Logistic Regression for Insured Mortality Experience Studies. Zhiwei Zhu, 1 Zhi Li 2

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1 Logstc Rgrsson for Insurd Mortalty Exprnc Studs Zhw Zhu, Zh L 2 Prsntd at th Lvng to 00 Symposum Orlando, Fla. January 8 0, 204 Copyrght 204 by th Socty of Actuars. All rghts rsrvd by th Socty of Actuars. Prmsson s grantd to mak brf xcrpts for a publshd rvw. Prmsson s also grantd to mak lmtd numbrs of cops of tms n ths monograph for prsonal, ntrnal, classroom or othr nstructonal us, on condton that th forgong copyrght notc s usd so as to gv rasonabl notc of th Socty s copyrght. Ths consnt for fr lmtd copyng wthout pror consnt of th Socty dos not xtnd to makng cops for gnral dstrbuton, for advrtsng or promotonal purposs, for ncluson n nw collctv works or for rsal. Zhw Zhu, Ph.D., s vc prsdnt of rsk modlng and analytcs at SCOR Global Lf Amrcas, zzhu@scor.com. 2 Zh L, Ph.D., ASA, CERA, s assocat drctor of fnancal rsk modlng at SCOR Global Lf Amrcas, zl@scor.com.

2 ABSTRACT Insurd populaton mortalty stmaton s ssntal to (r)nsurrs dvlopng lablty xpctatons and mantanng rurd solvncy captal. In practc, nsurd mortalty masurmnt nds to dal wth a broad rang of data and analytcal challngs. In ths papr, w ntroduc a logstc rgrsson-basd modlng approach for analyzng th U.S. nsurd mortalty xprnc, ncludng at advancd ags whr lss crdbl xprnc data ar avalabl. As a valdaton, w crat a vrson of ndustry basc xprnc tabls basd on th modl-stmatd mortalty and compar thm to standard ndustry xprnc tabls producd by th Socty of Actuars (SOA). Th concluson s that proprly dsgnd logstc modlng procsss can mor ffcntly utlz avalabl data to dlvr solutons for multpl nds, ncludng: a) tstng mortalty drvrs statstcal sgnfcancs n xplanng mortalty varatons; b) stmatng normalzd mortalty slops and mortalty dffrntals such as how mortalty ncrass by duraton or vars btwn undrwrtng classs whl product and attand-ag dstrbutons ar controlld; and c) addrssng analytcal challngs such as xtrapolatng for ultmat mortalty, smoothng btwn slct and ultmat stmatons, and constructng multdmnsonal basc xprnc tabls.. INTRODUCTION Th followng thr aspcts ar ually mportant n lf nsuranc ndustry mortalty studs. a) Mortalty trnd: how mortalty mprovs or dtrorats ovr tm b) Mortalty slop: how mortalty ncrass by ag or duraton c) Mortalty dffrntal: how mortalty, mortalty trnd and/or mortalty slop vary btwn nsurd sgmnts such as mals vs. fmals or prfrrd class vs. rsdual standard class Aspcts a and b ar rlatd but not duplcatv. Although th lf nsuranc ndustry collcts an normous amount of data, th collctons usually do not consstntly covr suffcnt tm prods for crdbl trnd analyss. Insurd mortalty trnds ar oftn approxmatd basd on gnral populaton mortalty-trnd studs. Ths dpndncy on gnral populaton studs for nsurd mortalty-trnd undrstandng s lkly to last at last untl th lf nsuranc ndustry stablshs an aduat nsurd-xprnc data rpostory, smlar to th Human Mortalty Databas for th gnral populaton, to support ts own comprhnsv and tru-xprnc data-drvn studs. As to undrstandng nsurd mortalty slops and mortalty dffrntals, thr ar nsurdspcfc challngs unlkly to b addrssd wth th gnral populaton data: dynamc nsurd sgmntaton, hgh mortalty dsparty among th sgmnts and th nd for multdmnsonal normalzaton. Frst, compard to th gnral populaton, th U.S. nsurd populaton s hghly unstabl. Insurrs constantly ntat rsk slcton fforts through mprovng undrwrtng, adjustng prcng stratgs, xpandng markts and dvlopng nw products, whch not only 2

3 attracts varous lvls of rsks but also causs currnt nsurds antslcton ractons such as polcy lapsaton and convrson. Ths rsk slcton and antslcton actvts form and rshap numrous nsurd cohorts. In addton to ag, nsurd cohorts may b dfnd by smokrs vs. nonsmokrs, prfrrd class vs. standard class, prmannt polcyholdrs vs. trm polcyholdrs, groups vs. ndvduals, or a mx of ths charactrstcs that usually ar nthr rlvant to nor capturd n any gnral populaton data. Scond, by dsgn, mortalty and mortalty-trnd pattrns dffr sgnfcantly among ths nsurd sgmnts. Compans oftn compt on proprly prcng by sgmnt. Thrd, th nsurd sgmnts oftn hav rlatvly small szs, short hstors and multdmnsonal charactrstcs (.g., mal prfrrd smokrs). Analyzng th tru valus of ach of th charactrstcs n dffrntatng mortalty and trnd rurs controllng th dstrbutons by th othrs (normalzaton). As th dmand for rsk managmnt and comptton for markt shar ncrass, th dmand for mor thorough undrstandng of nsurd mortalty ncrass. In ths papr, w apply a logstc rgrsson-basd modlng approach to analyz th U.S. nsurd mortalty xprnc basd on a larg amount of data collctd by a major consultng frm and a global rnsurr. Th adapton of th multpl-varabl modlng approach (Advancd Analytcs), th avalablty of a larg amount of polcy data (Bg Data) and th us of modrn computng tchnology provd many advantags ovr th convntonal nsurd mortalty study mthods, ncludng Emprcal data-drvn modls wth multpl xplanatory varabls (drvrs) Projcton of ultmat and advancd-ag mortalty by combnng past xprnc and modl xtrapolaton Smooth brdgng of slct and ultmat mortalty wth th modls lnk functons Drvaton of normalzd mortalty slops and dffrntals btwn polcy sgmnts wth modl coffcnts, whch s dffcult to do wth convntonal mthods Vrfcaton of rlablty of th ovrall study wth modl ft statstcs and not solly rlyng on th numbr of clams avalabl for crdblty vrfcaton Constructon of multdmnsonal ndustry xprnc tabls by usng th modls as prdctv modls, ovrcomng som of th waknsss of pvot tabls In summary, a logstc rgrsson modlng approach allows us of lss but mor rlvant data to addrss multpl challngs n uantfyng nsurd mortalty. Exampls and ntrprtatons of fndngs from our modlng procss ar rprsntd n scton 3. Th rst of th papr s organzd as follows: Scton 2 summarzs th data usd for ths study; scton 3 dscrbs logstc modls and how to modl mortalty slops and dffrntals; scton 4 rvws th ssu of dath cnsorshp by polcy lapsaton and a logstc rgrsson-basd soluton; and scton 5 dscusss th lmtatons and possbl nhancmnts n usng logstc rgrsson modls for ndustry xprnc studs. 3

4 2. THE DATA SOURCES Th Human Mortalty Databas (HMD) s our sourc for U.S. gnral populaton mortalty xprnc. At th tm of our study, th databas covrd 933 to 200. Th data s manly usd for comparson purpos, not for nsurd mortalty approxmaton. Th nsurd xprnc data usd n ths study wr collctd by a major consultng company and a global rnsurr. Th data fl conssts of xprncs from mor than 60 nsurrs wth xposur from 2000 to Th ncludd polcs wr ssud as arly as 92. A total of 74 mllon polcy xposur yars and.6 mllon dath clams ar avalabl for study. Actuars and analysts tradtonally us as much xprnc data as thy can gt to prform nsurd mortalty analyss, whthr th purpos s to valuat n-forc blocks (polcs ssud n th past) or to prc nw busnss. Ths smpl stratgy can b a doubl-dgd sword. Whl ncrasd volum of data may bnft th crdblty of analyss, t may also dmnsh th accuracy f lss rlvant data ar ncludd. Studs (.g., Vaupl 204) found that n som dvlopd countrs, lf xpctancy xpands about 2.5 yars pr dcad, whch mpls that usng polcs ssud 50 yars ago to stmat futur polcyholdrs mortalty can rsult n sgnfcant bas. Instad of trustng or subjctvly adjustng basd analyss fndng, w xplord usng lss but mor rlvant data to modl and to xtrapolat stmatons. It achvd satsfactory modl prformanc. For llustraton purpos, w lctd to apply th followng data-slcton crtra for stmatng mortalty that s mor rlvant to prcng futur fully undrwrttn polcs. Othr fltrs may b appld for dffrnt purposs. Polcs ssud snc 950 Fac amount $50,000 Th followng tabl summarzs th total and th fltrd study data. 4

5 Tabl 2.. Summary of th nsurd data Total data Slctd data Sx Attand Clam Exposd Clam Exposd /( ) ag count count count count /( ) 00 22,37 5,99, ,758, ,096 3,94, ,425, ,926 5,493, ,33, ,442 8,49, ,240 5,86, ,636 0,403, ,467 6,266, ,57,203, ,888 6,323, ,935 0,672, ,206 5,405, ,972 9,073, ,947 3,975, ,389 6,87, ,54 2,408, ,066 4,673, ,668,204, ,894 3,55, , , ,868 3,6, ,552 43, hgh 299,642 4,887, ,55 482, ,525 6,303, ,827, ,05 3,304, ,428, ,75 5,964, ,354 3,463, ,204 0,66, ,72 6,587, ,4 3,884, ,44 8,933, ,948 6,060, ,353 9,92, ,977 6,22, ,003 9,305, ,632 4,89, ,84 7,668, ,629 2,072, ,423 5,396, ,983 8,450, ,849 3,033, ,39 5,993, ,866,584, ,537 4,640, , , hgh 490,630 6,539, , , Nots: Th n th tabl s dfnd as th numbr of daths dvdd by xposur. In ths papr, mortalty, mortalty rat, dath probablty and dath rat all rfr to th sam, unlss spcfd othrws. Fmal Mal Th probablty of dath and th odds of dath / ( ) ar approxmatly ual for narly all ag groups bcaus. Ths mpls that many of th odds rato-basd ntrprtatons of th logstc modl can b rasonably ntrprtd n th trms of probablty ratos or mortalty dffrntals. S appndx A. Th followng two charts compar th fv-yar total mortalty of th gnral and th nsurd xprncs basd on th data w hav. Agan, Th gnral populaton data sourc s th Human Mortalty Databas. Th nsurd populaton data sourc s our total study data. Th nsurd data ar also splt nto two xclusv subgroups: prmannt and trm product. Ths mortalty rats ar drvd wthout normalzaton by any dstrbutons such as duraton, ssu yar and undrwrtng class. Accordng to th charts, for any gvn ag group, prmannt polcyholdrs hav about 50 prcnt hghr mortalty than trm 5

6 Mortalty Mortalty polcyholdrs. Latr, th dffrncs aftr normalzng by nn study varabls wll b uantfd wth our modls Chart 2.. Mal mortalty ( total) US Insurd Prm Trm Attand ag Chart 2.2. Fmal mortalty ( total) US 0.02 Insurd 0.05 Prm Trm Attand ag 3. MODELING INSURED MORTALITY WITH LOGISTIC MODELS Rsarchrs hav long bn usng statstcal modls to study gnral populaton mortalty. Snc th ntroducton of Gomprtz Law of Mortalty (825), th ffort of modlng th human mortalty trajctory by ag has only acclratd. Thatchr (999) provdd an xcllnt dscrpton and comparson of four mortalty-by-ag modls. Wth som smplfcatons n rducng th numbr of paramtrs and usng forc of mortalty as th dpndnt varabl, th four modls ar: (.) Gomprtz (825) μ α xp (β x) (.2) Wbull (95) μ = α x β (.3) Hlgman and Pollard (980) μ α β + β x 2 (.4) Kannsto (994) μ = α xp (β x) +α xp (β x) Of th four modls, only th Kannsto modl assums that forc of mortalty has a fnt asymptot. Thatchr s concluson was: Whn ths four modls ar fttd to actual (gnral populaton) data, thy ar all rlatvly clos to th data at ags whr most of th daths ar concntratd, and hnc rlatvly clos to ach othr. It s not surprsng h also confrmd wth varous populaton data (Thatchr, Kannsto and Vaupl 998; Thatchr 999) that th Kannsto modl fts and approxmats old-ag mortalty th bst. W modfy th Kannsto modl n two ways for nsurd mortalty study: us mortalty rat nstad of forc of mortalty μ as th dpndnt varabl and nclud not only ag but also many othr nsurd xplanatory varabls. Our logstc mortalty modl has a gnral form of ( x x... x * x x * x...) 2 2,2 2,3 3 (3.) or ( x 2 x2...,2x * x2,3x * x3...) 6

7 (3.a) ln( ) * x, j, j * x * x j... whr s probablty of dath n an xposur yar, gvn a polcyholdr survvd to th bgnnng of th yar x ar xplanatory varabls (.g., ag, sx, duraton, product) s th ntrcpt, to b stmatd wth xprnc data and maxmum lklhood mthod ar coffcnts of th xplanatory varabls, to b stmatd wth xprnc data and maxmumlklhood mthod (s appndx B). To dstngush from th logstc forc-of-mortalty modl or logstc µ modl (.4) by Kannsto (994) and Thatchr (999), lt us call our modl (3.) logstc modl. Accordng to Thatchr s llustraton, a smplfd Hlgman and Pollard modl (.3) wth only on xplanatory varabl, ag, s a spcal form of our logstc modl. A logstc modl has many advantags for nsurd xprnc studs. It modls mortalty that s drctly usd n busnss opraton and rsk managmnt. It can b flxbly confgurd for stmatng mortalty lvls, slops and dffrntals that ar ky mtrcs usd n busnss practcs (s attachmnt A). It prforms many othr analytcal functons such as normalzaton, hypothss tst, rsk scorng and xprnc tabl constructon that ar not smply to do wth convntonal xprnc study mthods (Harrll 200). It can b dvlopd wth wdly avalabl commrcal softwar systms such as SAS, SPSS and R. In addton to th dpndnt varabl, nn obsrvabl xplanatory varabls ar slctd as potntal ndpndnt varabls for our modl dvlopmnt. Gndr: mal and fmal Duraton: as contnuous varabl Issu ag (last brth): through 99 as contnuous varabl Smokr status: smokr, nonsmokr, unknown Product: prmannt, trm Undrwrtng class: prfrrd, rsdual standard, aggrgat (on class) Exposur yar: 2000 through 2009 as contnuous varabl Undrwrtng ra: Four ras dfnd by ssu yar to rflct ky undrwrtng volutons such as smokr and prfrrd ratngs Fac catgory: $50,000 $99,000; $00,000 $499,000; $500,000+ (nflaton adjustd) 7

8 Ths varabls ar slctd bcaus thy hav th last mssng valus and ar th most fruntly usd for prcng dcsons, undrwrtng adjustmnts and marktng stratgs. Unlk n gnral populaton mortalty studs, w chos polcy ssu ag and polcy duraton nstad of attand ag and calndar yar to rprsnt ag and tm. Th chosn par has bttr rflcton of nsurd charactrstcs and s th ky dmnson of nsurd mortalty tabls. Rcall that our logstc modl has th gnral form of (3.a) ln( ) * x j * x * x Th rght hand sd of th modl has thr componnts: ) th ntrcpt th man ffct componnt that s a wghtd sum of ndvdual xplanatory varabls, and 3) th ntracton componnt that s a wghtd sum of products of two or mor xplanatory varabls. Whn ntracton trms ar omttd, modls (3.) or (3.a) bcom (3.2) and (3.2a) ( x x...) ( x x...) ln( ) In ths modl, th modl coffcnts and * x can smply b transformd as stmatons for mortalty lvl, slops and ratos dpndng on how th corrspondng varabl s codd. Appndx A provds mor dtals on ths topc. Addng th ntracton componnt to a modl has th potntal to mprov modl ft. It also adds complxty to ntrprtng th modl coffcnts. From our tsts, w found that addng th ntracton trm mprovs our modl ft slghtly. For smpl ntrprtaton, n ths papr w only prsnt sampl modl (3.2) wthout ntractons. For a bttr matchd comparson wth Socty of Actuars (SOA) studs, w splt th slctd study data nto four substs and ft ach subst wth ts own modl (3.2). Th four substs ar mal smokr, mal nonsmokr, fmal smokr and fmal nonsmokr. Ths sparat modl dsgn allows ach modl s coffcnts to b stmatd ndpndntly from th othr thr modls, whch mans that ach of th four polcy groups can hav ts own mortalty lvl, slops and dffrntal factors wthout bng constrand by th othr thr groups. W us SAS softwar for our data prparaton and modl dvlopmnt. In th upcomng sctons, w hghlght th ntrprtatons and usfulnss of th thr typs of outputs from th SAS modlng procss., j, j... 8

9 3. Analyss of ffcts for th mortalty drvr sgnfcanc tst Of th nn xplanatory varabls, gndr and smokng status ar usd to splt th study data and svn ar lft to b ncludd n th modls. Tabl 3. summarzs th p-valus of th sgnfcanc tsts of th svn xplanatory varabls on ach of th four data sts. Tabl 3.. Analyss of ffcts Dgr Fmal Mal Pr > ChS (p-valu) of Frdom Nonsmokr Smokr Nonsmokr Smokr Duraton <.000 <.000 <.000 <.000 Issu ag <.000 <.000 <.000 <.000 Study yar Fac band <.000 <.000 Product <.000 Issu yar 2 < <.000 <.000 Class 2 <.000 <.000 <.000 <.000 Svral tms should b notd.. As xpctd, nsurd mortalty vars sgnfcantly statstcally by duraton, ssu ag, undrwrtng class, undrwrtng ra (ssu yar) and fac band for all four subgroups. Ths confrms that ths varabls ar among th most rlabl mortalty prdctors. 2. Study yar, or xposur yar, s ncludd as a placholdr for mortalty mprovmnt n th 0-yar prod covrd by th study data. Th corrspondng p-valus from th four modls mply that, aftr factorng out what hav bn xpland by th othr ght xplanatory varabls (ncludng gndr and smokr status), mortalty varaton xpland by xposur yar (or mprovmnt) s statstcally sgnfcant at = 0.05 only for mal smokrs. Ths may mply that mor mal smokrs casd smokng and rsultd n mor mortalty mprovmnt durng th studd prod. 3. Mortalty dffrntaton by product (btwn prmannt and trm polcyholdrs) s only statstcally sgnfcant for fmal nonsmokrs and mal smokrs, aftr controllng th othr ght xplanatory varabls. 4. At 95 prcnt confdnc lvl, all svn tstd varabls hav statstcal sgnfcanc n xplanng mortalty varaton n at last on of th four polcy groups. W dcd to nclud thm n all four logstc modls. Vnsonhalr t al. (200) analyzd prvat pnson plan xprnc data wth smlar logstc modls and only found on sgnfcant xplanatory varabl. Snc mortalty and longvty ar th two sds of th sam dath-rlatd rsk con, our fndng may suggst that mor potntal longvty rsk drvrs ar yt to b confrmd. 9

10 3.2 Odds rato stmat for mortalty slops and dffrntals Of th nn studd xplanatory varabls, thr (ssu ag, duraton and study yar) ar tratd as contnuous for thr rasons: ) to stmat smoothd rlatonshps btwn and ths varabls, 2) to allow th coffcnts of ths varabls to b transformd as mortalty slops, and 3) to nabl modl-basd mortalty xtrapolaton for oldr ags and latr duratons whr spars or no xprnc data ar avalabl. Th modld xtrapolaton can b usd as th ultmat mortalty stmat. Th valus of th othr sx xplanatory varabls ar catgorzd basd on data crdblty and rcodd as bnary varabls as dscrbd n appndx A. Thrfor, mortalty dffrntals ar obtand for ths varabls. Tabl 3.2 blow contans th odds rato stmatons (pont stmat columns) and thr 95 prcnt confdnc ntrvals. For th thr contnuous varabls, th odds ratos stmat avrag mortalty ncrass pr unt ncras n th corrspondng varabls. For th catgorzd varabls, odds ratos rprsnt th mortalty ratos as dfnd n th ffct column. Th 95 prcnt confdnc ntrvals provd a mans to vrfy th crdblty of th corrspondng slop or dffrntal stmat. Tabl 3.2. Odds rato stmats Effct Pont stmat Mal nonsmokr Mal smokr Fmal nonsmokr Fmal smokr 95% Wald confdnc lmts Pont stmat 95% Wald confdnc lmts Pont stmat 95% Wald confdnc lmts Pont stmat 95% Wald confdnc lmts Duraton Issu ag Study yar Fac $00k $499k vs. $500k+ Fac $50k $99k vs. $500k+ UndrW md vs. nonmd Product prm vs. trm Class on-class vs. standard Class prfrrd vs. standard As dscrbd n appndx A, odds(dath) = /( ) bcaus s usually vry small. Thrfor, odds ratos can b vwd as mortalty ratos n ths tabl. Also xpland n appndx A s that logstc modl coffcnts ar stmatd assumng th valus of all othr xplanatory varabls ar th sam (normalzd). Thrfor, thy approxmat normalzd mortalty dffrntals that may or may not appar to b consstnt wth rsults obtand from actual mortalty studs. Lt s tak th mal nonsmokrs modl as an xampl and ntrprt som of th odds ratos. 0

11 Duraton and ag slops: If vrythng ls wr ual, on avrag, mortalty ncrass about 4 prcnt pr duraton and about 0 prcnt pr ssu ag (odds rato =.4 and.0, rspctvly). Th 0 prcnt pr ssu ag ncras s known to b also tru for th gnral populaton (Thatchr 999). 2. If vrythng ls wr ual, thr s a statstcally nsgnfcant 0.2 prcnt annual mortalty mprovmnt (odds rato = 0.998, th 95 prcnt confdnc ntrval ncludng ). Ths fndng may sm to b nconsstnt wth th common thought of hghr mortalty mprovmnt. Thr ar thr possbl xplanatons. Frst, du to th short tm prod and nconsstnt data contrbutons from nsurrs, th study data may hav not capturd th tru nsurd mortalty mprovmnt. Scond, n th past dcad or so, U.S. populaton mortalty mprovmnt has bn lvlng off as shown n chart 3.2 (data ar from th Human Mortalty Databas; th ag rang rflcts th most commonly nsurd ags). Ths may also b tru of th nsurd populaton. Thrd, nsuranc undrwrtng has spcfcally targtd hgh dath rat causs, such as cardovascular dsass and smokng, and xcludd or dscouragd ths rsks bng nsurd, whch may hav rsultd n lss bnfts for nsurds from th advancmnt n mdcn, tratmnt and publc ducaton. Fourth, unlk a unvarat analyss that attrbuts all th mortalty 0.80% 0.75% 0.70% 0.65% 0.60% 0.55% 0.50% 0.45% 0.40% varaton to th sngl study varabl, a larg porton of th nsurd mortalty mprovmnt ovr th studd yars has bn attrbutd by th modl to othr varabls such as th ntroducton of prfrrd classs, trm products, and flattnd ag or duraton slops that do xplan many mor mortalty varatons. 3. If vrythng ls wr ual, compard to larg polcs wth fac amount at last $500,000, th polcs szd btwn $50,000 $99,000 and $00,000 $499,000 would hav 28 prcnt and 2 prcnt hghr mortalty, rspctvly (odds rato =.28 and.5). 4. If vrythng ls wr ual, mortalty of polcs that had mdcal xams at ssu s about 8 prcnt lowr than that of thos wthout (odds rato = 0.92, sgnfcant). 5. If vrythng ls wr ual, prmannt polcy mortalty would b about.3 prcnt hghr than that of trm polcs (odds rato =.03, nsgnfcant). Ths may appar Chart 3.2. U.S. populaton of ag Calndar yar

12 nconsstnt wth what s shown n charts 2. and 2.2. Kp n mnd that th dscrptv masurs n charts 2. and 2.2 ar obtand wthout controllng any othr varabls. Most of th dffrncs dsplayd n charts 2. and 2.2 may b causd by unmatchd duraton, ssu yar and undrwrtng class dstrbutons. Th logstc mortalty modl provds an ffctv mans to prform normalzaton. 6. If vrythng ls wr ual, th mortalty of th prfrrd class would b about 27 prcnt lowr than that of th rsdual standard class whl mortalty of th aggrgat class (on class plus unknown) s about 4 prcnt hghr (odds rato = 0.73 and.04). As mntond bfor, normalzd mortalty nformaton s ssntal n dntfyng undrlyng causs and avodng mscountng th mortalty dffrntaton valus whn sttng prcng factors. Fndngs of ths analyss can also b usful n valdatng ndustry tabls splt from an aggrgatd tabl, lk th Amrcan Councl of Lf Insurrs 200 Commssonr s Standard Ordnary (CSO) prfrrd class structur tabls. 3.3 Modl ft for ovrall study rlablty masurmnt Compard to halth or proprty and casualty nsuranc clams, mortalty clams occur at a much lowr fruncy and wth a much mor stabl pattrn. Rlatvly scarc clam counts and mor consstnt clam pattrns ld us to us all avalabl data for modl buldng, wthout sttng asd data for ovr-ft vrfcaton. On commonly usd modl-ft masurng statstc s c-statstc, or ara undr th rcvr opratng charactrstc (ROC) curv. Tabl 3.3 dsplays th c-statstcs for th four modls. Tabl 3.3. Modl ft Assocaton of prdctd probablts Fmal Mal and obsrvd rsponss Nonsmokr smokr Nonsmokr Smokr c Svral tms should b notd. Vnsonhalr t al. (200) analyzd prvat pnson plan xprnc data wth a smlar but smplr logstc modl (only on xplanatory varabl). Thr modl had c-statstcs n th rang of for most of th ag groups. Though w ar not masurng c- statstc by ag group, th comparson stll gvs a sns that our four modls hav rasonably hgh c-statstcs and ft th corrspondng data sts wll. An ntrstng obsrvaton s that th two nonsmokr modls hav lowr c-statstcs than th two smokr modls. If c-statstc s usd as a prdctablty masur, th prdctablty by th sam st of xplanatory varabls for smokrs s about 0 prcnt 2

13 hghr than for nonsmokrs. Ths 0 prcnt gan n dath prdctablty s lkly from knowng smokng status. 4. IMPACT OF DEATH CENSORSHIP BY POLICY LAPSATION Along wth th advantags mntond bfor, th adapton of a statstcal modl for nsurd mortalty study brngs a nw ssu that th convntonal dscrptv mthods do not nd to dal wth: dath cnsorshp by lapsaton. Thnk of a group of 00 currnt polcyholdrs. If 0 dd n th nxt 2 months but only fv gnratd clams and th othr fv dd aftr trmnaton of covrag, th dath rat of th group would b 0 prcnt but clam rat would b only 5 prcnt. Insurd mortalty, or clam rat, s condtond on polcy n-forc and only rflcts clam rsk. It s not uvalnt to mortalty masurmnt for a gnral populaton. Whn modls lk thos n (.) (.4) or our logstc modl ar usd for stmatng nsurd mortalty, or clam rat, thy do not rcognz or dscount polcy laps and tnd to ovrstmat clam rat. Ths ovrstmaton may not b a matral ssu for a mortalty dffrntal study bcaus dffrntal s usually masurd n aggrgat and by ratos. If ovrstmaton occurs to both th numrator and th dnomnator by a sam factor, th rato wll cancl out th ovrstmaton and rmans rlatvly accurat. Howvr, whn th modl s usd for ndvdual polcy or polcy group mortalty xtrapolaton, such as n xprnc tabl dvlopmnt, th ovrstmaton can b sgnfcant. On soluton s to modfy logstc modl (3.) and (3.2) to dscount possbl futur polcy laps from contrbutng clam. Cnsorng-basd adjustmnt: Lt s rsrv for dath rat and assum that ach nsurd polcy has thr obsrvabl statuss (and corrspondng probablts) at th nd of an xposur yar: laps ( l ), clam ( c ) and n-forc ( ) so that l + c + = 00%. Wth th sam xplanatory varabls x as usd n (3.2), w can us a multnomal logstc modl to modl th thr probablts as follows (s Hosmr, Lmshow and Sturdvant 203, chaptr 8, for mor dscrptons). (4.) c l ( x x...) ( x x...) l l c l 2 2 c ( x x...) l l c 2 2 ( x x...) l l 2 2 l l 2 2 ( x x...) c c c 2 2 ( x x...) c c c 2 2 ( l lx l 2 x2...) ( c cx c 2 x 2...) 3

14 Lt us call ths modl a logstc c modl to mphasz clam-rat stmaton. Th addd laps componnt l n (4.) plays a rol of stmatng th to-b-lapsd porton of xposurs and xcludng thm from contrbutng daths to clam-rat c stmaton. Asymptotcally, by comparng modls (3.2) and (4.), w hav (4.2) lm = lm ( c + l ) = duraton duraton whch mpls that (4.2) asymptotcally splts th total dath rat nto a clamd porton and a lapsd porton. As to th asymptot of th clamd porton, (4.3) lm c = duraton lm duraton ( l c l c ) ( )* duraton = ( ) l { 0 c l c l c l c For projcton purposs, l and c ar usually rlatd to ntal laps and clam lvls; l and c ar rlatd to laps and clam slops. A hghly smplfd ntrprtaton of (4.3) s that, dpndng on f th dath rat asymptotcally ncrass fastr than, slowr than or ual to th laps rat of a portfolo, th portfolo s clam rat wll approach 00 prcnt, 0 prcnt or somthng n btwn. It s undrstood that nsurd laps rats ar drvn by many long- and short-trm factors and do not ncssarly hav a rlatonshp as rgular wth duraton as clam rat or dath rat. Th laps componnt of modl (4.) may not hav as good a ft to nsurd laps xprnc. Howvr, t s rasonabl to vw th laps componnt of modl (4.) as an mprcal datadrvn adjustmnt for th unknown porton of th nonclam-gnratng xposurs. No mattr whch of th thr asymptots n (4.3) occur, th ovrall ffct of (4.) on c s to flattn th modld c slop by duraton and to rsult n lowr modld c than modld by modl (3.2), spcally for advancd ags or latr duratons. Modl (4.) allows c not to approach 00 prcnt, whch s not asly achvabl wth modls (.) (.4). Modl (4.) s spcally usful for stmatng ultmat mortalty. Du to a data-usag agrmnt ssu, w do not hav accss to th polcy-laps dtal for ths study and ar unabl to dmonstrat a ral output for modl (4.). A follow-up study s plannd. As an altrnatv, w appld som ndustry xprt opnons on nsurd ultmat mortalty to crat a smplfd vrson of modl (4.), usd th modl to produc modl-stmatd ndustry xprnc tabls, and compard th tabls wth SOA s 200 and 2008 Valuaton Basc Tabls (VBTs). Th rsult s vry postv (s th followng sx charts). Ths supports a pont w mad arlr: Usng lss but mor rlvant data may achv ual or bttr rsults than usng mor but lss rlvant data. 4

15 Mortalty (pr,000) Mortalty (pr,000) Mortalty (pr,000) Mortalty (pr,000) Mortalty (pr,000) Mortalty (pr,000) Bcaus ths altrnatv nvolvs varous subjctv assumptons, t s not prsntd n dtal hr. Howvr, w ar opn to nurs and ntrstd n dscusson Chart 4.. Mal, nonsmokr, ssu ags Exprnc x 200VBT 2008VBT Logstc Estmaton Duraton Chart 4.2. Fmal, nonsmokr, ssu ags Exprnc x 200VBT 2008VBT Logstc Estmaton Duraton Chart 4.3. Mal, nonsmokr, ssu ags Exprnc x 200VBT 2008VBT Logstc Estmaton Duraton Chart 4.4. Fmal, nonsmokr, ssu ags Exprnc x 200VBT 2008VBT Logstc Estmaton Duraton Chart 4.5. Mal, nonsmokr, ssu ags Exprnc x 200VBT 2008VBT Logstc Estmaton Chart 4.6. Fmal, nonsmokr, ssu ags Exprnc x 200VBT 2008VBT Logstc Estmaton Duraton Duraton 5

16 5. CONSTRAINTS AND POSSIBLE ENHANCEMENTS Among othrs, thr typs of bass can occur n an nsurd mortalty xprnc study: paramtr bas, samplng bas and data bas. A paramtr bas s a systmc bas that rflcts tchncal lmtatons of a study mthod (.g., usng a lnar modl to ft U-shapd xprnc). A samplng bas happns whn a substtut datast s usd to rprsnt a targt populaton but th substtut dos not hav th sam charactrstcs of th targt (.g., usng a small sampl to rprsnt a larg populaton, or usng past xprnc to approxmat futur outcoms). A data bas s th dscrpancy btwn data and actualty (.g., msrportd ags of daths or unrcordd laps). Som logstc modls paramtr bas (.g., a logstc modl ovrstmats clam rat c) and samplng bas (.g., uncontrolld company contrbutons causng nconsstnt rprsntaton of th ndustry) hav bn dscussd n th prvous sctons. As n any othr larg databas, nsurd xprnc data has plnty of data bass such as mssng data and nconsstnt data codng btwn compans, whch may comproms th ualty of logstc modlng or othr xprnc studs. Th followng ar a fw mor constrants of usng logstc rgrsson for nsurd xprnc studs.. Logstc or c modls may not ft nfant and pr-marrag attand-ag xprnc wll (paramtr bas). Mortalty s usually hgh n ths ags du to causs such as accdnts and sucds. As th xcss causs lvl off wth ag, mortalty rgrsss to a mor normal pattrn that fts bttr wth logstc functon. Th man strngths of logstc modls ar n aggrgatd mortalty slop/dffrntal stmaton and modl xtrapolaton. To mprov ft, a possbl soluton could b to furthr customz a logstc or c modl wth som spln or localzd rgrsson mthods to ft th ags that hav lss rgular mortalty pattrns. 2. Whn scarc xprnc data ar avalabl, such as at vry old ssu ags or latr duratons (data bas), a logstc functon wll b th prmary drvr for stmatng modld or c. For mor accurat stmatons, calbratons wth xprt knowldg ar usually ncssary. 3. Shock laps and shock mortalty that occur at th nd of th lvl prmum prod or durng rar vnts lk pandmcs cannot b ft or rflctd wll by a contnuous functon-basd modl (paramtr bas). At a mor granular lvl, modlng ssus such as uantfyng th nd of th lvl prod ffct for a spcfc portfolo wll nd mor than a logstc mortalty modl. Howvr, at an ndustry aggrgatd lvl and for constructng nsurd mortalty tabls, our study shows logstc modls dlvr rasonabl rsults. 4. Th currnt lack of a consstntly collctd long-trm nsurd xprnc databas s lmtng th optmzaton of any ndustry xprnc studs, ncludng logstc mortalty modls (samplng and data bass). For xampl, not all compans and not all product 6

17 nformaton ar consstntly or proportonally prsntd n an ad hoc ndustryxprnc data collcton. Spcal cars ar ncssary n ntrprtng modl outputs that mplctly assum th consstncy. As data-procssng tchnology and analytcal mthodology advanc, t s our hop th ndustry wll stablsh a mchansm to consstntly collct comprhnsv xprnc data for n-dpth xprnc studs. In summary, logstc rgrsson modls hav many strngths and much potntal for nsurd mortalty xprnc studs, ncludng Tstng for statstcal strngth of mortalty drvrs n xplanng mortalty varatons wth ffct analyss Gnratng normalzd mortalty mtrcs such as slops and dffrntals wth odds rato analyss Extrapolatng for advancd ag or ultmat mortalty wth modld stmaton Brdgng or smoothng btwn slct and ultmat mortalty wth modl lnk functon Quantfyng ovrall study rlablty wth modl ft statstcs Hlpng construct multdmnsonal xprnc tabls by usng th modl as a prdctv modl Bng mplmntabl wth wdly avalabl softwar systms ACKNOWLEDGEMENT W would lk to thank Davd Wyld, Gorg Hrschnko, Mk Falor, Jun Han, Dara Osspova- Kachakhdz, Dors Azarcon and Pat Brsna for thr nsghtful nputs and hlpful suggstons. 7

18 REFERENCES Gomprtz, Bnjamn On th Natur of th Functon Exprssv of th Law of Human Mortalty and on a Nw Mod of Dtrmnng Lf Contngncs. Royal Socty of London Phlosophcal Transactons, Srs A 5: Harrll, Frank E. Jr Rgrsson Modlng Stratgs: Wth Applcatons to Lnar Modls, Logstc Rgrsson, and Survval Analyss. Nw York: Sprngr-Vrlag. Hlgman, Larry, and John H. Pollard Th Ag Pattrn of Mortalty. Journal of th Insttut of Actuars 07 (0): Hosmr, Davd W., Stanly Lmshow, and Rodny X. Sturdvant Appld Logstc Rgrsson. 3 rd d. Hobokn, N.J.: John Wly & Sons Inc. Human Mortalty Databas. Unvrsty of Calforna, Brkly. Kannsto, Vano.994. Dvlopmnt of Oldst-Old Mortalty, : Evdnc from 28 Dvlopd Countrs. Odns Monographs on Populaton Agng. Odns Unvrsty Prss. Odns, Dnmark. McCullagh, Ptr, and John A. Nldr Gnralzd Lnar Modls. 2 nd d. London: Chapman & Hall. Olshansky, S. Jay On th Bodmography of Agng: A Rvw Essay. Populaton and Dvlopmnt Rvw 24 (2): Socty of Actuars. 200 Valuaton Basc Tabl (VBT) Rport & Tabls. Socty of Actuars Valuaton Basc Tabl (VBT) Rport & Tabls. Thatchr, A. Rogr, Vano Kannsto, and Jams W. Vaupl Th Forc of Mortalty at Ags 80 to 20. Odns Monographs on Populaton Agng 5, Odns Unvrsty Prss, Odns, Dnmark. Thatchr, A. Rogr Th Long-Trm Pattrn of Adult Mortalty and th Hghst Attand Ag. Journal of th Royal Statstcal Socty: Srs A 62 (): Vaupl, Jams Th Advancng Frontr of Human Survval. Gnral Ssson prsntaton, Lvng to 00 Symposum, Jan. 8, Orlando. Vnsonhalr, Charls, Naln Ravshankr, Jyaraj Vadvloo, and Guy Rasoanavo Multvarat analyss of Pnson Plan Mortalty Data. North Amrcan Actuaral Journal, Vol 5, Issu 2, Wbull, Walodd A. 95. A Statstcal Dstrbuton Functon of Wd Applcablty. Journal of Appld Mchancs 8 (Sptmbr):

19 Appndx A. Logstc modl coffcnt ntrprtaton Consdr a logstc modl (A.) ln( ) * ag 2 * sx wth two xplanatory varabls: x ag as contnuous and x2 sx as a bnary varabl havng mal and fmal two-valu catgors. For th catgorcal varabl sx, thr could b many dffrnt ways to cod th varabl for analyss. Th most commonly usd codng schm s rfrnc codng: Cod on catgory as and th othr as 0 and call th catgory 0 th rfrnc catgory (.g., for fmal and 0 for mal and mal s th rfrnc catgory). Rfrnc codng s usful whn th prmary goal of a study s to compar mortalty btwn two sgmnts of polcs. Undr ths codng schm, w can calculat th dffrnc of log of odds btwn fmals and mals for th sam ag (controllng ag), (A.2) fmal mal ln( ) ln( ) 2 fmal mal or (A.2a) 2 fmal mal ( )/( ) fmal whch s th odds rato of dath btwn fmals and mals. For th contnuous varabl ag, f w tak th dffrnc of log of odds btwn any ag x and x + for th sam sx (controllng sx), w can drv ag x ag x (A.3) ( ) /( ) ag x Ths s th odds rato of dath whn ag ncrass by unt. mal If w st ag = 0 and sx = 0 (or mal) and consdr ths as th ovrall rfrnc group, w hav mal, ag0 (A.4) mal, ag 0 ag x In summary, (A.2), (A.3) and (A.4) llustrat how th coffcnts of a logstc modl can b ntrprtd as odds ratos undr th rfrnc codng. Th xponntal of a bnary varabl s coffcnt rprsnts th odds rato of th nonrfrnc catgory vs. th rfrnc catgory. Th xponntal of a contnuous varabl s coffcnt rprsnts th odds rato whn th varabl valu ncrass by unt. 9

20 Th xponntal of th ntrcpt rprsnt th odds of th ovrall rfrnc subst that hav valu 0 for all th xplanatory varabls, n ths cas, th mals of ag 0. Through varabl transformaton and rcodng, w may choos any catgory as th rfrnc. In a mor gnral stuaton, f a catgorcal varabl has k catgors of valus and k > 2, w can rplac t wth a st of k bnary varabls and rtan th rfrnc codng advantags. For xampl, f n modl (A.) sx has thr valus: fmal, mal and unknown, w can rplac sx wth 3 = 2 bnary varabls y and y2. And th thr sx catgors can b rprsntd by th pard (y, y2 ) as: y y2 fmal 0 mal 0 unknown 0 0 Ths mans that y srvs as fmal ndcator, y2 as mal ndcator and th par of (0,0) as th rfrnc. Modl (A.) s rformattd as (A.a) ln( ) * ag 2 * y 3 * y2 Ths modl has only contnuous and bnary xplanatory varabls. Its coffcnts can b ntrprtd as summarzd bfor. Thr ar also othr usful codng schms for catgorcal varabls, undr whch th modl coffcnts can b ntrprtd dffrntly. For xampl, th dvaton from mans codng cods th bnary varabls wth valus of and nstad of and 0. Wth ths codng, th rfrnc catgory s always th total controlld man and th coffcnt of a bnary varabl stmats th odds rato btwn th rprsntd varabl catgory and th ovrall man. Ths codng schm s vry usful for comparng mortalty of a sgmnt rlatv to th ovrall mans. S Hosmr, Lmshow and Sturdvant (203, chaptr 3) for mor dscusson. 20

21 Appndx B. Logstc modl coffcnt stmaton Consdr logstc modl, (B.) Lt y b th dath ndcator, wth valu for dath and 0 for n-forc, X dnots th vctor of xplanatory varabl X = { x, x2,,xk}, and β = {,, } ar th coffcnts. Thn, = Prob(y = X) s a functon of β whn a sampl valu of X s gvn. Suppos w hav a sampl of n ndpndnt obsrvaton pars (y, X), =,, n. Snc th lklhood of on obsrvd y gvn X s th jont lklhood of all n obsrvatons s th product of ths lklhoods: (B.2) To solv for th β that maxmz th lklhood functon (B.2), t s uvalnt and asr to solv for β that maxmzs th log lklhood. (B.3) y ( ) ( x x...) ( x x...) y l() n y ( ) y L( ) ln( l( )) { y ln( ) ( y )ln( )} k Unfortunatly, th maxmum lklhood stmat of β cannot b wrttn xplctly. A Nwton- Raphson mthod s usually usd to solv tratvly for th valu of β that maxmz (B.3). Consult McCullagh and Nldr (989) for dscussons of th mthods commonly usd by statstcal modlng computr softwar. 2