12+O(6s)andthatamore"uniform"typeofcancelingtakesplaceherethanthecancelingthattakesplacewhensimpleinterestisused.
Thisuniformityisattributabletothefactthattheaveragingtakesplaceovereachcashflowseparately(thatis,overstatedpremiumsduringpartoftheyearaveragewithunderstatedpremiumsduringanotherpartoftheyear,etc.
)ascomparedwithsimpleinterest,whichaveragestheoverstatementofsomecashflowsagainsttheunderstatementofdifferentcashflows.
Ofcourse,someoftheaccuracyofthelinearaveragingapproximationalsomaybeduetotheparticularexampleitself,inthatanequivalentamountofexactnessmaynotoccurinotherexamples.
Inapplyingtheaboveapproximationstocalendar-yearassetshares,itisrecommendedthattheapproximationsnotbemadeuntilafterthe318ASSETSHAREMATHEMATICSevaluationoftheinnerintegral.
UsingMr.
Huffman'snotationfromSectionV,11--~I'-I-C'=ff(1+i)t-'-°f(s)dgdz001I--*=ff(t+O'-'-'dzf(s)~g(28)001,=-60f[(1+i)t-"--1]f(s)dg.
Bydelayingtheapproximationuntilthelaststepin(28),oneisable.
tochoosewhetheralinearorquadraticapproximationwillbeused,therebyobtainingthemostaccuracyfortheeffortsinvolved.
Approxi-matinglinearlyinthefirststepwouldproduceaquadraticinthelaststep,whichwillinvolvethesameeffortsofcalculationasanyquadraticap-proximationbutgenerallywithonly"linearapproximation"accuracy.
Otherapplicationsoftheaboveapproximationsincludecontinuousanddiscreteinsuranceandannuitypremiumsandreserves.
Inanygivenapplicationthechoiceofapproximation,whetheritislinear,quadratic,orofhigherorder,willdependonpracticallimitationsinherentindevelopingthenecessaryfactors(C,D,E,etc.
)aswellasonconsiderationsofaccuracy.
JAMESA.
TILLEY:Theprir/cipalcontributionofMr.
Huffman'spaperisthesystematicapproachtoassetshareandmodel-officecalculations.
Ifeel,however,thattheauthorhasplacedtoomuchemphasisonthephrase"Stieltjesintegralinterpretation.
"Everyactuaryknowsthatassetfundsarecalculatedbyaccumulatingallcash-inflowand-outflowitemswithinteresttoanap-propriatepointintime.
Toperformthecalculation,theactuarymustmakeassumptionsaboutboththeamountandtheincidenceofeachcash-flowitem.
Ifthecashflowsoccuratdiscretepointsthisaccumula-tioncanbeexpressedasasum,andifthecashflowsoccurcontinuouslytheaccumulationcanbeexpressedasanintegral.
Eventhediscretecasecanberepresentedasanintegralifthe"incidencefunction"isdefinedintermsofstepfunctionswithdiscontinuitiesattheoccurrencesofcashflow.
Mr.
Huffmanfocusesontheimportanceofidentifyingthevariouscash-flowitemsandtheirpolicy-yeardistributionfunctions.
Undertheassumptionofsimpleinterestfromtheoccurrenceofcashflowtotheendofthepolicyyear,anyfinancialvariablecanbeexpressedsuccinctlyasaDISCUSSION319sumoftermsinvolvingatmostthesecondmomentsofthevariouscash-flowdistributions.
Asaminortechnicalpoint,theassumptionofcom-poundinterestwithinapolicyorcalendaryearleadstoexpressionsinvolvingallthemomentsofthecash-flowdistribution.
However,thehigher-ordermomentsdonotresultinsignificantcorrectionstotheoriginalcalculation.
Inlieuofamomentexpansion,integralsoftheform1f(1+01-°f(s)dg(s)0canbeevaluateddirectly.
However,thiswouldresultintheauthor'streatmentlosingmuchofitssimplicitywithlittlegaininaccuracy.
ItshouldbepointedoutthattheresultsofMr.
Huffman'spapercanbeobtainedbytraditionalactuarialmethods.
Inthecalculationofpolicy-yearassetfunds,onlythezeroandfirstmomentsofthepolicy-yearcash-flowdistributionappearintheequations.
Hence,theentiredistributionforaparticularcash-flowitemcanbereplacedbyasinglespikeofap-propriateheight(zeromoment)attheappropriateduration(firstmoment).
Forexample,deathsskewedslightlytowardtheendofthepolicyyearcanbelumpedtogetheratameanfractionalpolicydurationslightlyafterthemiddleofthepolicyyear.
Thisistheconventionalap-proachtopolicy-yearassetsharecalculations.
Whenitcomestocomputingcalendar-yearresults,Mr.
Huffmanshowsthatinvestmentincomeforaparticularcalendaryeardependsonthesecondmomentofthepolicy-yearcash-flowdistribution,thatis,onthedispersionofthedistribution.
Thus,whencomputingcalendar-yearresults,itisnotcorrecttheoreticallytoreplaceacontinuousdistributionofcashflowbyasinglespikeataparticularpolicy-yearduration.
Theproblemisthat,ingeneral,apolicyyearextendsfromonecalendaryearintothenext.
Theendofthecalendaryeardividesthepolicyyearintotwopieces:an"alpha"portionanda"delta"portion,borrowingstandardno-tationfromthetheoryofmortalitytableconstruction.
Thepolicy-yearcash-flowitemcanbereplacedbytwospikes,oneineachofthealphaanddeltaportionsofthepolicyyear,witheachattheproperdurationtopro-duceexactsimpleinterestforcalendaryears.
Thisistheapproachusedinmortalitytableconstructiontoensuretheproperexposuretoriskofdeath.
(AUTHOR'SREVIEWOFDISCUSSION)PEYTON.
]'.
HUFFMAN:ManythankstoMessrs.
Chouinard,Evans,Metz,Reitano,andTilleyfortheirvaluableanddiversediscussions.
320ASSETSHAREMATHEMATICSProfessorChouinard'scommentsonthedistributionofissuesprovideagoodintroductiontoaknottyproblem.
Modelingapplicationsoftenin-volvenonuniformlydistributedissues.
Indiscriminatelyassumingauniformdistributionofissuescanleadtotimingdifferencesandmis-leadingresults.
Thecontinuouslyincreasing(decreasing)salesapproachwillworkwellfornewsalesforecasts,providedthat"continuouslyincreasing"iscon-sistentwiththeexpectedsalespatternandprovidedthateachyearhasafullyear'ssales.
Difficultiesariseintryingtoadaptthisapproachtohistor-icalissuepatterns.
Thestep-functionapproachproducesintegralsthatarenotevaluatedconvenientlyonacomputer.
Thepolynomialapproachappearstobethemostpromising.
Unfortunately,thequadraticR(z),whichrequiresonlytheadditionofathirdmoment,willproducenegativevaluessomewhereontheunitintervalwhenevertheaveragedurationofthedistributionunderconsiderationisoutsidetherange[7,]].
AfourthcandidateisthepairoffunctionsR(z)=zkandr(z)=kzk-l.
Theparameterkmaybechosensothattheaveragedurationofr(z)/R(z)matchesthatofthedistributionbeingapproximatedbysettingk=D/(1-D),whereDistheaveragedurationofissue.
Theresultingintegrals,unfortunately,arealsodifficulttoevaluate.
Itap-pearstomethatanapproachtononuniformdistributionofissuesthatwillbesuitableforautomatedmodelingapplicationswillrequireanap-proximationofthecashflow(aswellasthedistributionofissues).
Ataminimum,theapproximationwouldneedtoreproducethecashflow'sfirstandsecondmoments.
Forexample,thebetafunction,B(p,q,t)=r(p+q)tp_l(1_t)p-'r(p)r(q)maybesoadaptedbysettingp=T(1--M)/(M--T2)andq=(1--T)(1--M)(M--T2).
Approximations,ofcourse,mustbetestedthoroughlybeforebeingused.
ProfessorChouinardalsodemonstratesthattraditionalnonannualmodecalendar-yearassetsharesareconsistentwithStieltjesassetshares/funds.
Mr.
EvansappliestheStieltjesmethodusingcompoundinterest,ratherthantheinterestassumptionusedinthepaper.
Theresultingformulasareusefulasacomparisonwiththoseinthepaper.
Theformulasinthepaperhavetwoadvantages.
First,thewithdrawalsneednotbedistributeduniformlyovereligiblewithdrawaldates.
Second,theinterestandinsurancecash-flowelementsareseparated.
ThelatteradvantageisparticularlyusefulformodelingapplicationswhereitisnecessaryonlytoDISCUSSION321calculatetheinsurancecashflowsandmomentsofaplan-agecellonce,eventhoughitisusedforseveralissueyears.
AsDr.
Reitanodemon-strates,thedifferencebetweencompoundandsimpleinterestissmall.
Mr.
Metzdemonstrateshowanactuarymightmodifythepaper'sformulastosuithisparticularcircumstance.
Inpractice,assetsharesgenerallyincorporatedozensofcash-flowcategories.
InapplyingtheStieltjesapproach,theactuarycanmakemodificationsmoreeasilytorecognizethecharacteristicsoftheblockofbusinessunderstudy.
Dr.
Reitano'sexpositionofStieltjesintegrationisawelcomeaddition.
TheStieljesintegralwasintroducedfirstin1894inT.
J.
Stieltjes'Re-cherchessurlesfractionscontinues.
ItwasageneralizationoftheRiemannandDarbouxintegralsandsubsequentlywasitselfgeneralizedastheLebesgue-Stieltjesintegral.
TheStieltjesintegralcarrieswithitcon-siderablestructureandiseasytoapply.
Inthesectionofhiscommentsentitled"ApproximatingCompoundInterest,"Dr.
Reitanodescribesaninteresttreatmentthatreducesthe"error"introducedbyusingtheinterestassumptioninthepaperratherthanthestandardcompoundinterestapproach,Itisextremelygratifyingtoseethemathematicalstructureofthispaperusedsoelegantly.
Inaddition,itiscomfortingtofindthattheresultingvalueofF20(77,117.
89)isclosetothevalueofF20showninTable2ofthepaper(77,136).
AsDr.
Tilleypointsout,manyoftheresultsofthispapercanbeobtainedbytraditionalactuarialmethods.
Thesemethods,however,tendtobeadhocandinformalinnatureandusuallyarebasedongeneralreasoning.
TheStieltjesintegralformalizesthestructureofthecashflowsandprovidestheactuarywithasystematicmethodofapproachinganycashflow.
Dr.
Tilleysuggeststhatthecashflowsofagivencategorywithinapolicyorcalendaryearmaybereplacedbyasingle"spike"cashflowattheappropriateduration.
Formanypurposes,thisisacceptable.
Itshouldbeborneinmind,however,thatthecashflowsactuallydonotoccuratthatmoment.
Inthecaseofcash-valuesurrenders,theaveragedurationofcashflowandtheaveragedurationofterminationarenoteventhesame.
Thekeypointtorememberisthatthefirstandsecondmomentsdonotdescribethecashflowsfully.
AusefulalternativetoDr.
Tilley'sspikeapproachistoallocateeachcash-flowcategoryCtothebeginningandendoftheyearintheproportions1--TandT.
Thisapproachiscon-venientwhennew-moneymethodsareused.
Investmenttransactionscanbelimitedtointegralpolicy(calendar)durations.
rTheobjectiveofthispaperistoshareamoregeneralandmoremathematicalapproachtoassetshares.
Thepaperisfullycompatible322ASSETSHAREMATHEMATICSwithexistingmethods.
Itprovidesabenchmarkagainstwhichapproxi-matemethodsmaybemeasured.
Themethodspresentedarethoroughlypractical.
Theyhavebeeninuseforoverfouryears,duringwhichtheyhavebeenthebasisofwellover100,000assetshares,profitstudies,andmodelsofindividuallife,health,andpensioncoverages.
Hopefully,otherswillfindtheStieltjesassetshare/fundequallyuseful.
Inadditiontothefivediscussants,IwishtothanktheactuariesoftheContinentalAssuranceCompany,especiallySamGuttermanandLindaBronstein,andmywife,MariaHuffman,fortheirhelpwiththispaper.