The Financial Economics of Universal Life: An Actuarial Application of Stochastic Calculus. Abstract

Transcription

1 Th inancial Economic of Univral if: An Acuarial Applicaion of Sochaic Calculu. John Manir MMC Enrpri ik CE Plac, 161 ay Sr, PO ox 501 Torono, ON, M5J S5 Canaa Phon: ax: Abrac Thi papr u anar financial conomic ool o analyz h rucur of a impl quiy inx univral lif conrac. Th analyi para h quiy invmn rik from h inr ra rik ini h policy by rmining h rplicaing porfolio of bon an quii ha mach h cah flo gnra by h inuranc conrac. Th main rul i ha h inuranc conrac can b brokn ino a linar par, for hich h rplicaing porfolio can b conruc in nially clo form, an a non- linar par hich rquir numrical or ohr mho for oluion. A rla rul i ha h linar par of h oluion i inpnn of h ochaic proc riving h quiy a bu h non-linar par o pn on h ail of h ochaic proc. cau an imporan lmn of h analyi can b carri ou in clo form h mol i a cania for u a a ca uy in h applicaion of ochaic calculu o acuarial problm. Th papr yl of prnaion hrfor rflc h ca uy forma.

2 1 Inroucion Univral if ha bn an imporan lmn of h Norh Amrican lif inuranc mark for ovr 0 yar. or many compani in h inury oay h prouc i h cor of hir n buin opraion. Dpi h prouc imporanc rlaivly lil ha bn rin abou h funamnal financial conomic of univral lif. Thi papr i a conribuion o h filling of ha gap. On poibl raon for hi iuaion i ha non of h valuaion mho currnly u in h Uni Sa US STAT & US GAAP focu on h prouc financial conomic ircly. Hovr, n Canaian GAAP valuaion chniqu ar mor cah flo ba an a financial conomic vipoin can yil ignifican inigh ino h Canaian financial rporing proc. Th financial conomic vipoin i alo ircly rlvan o h approach bing vlop by h Inrnaional Acuarial Aociaion for u a par of h mrging Inrnaional Accouning Sanar. ar ho ar familiar ih h xiing accouning mol ill fin omhing n in h financial conomic vipoin oulin in hi papr. A ky rul i ha h policyholr accoun balanc i no ncarily h corrc maur of h inurr financial xpour o h invmn rik bing pa hrough o h policyholr. olloing hi inroucion h papr ha hr main cion follo by a concluion. Th hr main cion ar. ormulaion of h problm: Sochaic iffrnial quaion for h accoun balanc, princy bonu an policy rrv ar vlop. Th valuaion problm i formula a a parial iffrnial quaion PDE ih appropria bounary coniion. Th linar oluion: Th PDE for policy rrv ami a linar oluion hich aifi h bounary coniion a mauriy bu no a fun xhauion. Th linar oluion i imporan for unraning h ynamic of h rplicaing porfolio vn hough i i only an approximaion. Propri of h linar oluion ar icu in conirabl ail. Daling ih un Exhauion: Giranov Thorm i u o g a horical xprion for h ajumn rquir o g from h linar oluion o h xac oluion of h valuaion problm. Thi rprnaion allo u o conclu ha h linar oluion i mo accura in h larg accoun balanc limi.. ormulaion of h Univral if aluaion Problm 1

3 Thi cion aum h rar i familiar ih boh ochaic calculu concp an lif inuranc in coninuou im. I i ncary o ar by inroucing a ignifican amoun of noaion..1 Evoluion of h Accoun alu I b an xrnal quiy inx, hich h inurr ih o cri o i Univral if policy. I i aum ha h inx follo h anar log normal rurn mol i.. A funamnal characriic of Univral if i h mainnanc of an accoun valu o hich h inurr cri prmium an inr an uc xpn an inuranc charg. Th accoun valu i u o rmin ah bnfi, urrnr bnfi an lap bnfi. In hi papr ill b aum o follo h Io ochaic iffrnial quaion m σ z ε g. In h fir rm i a rminiic pra ha i uc from h inx rurn o rmin h amoun of inr cri o h policyholr. Th con rm i h prouc of a Co of Inuranc COI ra an h n amoun a rik. Th n amoun a rik aum h ah bnfi i a fix ac Amoun plu ε im h accoun valu. Th paramr ε ill b 0 or 1 pning on h yp of conrac. Th final rm in h ochaic iffrnial quaion for rprn h criing of prmium h ucion of non-pra rla xpn. I ill b aum ha prmium ar a knon rminiic funcion of im i.. g g an ha h xpn charg i alo a rminiic funcion of h form λ g f. Collcing rm in h quaion abov can ri m ε g σz 1 hr hav u h noaion ε 1 ε.. Evoluion of h Accru onu I I m σz Mo Univral if conrac in Norh Amrica oay offr a princy bonu. Thi i a ranom variabl hich accru uring h lif of h policy an i cri o h accoun valu a pcifi policy uraion τ,..., τ. Th bonu i a prouc faur n

4 ha ncourag favorabl policyholr bhavior an i alo play an imporan rol in h al illuraion proc. or hi uy ill aum ha h bonu volv accoring o a ochaic iffrnial quaion of h form α m σz ε. Th fir rm rprn a bonu accrual ha i proporional o h accoun valu. Thi bonu bhav omha lik an aiiv incra o h cri inr ra. Th con rm accru a bonu proporional o h cri inr hil h hir rm i ign o rfun a proporion of h Co of Inuranc COI charg. y collcing rm hi quaion can b rrin a α m ε σ z. I oul b unuual for any on policy form o offr mor han on bonu rucur a h am im. Hovr, h iffrn bonu rucur ill affc h rplicaing porfolio in iffrn ay o ill carry all hr yp forar in h analyi. onu accrual ar ofn o zro uring h fir 10 policy yar an hn ar o pay a rgular inrval, uch a vry fiv yar, hrafr. Typical valu for h accrual paramr migh b α 0 for yar 1-10 follo by accrual ra of 1%-% ih h fir bonu acually bing pai a uraion 15. Whn proporional bonu ar u h bonu paramr i ofn in h 10%-15% rang onc i bcom non zro. Whn h rurn of COI bonu rucur i u h paramr uually vari bn 0 an 0%. Th ar no h only bonu rucur in u in h mark plac. To ummariz h icuion o far, hav a ym of ochaic iffrnial quaion for h join voluion of h accoun valu an bonu givn by quaion 1 an rpcivly. Whn h accoun valu i poiiv h quaion ill hol a all im xcp ho im τ hn h bonu i cri. On a bonu paymn a hav τ τ τ τ 0 If h accoun balanc bcom ngaiv ay ha h policy ha nr fun xhauion. Wha happn hn pn on h inurr aminiraiv pracic. In hi papr ill aum ha hn fun xhau all accru bonu ar forfi an 3

5 lif inuranc covrag rmina unl h policyholr lc o pay ufficin prmium o covr h ongoing co of inuranc an xpn charg ha kp h accoun valu a 0 i.. 0 an 0 from ha poin on..3 Parial Diffrnial Equaion for h rv Thi cion ill u h principl of ochaic calculu, financial conomic an raiional acuarial cinc o riv a parial iffrnial quaion for a rrv,,. In hi conx h or rrv man h valu of a porfolio of b an quiy inx invmn ho cah flo allo h inurr o m h inuranc conrac cah flo obligaion. Whn an appropria lvl of conrvaim i inclu in h aumpion many acuari oul call h quaniy h fair valu of h inuranc liabiliy. Wha follo i a hr p proc hr conir h chang in liabiliy u o acuarial coningnci h chang in liabiliy u o financial iu h chang in a u o financial iu an policy mchanic Whn h hr p ar compl ill arriv a an quaion hich nially ay ha h xpc chang in liabiliy i qual o h xpc chang of h a in h rplicaing porfolio. Th analyi bgin by looking a ho h rrv chang hn h acuarial coningnci of ah, urrnr an parial ihraal ar conir. or impliciy ill ignor ohr prouc faur uch a policy loan. Dah nfi: W ill aum a forc of moraliy an a ah bnfi qual o h fac amoun plu ε im h accoun valu D ε. Th xpc chang in liabiliy u o moraliy for a im incrmn i hrfor qual o h chang in liabiliy ε im h probabiliy of ah ε. Surrnr nfi: W ill aum a forc of urrnr an a urrnr bnfi qual h accoun valu l a uraion pnn urrnr charg i.. h cah valu i of h form C, max0, SC. Th conribuion of h urrnr coningncy o h xpc chang in liabiliy for a im incrmn i hn max 0, SC. Parial Wihraal: Mo Univral if conrac allo for om ihraal from h accoun valu ihou cauing h policy o rmina or incurring a urrnr pnaly. 4

6 Thi i ofn a ignifican bnfi a lar policy uraion if amoun can b ihran, ax fr, o hlp financ rirmn. A full analyi of all h iu urrouning parial ihraal an policyholr bhavior i vry complx bu a raonabl aring poin i o aum hr i a forc of ihraal hich riv h incinc of parial ihraal vn an h vriy of parial ihraal i an amoun x hr x ha a probabiliy iribuion f,x on 0,1. Unr h aumpion h xpc chang in liabiliy u o parial ihraal, in a im incrmn, i Thi i ill oo complx o ork ih o ill mak h aiional implifying aumpion ha h xprion abov can b approxima by uing a Taylor xpanion 1, x,,, x x. Whn hi approximaion i u h xpc chang in liabiliy u o parial ihraal bcom Whr an 1 { x, x, } f, x x,, 0 κ xf, x x υ x f, x x 1 κ κ υ ar h fir an con momn of x rpcivly. Eima of h quanii oul hav o b vlop from xprinc ui. Th con p in h analyi of h liabiliy o conir ha happn if non of h inuranc coningnci occur, an vn ih probabiliy 1. Th chang in liabiliy ill b rivn by h voluion of h accoun valu an accru bonu. Th chang in liabiliy i,,,,. W can no u h ool of ochaic calculu o ri hi chang in liabiliy a 5

7 6 Whn combin hi ih h acuarial analyi on arlir conclu ha h oal chang in liabiliy ha a man givn by an h niiviy o a chang in h ranom lmn riving h quiy inx i W ill combin all h rul ino h on amn, accura o fir orr in, or Thi quaion capur h man of h chang in liabiliy an ranom flucuaion u o mark movmn. I o no capur h ranom flucuaion u o variaion in acuarial xprinc or policyholr bhavior. W no xamin h a ynamic. W ill aum oal a backing h liabiliy in h amoun ih an amoun Q inv in h quiy inx an h balanc -Q inv in a b a hich arn a forc of inr afr alloanc for faul an invmn xpn. Th chang in a valu A ha corrpon o h chang in liabiliy can hn b rin a z m g m SC 1 max0, σ σ σ υ σ ε α κ ε κ ε z m g m SC 1 max0, σ σ σ σ ε α ε υ κ κ ε z m g m 1 σ σ σ σ ε α ε 1 max0, E SC E υ κ κ ε. z σ

8 A Q m σ z Q g Th rm g- rprn h incra in a u o h rcip of gro prmium g h am prmium ha go ino h accoun valu l an xpn alloanc. Th xpn alloanc ill gnrally b iffrn from h xpn loa λ g f uc from h accoun valu. Hr ill aum a combinaion of pra rla an fix xpn of h form λg f hr i a pra rla xpn hich ill allo for im uch a a ba commiion, Canaian invmn incom ax, or any ohr xpn hich h inurr choo o rcovr hrough h pra mchanim. Th a an liabiliy analy ar no compl. Th final p i o man ha h chang in a mach h chang in liabiliy for any quiy rurn i.. or A ε m ε κ g α m ε 1 σ υ σ σ σ z Q m σz Q g. Equaing h cofficin of h ranom lmn z on boh i of h hi quaion yil, o ha h quiy raio Q i givn by max0, SC κ σ z σqz, Q. Thi i a anar chniqu of financial conomic for rmining h appropria hging ragy. 7

9 Wih hi rul can no qua h man an ri E E A a ε κ g α ε 1 σ υ σ σ 3 ε max0, SC κ g λg f. Thi i h unamnal Parial Diffrnial Equaion ha hav bn orking o formula. I i h amn ha h oal xpc ra of chang in h liabiliy, u o boh acuarial an financial iu, i qual o h xpc ra of chang in h a, coniion on h invmn rik bing hg accoring o h ragy Q.. W no ha hn ha rik i hg h xpc rurn on h quiy inx m rop ou of h analyi an i rplac by h forc of inr. Thi i an xampl of h rik nural valuaion principl a ork. I i alo uful o no ha h funamnal quaion allo for uncrainy in boh h conomic nvironmn an policyholr bhavior..4 ounary Coniion Th fac ha a funcion,, aifi quaion 3 riv abov o no ncarily man ha i i h oluion o h valuaion problm. Thr ar an infini numbr of funcion hich aify any givn iffrnial quaion. Th ay o iola h oluion ha olv h problm i o pcify bounary coniion. A h nam impli bounary coniion rfr o h funcion bhavior a crain xrm poin of im or pac. or hi problm hr ar four rlvan bounary iuaion o icu alu a Mauriy: If h conrac maur or no a policy uraion T.g. ag 100 hn h rrv mu accru h mauriy bnfi. W ill aum ha h policy pay h fac amoun plu h accoun valu plu any accru bonu a mauriy. Th rlvan bounary coniion i hn T,, alu bfor an afr a princy bonu i pai: If a princy bonu i cri a uraion τ hn h rrv ju bfor h paymn i τ,, hil h valu ju afr h paymn i τ,,0. or conincy h valu mu b qual o a bonu paymn a 8

10 τ,, τ,,0 alu a un Exhauion 0: I i poibl ha a combinaion of invmn rurn, parial ihraal an prmium paymn la o a cnario hr h accoun valu xhau bfor h mauriy a. Whn hi occur, or vn bfor, mo inurr noify h policyholr ha h inuranc covrag ill lap unl aiional prmium ar pai. Th policyholr no ha an opion o coninu h conrac or no. or h proporion 0< π <1 of policyholr ho lc o coninu h conrac ill aum hy pay a prmium ĝ ju ufficin o kp h policy inforc. Thi oul imply g ˆ λgˆ f. l Givn a uiabl forc of lapaion, h rrv for hi iuaion can b rin a l or h rmaining proporion 1 π hich lap an appropria liabiliy o hol oul b gˆ /1 / 1. Thi liabiliy rflc h inury pracic of alloing h policyholr a 30 ay grac prio o pay h prmium. Thu, if ah occur in ha prio, h ah bnfi pai i h ac Amoun aju for h prmium ĝ /1 ha oul hav bn ufficin o kp h policy in forc for 30 ay. Puing h o pic oghr g h bounary coniion for 0 a,0, X hr h xhau valu X i givn by gˆ l X π l 1 π gˆ /1 /1. In many pracical iuaion hi bounary coniion oul no b marially iffrn from zro. Hovr, hr ar om iuaion, uch a Canaian prouc ih lvl co of inuranc cal, hr h choic of aumpion govrning fun xhauion can b vry imporan. alu in h larg accoun balanc limi. oh US an Canaian ax auhorii pu limi on h amoun of mony ha can buil up ini an inuranc policy ihou riggring avr ax conqunc. Thi uually forc h policyholr o ihra fun, ruc prmium or incra h ah bnfi o ha h accoun valu ay 9

11 ihin h allo rang. W ill no amp o ar hi iu any furhr in hi papr bcau h ail vary by ax juriicion. Thi compl h formulaion of h parial iffrnial quaion an bounary coniion for h quiy inx Univral if problm. Th rmainr of hi papr focu on chniqu for xracing uful informaion from hi formulaion. 3 Th inar Soluion 3.1 ormulaion Thi cion implifi h problm of olving h funamnal iffrnial quaion 3 by auming a oluion of h form,, H K hich i linar in h variabl an. Thi implificaion i juifi by h fac ha h funamnal quaion 3 i, almo, linar in ho variabl an hnc houl hav a linar oluion. Th only non linar rm in 3 ari from h fac ha cah valu can no b ngaiv. W ill call H h hg raio, h bonu raio an K h inuranc rrv rpcivly. Aocia ih a oluion of hi form i h quiy xpour Q. H. Th oal rrv hn compo a ino quiy an b componn a follo:,, H K H K W immialy ha h muliplicaiv bonu rucur 0 rquir a loan in h amoun from h b i o quiy i of h conrac. Th analyi ha follo i ign o rmin h funcion H, an K o ha h a porfolio abov i capabl of rplicaing h policy cah flo, unr h implifying aumpion hav ma. Whn hi program i compl ill ha h A/ M problm ha bn analyz ino hr iffrn porfolio rquirmn An quiy fun in h amoun Q H, A fun inv or borro hor in h amoun, A porfolio of longr b inrumn in h amoun K. W ill rfr o h a h quiy fun, h hor fun an h long or inuranc fun rpcivly. Th cah flo aocia ih h long fun ill urn ou o b vry imilar o h cah flo aocia ih raiional inuranc prouc. W ar by rriing quaion 3 a 10

12 11 Th linariy of h oluion ha allo u o rop all h con orr rm an ar alo ignoring h cah valu floor of zro. Subiuing,, K H fin Collcing all h rm proporional o, an 1 rpcivly hi bcom If hi quaion i o hol for all valu of, hn ach quar brack in h quaion abov mu vanih ilf. Thi la o h folloing ym of orinary iffrnial quaion for h quanii H, an K. Th bounary coniion for h original parial iffrnial quaion 3 can b inrpr for hi ym of orinary iffrnial quaion a follo: alu a Mauriy: T,, impli HTT1 an KT. alu on onu Paymn Da: Th rquirmn ha h rrv b coninuou a bonu paymn a,..., 1 n τ τ i τ τ τ τ τ K H K H hich clarly impli τ τ H.. f g g SC g λ κ ε ε α κ ε. f g g K H K H SC K H H g K H λ κ ε ε α κ ε 0. SC Hg g H H K K H H κ ε α κ ε 0. * SC Hg g H H K K H H κ ε α κ ε

13 alu a 0: Th linar oluion i no flxibl nough o fi h bounary coniion,0, X. Thi i h main ourc of rror in h linar mol. W can xpc ha h iffrnc bn h rquir valu X an h valu K aain by h linar oluion i a maur of h rror in h linar mol. Thi iu ill b icu in mor ail in h cion 4 of hi papr. 3. Analyi of h onu aio W can u h bounary coniion on bonu a, an quaion * o unran h bhavior of h bonu raio bn bonu a. If li bn o bonu a, τ hn * an h bounary coniion allo u o ri τ k 1 k H τ Th bonu raio a im i hrfor qual o h hg raio a h nx bonu a icoun o im ih inr an conrac princy. Thi i a logical rul bcau i ll u ha h liabiliy, for an accru bonu, i qual o h rrv H τ k hich mu b ablih on h bonu paymn a icoun for inr an princy. No ha hil hi choic of bonu raio guaran ha h rrv i coninuou acro bonu paymn a h bonu raio ilf ill uually hav a iconinuiy a bonu paymn a. S h char ha follo. 3.3 Analyi of h Hg aio H k τ k A ignifican inigh ino h naur of h hg raio H can b obain by a impl chang of variabl. W ill fin h xc pra funcion ξ by H 1ξ. No ha h bounary coniion HT1 impli ξ T 0. Whn u hi chang of noaion in h fir of quaion * fin ha ξ aifi h iffrnial quaion. ξ ξ κ ε α ε ε Solving hi iffrnial quaion g h rul hich ho hy h xc pra funcion bar ha nam ξ T ε κ α ε ε. Whn ε 0 aily ha h xc pra funcion i h prn valu of h pra rla loa in h conrac l h prn valu of h pra rla co ih icouning aking plac uing accoun valu princy an pra loa inr. Th 1

14 pra rla co ar h xplici pra xpn an h co of accruing h princy bonu α. Th co ar off by h xplici pra rla loa buil ino h ra criing ragy. I i alo inring o no ha h xc pra i affc by h inr ra aumpion only inircly hrough i impac on h co of bonu accrual. If h prouc ha no princy bonu h xc pra funcion oul b inpnn of h inr aumpion. Whn ε 1 hr ar a numbr of aiional iu affcing h xc pra ξ. Th analyi ll u ha xprinc moraliy i acing lik a pra-rla loa an h co of inuranc moraliy bhav a a pra-rla co. Tha xprinc moraliy i a pra rla loa i raonabl bcau ε 1 impli ha h accoun valu i no pai on ah o h quiy porion of h conrac xprinc a gain on ah. W ill xplain hy hi loa i off by h co of inuranc moraliy acing a a pra rla xpn hn xamin h inuranc fun cah flo. I ill b n ha hi i an inrnal ranfr bn h quiy an b porion of h conrac hich i rquir o hg h quiy rik. Th rm ε appar a a pra rla loa hn ε 1 an h co of inuranc rfun bonu rucur i in plac. Thi i alo par of an inrnal ranfr rquir o hg h quiy rik. inally, h rm ε appar in h li of quanii u o icoun h pra iffrnial. Thi rflc h fac ha accoun valu accumula ih boh inr an co of inuranc moraliy hn ε 1. W can no unran h naur of h quiy xpour rquir o mach h quiy inx Univral if inuranc policy. Th oal rquir quiy xpour i Q 1ξ Th fir par of h rquirmn 1 ξ rflc h pra rucur of h prouc. If pra rla loa xc h pra-rla co hn h rquir xpour i l han uniy. Similarly if h pra co xc h pra rla loa hn an amoun grar han h accoun valu mu b hl. Th con par of h rquirmn i connc o h paricular rquirmn of a princy bonu hich i proporional o h cri inr. Thr ar o apc o hging h quiy rik aocia ih hi bonu rucur 13

15 y borroing an amoun from h b i of h conrac an inving hi amoun in quii h inurr ill gnra an incrmnal invmn rurn of m σz In aiion, h pra rucur provi for a rla of Aing h o pic oghr fin h oal amoun rla from h quiy i of h conrac i m σz m σz Th fir rm on h righ rprn h co of accruing h bonu hil h con rm i h inr rquir o pay for h loan akn from h b i of h conrac. Th char blo plo a numrical xampl of h funcion Q, H an uing raliic acuarial aumpion for moraliy, urrnr an inr. Th accru bonu in hi xampl i bing rivn by h aumpion ha 0 for h fir 10 policy yar an 10% hrafr. Ohr ky aumpion ar a pra loa of.5 %, a pra co of 1.0% an an inr ra nvironmn of 6.0%. inar Mol Dmonraion aio % 110.0% 100.0% 90.0% 80.0% 70.0% 60.0% 50.0% 40.0% Policy Yar Equiy aio Q Hg aio H onu aio A h plo ho, h hg raio H gra moohly from an amoun ju ovr 80% a h bginning of h policy o 100% a h n. Thi i conin ih h unraning of H a 1 l h xc pra funcion. Alo no ha h lop of h hg raio curv chang abruply a uraion 10 hn h princy bonu ar o accru. 14

16 onu raio xhibi a a ooh parn ih an ampliu ha gro ovr im a h impac of incraing moraliy affc h icouning proc. Th quiy raio i imply h combinaion Q H xcp on bonu paymn a hr h plo rflc h valu ju afr h bonu ha bn pai. No ha h rquir quiy xpour i l han h accoun valu for h fir 0 yar of h policy. I i alo inring o no ha if a policy u h aiiv bonu rucur ih α. 10 * bp in yar 10 an lar i ill hav h am hg raio a h xampl in Char 1. Hovr, i oul hav QH a h maching invmn ragy. Holing quii qual o 100% of h accoun valu in hi ca oul coniu a ignifican ovr xpour. 3.4 Analyi of h Inuranc rv K Th final lmn of h linar mol i h inuranc rrv K. rom quaion * i aifi h iffrnial quaion K K H H g Hg SC. Thi quaniy can hrfor b unroo a a raiional acuarial prn valu K T T H H g Hg SC. Th fir rm of hi rul H rprn h iffrnc bn h ah bnfi ha mu b pai an h hg raio mulipli by h co of inuranc charg uc from h accoun valu. Whn h n amoun a rik i qual o hi i raonabl bcau h rrv rla, from h quiy fun hn h COI charg ar uc from h accoun valu i H. Thi i no inconin ih h fac ha h nir amoun i a rvnu. Whn h N Amoun a ik i qual o - h rrv rla by h co of inuranc ra i only H hra h formula for K rquir a ranfr of H. Th iffrnc H i h pra rla co ha appar in h analyi of h xc pra funcion hn ε 1. No ha ihou hi ajumn h cah flo for h inuranc fun oul no b rminiic. Th rm H rprn h iffrnc bn acual, non pra rla xpn an h rrv rla hn h xpn loa i akn from h accoun valu. Similarly, h rm g Hg rprn a coninuaion of h parn alhough in hi ca ar concrn ih a cah flo going ino h accoun valu rahr han a charg coming ou. 15

17 Th la o rm in h finiion of K rprn h co of accruing a co of inuranc rfun princy bonu off by h incom xpc o com from urrnr charg. No ha h inuranc rrv mu provi for h co of h rfun ba on h full fac amoun vn hn ε 1. Thi i conin ih h arlir rul ha ε i pra rla incom o h invmn i of h conrac. Th la rm in h formula for K i h valu of h nomn bnfi. 3.5 Summary of h inar Mol Th folloing abl ummariz h rul of h linar mol an ho ho h policy bnfi an xpn hav bn pli bn h hr fun yp crib in h inroucion o hi cion. 16

18 inar Mol: Summary Cah lo Analyi Invmn un Inuranc un onu un Toal Dirc Cah lo Dah nfi ε 0 ε Enomn Surrnr nfi SC 0 SC Wihraal κ 0 0 κ Expn λ g f 0 λ g f Gro Prmium Inr un Tranfr Expn Charg COI Charg Hg 1 H g 0 g H λ g f H λ g f 0 0 H H 0 onu Accrual Charg α ε m σz α 0 ε m σz N orroing onu Paymn rv 0 0 τ H τ 0 τ H τ 0 H K H K Thr ar a numbr of obrvaion ha can b ran from hi abl. Th oal cah flo of h policy ha bn compo ino hr fun. Th cah flo for a givn fun coni parly of inracion ih h oui orl h irc cah flo an parly of ranfr o or from h ohr fun. Th cah flo for h inuranc fun ar complly rmin by h acuarial aumpion an ar no affc by ranom movmn in h quiy inx. rom a financial prpciv h cah flo ar nially fix o h inr ra aumpion i h crucial lmn in rmining h valu of h inuranc fun. To h xn ha h hg raio i clo o on an h accoun valu ucion for moraliy an xpn ar clo o h acual amoun rquir h rrv buil up in h inuranc fun migh no b larg. Hovr, if hr i a marial iffrnc 17

19 bn h lop of h acual an COI moraliy cal, a i ofn h ca, h inuranc rrv can b marially iffrn from zro. Th cah flo in h quiy fun ar a combinaion of ochaic bnfi combin ih rminiic ranfr bn i an h inuranc fun oghr ih ochaic ranfr o or from h bonu fun. Th crucial lmn riving h valuaion of hi componn ar h pra rucur an accoun valu princy. Th inr ra aumpion affc h valu inircly hrough h co of bonu accrual. Apar from hi iu h linar mol uccfully pli h inr ra an quiy rik ino o iinc lmn. Th cah flo moving in an ou of h bonu fun ar highly ochaic y i i par of h b i of h conrac. In h ca hr 0 h cah flo ino h bonu fun coni of bonu accrual charg in h amoun α ε floing ino h fun hich accumula, a h inr ra, o h bonu paymn a. Whn 0 h mchanic ar qui complica ih h borroing cah flo an inr paymn moving back an forh. Sinc h fuur bonu fun cah flo ar unknon a h valuaion a h inurr can no lock in a yil curv a ha im an b ur ha h ill alay b abl o pay h bonu. A rigorou icuion of h bonu fun iu hrfor rquir a ochaic mol of boh quiy rurn an inr ra hich i byon h cop of hi papr. A l rigorou bu mor pragmaic approach o unraning h bonu fun i o no ha coul u a iffrn inr aumpion r o valu h bonu fun. I ill b conrvaiv o u a lo ra r hn h bonu fun i poiiv.g. hn 0 an a high ra r hn h bonu fun i ngaiv hich oul uually b h ca hn conciliaion o h vnu Mol Mo acuari ar familiar ih h rvnu mol of Univral if hich xpr h liabiliy for a Univral if conrac a h policyholr accoun valu l a prn valu of fuur margin. Thi i h mol hich AS 97 ak a a aring poin. Th linar cah flo mol vlop abov can b rconcil o h rvnu vipoin by uing h iniy H 1ξ o rri h linar cah flo rrv a H K 1 ξ T T H H g Hg SC. 18

20 Which can b furhr xpan by riing T SC T T ξ ξ g T 1 Th fir lin in h larg brack abov can b inrpr a h prn valu of moraliy, xpn an urrnr charg margin. Th con lin i h prn valu of fuur inr pra margin hil h la lin rprn h inr an princy gain on h currn accru bonu off by h co of ho fuur bonu accrual hich ar no buil ino h pra rucur. No ha h moraliy, xpn an urrnr margin ar icoun uing h inuranc fun inr ra a ar h fuur COI bonu accrual charg. Th proc for icouning h fuur pra margin ar ih h prn valu of all pra for amoun currnly in h accoun valu ξ hich i hn aju for h pra impac of fuur prmium, xpn loa an COI charg. A high lvl ummary of h abov icuion oul b h amn ha rvnu ar no cah flo bu hr i a of cah flo ih h am prn valu a h of rvnu. Th icouning proc ugg by h linar mol o no agr ih h icouning proc mana by AS 97 for US GAAP accouning. 3.7 Acuarial ik in h inar Mol Th linar mol crib abov i ign o hg aay h rik aocia ih ranom movmn in h quiy rurn. I o no hg h rik ha policyholr may no bhav a aum. Thi cion brifly look a h iu of acuarial xprinc gain an lo in h linar mol u o uncrain policyholr bhavior. or xampl, if hav an unanicipa ihraal W h lo incurr i nially h prn valu of all fuur pra aocia ih ha ihraal. W, W,,. W 1 H ξ W. A pracical implicaion of hi calculaion i ha if an o buil om conrvaim ino h xpc ihraal aumpion houl o o in uch a ay ha 19

21 E ξ W < 0. Thu if ξ > 0 houl uparly bia h valuaion ihraal ra o ha E W < 0. Similarly, an unanicipa prmium paymn g, rul in a lo qual o g λ 1 H 1 λ g λ λ ξ 1 λ. Th prmium paymn aumpion ill hrfor hav an lmn of conrvaim if E g λ λ ξ 1 λ < 0. Thi man ha h ign of h bia in g i rmin by h ign of λ λ ξ 1 λ. In gnral i i no poibl o hg boh h financial rik an h acuarial rik a h am im unl ξ 0 an λ λ. I i poibl o vlop mor ophiica mol of policyholr bhavior ihin h conx of h linar mol bu ill no o o in hi papr. 4 un Exhauion an h Giranov Thorm Thi cion al brifly ih h rror in h linar mol u o h fac ha i fail o aify h corrc bounary coniion hn h fun xhau a 0. W ar by rcalling ha h oluion o h funamnal parial iffrnial quaion 3 ε κ g α ε 1 σ υ σ σ ε max0, SC κ g λg f can b rin a h man of a ochaic ingral,, E min T, x r min T, x,, 4 E min T, x min T, x r ε max0, SC κ λg f g Hr h xpcaion i akn ih rpc o a maur in hich h accru bonu an h accoun valu aify h ochaic iffrnial quaion an α m ε σ z. ε g σz κ υ z 0

22 rpcivly. Th ranom lmn inpnn of z. z riving h parial ihraal proc i Th ingral conir h prn valu of ah bnfi, urrnr bnfi, ihraal bnfi an xpn off by gro prmium. Th uppr boun on h ingraion inica ha h icouning op a h arlir of conrac mauriy T or h im of fun xhauion x. In gnral i i no poibl o carry ou h ochaic ingraion in clo form o ha numrical mho, uch a mon carlo imulaion, mu b u hn prci rul ar n. Hovr, i i poibl o unran h rlaionhip bn h xac rul 4 abov an h linar mol crib in cion 3 of hi papr by noing ha h linar mol ilf can b vi a h man of a ochaic ingral H K E E min T, x min T, x r min T, x r ε Subracing quaion 5 from 4 g inar min T, x,, SC κ λg f 5 g,, H K E E min T, x min T, x r min T, x r max0, SC. X x x x K x 6 Thi rul ho ha hr ar o ourc of rror in h linar oluion 1. Th rror hich ari from h bounary coniion a 0.. Th rror hich ari from h cah valu floor. oh of h rror bcom mallr a h accoun valu ri o can conclu ha h linar mol i h larg accoun balanc limi of h xac rul. Iu uch a fun xhauion an cah valu floor mak h xac oluion of h valuaion problm omha non-linar bu h iu bcom l rlvan a h accoun balanc g largr an h problm bcom linar. Unr h coniion h inigh gain from h uy of linar mol ill apply. Whn h accoun balanc i mall h naur of h rplicaing porfolio can chang. rom quaion 6 can ha a ky iu ill b h rlaionhip bn h xhau valu Xx an h amoun naurally provi by h linar mol Kx a h poin of fun xhauion. If X>K for all policy uraion can xpc ha h 1

23 linar mol ill unr rrv a lo accoun valu an if X<K h linar mol ill b conrvaiv. Th concluion can b valia ih numrical xampl. 5 Concluion Th papr ha ui h financial conomic of a impl quiy inx Univral if policy. Th main rul i h icuion of h linar mol hich hav hon o b a goo approximaion in h larg accoun balanc limi. Th analyic racabiliy of h linar mol allo u o unran h ky iu unrlying h conrucion of a rplicaing porfolio of quii an b inrumn. Th main iu riving h quiy rquirmn i h pra rucur or parn of pra rla loa an xpn. Db inrumn ar rquir o al ih any mimach bn acual moraliy an xpn an h amoun rla from h invmn i of h conrac hn charg ar uc from h policyholr accoun valu. Univral if prouc r originally ign o pa h invmn rik of a lif inuranc hrough o h policyholr. Whn vi from h prpciv of a rropciv accouning ym i oul appar ha an inurr can achiv hi pa hrough goal by inving o a o mach policyholr accoun balanc. Th propciv financial conomic vipoin akn in hi papr ho ha hi n no b h ca. Sinc mo prouc ar ign o hav mor pra rla loa han pra rla xpn hy uually hav quiy rquirmn ha ar l han h accoun valu. Th common inury pracic of maching accoun valu hrfor la o favorabl mimach hn h quiy mark ri an conomic lo hn h mark clin. inally, i houl b no ha hil h icuion in hi papr ha focu on quiy inx prouc h main concluion rmain vali for any form of univral lif. Th rul of h linar mol o no pn on any aiical propry of h rurn gnraion proc. Thu h inigh rgaring h rplicaing porfolio in h larg accoun balanc limi ar vali for any form of univral lif.