APPLICATIONS OF OPERATIONS-RESEARCH TECHNIQUES IN INSURANCE

Arnold F. Shapiro

ABSTRACT

A literature review Wal conducted aI part of a Itudy of application. of

operation.-relearch techniquu in inlurance. Thil paper provide. a cunory

overview of lome of the literature reviewed.

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APPUCA110NS OF 0~110NS-RESEAR.CH TECHNIQUES IN INSURANCE I

AnoId F. SbapIN

...... PeuylftDla State Ualvenlt, UDlnnity Park, PA 16102, USA

ABSTRACT. A literature review w .. eoadueted .. put of a .tud, of app1leatlou of operatlou-nMucb tecbDlqu.. ID luunDee. Tbb paper JWOYIda a CVIOI'J overview of 80me of the Ilteratun reviewed.

I. INTRODUC110N

nus paper Is based UPOD a project wbleb arose beeaUN of a eoaeerD voiced by actuarial atudeata aDd mu, praetlc:lDa actuarl.. that tbey bad a problem COD­ceptuall .... practical app1lcatlou for aperatlou ReMucb, OR, ID their dall, work, particularly app1lcatlODa tbat an uDlque to the IDIuruce bualD.... Tbb obviously Is a serious problem. Wblle 80me .. pecta of OR ma, be a .. tbetle, lbe primary thrust of ~ IUbject Is lbe deve10pIDeAt of too1l aDd tecbalquea to 801ve practical proble..... 'I'hlI focus c:u be Io.t U the area Is viewed .. a"tract or priIDarl1, academic • .

1'be purpoM of 6.. project w .. to help alleviate W. problem by providlDa a reriew of app1lcatlou of OR tecbDlquea ID lDIUruce. A major NIOUI'ce, ID W. reaud, w.. jounaalI ID operatlODl NIeU'Cb, actuarial lCieDee, luuruee, aDd related fleld.a of .......... 'I'hlI paper pva a cunory overriew of lOme of the lit­erature reviewed.

1nu. research w .. fUDded ID part by a Fut from lbe Actuarial EducatiOD ud Retearcb Fuad.

Zsome would reprd W. metbodolOlY-orieated riew of OR .. too oarrow. Jewell (1980, p. 113), for eumple, would prefer to Itraa the SJltem bui1dlDi opportUDltl .. aDd an .. for eODItruetive IDteraetioa wltblD luuruce, ratber tbaa too1l aDd tecbDlquea of OR.

Itt. o-raael. (<<b.}.I_-.d Ririe rr.-"./~/fJ. C/9ll6 by D. RIiMI ~c-,,-y.

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A. F. SHAPIRO

2. PRIVlOUSSTUDIES

n... ..... ben a DUlDber of .tucU .. 01 tIa. appUeatlo. 01 OR tec:laDlquea. IalDpl .. laelud. 1\Rbu (1972), "abo .. l ud Val.Dt. (1976), ud lAcSbetter aDd eo. (1977). na... .• tudl .. reported OD t ... nault. of .un.,. of lap eOlDpul .. lad ........ , but DOt llmlted to, .olD. lDaunra.

n.. aJao ..... ben a DUlDber of .tudles wble" ..... d.alt apedfleaUy with tbe lateraetlOD of OR t.elaDlqu.. ud tbe lDIuI'uc. ladUitry. Th... ,.Derally ..... taIteD two dlatlaet routeaa .tudl .. of OR appUcatlODa directed at u lDIu­rue. audlaee ucI .tudl.. of OR appUcatlODa la lDIunDce directed at u OR. audiace. ColDlDoaly referred to e .. lDpl .. of tIae 10l'ID8r are the .tudles of Zubay (1965, 1969)' Dna ..... (1961), ud Wade et. a.. (1970). ExalDpl .. of tIae later are fOUDd lD SultOD (1965), HalIDatad (1974), aDd J.w.ll (197').

A DUID_ of theM .tudiH reported tlaat tIaere wu DOt a peat deal tlaat ape­cUlcally Nlated OR to luuruc. altuaUou, aDdlDUy of tile appllcaUoaa alluded to bad DOt yet ben IlDplelDeDted. Put of tbe focUi of tbla .tudy la to ... Iaow the literature .... c ..... ed lD theM rep.rda.

3. GAME THEORY

GalDe tlaeory lavolv .. eOlDpetltio. or cODfUet betweea two or more declsioD makers, ud Ia co.cened witb JlftSCriblq beat .trat .... SpecUic iuuruce .ppllcatlODa laelude topla lUcia u lDIUruce ~ tal.., UDderwritlDa, IDU­.. emeDt, upeDM a11ocatioD, relDlul'uce, aDd .... lDlum deterlDlDaUoa.

3.1. lulwuce CoaauaaptlOD

wuu. .. (1960) dUcuaed tile Ute of pure .u.t ..... lD plDe tlaeory few tbe evaluatloo of lDa1Ruce cODlUlDptioD alteraatlvea. The ualyala w •• baaed oa lou la utlllty auociatecl witla tile d.eclaioD of wbetber or DOt to buy fire lDauruce. The _tloa of • worry fact. wu used to belp e.plaiD lDauraace purchases.

Furtb ... lDatabt lato the deve10pmeat of tbla model wu coatelaed ID WWlama aDd Dlcbnoo (1966) aDd aD elDplricallaveatlptloD of the model wu reported ID Net ... aDd wuu. .. (1973).

3.2. Optimal Iuuraac:. CoafUct

WbU. two-penoa, sero-sum pm .. have .. rioUi abortcomiDp la IDUY pr.cti­cal altuatlou, oWlDa primarily to the lack of total kDowl.qe aDd sero-aum COD­ditioaa, they CaD provide _ful lDodels la the proper circumatuce.. They .. t tbe atap f. MUl ... (1972), for example, wbo used expected utlllty muimla.tioD to abow tile cODfllct betweea tbe lDaurer aDd lDaured lDaofar u the optimal type of lDauraace CODtr.ct la cODcened.

3.3. Iaauruce Sal .. G.me

Brag (1966) eDvIaloDed u lDauraace sales pme played by three players: the

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APPUCAT10NS Of OPERA~RESEARCH TECHNIOUES IN INSURANCE

lIuu'e, tbe ..... t, ad tU buJ-. ".. lint ..... of tM p .. SaYOl .... coalI-tloll ... t ... tM IuUN' ad tbe ..... t aplut tM buyer, .bIle tM aut SaftlYed • CODfllet ... t .... tM ..... aDd tM ..... t.

".. _tlal fat .. of tM flNt ...... u aD .tt •• pt to muI.I •• tM ...,.taal upectatloa of • polley .... , ....... tM ...... u tM ..... Ium 1eM coet otber thaD coaa.IulOll, aDd tbe • ..,.tul .apectatloa .u tM JII'OCIuct of tile ......... ad tM ..... bIlIty of tile ..... t • p .. pric.. ".. Sa ... r ...... t COll-filet reYOIYed .bout tile JII'Oportloa of tile ..... .bleb IbouId ... allocated to com.l8IIoaa, ad .u •• ed u • t.o-perIOD cooper.tl ...... .

3.'" W.IuunDc. UDClewrtt",

Lema," (1980) .ddNued tbe pI'Oble. of IaauraDc. UDderwrltlDa wlthlD tbe coatut of • t.o-perIOD DOD-cooperatl •• p ..... t .... aD Iuunr (play_ 1) ad potatlal polleyboJden (play .. 2), .ho .... Sa eltber • baaltlly 0&' DOIIbeaJtby .t.t •• n. mlalmu criterloa, .bleb ...... ppoees mal.YOleDt .ppJlcaab, aDd tM Bay. crtterlaa, .bleb ... m .. tile appUcaDt baa adopted. flud priori .tnt., .ere coaalderecL

3.5. Euc:utlv. Ca ••

0.. of tile earlier .od .. of ..... fO&' a JII'Operty ad liability luuraDc. compaay .u tile uec:utl ...... fO&' offlc ... ad middle ....... ut augested by McCulDeu (1960).

T1ut laput. to tbe .odel lac1uded bulc decla1ou, auumptioas, aDd data .1dIe tile output. of tile .odel .ere Nport. to tbe play ... aDd aD aaalysla of tbe effect of tbelr declIlou. AIau.ptlou lacJucled lbOR tbat .ere UoWll to lbe play ... aDd tboM .bleb deIcrlbed the uvlroDlDnt aDd .ere DOt UoWll to tile play .... T1ut data lac1uded IUda lDformatloa u ecODOllilc .ctlnty, UDderwrltlDa .xperteDc. nuctuatioas, aDd cc.t of tralDIaa apDts. T.o IhDltatloas of tile ......... that DO .... tllluta .ere lavolvecl aDd rat ... huNaus did DOt play • role.

3.6. eo.t AlJocatJoa

Lemaire (19") dIIcuaecI tile appllcatloa of .... tbeory to tile JII'Oble. of aUocatlDa .......... a.OII8 tbe depart.uta of aD luuraAc. cOllipuay .beD coo~ .. atloa IeacIa to ecoaoillies of 1ICale.

Be lint abo.ed that the co.t alloc:atloa problelll .u Idutlcal to til. ftlue of a .. me wltIl trauterabl. utWtles, aDd tbeD dbcuuecl lb. attribut .. of four co.t alIoc.doG •• tbocla buecl OIl "III. tbeory. T1ut criteria advocated .... coUec­tlvely ratloDal, la tbe ...... that DO departllleDt. subaltae aDother, .oaotODlc cost., la tbat all departillut. coatribute to aD lac:reue la 810W co.t., aDd addi­tivity, la tbe ...... tbat a subdivilloD of a departllleDt does DOt affect tbe COlt aUocatlOD.

3.1. ReclprocallWtuuraDc. T .... tlel

A slmple .odel for ualyslDa a Nclprocal NluuraDce treaty .u dlIcuued

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A. F. SHAPIRO

.., Iorcb (1960) wltbID tile coateat 01 a two-penoa cooperatlv •• ame. After reviewiq bulc ulo .... relitiDI riak altuatlou &Del Ibowtaa tbat the quota Ibare treaty wW .... t reduce. tbe YUiaDce of tbe portfolio 01 .. cia compuy wbeD tb. utUltylDd_ de,.... 0Dly oa tb. free ..... rv .. &Del tb. variaDce of tbe riak dlatrl­butloa, be IOU oa to dlacuu the aature of tbe quota ceded &Del tbe free reaerv .. truafernd.

3.1. The Actuary AlaiDat Nature

BUblmUD (1976) iDveat.ated a .am. wblcb pltted the actuary .. aiDat aature lD the coat_t of .. tlmatiDI a pure pnmlulD froID sample lDformaUoD. The altu­atioa lDvolnd tb. actuary wbo cbo .. a pair of .. Umatora, aature who cboM a parametrlc famlly of dlatrlbutiOD fUDctiou &Dei Itructure fUDctiOD, &Dei a quad­ratic lou fUDctloa. Both tbe DOD-parametric c.... wbere the parametric famUy wu IIOt IuIowo, ud the parametric c .... where the parametrlc famUy wu bo,"" were cODlldered.

.. STOCHASnC DOMINANCE

Stocbaltlc domiDaDce (SD) provides a meaDS for prefereDce orderma wbeD UDcertalD altematives are lDvolved. The three orden of SD are baaed OD lDcreaa­iDlly restrictive .. ta of utUlty fUDctioaa: iDcreaaIDs. risk adverse. aDd decreulDl a""ute risk adverse. PoteDtial areu of appHcatiOD lDc1ude optimal iDaurance coveraS" aDd relDsurance.

4-1. Insurance &Del JUak SbariDI

Doberty (1977) used secoDd order SD to develop a model for iDvest.atlq tbe appropriateaeu of a deductible from the VaDtalepoiDt of an lDdividual risk-ad­verse iDaured. He later uteDded tbla approacb (Doberty. 1980) to model the PH­fereaces of both the iDaurecl aDd tbe iDsurer lDsofar u the use of deductibl.., coiDaurance. the francbiae form. &Del firat lou.

4-2. A ReiDauranc. Example

G&Ddbl et. al. (1911), diac:uued a limple reiDauranc. applicatioD wbere tbe maaqer of a Itock compuy mUit choose bet WeeD two portfolioa, ODe cODtalDlDg reiDaurance, &Dei the other DOt. The paper described lD detaU tbe characteristica of firat ... coad, aDd thlrd-order atochaatic domiDaDce. &lid d.lacUlled tbe superi­ority 01 Itocbutic domiDaDce over mean-variance. coefficieDt of variatiOD. aDd expected utiHty models.

5. ~ARPROG~NG

LiDear programmma baa 10DI heeD recognized as ODe of tbe mOlt importaDt techaiques of OR because of its venatUity and power lD resolviDa problema lDvolvlag tbe allocatioD of scarce resource.. Specific appHcatioaa lD iDaurance iDclude lDsurance purchase optimizatioD, iDaurer profita muimizatioD uui tbe boUDdiag of atop-lou premiulDl.

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APPUCAnONS OF OPERAnONS-RESEAROI TEOINIOUES IN INSURANCE

5.1. LIf. ruur .. ce Purcba'!S

A penuIal JII'Obl.m ID th ... 1. of Uf. luwuc ... th. optimal comblDattoa of ftrioua typn of m. luuruc. poUd.. aDd altenat. baveatm.nt.. Schleef (1910) Ibo.ed bow a UDur JII'OIl"ammlDa mod.l could be UMCl to belp molve tbla JII'Obl.m.

The objective of the model ... to muiml.e tbe ........ t value, adjusted for tbe maralDal tax rate, of future ca. no •• due to caab value NCOVWY, loaDs, ud otMr bav_tment.. The couualDt. of the model baclucled a budaet cODItrabat, .bleb JII'Ovlded for the JlaJlll-t of .... mlum ud lou baterat aDd recoplzed alterutlve bav_tment caab no ... a deatb beD.flt coutrabat, .blch provided for the desired level of death beDefits, ud DOD-D.ative coutrabat. OD the alteru­ttve bav .. tm_t fWld, the caab value, the loaD balaDce, the face amOUDt, ud all cIedIloa variables.

S.z. Ptooflta of Ptooperty aDd IJabUlty IDIUNn

HofflaDder aDd Drudell (1969) used a UDear JII'OIl'ammlDa model to mmmize the profit. of a property ud HabUlty iDlmer .ubject to operatioDal aDd resula­tory cODStrabat .. The purpose of tbelr model .a. to fbad tbe optimum allocation of .... t. 10 that profits could be maximized.

5.3. Stop-Lou Premium.

ODe .ay of reiDSuriDa is througb a .top-Ioll arrangemeDt, UDder whicb the reiDSurer pays that portioa of the lou ba exc ... of a .pecified amount. A con­cera, ba tbls reaard, is the maximum value for a .top-lou premium.

Taylor (1977) used a UDear PI'OFammlDa technique to help isolate the upper boUDds OD .top-lou .... mlum.. His approach w .. to ,eaerallze from u Wustra­tiOD .blch YalicIates that if there was a discrete claim distribution, the maximal .top-lou dlstributloa bas lum ... concentrated at the two extreme point. of the range.

6. GOAL PROGRAMMING

The eueatial feature of loa! programming is that it provides u opportunity to auip priorities to cODflictiq objectives ADd thea minimize. deviatioDS from those objectives. The method provides u opportunity for three types of analysis: mput requirement., liven a ,oal; relative goal aUammeDt, giveD resources; ADd a test for u optimum lOlutiOllo An importut beDefit of the model is it. simulatiOD capabilities.

6.1. lDSuruce Ageney MaulemeDt

Gleuon aDd Lilly (1977) examined ,oa! programmiDg a. u aleDey decisioD­makiDi tool m the COD text of property ud casualty iDSuraoce a,eDey decisioDS reprdiDa number of iDSurera to represeDt, co.t reductioD efforts, &Dei expaDded

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A. F. SHAPIRO

co •• ercial lID-. Rat_ tbaD cIea1 willa all poalble pall that .spt coafl'ODt lUCIa u apDcy, tHy ll.lt tIae ... l".. to CO •• OD pia.

6.2. Profit Maxl.lsaUOD lD Property .Dd Llabllltl .. ColDpaDl ..

lClock aDd Lee (1974) .rped tbat • llDHr "".IDID" .pproacb to JII'Oflt­.bUlty, .V.D U It .. .xt.Dded to lDclud. otber utWty fUDctlOD cbar.ct.rlatlca, wbU. tbeoreUcally CorNct, .. IlDpr.ctlcal. They .... ted IOAI "".IDID" •• u alt.rutl ....

The .utbon .pplled • aoal-"".IDID" lDocIe! to tbe H-D .ocI.l (Hofflud_ • DraDd.ll, 1969), witb 10.. lDoclUlc.tlou, to truaforlD It froID • llD.u JII'O­...... forlD.t. They klntUled IIa aoal cODltr.lDta wbleb ue, lD decre ....... prloritylllquidlty, fIDucial .treqtb, JII'Oflt, dlatrlbutlOD of ... t portfollo, dlatri­butlOD of )ll"elDlulD portfollo, aDd UN of c.padty. The ayat.1D coutr.lDt. lDYOlve pollcybold ... • .urplus, c.pltal, )ll"elDlulD' writteD .. • lDulUple of pollcybolden' .urplus, total .... t .... lDultlple of )ll"elDlum volulDe, )ll"elDlulD baluc ..... JII'OportlOD of )ll"elDlulD volulDe, dlatributloa of .... ta, JII'OporllOD of lDiacel1ueous llabUlUea, JII'OpcwtlOD of lou NMrV", JII'OportlOD of UDeanecl )ll"elDlum .... " ... total ..eta, aDd equality of ..eta aDd llabUltia plus Det wortla.

ODe of tbe obvious que.tlou to be NIOlved lDaofu .. aoal "".IDID" .. cODcened, .. tIae allDllarlty bet WeeD It aDd llDeu "".IDID". Ia tb .. regard, DnDden (1977) delDODItr.ted tbat tbe loa! Jll'0II'.1D1Dm, lDociel of Klock aDd Lee pr'Ocluced multi whicb were equivaleDt to tbe H-D lDocieL The bula of tbe co.­parlIOD w ... fonlulatlOD of aoala lDto llquidlty, .t.bWty, aDd JII'Ofit.

6.3. Capital Budsetm, lD Property L1abWty ColDput ..

Lawruce aDd Reev .. (1982) forlDulated lIae c.pltal budsetm, lD • JII'Operty llabUlty luuruce cOlDpaDy". 0-1 aoal "".IDIDm, JII'OblelD lDvolvlDs lDultlple aoala aDd lDdlvlalble .ctlvlt.... They viewed the .ocIe1 .. bem, 1D0re repr __ -t.ttve of reality aiDce It w .. DOt cODltr.lDed by • aiDsl. objective or cODtiDuous .ctlvltl ... wblle bem, US)' for declaioD lDoen to UDderatud aDd UN.

For the purpose of exa.ples, .. vu JII'OJect. we.. cODIiderecl wblcb covel' three time periocla aDd • Dumber of NIioDI of oper.tlou. Tbe cODltralDt. of tbe model lDcbadecl MY_ .. ta of aoal coaatr.lDta conespoadiDs to the objectives. AdditioDally, theN were coutralDta to reflect lDdlvlalbUlty, aiDsle periocl _ture, aDd ladepad_ce UIIl/or clepeDduce of tbe JII'OJecta.

7. QUADRAnC PROGRAMMING

Quadr.tlc "".lDmlDs .. cODcel'Ded with optilDblDs • quadr.tlc objective fUDcta subject to llD .... coaatralDt.. Are .. of appllc.tloD iDelude portfollo .... -lyala, from both a fIDucial &Del lDauruce perspective, aDd lDterutioDal rewu­r&Dce.

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"ppUCAnONS OF OPERAnON5-RESEAROI TECHNIOUESIN INSURANCE

7.1. '-tfollo AaalyalI

ODe of tbe IDOit COlD moD .... of quadratic JII'OFaIDIDlDa II Ia tile NIOlutlOD of quHtiou "lat" to pordoll .. n. oripa of lbll approacb w .. tile w.k of Markowit. (1952), wbo ..... t .. uslaa JWObabUltJ .. tllDat .. of future MCUritJ ,.., .... c. to cleYe10p aD .fllcleat .. t of poetfo1lOl_ wblcb could tbea be matched wilb aD laV .. t.'1 ,"feNDc ••

Sbarpe (1963) dlacuued aD .fllcleDt NIOlutlOD of tb. pi'Obl.ID wbleb peatlJ ailDpllfled tile uaalJlla. .

7.2. Pwtfo1lo Tb.., aad NOD-W. IuUNl'I

Markle aDd Hoff1aader (1976) applied tb. Markowit. lDodel to DOD-Uf. lDIuI'­en uad_ tb. UlUlDptlOD that tb. loa! II to IDuilDlze "tuna for PVeD I.vell of rlIk IUbJ.ct to ... tltutloaal IOlveDCJ coutralat.. n. objective fUDctlOD II tile uadenrritlDa aad IaV .. tlDeDt pi'Oflt ..

For tbla puI'pOM, tb. overall "tun for a PVeD llDe w .. UlUlDed to be aD av.... "tun f. that llDe oJ- tile period of lav .. tlptioe, .. WeN tile expected "tuna fl'OlD aecuritiea.

7.3.lIltenatiooal a .... uraDce

Louberp (1983) uaed quadratic JII'OIrammiDI to mod.l latenatioaal "Iaau­raDce operatlou witbJD a meaa-varluac. cODtext. III additiOD to adclreaiDI ItaD­cIard UlUmeDt ....... t IUcb aD applicatioD for aD laaurer Ia leDeral, be dealt with questiou lUa .. tile DODDormalltJ of ucbaap rates aDd tbe distributlOD aDd .. tlmatloD of tile "tura of tile .xc ... of 1011 treatJ.

7.'- Llf.luuraDc. CompoJ IIlvestmeDb

Tb. COlD mOD approacb to lb. uaalyais of altenatlv. iDvestmeDt opportUDlties Is baled OD quadratic JII'OIralDmlDa mod.1I because tbe objective fUDctlOD iDvolves the variuc. of dlstributlODl, a quadratic.

BI'oclt (1983) abowed bow to develop a llD.ar JII'OIralDmiDI alterutive bued OD tile meaD abaolute deviatloD of nturu, ratb. tbaa tbe variaace of "turu. Th. objective w.. to mlDimize risk, .. mea.ured by meaD absolute deviatiOD, subject to iDtra-temporal uad iDter-temporal coastraiDt.. SiDce tb. objective fUDctiOD ... DoDliDear, ioal PI'OII'ammiDI ... resorted to for aD actuallOlutioD.

huce-c:oastraiDed JII'OIrammiDI bas alao heeD used to adclrea tbls pro~ lem. However, the UN of a cbaDce-c:oastraiDed JII'OIrammiDI model to lavesti­pte tbe risks uaoclated witb iDve.tmeDt uad UDcierwritiDI nturas bas heeD cri­ticiaed by KabaDe (1977), wbo arped that a full quadratic PI'OII'alDlDiDlmodells 1 ... restrictive uad pi'Ovldes simUar result., PVeD tbe lime UlUmptioas.

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A. F. SHAPIRO

I. DYNAMIC PROGRAMMING

o,ulllic propalilmlac •• t.Dds tbe .lacl .... t... uauillptioa 01 1IDeu JII'O­aralllllllac to a lIlultistap .DriI'oDIIl.Dt. aDd is CODcenaed wilb the overall ."ec­U ..... 01 MClUeDtlal, IDtenlated decisloaa over lb. planD'. borlsoa. ANu 01 appllcaUoa IDclud. ut. lDauraac. JWOII'alllllllac. rlak 1Il ...... IIl.Dt. aad ao-c:lailll 1I11l1ts ID autolllobil. lDauraac ••

1.1. o,ulllic W.lDauraDc. Propalllllllac

ODe 01 lb. early attelilpt. to brlas dyaamlc JWOII'ammlac to bee oa lbe probl.m 01 cbooal. aa opUmal ut. lDauraDc. propalll wu lb. .tudy bJ Bellb (1964). 11ae .tudy wu aal .. bJ CUft'eDt .taadarda, but thelDternlaUoaahlp 01 the variabl .. is u releYallt today u whu they were flnt doae.

8.2. Rlak ........ lIleDt

LWy aacI GlealOD (1977) diac:uaaed the UN 01 dyaamlc propammlac u aa aid to lb. rlak aaaaapr ID equipmeat Nplacemeat decialoaa wheN the pI is to mla­lmls. poteatlalloaea.

The problem is that the riaIt maDapr must balaace the replacemeDt of the equipmeDt whUe It .till has ecODomic value agalDst poteDtial direct aDd lDdirect losses iDvolviDa business iIlterruptioD lou, boclUy IDJury aacl property dam ... lou, replacemeat cost of dam .. e to maaufactUl'el', worken' loa, aacl loss of CUfteDt boiler.

Backwud dyaamic prosrammUta wu used to resolve the problelll, where the at .. _ were the yean of the pJaDDiDg horisOD aacl lbe atat. witbiD the ataps were the yean of purc:hue 01 the boiler.

8.3. Optimal No-CIalm Umits

The problem commOD to all motOl'iats is the optimal choice of wheD to make a claim. MotOl'iats have aa iDformal aet of rules which they follow. UDder which claims ue made if the uteDt of the aeeldeat exeeeda some miDlmal amOUDt, aDd DO claim is mad. othenriae. Thls is a .tudud type of problem ill Markov ChaiDs, aiDe. It lDterntlatea lDauraac. cl.alma duriDa a PVeD perlod aacl lDauraace premi­ums d1lr'iDa au'-equeat perlocla.

Manla-Lof (1973) diac:usaed luuraaee wilb a boDUS systelll UDder which aaaual prelilium. depeDd oa the boDUS cla.. to which the iDsured beloap. An optimal declaiOD rule wu developed buecl upoa aa adapted policy iteratloa which reduced the at •• of the system which Deeded to be solved at each Iteratioa.

Similar probleill. were resolved by HaehllDa YOD Laaseaauer (1974) aacl aa.t­iDp (1976) who dealt with determiDlDg whether to file a claim after aa aecldeDt wheD there wu mON thaa ODe possible aceldeDt duriaa a siVeD policy yeu.

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~(,A nONS OF 0PEItAn0N5-IlESUItCH TECHNIQUES IN IN5UltANCE

t. INYEHTORY MODEL

....... tOl'J mod_ ... tM optbaal lIaluc. ...tw.. tbe coet of Iaoldlal .ftatOl"J ucI tIae coet of procurIDa It. SpecUlc luuruc. appllcatlou lDclud. the aaatat ...... c. of c .... NMrY .. aDd th. JII'OCUNm.at of luuruce COYerap.

,.llMUnr Cub R..."..

Wblt. aDd NoniU (1965) formulated tb. cODtrol of cub reserv .. of U ......,. .. u lDveatory coatrol probl.m WbeN .tor ... aDd peaa1ty co.t. aN w.U defiDed. The optimum valu. for - the ,Ibort-term .ffective buIt baluc. w .. rMOlved ..... dJUmlc JII'OFalDlDla&.

9.Z. OptlmallDauruc. Cover ...

SlDltb (1968) .avWOHCl the optimal lDIuruce cover ... la tIae cODtat of u optllDal lavatory Itocbp UDder UDcertalaty. SpecUlcaUy. U u lDIurable 1011 (delDaDd) 11 .Jtceeded by tIae luuruc. covera .. (lav.atory level). uceulve lDIu­ruc. co.t (lavatory boletln, co.t) 11 lacurred. Coavenely. U luurable loa 11 ..... ter thaD luuruce co ....... UDNCoverable louea IDUit be ablorbecl by the iasurecL The probl.lD. thea, becom .. ODe of cbooIiq OptilDal luuruc. (lavea­tory) levell.

10. NETWORK MODELS

The .taadard appllcatlou of aetwork modeb for the adlDiDiatratloa aDd COM­diDatiOD of proapect. aDd Ol'IaaisaUoaa were euily traaaferable to the iDsuruce eaviroalDeat. .. were the plannin, aDd CODtrol tecbDiques of PERT aDd CPM. SpecUic .. lDplea of appllcatlOD aN cub now lDaDqelDeDt aDd buIiDeu later­ruptiOD dowaUID ••

10.1. Cub Flow ........ mat

NaDAlelDeat of a property-cuualty lDauruce cOlDpaDy baa two prilDary res­poDlibWUea. The flnt respoulbWty Nwlv .. about the selectJoa of aD aPJll'Opri­ate COlDpoaltJoa for the luuruc. portfolio. CouideratiODlla tbla area laclude which u.a.. to iDauN, the alDout of cover .. e to be provided la each lIDe, aDd lDuk.tla& coaatderatiODl. The other respoDlfbWty revolves about tile lDIurer'. investlDeDt portfolio. Here, coaalderatiODl iDclucle the proper IDb of securitia, appropriate lDaturity data, the Umit OD iDvestmeat iD aDy particular security, aDd 10 OD.

Crum aDd Nye (1981) appUeda aetwork model to tbe problem of maDaM the overaU cubflow of a medium-abe 1Daurer. The thrust w .. to coordiDate the iDluraDce uaderwritiDa operatiODl aDd the iDve.tmat portfoUo punuiu.

10.Z. BUIlD ... lDterruptioD DowDUme

Close (198Z) dbcuued tile appllcaUoa of PERT to the reductioD of dowatime due to buIlD ... iDterruptiOD. The ~tem cliIcuaed ... appUed to the altuatlOD

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A. F. SHAPIRO

wben a fIN loa occunoed, aDd a n.twork wa. liv.n abowiDa tb •• t .... betweo th. fIN loaa aDd the ... umptlon of tb. former l.v.l of productlOD.

11. QUEUING MODELS

The numerous appllcatlou of queuiDa tbeory uaed to molv. waitiDa liD. probl.m. of buaiDesa are dlNctly applicabl. to tb. iDauranc. ladustry. AD app1l­catlOD, bowever, wbicb la unique to iDaurance ha. to do witb tbe ruiD of aD iDau­ruc. ut.rprlae.

11.1.lDaurer RuiD as a Qu.ulq Problem

UDder a proper aceDario, tb. 101veDCY of u iDaurer may be viewed a. a queu­iDa problem, wbere tbe probabillty of not beiDa ruiDed by lOme time t, u(U,t), la .... tla11y th. same probabWty a. that iDvolvlDa a cuatomer waitiDa 1 ... thaD lOme time U, liven that th. cuatomer 10iDed tbe queue t periocla after tbe aerver was free. A dlacuIaiOD of tbla aspeet la liven by Seal (1978, Chapter 2).

12. MARKOV PROCESS

A Markov process provide. a dynamic sy.tem under which oDly the immediate pa.t la reIevut to the prediCtiOD of future behavior. It. applicatioD to life table. la u obvious ODe.

12.1. WorkiDa Life Table.

SiDc. workiDa life tables evolve from the dynamic. of labor force partictpa­tioD, Roem (1977) advocated that such table. be produced ualq the theory of cODtiDuoua-time Markov chaiDa. The traasitioas iD this caae are due to death, acceuiOD to the labor foree, ud aeparatioD from the labor force. His discussioD of u applicatioD of such a modella baaed OD a previous .tudy by him aDd Fong (1976), whicb cODtaiD tbe formal detaill of the model.

Similarly, Braun (1978) empha.ized stochastic .table population theory wbere tbe fore .. of fertillty aDd mortality depend OD age, parity, and place of resi­dence.

13. SIMULAnON

Because of the easy accesa to computers, the use of simulation within iDau­rance and related ladustries bu become commoDplace. Parallel1iDg this applica­tiOD of computers and simulatioD, ha. beeD the Deed for the development of models whicb adequately eDcompass particulars of 'pecific are ... itbiD the ladustry. A number of tbese models are discussed by Pitacco elsewbere in tbis volume. RepreseDtative ·eample. of areas of applicatioD include variable life insurance, merit-ratiDa, and reinsuraDce.

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APPLlCAT10NS OF OPERAT1ONS-RESEARCH TECHNIQUES IN INSURANCE

lJ.l.lqulty I.IaIbd L1f. ~ .. c. PoUdee

....... &ad Scb.ut. (1979) UMcIHuJatloa to apt .. rI8k-ndudaa iIwnt­m.t atratep .. UIOClated wttll equlty lIaked HI. luuraace poIlclea. TbeN an Impllcatlou r. til. d.velopmat of a BayesiaD approaeb r. flDdlaa _1 .... t .. of the mod.l'a parameten. .

13.2. Merit-RatlDa PIau

Ferreira (197.) modeled a driver'a .... ua1 premium .. a MuItov proc_ &ad ued a compowad PoluoD accldeDt model &ad WODte Carlo aimulatloD to compute 50 yeu premium blatories lD order to lDv .. ttaat. vuloua merit-ntiDa piau.

13.3. IuuNr &ad RelDauNr IDteractioo

GaUt. aDd BroWD (1981) dIacuued tbe qualitative _tUN of a Hulat_ model r. lDauraDce aDd reluuruce operatloaa. Altho. the relatloublpa .... DOt apeclflcall)' deflDed, importaDt overall coulderatlODa .... dellD .. ted. The bulc compoDeDta of the mod.l were the aurplua, capital, UHal'Decl premium reterYe, aDd tb. lou reserve ..

The mod.l uaed a Oo.~veD .,.Item .bicb fbed the variables at lOme stutlDa polDt, &ad the 00. of variabl.. therefrom, lD order to determiDe the atate of the lD1tlal .ariables at lOme later time.

14. OR SYSTEMS AND MODELS

OR authorities vie. the -.,..tema aPJWOAeb-, .hieb coc.dlDat .. overall rela­tlouhlpa aDd lDterdiacipllDary t.ama, .. the rWldameDtal thrust or OR. Repre­"Dtative appUcatlou lDclude tbe an.. of populatlOD p1UDtDI, the lDauruce lDduatry aDd .orkmeD'a compeuatioa.

Rebake (1970) dlacuued the role of model build'Di lD populatloo plaDD'Di lD UDderdeveloped COUDtri... The problem ... that there ... a coafUct bet.eeD DatiODal aDd family loala, sparee releVUlt lDfor .... tloa, aDd HlDitecl resourc ... WlthlD thla frame.ork, tbe role of OR ... to ualyae the declaioD proc ... aacl orpDise profeuioulactivltles lD thla area.

14.2. The lDaurucelDduatry

PeDtlkalDeD (1983) dlacuuecl a leDeral-purpoae model for atudylDa the 101-veDey coDditioD of a broad rUle of lDauren. The model lDclud .. provlalODI for Itate variables aDd parametera, auch .. portfoUo mb, raerv.., ud .... ta, eXOILeDoua factora, auch .. builD ... eycles aDd lDflatlOD, aDd builD ... atratealea, such .. reteDtloD levels, lDveltmeDt poUey aDd sal .. effort. Ala importat attri­bute of tbe paper la that it cllac:uued maD, faceta 01 the model .hieb requlre coaaideratloa.

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A. F. SIIAPIRO

1".3. Workm.D'1 CompeuatiOD luuruc.

J.w.n .1. al. 097'" dilcUlHd the multidilclpllDary Datun of a com .... h.D­liv. JII'OJec:t lDvolviDa workmu'l compe .... tiOD lDIuruc.. Each pha .. of the pro­J.ct II dlIcuued, u .... the lDterrelatioDlhi .. of the pba .... Th. Itudy .mbodl .. the lpirit of model build1n, which OR authorlti .. It rna.

15. COMMENT

ThlI articl. provides only a cunory overview of the literatun pertainma to appllcatiou of OR techDiquei in lDIuraDc.. NODethel..., it II hoped that it lti­mulates dlacuulon aDd provides d1rec:tiOD aDd lDIlaht into further mearch in thla area. To the utut it hal met these criteria, it wW have aerved itl purpoae.

16. BIBUOGRAPHY

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