A Dynamic Solvency Approach for Life Insurance

Transcription

1 A Dyamic Solvcy Appoach fo Lif Iuac Roa Cocozza, Dipaimo di Ecoomia Azidal Uivià di Napoli Fdico II Emilia Di Lozo, Dipaimo di Mamaica Saiica Uivià di Napoli Fdico II Abac. Th pap iviga ik maagm poc i lif iuac, i a ppciv coi wih h famwok of Solvcy II. Th pap a wih h bakdow of h bui dyamic. Thi aalyi povid fo a compl dpicio of ik ad valu div wihi lif bui. Th copodig map i h pu io h olvabiliy co, i od o fomally idify h quilibium codiio. Coidaio abou h chical quilibium of a iuac pofolio ad h fiacial gulaio lad o a dyamic ym of olvcy am. Th fomal modl i applid o a lif auiy coho i a ochaic co i od o mplify h poial of h modl, pcially fd o h d o fam olvcy am i a dyamic ppciv. JEL claificaio: G, G8 Kywod: Lif iuac, fiacial ik, iolvcy ik, capial adquacy, fiacial gulaio. Alhough h pap i h ul of a commo udy, h fi h cio a by R. Cocozza ad h maiig h a by E. Di Lozo.

2 . Ioducio Solvcy of iuac compai ha cly udgo a wd i o oly by EC iiuio ad compai, facig a pudial gulaio fom poc, bu alo by iaioal aociaio i a boad co, bcau of a pojc - dvlopd by h Iaioal Accouig Sadad Boad (IASB) - o popo a accouig adad fo iuac wih h aim of hacig udadabiliy, lvac, liabiliy ad compaabiliy of gal pupo fiacial poig fo iuac woldwid. Moov, h wid dicuio o capial adquacy pakd by h w Bal capial accod (BIS 00) add h d fo aifacoy ium fo pudial upviio of iuac compai ad fo coicy wih oh co, maily ha of bakig. Th cicumac, coupld wih h pi woldwid fiacial difficuli compai a facig, hav giv i o a makabl covgc of diff ad lva olvcy opic, ivolvig copoa, acuaial ad poig iu 3. Wihi hi dba, h pap add om mhodological iu cocig h olvcy of lif iuac compai, wih mai fc o ivm ik, ad ugg a cocpual famwok fo h ik ym a wll a fo h olvcy am. Thi famwok coiu h bai fo h dvlopm of a fomal modl fo h appaial of h chical quilibium of a iuac pofolio, applid o a lif auiy coho i a ochaic co cluivly fo h pupo of illuaig h poial of h modl ilf. Fom a ciical ppciv, aio i focud o boh h ik ym of iuac udakig ad h dichoomy bw aic ad dyamic ym of olvcy am, fo which h dicimiaig faco i pd by mployd a wll a poially ploiabl ik idicao. A h d of 00 h Euopa iuac lgilaio ha b vid ad updad, wihi a mo gal fom co gadig h olvcy of iuac compai, kow a Solvcy I.Thi amdm plud o a wid fom pojc (Solvcy II), alady ad ad aimd a viwig h gim i h ligh of cu dvlopm i iuac, ik maagm, fiac chiqu ad fiacial poig ad a ablihig a mo ffciv olvcy ym.fo dail Lodo Wokig Goup (00) ad KPMG (00). Th IASB pojc ad i 997, wih h objciv of dvlopig a adad capabl of allowig compaio of fiacial am.th fi ag of h pojc dd o Mach 004 wih h publicaio of h Iaioal Fiacial Rpoig Sadad 4 Iuac Coac. 3 Amog h oh, KPMG (00); IASB (999); Iiu of Acuai ad Faculy of Acuai (Hai al.00); Iaioal Acuaial Aociaio (IAA 00); Iaioal Aociaio of Iuac Supvio (IAIS 000, 00).

3 . Solvcy, capial ad pudial upviio A iuac compay i olv if i i abl o fulfil i obligaio ud all coac ud all aoably foabl cicumac (IAIS 00). Nvhl, i od o com o a pacicabl dfiiio, i i cay o mak cla ud which iuaio h appopia of h a o cov claim i o b coidd. Th quio, fig o h valuaio pupo, do o hav a aw vig all, ic i dpd o may iu. Idd, i i lva whh h compay i dmd a a clod opaio o a goig coc, hu icludig oly wi bui (u-off bai) o alo fuu w bui (goig-coc bai). Addiioally, i dpd o h aim of h valuaio ha i h m fiacial pog of h compay o i abiliy o m claim ad oh obligaio i all bu h mo m cicumac. Th fi opic dfi h lva ik faco: o a goig coc bai alo flucuaio aoud h pcd valu of h w bui will b ak io accou (hu allowig fo ig wihi diff pool), whil o a uoff bai oly flucuaio wihi h igl pool will b coidd (hu cludig claig). Th cod o dfi h ampliud of h valuaio, which i pobabiliic m would copod o h cofidc lvl choic (fom imdia lvl o m v). I gal, gulao could valua olvcy o a u off bai ad/o o a goig coc bai, a hy a boh igifica, alhough h la appoach i a mo aliic dpicio of h auy, which omally amog diff pool, whil h fom i mo pud, giv ha i valua h quilibium wihou ay ogou ad aiv uppo fom vual bui. Thfo, olvcy valuaio i a poc, who logical paadigm hould quially coi of h mai p: lva ik cogiio, ik maum ad dfiiio of capial quim o abob occuig lo. Ufoualy, h appa a of h pocdu complly fad ou wh movig fom h gi o h mpiical implmaio, wh chical oubl coupl wih dalig difficuli cocd o capial quim co bo by imdiai ad by h ym. Cocig h fi iu, h ik ym ad h laiv bakdow i o alway poply ad; whil wih fc o h cod apc, appaial mhodologi ca doub o hi ffciv ad fficicy.. Rik cogiio fo lif iu Th aim of hi cio i o povid a mhodological iigh io ik cogiio ad a ik aalyi famwok, ad o a ik bakdow, ic ay poibl ik cagoizaio i uiabl fo h igl pupo ad o ca v all. 3

4 I gal, h mai ik fo a fim i ha vu pov o b uabl o cov p. If hi valuaio alo coc h hahold, h capial ivd i o adqualy muad. Thi migh b addd a quiy ik. Thi vy boad dfiiio mbac all ik ad povid fo hi maum hough h vaiac of h pcd pofi ov a im buck (ik hoizo, accouig piod ad o o), o alo ov h i duaio of h bui. Pmium ad claim pcivly a ypical iuac vu ad p: hfo quiy ik m fom h poial iqualiy amog h lm, wih h fuh difficuly ha vu hav o b imad bfo p bcau of h ivd cycl. Thfo, all h faco ha ca giv i o h iqualiy - ad i fac o a lo - a cucial ad dfi h whol ik ubym. I hi ligh, h diffc ca ai fom boh icoc -a ig ad icad -po p, hu ivolvig all h bui fucio. I oh wod, h divgc may b dmid by h (i)abiliy/(u)faibiliy o ima o oly pu pmium bu alo loadig i accodac wih p ad pofi ad, a h am im, by h (i)abiliy/(u)faibiliy o kp ho p low, wih fc o boh pcaio ad vaiac. If w gad h lif bui o a u-off bai ad coca oly o h dmia of pu pmium, h ik ym ca baically b dcibd a a ma of a: i ad moaliy. I oh wod, fom a gal poi of viw, lif iuac bui i chaacizd by wo mai ik faco: dmogaphic ad fiacial 4. Dmogaphic ik ai fom h v ha aumd fquci ca diff fom acual occuc of lva oucom 5. Likwi, fiacial ik - cocd wih h implici guaa of a a of u buil i mo polici - oigia i h ca of a divgc bw h acual u o a puchad wih wi pmium ad h a of i fomly applid o pmium aig.. Rik maum ad capial quim Th cod p i h olvcy am i, a ad, h ik maum pha, which hould ul i a fai paio of h hazad icumb o h 4 Babbl al.(997) dfi h ik ha h fim i payig oo much fo h fud i civ, o alaivly h ik ha h fim i civig oo lil fo h ik i ha agd o abob a h acuaial ik. 5 Th IASB add h v ha umb of iud v will diff fom pviou pcaio a occuc ik, which i acibd o h mai faco: modl (icoc modl), paam (icoc ima) ad poc (adom aiical flucuaio).th qualificaio could b alo fid diiguihig bw faul du o avoidabl iaccuacy ad ho aiig fom uavoidabl flucuaio.i a acuaial ppciv, h occuc ik i h iuac (o udwiig) ik. 4

5 iuac bui. Th maum ym hould b hc capabl of aig h dag poial ad abl o limi i hough capial quim. I hi pc h cu mhodologi ca om doub o hi ffciv ad fficicy ad h cop of hi cio i maily o focu h aio o om ciical iu 6. Th a ially wo mai yp of olvcy mhodologi ud by upvioy auhoii: fid aio ad ik-bad ym. Th fid aio ym, icludig ha adiioally implmd i EU coui, i a fomulaic mhod which amp o calcula a olvcy magi quim hough a fid pcag of a ik pou poy, uually a fiacial am im. I h EU modl fo lif compai, fo ampl, h book valu of h mahmaical v i gadd a a fiacial ik poy, whil h amou of h o-gaiv capial a ik i coidd a iuac ik poy. Th quid olvcy magi i h aggga of a fid pcag of h wo poi. Th wo poi a ducd i valu accodig o p gulaoy boudai, i od o limi h iuac covi. Alhough vy impl, ipiv ad o dicioay, hi appoach do hav diadvaag. Apa fom h lvac giv oly o cai yp of ik (i.. moaliy ik), i do o flc h compay-pcific ik pofil fo icd iuac allowac 7. I addiio, h dic popoioaliy likd o fid coffici of capial quim o v ad capial a ik aum ha high valu of h im auomaically accou fo high ik pou. Thi auomaic pocdu i o ay h la aiv, if o uaf ad ufai. Such a dic laiohip could b olabl if h iuac pofolio (i.. h ik pool) did o hav h ial chaaciic of qualiaiv ad quaiaiv homogiy. A h am im, i i miladig if h lag v coicid wih pool ha a o oly homogou bu alo vy lag, idd o lag ha h oigial uiv ca b plicad wih gowig pciio. Likwi, h amou of v i oly a ough ima of h ivm ik pou, h la big codiioal upo h a (ivm) ad liabilii (v) mimach ad upo a fau. Hc, a capial quim dicly popoioal o mahmaical v ad capial a ik hough a fid aio o oly capu magially h pcific ik pofil of h compay, bu ca alo giv i o gulaoy abiag a wll a povid iciv fo ud-vig. O h oh had, ik-bad appoach a foudd o a ad hoc valuaio of ik compo i od o calcula capial quim flcig h iz ad ovall ik pou of a iu. Th mo impoa of h ym i h ik- 6 Fo a compl compaaiv aalyi KPMG (00). 7 Th la diciv (00/83/EC) cilig fo iuac allowac fo lif auac ad auii (5% fo mahmaical v ad 50% fo o-gaiv capial a ik). 5

6 bad Capial (RBC) implmd i h US ic h aly ii by h Naioal Aociaio of Iuac Commiio 8. Th objciv i o calcula a capial quim fo ach of h mai ik facd by iu, which fo lif bui a dividd io a ik, iuac ik, i a ik, ad bui ik. Th i o doub ha h backig ik ym i fa mo comphiv ha EU appoach a wll a ha h valuaio pocdu i mo coi wih h pcific compay ik pofil. To a wih, h a ik, dfid a ik of dfaul fo affiliad ivm ad db a ad ik of lo i mak valu fo quiy a 9, i paaly dcd fom i a ik dfid a h ik of lo du o chag i i lvl likd o a mimach bw a ad liabiliy cah flow 0. Scodly, iuac ik (udwiig ik), fig o h c claim fom boh adom flucuaio ad iaccua picig fo fuu lvl of claim, i valuad a a pcag of h capial a ik cald by iz. Fially, h i h icluio of a wid ag of gal bui ik facd by lif iu (bui ik) accodig o pmium icom. Fo ach ik diff faco a applid o h copodig im o h fiacial ifomaio o p h ik poial a likly lo. Effc of pofolio agggaio ad colaio amog vaiou yp of ik a coidd - bu o uly maud - by a covaiac adjum, addig ogh im ha a blivd o b colad, o ha wha i lf a goup of ik im blivd o b ubaially o colad o ach oh. Th covaiac adjum h qua h ulig goup, add h ulig qua ogh ad ak h qua oo of h um of h qua. Fially, h RBC i calculad a h um of h oal ik of h covaiac adjum. Boh mhod, alhough diff i calculu, hav a aalogou hoic backgoud, which i vy imila o h bakig appoach. Oc om ima of h poial lo ha b, a capial quim i divd fom, i h amp mo o l fomalizd o kp h ui (i.. iolvcy) pobabiliy wihi a lvl 8 Caada implm a imila ym dod a Miimum Coiuig Capial ad Suplu Rquim. 9 Off-balac h im (o-coolld a, divaiv ium, guaa fo affilia ad coig liabilii) a icludd i hi ik compo.all iuac compai a ubjc o a a cocaio faco ha flc h addiioal ik of high cocaio i igl pou. 0 Th faco i hi calculaio p h uplu cay o povid fo a lack of ychoizaio of a ad liabiliy cah flow.th impac of i a chag i ga o ho poduc wh h guaa a mo i favo of h policyhold ad wh h policyhold i mo likly o pod o chag i i a by wihdawig fud fom h iu.thfo, ik cagoi vay by h wihdawal v (i.. whh h i ubaial paly fo wihdawal). 6

7 ckod accpabl by h ym. Th lvl of h fomalizaio, ha i o ay h adopd valuaio modl, do mak h diffc. I hi pc, h wo mhod a qui imila, ic i boh ca h poial lo i o uly imad, bu oly dmid by paam ifd fom obvaio of lva co. Thfo, h lvl of capial quid, alo fo upvioy ivio, i accodig o a pagmaic dfiiio of olvcy alog wih iduciv mhod. A a alaiv, h i h poibiliy of buildig up a compl diibuio of h compay ul ad h of h pobabiliy of ui ud a dduciv mhodology, capabl of ig a fomal laiohip bw capial quim ad ui pobabilii. Thi pobabiliic appoach, mo compl bu alo mo accua, ha wo mai fom: imulaio-bad ad aalyical. Th imulaio-bad appoach amp o cov h full ag of ik vaiabl ampld fom aiical diibuio i a imulaio pocdu, coidig a wid ag of oucom, liklihood of adv dvlopm ad iacio of ik vaiabl. Nvhl, h ul of ucciv ial do o ally hav o plica h vy pobabiliy diibuio ilf. Rah, h diibuio i complly modlld i h aalyical appoach accodig o ochaic poc ha a abl o plica h divig dyamic of h modl. Naually, h dduciv mhodologi hav may vid advaag, ic hy poduc oupu ha i lva ad maigful (Babbl al. 998, 00; Hai al. 00; KPMG 00) ad, la bu o la, hy a coi wih h Bal appoach, by viu of big acually ial modl, who ffciv applicaio hould, of cou, b codiioal upo a validaig pocdu. 3. A famwok fo h quilibium appaial Rik cogiio i fucioal o ik maum ad i boudai: h lvac of a ik puh owad om maum ym. To hi d, i i oppou o vi h codiio of h quilibium fo a lif iu, lighig h lva ik faco. L u pocd alog h choological ad logical quc of h iuac udakig. Th fi hazad of h co bui m fom h qualiy of h ifial poc ad, hfo, fom h vual dicpaci bw fuu ad acual p h iu ha o ba. Th fi difficuly i claly lad o h ca of acual pi occuc high ha h oigial imad fquci, which could giv i o a igifica pauciy of availabl a, flcig a iadqua Th appoach bfi alo fom caio-aalyi, ha a pojcio of h compay fiacial am, wih h d of modllig h whol pfomac ud diff codiio ad impoig a capial lvl adqua o h poibl caio (maily o h wo ca). 7

8 valuaio of chical v. Likwi, h would b a caciy of availabl a, hould h acual a of u b low ha h aumd i a. Thfo, h wo codiio hav imila lvac ad gh ad flc h wo mai ik faco of h lif-iu bui, ic vy chagig paam i a picig fomula i a ik faco. Fo ampl, l u coid h pmium (P ) fo a immdia mpoay () uiay auiy P = p δ()d 0 = wh δ() i h iaaou a of i obvd a h bgiig of h bui, ad p i h pobabiliy of uvivig ya af aaiig ag. Fomula impli ha h fial quilibium (im ) of a coho of c idical polici i coaid by cp δ()d 0 = N () δ()d 0, () wh δ() i h iaaou oal a of u o a puchad wih wi pmium obvd a h d of h bui, ad N () i h acual umb of uvivo a ag. Fomula () compu h ul of h pofolio, ha i o ay wha acuai would qualify a uplu, accoua a icom, ad coomi a pofi. If i pov poiiv, h bui poducd bfi ad, of cou, lo, if gaiv, whil h ull lvl h miimal quilibium (i.. iolvcy hhold). By viu of, iqualiy () ca b wi a = which hold a la if [ δ()d 0 δ()d [c p δ()d 0 ] δ()d 0 N () 0 (3) 0 ] δ()d (l(n ()) l(c p )) 0. (4) I hi ca h miimal quilibium i dicly dpd o h igl piod diffial bw h igal of h iaaou oal u o a puchad wih wi pmium ad ha of h oigial i a, ad bw h logaihm of h acual umb of uvivo ad h pcd umb of uvivo. Th ul () A Pak (997c) a vy claly hi a compa i icom, capial gai ad lo. 8

9 will hibi a uplu (dfici) - ad vually a pofi (lo) - whv fomula (4) i poiiv (gaiv) fo ay combiaio of h fou lva vaiabl, h ul big qual o h igl piod diffc capializd a h a of u o a fo h idual duaio of h policy a how by fomula (3). Naually, may dyamic ca imulaouly coibu o h diffial ha dpd o ik faco likd o boh h a i which pmium a ivd ad h valu of liabilii fo which capializd pmium a dfd. Togh wih h dmogaphic dyamic, h mo impoa faco i h au of h a: if h a fiacial, h ik facd will b maily fiacial, 3, hy will dpd dicly o h a yp ad will o hav ay auoomou lvac. Bid, h cu of h poblm i h diffc bw h oal a of u o a ad h a of i oigially applid i pmium calculaio, o ha i ca b pcily addd a ivm ik, i od o highligh h compoi au of lva ik div. A h am im, oh faco ca coibu o h diffc uch a h qualiy of h ik maagm poc, wih fc o boh divificaio ad ik poolig 4. Thi impli ha h lvl of h ul ad i vaiabiliy i icly dpd o idividual compay lm ha ivolv boh ogou ad dogou faco 5. I od o gai a iigh io h divig faco of h ul, l u aaly i im aificaio. A h d of h fi piod h quilibium i giv by h abiliy of h wi pmium ivd fo o ya o cov boh h cu claim ad h p valu of fuu ouflow (mahmaical v), ha i cp δ()d 0 N N = p δ()d 0. (5) Thi laio i abl o p a h am im h quilibium o h balac h ad o h icom am, ic i i capabl of focuig h aio o h opaig icom, which i ou icd ca i wha ally div h chag i h woh (W ). A a ma of fac, wi pmium plu ivm icom (fi m of 3 Fiacial ik i h ik of a poibl fuu chag i o o mo of a pcifid i a, cuiy pic, commodiy pic, foig chag a, id of pic o a, a cdi aig o cdi id o imila vaiabl.logically, alo i hi ca i would b poibl o diiguih modl, paam, ad poc, a fo occuc ik. 4 Of cou, h maagial aa ad h ik faco ivolvd will b mo umou a w mov fom pu pmium o offic pmium, ha i o ay fom a icd o a compl aalyi of h bui. 5 A a coquc, h ik am ym hould b abl o valua alo h pcific ik compo, ha i o ay hould b bad o ial modl. 9

10 (5)) miu claim (cod m) a h liabiliy-div a, h fial v (hid m) i h copodig liabiliy o ha h diffc i h capializd woh (cf. Figu ). A h am im wi pmium of h fial v a h ad pmium, which ogh wih h ivm icom ad h icud claim mau h opaig icom o a accual bai (cf. Figu ). Similaly a h d of h cod piod h quilibium i giv by [ ] cp δ()d 0 N ) δ()d N () N () = p δ()d 0 (6) ha oc agai ca b ipd a a aic codiio of quilibium o h balac h ad a a dyamic codiio of quilibium o h icom am ad ca b dividd io h capializd ul o h fi piod plu h ul o h cod piod. Similaly fo h ya h gal pio fo h quilibium codiio i [ ] cp δ()d 0 N () δ()d (7) = [ ] N () N () p δ()d 0 = whil a h d of h bui (im ) h ul i giv by h compac fomula (), fom which i ca b ifd ha h ul dpd o h u o h a alog h whol piod ad o h icom accud i ach piod. Thfo, olvcy i poply h abiliy o comply wih h o-gaiv laiohip ad ca b fomally pd a h pobabiliy of pcig pmaly, ha i o ay, i ach piod, h gal quilibium codiio pd by ([ P cp δ()d 0 = = ] N () δ()d (8) [ ] ) N () N () p δ()d 0 = ɛ Hc, iqualiy (7) p h quilibium imulaouly fom bui ad acuaial ppciv ad ca b ud fo pudial gulaio if a pobabiliy lvl fo i o gaiv valu ca b (fomula (8)). Th lvl of hi pobabiliy i a poliical quio ad h lvl of h capial adquacy, which acually f o a magi adqua o kp h pobabiliy of iolvcy wihi a limi which i coidd 0

11 baabl, wih fc o boh capial co bo by h imdiai ad h ik lvl facd by policyhold. Thi famwok, which i of cou a miimal bakdow of h ik ym facd by lif-iu, ha h advaag o ligh om fudamal logical ad mhodological iu: a) gaiv lm of h iu pofolio, ha i o ay chical v, a pod o ypical ik faco mmig fom h qualiy of h ifial poc (modllig ik, logviy ik, i a ik); h ik faco, dfiabl a liabiliy ik div, lad o a ica i h chical v high (low) ha ha pviouly ad ad ai fom a ica (dca) i h pcd moay valu of h coig liabiliy (iuac ik) ad/o fom a dca (ica) i h dicou a applid fo h v valuaio 6 ; b) poiiv lm of h iu pofolio, ha i o ay ivm copodig o v, a pod o ik faco mmig fom h lcd yp of ivm (mak ik); h ik faco, addabl a a ik div, giv i o acual vu low (high) ha ho pcd ad com fom a dca (ica) i h ivm icom (ivm pocd, valu adjum, alizaio valu); c) h bld of a ad liabilii wih u o pfcly (poiivly) colad yild a adddum o h pofolio vaiac ha i ubaially dpd o h colaio amog h ik faco iflucig boh id of h balac-h. Th ik faco, ha ca b lablld a pofolio addiioal ik div, giv i o a low o high chical accou balac (icom am ul) ha pcd. I follow ha h baic ik ym ca b dividd io wo mai goup: h odivifid ik aociad wih holdig of a ad liabilii 7 ad h addiioal ik fo pofolio mi (i.. idividual vaiac, pofolio wigh ad colaio coffici). Thfo, whv h a imila ik faco iflucig boh poiiv ad gaiv lm, h ffc poducd by ho faco o h valu of h pofolio will diff fom h ffc poducd o h compo if h colaio amog ik faco i o pfc. Thi impli ha i a flucuaio affc boh h 6 I h EU gulaio h a wo mai opio: h fi f o a kid of mak a bcau of h ducio caid ou ud h Euopa ul govig h mak a i od o obai h chical a; h cod f o a dicou a dpdig o h yild of compay a.nih of h opio i i li wih h cu IASB pojc; i h pou daf fo iuac coac i i ad ha h aig poi fo dmiig h dicou a fo iuac liabilii ad iuac a hould b h p-a mak yild a h balac h da o ik-f a. 7 Th m odivifid appli h o h wo id of h balac h gadd a igula compo of a wo-a pofolio, alhough hy ca oigia fom a pop divificaio agy.

12 ivm icom ad h chag i h chical v, bu hi impac do o caily off if h laiciy of h lva valu i o idical ad/o if h valu of h poiio i o pfcly balacd. I oh wod, if h yild cuv i o fla, (7) bcom cp h=0 N () (h, h, h ) N (k) k kp k= h=0 k= h=k (h, h, h ) (9) v(, h, h ) N () 0, wh v(, y, z) i h valu a im y, quod a im, of a coac which guaa a moay ui a im z. Fo vy fid valu of ad y, v(, y, z), coidd a fucio of z, giv h m ucu of pic a im y of coac udwi a im. If y>, w hav h fowad m ucu; ify = w hav h po m ucu. Coily (, y, z) = /v(, y, z). A a ul, h i a la fom a hoical ppciv h poial fo a ica i h chical v aiig fom a dca i h a applid fo h valuaio o off by a poiiv ffc i h ivm icom. Thi i h ca wh h laiciy of h v ad ha of h cocd ivm a o pfcly machd, a wll a wh h copodig mak valu a diff. Th impac of h hazad will b, h, hacd o livd by colaio ad by pad bw h oal u o ivm ad h a applid i h v valuaio ad by h im aificaio of hi dpau. I oh wod, would h lva a b h am fo boh id of h balac-h alo wih fc o hi pciv duaio, h ally would b h cop (aioal) fo aalyzig a uiqu ik faco which would impac o h valu of h poiio. Nvhl, a vaiy of gulaoy coai, uch a h ivm ul o accouig pcipio 8, foc h wo id of h balac o b pod o diff ik faco alo wih fc o duaio. Thfo, h i a diff impac of h i a ik o h a ad liabiliy pofolio - ad o h fim pfomac - which i cocpual diff fom h olly vaiaio of h ivm icom. Th i hfo boh h hoical oppouiy ad h 8 Ialia gulaio, fo ampl, a compl ym of cilig fo a allocaio.thfo, pofolio lcio i ogly biad ad v dcivd whv h ovall a wigh, fid by law, pv fom pickig h opimal ivm pofolio fo h igl coho of polici.thfo, a a paado, ivm ul could ga a ub-opimal allocaio, hu givig i o couiuiiv ul.similaly, h gulaoy pcipio cocig h a of i o b applid i h v valuaio could ga om opical illuio abou h icom ha ca b diibud.

13 pacical cop fo valuaig h chical quilibium of h pofolio wih fc o boh compo ud a poply dduciv mhodology. 4. A alaiv modl I hi cio w p a modl fo valuaig ad quaifyig iolvcy i h ca of a pofolio of lif polici. Th mhodological ba a i h udy of h cumulaiv pobabiliy diibuio fucio of h pofolio v; i fac kowig h upp ail of uch diibuio allow h acuay o ima if h valu of hi fuu obligaio cd h pogammd v fud. Th mahmaical caio, i which h iolvcy poblm i famd, povid a aalyical mhodology, ap o a ic appoach o olvcy am. Thi i v mo uful, oc w call ha h c acuaial liau how ha h iolvcy poblm i o alway aalyzd poply by imulaio chiqu o caio ig mhodologi, ic vagu of h pciio lvl, log imulaio im ad ilocuoy ul of mpiical chckig do o allow o implm igifica Noaio ad plimiay ul L u coid a pofolio of c idical ya mpoay lif auiy-immdia, ach policy big of ui payabl a h d of ach ya whil h lif agd () uviv. L T i () b h adom vaiabl pig h fuu lifim of h ih iud (fo vy i). Th popciv lo a im, fo h ih policy, i dfid a h diffc L (i) a im bw h p valu of fuu bfi paym ad h p valu of fuu pmium paym, aumig ha T i () >. L K i () do h cua fuu lifim of (), ha i h adom vaiabl pig h umb of compl fuu ya livd by () i[0, ]. Th L (i) = K i () j= j δ()d, 9 Pak (997b) compa h mhodologi (acabl modl, ochaic imulaio, caio ig) o iviga h ui pobabiliy fo a pofolio of lif iuac coac wih o wihou iuac: imulaio val hmlv o aily plicabl by oh acuai ad gulaoy auhoii ad d log uig im o obai a ufficily accpabl appoima diibuio; o h oh had caio ig cau udimaio of h iolvcy ik. 3

14 wh δ() i h foc of i. A wll kow, h pmium v a im i dfid a h codiioal pcaio of L (i), giv ha T i () >. L u do by L h popciv lo fo h i pofolio. I paicula, if N () i h umb of uvivo a im i h coho of h c iud agd a iu, i hold L = N () p δ()d, (0) o, by a quival oaio, = = L = c δ()d K i () >, c big h umb of uvivo a im. I h followig w coid h adom vaiabl L/c, pig h avag v p policy, fo which h followig ul hold. Thom 4.. L h adom vaiabl K i () b idpd ad idically diibud ad idpd o h vaiabl δ(); h,wh h oal umb of polici a iu d o ifiiy, L/c covg i diibuio o h adom vaiabl Λ = p p δ()d = Poof: Coidig h dfiiio (0), i i ay o obv ha N () i a biomial adom vaiabl, which aum h valu 0,,..., c, ach lif iud havig h am pobabiliy p of uvivig uil h ag. Sic ( lim P N ) () p ɛ =0, c c w dduc ha ( N () c p ) = p δ()d [ covg i pobabiliy o zo, bcau E = p ] δ()d <. Th L/c covg i diibuio o Λ. I i poibl o pov h popoiio alo coidig h dfiiio fo h v, ad obvig ha c i a biomial adom vaiabl, hc ( lim P c c c p p ɛ ) =0. Rmak 4.. Th adom vaiabl Λ appoima h avag v a im, i h ca of a vy lag pofolio. I hi caio h poolig ffc lad o 4

15 h adom dviaio of h umb of dah com u, o h iuac ik ca b glcd, whil h fiacial ik play a fudamal ol i h global pofolio iki Iolvcy mau I ou oaio, fo vy fid valu of, y(u) p h foc of i accumulaio fucio, ha i y(u) = u δ()d, u 0. Fo a fid E w idica by χ E i chaaciic fucio, ha i { if E χ E () = 0 ohwi. Followig a mhodology popod by Pak (994) ad uccivly dd by Coppola al. (003) i h ca of lif auiy pofolio, w g h followig ul Popoiio 4.3. L u coid h adom vaiabl Ψ m = m i= ip y(i). If {δ()} N i a Gauia poc,h diibuio fucio of Ψ m i pd by h fomula: F Ψm (z) = g m (z,y)dy, () wh wih g m (z,y) = f y(m) (y y(m ) = )g m (z m p y,)d (3) g (z,y) =χ {ζ:ζ p y }(z)f y (y), f y() big h pobabiliy diy fucio of y(). Poof: L u g m (z,y) =P (Ψ m z y(m) =y)f y(m) (y), (4) wh f y(m) i h diy fucio of y(m). Th, h diibuio fucio of Ψ m i giv by F Ψm (z) = g m (z,y)dy. (5) 0 Obviouly h dmogaphic chag (moaliy/uvival) a vy impoa i h ca of mall pofolio; moov i a wid ppciv h mahmaical modl could icopoa alo oh ik faco, uch a lapaio, p, akig io accou poibl laiohip bw lap a ad a of u. 5

16 I od o valua h igal o h igh had id of (5), w coid a umical pocdu popod by Pak (994) ad (997a) ad vid by Coppola al. (003). I paicula, by uig kow popi of codiioal diy fucio, w g g m (z,y) =P (Ψ m z)f y(m) (y Ψ m z) = = P (Ψ m z m p y )f y(m) (y Ψ m z m p y )= = P (Ψ m z m p y ) f y(m) (y y(m ) =, Ψ m z m p y ) f y(m) ( Ψ m z m p y )d Fially, mmbig fomula (4) ad h makovia popy of h poc {y()}, w ca wi g m (z,y) = = f y(m) (y y(m ) =, Ψ m z m p y ) g m (z m p y,)d = f y(m) (y y(m ) = )g m (z m p y,)d Moov, if m =Ψ = p y ad, by viu of (5), g (z,y) =P (Ψ z y = y)f y (y) = = P (p y z y = y)f y (y). Th w obai g (z,y) =χ {ζ:ζ p y }(z)f y (y). O h bai of Popoiio 4.3, oc w obv ha Λ = p Ψ, w ca a h followig ul ivolvig h diibuio fucio of Λ. 6

17 Coollay 4.4. Ud h hypoh of h Popoiio 4.3,h diibuio fucio H Λ of Λ i giv by ( ) u H Λ(u) =F Ψ p fo vy u R. 5. Th ochaic i a I od o p a poibl caio fo h foc of i i h piod ud coidaio, w aum ha h dyamic i im of uch a i dcibd a h um of wo compo (cf. Di Lozo al. 999) δ() =()X(); (5) () i a dmiiic compo obaid o h bai of h cu lva a ; X() i a ochaic compo pig ay dviaio phomo. I paicula w uppo ha {X() : 0 < } i a Oi-Uhlbck poc, wih paam β>0 ad σ>0, ad iiial poiio X(0) = 0, ivolvd by h followig ochaic diffial quaio dx() =βx()d σdw(), wh W () i a adad Wi poc. Th h dicouig poc i giv by φ(0,)= ( ()d X()d 0 0 ), X()d 0 big log omally diibud wih paam E[ X()d] ad 0 va[ 0 X()d], wih [ ] E X()d =0, 0 [ ] va X()d = σ 0 β σ β 3 [34β β ]. Chooig a paicula modl fo h h i a i aually dpd o h aim of h valuaio poc: ayway h modl popod ovcom h difficuli cocd o h choic of h mahmaical modl dpicig h voluio i im of h i a, ic i i flibl ad uiabl o h a lag cla of modl, i.. h makovia o. 7

18 A i i wll kow h auocovaiac fucio of h ochaic dicouig poc i giv by cov [ h X()d 0, k ] X()d 0 = = [va[ h X()d]va[ k [ X()d]] 0 0 cov( h X()d, k ] X()d) A umical ampl Now, o h bai of Coollay 4.4, w ca calcula h cumulaiv diibuio fucio of Λ i = 5 i h ca of a pofolio of mpoay lif auii wih =7, ach policy big iud o a po agd =50. W fi a coa dmiiic compo = 0.09, ad h paam β = 0., σ = fo h ochaic compo. Th ul a collcd i Tabl. Th abl how h bhaviou of h upp ail of h diibuio of 5 Λ. Fo iac, fo a fid avag v qual o.7576, h iolvcy occu wih pobabiliy 5.%. I oh wod, h valu a im of h iu fuu obligaio (ha i h valu a im of h iu db poiio) i ga ha h pogammd v fud wih pobabiliy 5, %. 7. Cocluio ad fuu ach li Th aicl focu o h compl ad compoi olvabiliy poblm fo a lif iuac bui. A fi, af a ovviw of h ik map, i diplay h cucial ol of a accua maum of h vaiou ik compo, i ligh of h lva implicaio i calculaig h olvcy magi, a wll a h o ivial cocio wih h pudial upviio. Fom ou uvy of h mai mhodologi culy adopd by upvioy auhoii i olvcy am, h d ai o ba h ik maum ym o a ic dfiiio of h diibuio of h compay ul, i od o dduc, by ma of acabl modl, paam idicaiv of (i)olvcy. Agai hi backgoud, a iovaiv aalyical mhodology ha b fomalizd, lagly bcau of i fomal foudaio wih pc o imulaiv ad/o caio ig pocdu. W hav how ha i i poibl o obai pobabiliy diibuio of mai paam lad o a iuac policy pofolio. Th mhodology ha W aum h uvival/moaliy pobabilii pod i h Ialia Moaliy Tabl 98-Mal. Th ul a pfomd by ma of Mahmaica. 8

19 b applid o h v of a lif iuac pofolio, mo pcily o a mpoay lif iuac pofolio coho, o dlia h poial ivolvd i h modl. I paicula, h upp ail of h diibuio of h pofolio v ha b dducd, hu obaiig igoou ima of h iu capaciy o fac hi fuu obligaio, i a caio ivolvig volailiy of i a. Th abov modl could giv i o may diff applicaio ad fuu ach li. A fi, i i o coaid by h choic of a pcific ochaic poc, bu i ca b ufully adopd wih a lag cla of poc ad ca b calibad wih div paam. I hi co a iig fuu iu, which i byod h cop of hi pap, migh b h valuaio of diff gulaoy gim aimd a aig h copodig pobabiliy of iolvcy. Fuhmo, fom a mo pacical ppciv, h dicpacy bw accouig olvcy ad coomic olvcy could b ivigad. Fo ampl, h aalyi of vaiou ul, cocd wih div poc ad paam dcibig h i a dyamic, could b gadd a a mau of h iqualiy bw h book valu ad h cu valu of h imdiaio pofolio. Fially, h modl could b dd o o-homogou pofolio by iig h colaio amog commo ik div. Som oh aa of i could b plod. Th mo impoa, which will b udid i fuu ach, coc whh h i a igifica quaiaiv iqualiy bw h u of a imulaio-bad modl o h adopio of a aalyical appoach. Th aw o hi quio i of cou codiioal upo h choic of coi mau, ha i o ay of coi caio, o a o guaa a mo maigful compaio. 9

20 Rfc Babbl, D.F., Gold, J., Mill, C.B., 00. Fai valu of liabilii: h fiacial coomic ppciv. Noh Amica acuaial joual 6, -7. Babbl, D.F., Mill C.B., 998. Ecoomic valuaio modl fo iu. Noh Amica acuaial joual (3), -7. Babbl, D.F., Saomo, A.M., 997. Rik maagm by iu: a aalyi of h poc. Joual of ik ad iuac 64, Bg, A.N., Hig, R.J., Szgo, G.P., 995. Th ol of capial i fiacial iiuio. Joual of bakig ad fiac 9, BIS Bal Commi o Bakig Supviio, 00. Th w Bal capial accod. BIS, Jauay. Cocozza, R., 000. La gio dl ichio di ao di i ll aicuazioi dl amo via. CEDAM, Padova. Coppola, M., Di Lozo, E., Sibillo, M., 003. Sochaic Aalyi i Lif Offic Maagm: Applicaio o Lag Auiy Pofolio. o appa o Applid Sochaic Modl i Bui ad Iduy 9, pp Di Lozo, E., Sibillo, M., Tio, G., 999: A Sochaic Modl fo Fiacial Evaluaio. Applicaio o Acuaial Coac. Applid Sochaic Modl i Bui ad Iduy 5, pp Giad, L.N., 000. Mak valu of iuac liabilii: cocilig h acuaial appaial ad opio picig mhod. Noh Amica acuaial joual 4, 3-6. Elo, E.J., Gub, M.J., 99. Opimal ivm agi wih ivo liabilii. Joual of bakig ad fiac 6, Hai, C. J., Blham, D. J., Byo, N. M., Gog, C. M., Ha, D. J. P., Smih, D. A, Thompo, S., 00. Fai valuaio of liabilii. Iiu of Acuai ad Faculy of Acuai, Novmb. IAA Solvcy Wokig Pay, 00. Rpo of olvcy wokig pay ppad fo IAA Iuac Rgulaio Commi. IAA, Fbuay. IAIS Solvcy & Acuaial Iu Subcommi, 000. am ad acuaial iu. IAIS, Mach. O olvcy, olvcy IAIS Solvcy & Acuaial Iu Subcommi, 00. Picipl o capial adquacy ad olvcy. IAIS, Jauay. 0

21 IASB Sig Commi, 999. Iuac iu pap. IASB, Dcmb. Kigh, K., 000. Mahmaical Saiic. Chapma & Hall/CRC, Boca Rao, Floida. KPMG, 00. Sudy io h mhodologi o a h ovall fiacial poiio of a iuac udakig fom h ppciv of pudial upviio. Euopa Commiio, May. Lodo Wokig Goup, 00. Pudial upviio of iuac udakig (Shama Rpo). Cofc of iuac upvioy vic of h mmb a of h EU, Dcmb. Mla, J.L., Sag, A.P., 973. A Ioducio o Pobabiliy ad Sochaic Poc. Pic-Hall, Nw Jy. Pak, G., 994. Limiig Diibuio of h P Valu of a Pofolio. Ai Bulli 4, No., pp Pak, G., 997a. A Pofolio of Edowm Polici ad i Limiig Diibuio. VIII Iaioal Sympoium o Applid Sochaic Modl ad Daa Aalyi, Aacapi, pp Pak, G., 997b. Iolvcy Rik: Tacabl Modl, Simulaio ad Scaio Tig. Tchical Rpo. Pak, G., 997c. Sochaic aalyi of h iacio bw ivm ad iuac ik. Noh Amica acuaial joual (), Vadhoof, I.T., Alma E.I., (d.) 998. Th fai valu of iuac liabilii. Kluw Acadmic Publih, Boo. Dodch, Lodo.

22 Tabl. Cumulaiv diibuio fucio of 5 Λ(u) u F Λ

23 Figu Syhic Balac Sh Ed of ya A (ivm) ( W ) [ ( cp W ) ( ) δ d 0 cp N δ d δ d [( ) 0 ] cp W N N δ d d δ 0 N ] N = δ d δ d cp W 0 N = Liabilii (poviio) N woh δ d δ d N p W 0 = δ d δ d N p W 0 = δ d δ d N p W 0 = δ d W 0

24 Figu Syhic Icom Sam Ed of ya pmium cp Rvu ivm icom ( ) ] [ 0 d W cp δ ( ) 0 d d N W cp δ δ ( ) = 0 d d d N W cp δ δ δ ( ) = 0 d d d N W cp δ δ δ claim N N N N Ep chag i MP = d p N δ = = d d p N p N δ δ ( ) = = d d p N p N δ δ Icom W W d 0 δ W W d 0 δ W W d 0 δ W W d 0 δ