The Value of non Enforcable Future Premiums in Life Insurance i. Pieter Bouwknegt. Nationale-Nederlanden. Actuarial Department.

Transcription

1 The Value of non Enforcable Fuure Preius in Life Insurance i Pieer Bouwkneg Naionale-Nederlanden Acuarial Deparen PO Box AT Roerda, The Neherlands Tel: (0031) E-ail: Pieer.bouwkneg@nn.nl This version: 13 March 2003 Absrac: The valuaion of fuure preius in life insurance us ake ino accoun ha he preius are no enforceable by he insurance copany: he policyholder can sop o pay he preius a his discreion. This can be a valuable ebedded opion in he policy. The valuaion can be based on he classical acuarial ehod, using a fixed percenage of decreens. I is also possible o assue ha he policyholder behaves econoically raional and apply a coningen clai analysis for he valuaion. The curren proposals (DSoP) for he new accouning sandards for insurance conracs leaves he choice beween boh ehods open. In his paper a odel is se up o analyze his feaure of insurance conracs. Boh ehods will be copared, and i will be seen how oraliy and policy condiions can influence he raional behavior of he policyholder. I becoes clear ha he release of profis fro a conrac will be influenced significanly by he odel used o value he fuure preius. Keywords: Life Insurance; ebedded opions; oraliy; policyholder behavior;

2 The Value of Fuure Preius in Life Insurance There is a growing ineres for he valuaion of life-insurance policies using principles fro financial econoics, ofen called fair value. Fair value for life insurance can be regarded as he inroducion of echniques fro he financial econoics in acuarial probles. Fair value is defined in he DSoP (3.15) as "he aoun for which an asse could be exchanged or a liabiliy seled beween knowledgeable, willing paries in an ar's lengh ransacion", IASB (2002). The ipleenaion of fair value will lead o changes in he values used for he balance shee and in he profi and loss accoun (P&L). A lo of aenion goes o he calculaion of he fair value of life insurance liabiliies because his is a ajor ie on he balance shee and ypical for insurance operaions. As here are no observed arke values for he insurance liabiliies, he value us be calculaed using a odel ha is consisen wih arke prices and valuaion echniques (for insance: no arbirage). For policies wih no fuure preius his sees relaively easy: only he fuure ougoing payens due o he paid up policy us be valued. These payens are coningen on he policyholder's fuure lifeie, no on he fuure sae of he econoy (disregarding he possibiliy o lapse). Wih preiu paying policies he siuaion is ore coplicaed, as he fuure insurance benefis also depend on he fuure preiu incoe due o he sae policy. The policyholder has ofen he righ o sop paying preius: in ha case he policy is convered in a paid-up policy (his conversion will be called "o pup" in he res of he aricle) and he insured aoun will subsequenly be decreased using he ariff of he original policy. In soe cases his represens a valuable opion o he policyholder, especially if he ineres raes in he arke are uch higher hen he guaraneed ineres raes in he policy. In he lieraure his opion is ofen ignored or overlooked. This akes he valuaion of he fuure preius and benefis ore difficul, as hey are dependen on he sae of he econoy. The odeling of hese ebedded pup-opions is quie coplicaed: If a policy is puped here is no righ o recoence in he fuure wih he original preiu-ariff. So he pup-opion of a policy only can be exercised once. This aricle addresses he issue of he fuure preius in fair value calculaions. Oher policyholder opions are negleced on purpose in order o clarify he analysis. The inclusion of he righ o lapse a policy is sraighforward bu does no give addiional iporan insigh. Two odels are analyzed; he firs, classical, odel assues a fixed behavior of policyholders independen fro he econoy. The second, new, odel assues econoic raionaliy for he decision o pay he preius. The odels include oraliy in is analysis, adding his ypical life insurance feaure in order o coe closer o he acuarial world. The paper is organized as follows. Firs a shor oversigh of relevan lieraure is given, and hen he insiuional fraework for he righ o sop paying preius is se ou. Then follows he odel as used for he analysis and a nuerical exaple of he odel. Hereafer we give soe applicaions in he field of oraliy selecion and he inroducion of a penaly when he policy is puped. Afer his we give soe consideraions on he real behavior of he policyholder and a replicaing invesen sraegy. We conclude wih soe final rearks. The Inernaional Accouning Sandards Board (IASB) for lised life-insurance copanies in he European Union will define he new regulaions. The curren Draf Saeens of Principles (DSoP) clearly go ino he direcion of Fair Value valuaion principles (even if his is done under oher naes like Eniy Specific Value). In he second place, soe insurance

3 supervisors (like he Duch and Danish) are also oving in he direcion of he use fair value principles for he evaluaion of he financial posiion of insurance copanies. This no only can be aribued o a desire of consisency wih he new IASB rules, bu also is ainly caused by a desire o have ools for risk easureen wih ore power hen he presen accouning. The curren accouning rules can poin o dangerous siuaions when i is already oo lae o reac. For insance a peranen downward shif in ineres raes will only be seen in he long er, when he old bonds have o be reinvesed and he coupons are lower han previously. The valuaion of ebedded opions in financial producs assues ofen raional behavior of econoic agens, even if experience can show ha he real behavior of he cliens differs fro ha. This is he sandard procedure for he valuaion of Aerican pu opions. For orgages siilar probles are encounered, as prepayen opions are in a lo of jurisdicions valuable o he clien. The valuaion of he prepayen opion using fixed behavior can ge very isleading when he behavior of cliens change (for insance due o ore aggressive copeiion) or when i is based on experience of he pas, when he pas experience did no include a sharp fall in ineres raes. In general acuaries rely ore on projecions based on own experience hen on odels assuing econoic raionaliy. Sid and Wolhuis (2001) inroduce in heir odel of a life insurance copany a lapse frequency ha is only dependen on, no on he econoic environen a ha ie. This procedure is also followed in Gerber (1990), who inroduces a decreen force only dependen on ie raher hen he econoic environen. Bouwkneg and Pelsser (2002) show for a siplified policy wih profi sharing how a fair value of his liabiliy can be calculaed. The cerain cash flows (preius, guaraneed benefis) are valued wih he er srucure of ineres as derived fro governen bonds. The valuaion of he profi sharing is done using risk neural valuaion echniques. For he paricular profi sharing under consideraion i is possible o use swapions wih well-known analyical valuaion forulae. The preiu payens are (iplicily) considered o be sure, and o siplify he analysis he auhors disregard coon life insurance feaures as oraliy and ebedded opions like he righ o lapse or pup a policy. The opion o lapse a variable annuiy is considered by Milevsky and Salisbury (2001). Their analysis boh includes oraliy and he econoy. Howeverhe produc hey analyze has an iporan feaure uncoon o a lo of insurance arkes: an annuiy ha can be lapsed. They conclude ha he opion o lapse he annuiy can be very valuable o he policyholder, and for he insurance copany i is iporan o consider he opion wih grea care. The opion has an Aerican characer, as he policyholder has he righ o lapse (only one ie) during he whole period of he annuiy. Their analysis o lapse he policy differs fro he analysis in his paper, as i only considers he ineres raes a a cerain ie, no including he opion o lapse in he fuure. In he siuaion in his aricle he ineres rae environen is ore coplex because he ineres raes of fuure (preiu-) oens are also included in he analysis. Grosen and Jorgensen (2000) consider a sylized single preiu policy wihou oraliy and wih soe kind of profi sharing. They value he profi sharing and show ha he opion o lapse he policy can be very valuable in cerain circusances. The value of he lapse opion is highly dependen fro he guaraneed ineres raehe profi sharing and he arke ineres rae. The odels we sudy are differen, as hey exclude profi sharing bu include he righ o

4 pay preius in he fuure a he curren ariff, i also includes oraliy. Also in his paper an arbirage free sochasic ineres rae environen is included. Wallace (2002) copares US-GAAP and fair value accouning. She deonsraes ha he release of profi will differ significanly under fair value copared o classical accouning rules. She considers a liabiliy consising of cash ouflows. She underlines he possibiliy o define he iniial value of a conrac o equalize he iniial single preiu, by adjusing he arke value argin (MVM). The issue of econoic raional behavior is no addressed. In he financial lieraure here exis producs wih a coparable one-sided righ o pay he preiu like a noral life insurance policy: he insallen opions ii. Wih his opion he clien pays he preiu for his opion in insallens. If he sops paying preius he opion canno be exercised anyore. So if he opion ges far ou of he oney i ay be ineresing o sop paying he insallens. I will be clear ha he oal preiu of he opion will be higher if he righ o sop paying i is included. The issuer of he opion (=receiver of he preiu) will no receive he preiu in siuaions ha are very advanageous for hi. In he oher cases he opion will be coninued, so he loses on one side. The valuaion of hese (Aerican) kind of opions is raher coplex, as siilarly boh he underlying opion and he preius us be considered, in heir uual relaion. For noral opion valuaion only he opion iself is considered. Copared o he here presened analysis he ain difference wih he aforeenioned opions lies in he inclusion of ypical insurance feaures like oraliy, he exisence of paid up policies. The principles of valuaion however can be he sae. The os iporan conribuion of his paper lies in he cobinaion of boh classical acuarial accouning ha is used for seing he preius and he calculaion of he paid up value of he policy wih financial econoic echniques for odeling he behavior of he policyholder. In doing so i ries o bridge he difference beween boh sides. A second conribuion is in he ephasis ha is laid on he one sided characer of preiu payens, as he policyholder has he opion o sop i as his discreion. This righ can be very iporan and is ofen overlooked iii. In his way he paper exends he analysis of ebedded opions in life insurance. A hird conribuion is in he view ha he own knowledge of he healh of a policyholder now also is seen as a kind of opion, because i uses inforaion asyery. A fourh poin is ha he eergence of profi in he here presened way differs draaically fro wha a he oen is hough o be he expeced profiabiliy under Fair Value accouning iv. Insiuional Fraework In os counries (for insance The Neherlands) he preius for a life insurance policy are no direcly enforceable by he insurance copany. The policyholder has he righ o sop paying preius if he wishes so; he insurance copany can no go o he cour and enforce hi o coninue o pay. The ain reason for his is ha he life insurance policy is seen as a kind of invesen fro he policyholder, and like a savings accoun or oher fors of invesenhere is no clai on he policyholder: he is free o spend his oney as he wans. In he curren version of he DSoP here are wo views on he way o rea policyholder behavior. Aricle 4.35 saes: "...should assue ha policyholders will exercise lapse decisions in he way ha is leas favorable o he insurer ". This is consisen wih he financial econoics approach as presened in his aricle. However, in 4.36 anoher possible approach is enioned: "An insurer will esiae he probabiliy of differen lapse decisions...and hen use he expeced presen value... ". This approach differs fundaenally fro he previous one, as he fuure expeced behavior of he policyholder is cenral in he

5 analysishis ehod is he acuarial approach o he proble. Oher aricles in he DSoP elaborae on he 4.36 approach as for insance he presen value of fuure preius us be disclosed. This is ipossible wih he ehod using econoic raionaliy of he policyholder. I is our view ha, especially for supervisory purposeshe "unfavorable-behavior" approach should be applied, because i is inherenly safe and i does diinish he possibiliy o influence he reserve seing process by using favorable assupions. Using he earlier enioned fair value definiion, i is defensible o use he "bes-esiae behavior" approach because in a ransacion beween willing paries he non raional behavior of he policyholders will probably be included in he ransacion price. General Model In he odel he following variables are used: P : he annual preiu ariff SP y : he single preiu as calculaed on he ariff for he insurance of one uni for a person wih age y IA : he insured aoun ä x: : he value as calculaed on he ariff of an -year annuiy (payens in advance) for a person wih age x x : he age of he insured a he sar of he policy n : he axial duraion of preiu payens : he paid up insured aoun of he policy jus before he h preiu payen : he increase in paid up insured aoun due o he h preiu payen q y : he bes esiae one-year oraliy rae for a person wih age y α : he correcion for oraliy ariff q y : he one-year oraliy rae for a person wih age y as used in he ariff p y : he bes esiae -year survival probabiliy for a person wih age y z n, : he zero ineres rae for an n-year zero loan wih no credi risk a ie D n, : he value of a n-year discoun bond wih no credi risk a ie E y : he single preiu for one uni insured of a pure endowen, payable over years for a person now aged y a ie as calculaed on arke raes and bes esiae oraliy probabiliies SP y, : he single preiu for one uni insured aoun as calculaed on arke raes for ineres and wih he bes esiae oraliy for a person aged y a ie β : he reducion in he paid up insured aoun if he h preiu is he firs preiu ha is no paid s y : he probabiliy ha he policy will be pupped for a person aged y r y : he bes esiae probabiliy ha he policy sill is in force afer years for a policyholder wih age y VB, : he value of he fuure policy benefis as easured a ie jus before he h preiu payen VP, : he value of he fuure preius jus before he h preiu payen as easured a ie IF TR, : he echnical reserves jus before he h preiu payen as easured a ie for an in-force policy, easured on he basis of fixed policyholder behavior

6 TR, : he echnical reserves for a paid up policy ha is puped afer he h preiu payen as easured a ie, easured on he basis of fixed policyholder behavior VF, : he value of he fuure preius inus he relaed benefis jus before he h preiu payen as easured a ie PP, : he value of he h preiu payen a ie disregarding he value of fuure preiu payens FV, : he value of he righ o coninue paying preius in he fuure jus afer he h preiu payen a ie r(s) : he insananeous ineres rae a ie s RV, : he value of he reducion in insured aoun if he h preiu is he firs preiu ha is no paid VP, : he value of he righ o pay he h preiu a ie IF RT, : he echnical reserves jus before he h preiu payen as easured a ie for an in-force policy, easured on he basis of dynaic policyholder behavior Tariff calculaions The calculaions of he values, insured aouns and preius are perfored on classical acuarial ehods, using ariff raes ha can differ fro arke raes and wih a general oraliy assupion. For sipliciy of he analysis only ineres and oraliy are considered, disregarding coss, orbidiy and profi sharinghough hese facors can be included. The annual preiu for he insurance will be: P = ariff ( SPx. IA) / äx: n So a ie, jus before he h preiu payenhe paid up insured aoun is: ariff ariff = ( SPx +. IA ä. P) / SP x+ : n x+ The increase in he paid up insured aoun in year ( ) is financed wih he h preiu payen: ariff = P / SPx + = + 1 Finallyhe full insured aoun IA will be financed wih he oal of n preius: n i= 1 i = IA This analysis is fully perfored using classical acuarial forulae. I is iporan o noe ha he ie does no ener in he forulae. All calculaions can be perfored a he sar of he policy; here is no influence of he real econoic environen afer ha dae on he values calculaed above. For he value of his is reasonable because i is a conracual righ.

7 Moraliy and ineres The odel uses a single paraeer for he bes esiae oraliy copared o he ariff oraliy α. If α differs fro 100% (i.e. he ariff oraliy is differen fro he bes esiae oraliy) his can be aribued o soe consideraions of he life insurance copany issuing he policy. These consideraions can include an aggressive pricing policy or he use of conservaive esiaes in order o axiize profi. I is also possible ha he policyholder has ore knowledge of his individual oraliy probabiliies. A sall α (for insance 0,1) poins o a very healhy person, a large α (for insance 2) indicaes serious healh probles. ariff q x+ = α qx+ = px ( 1 qx+ i 1 ) i= 1 The value of a cerain payen in -years ie as easured a ie is he discoun bond D, : D = z ) ( 1+, The os eleenary acuarial single preiu is defined in an analogous anner copared o he classical acuarial forulae: x = E D., p x Using he real arke ineres rae as observed a ie his is a clear breach wih he classical odel and akes he connecion wih he financial econoy. For he sake of sipliciy we resric ourselves o a pure endowen, exensions o ore coplex ypes of insurance can blur he view of he underlying processes. = n Ex+ SP, Reducion of he puped policy In he odel we include he possibiliy ha he insurer applies a reducion o he paid up aoun when he policy acually is puped, for insance calculaed on a Zillerized basis. The reason for his can be ha in his way he iniial coss are (parly) recovered, or i is a copensaion for he loss of possible fuure profiabiliy. In he odel his reducion is given by he facor β. The value of his reducion akes ino accoun he reducion in insured aoun and he hen prevailing arke raes for ineres and he acual oraliy. The value of he reducion when he h preiu is he firs no o be paid a ie is: RV = β.. SP Fixed behavior In his secion we will consider a odel ha assues a fixed behavior of he policyholder, independen fro he econoy. The behavior of he policyholders is assued o be fixed in ie, a cerain percenage of he policyholders will pup heir policy in a year. In os ebedded value calculaions a fixed lapse and pup-rae are assued for he projecion of fuure cash flows. For he policy under consideraion we assue only he possibiliy o pup he policy exiss, wih a predeerined pup-probabiliy of s x+i for a person aged x+i.

8 For convenience we inroduce he variable i r x ha defines he probabiliy ha a policy ha is in force a age x is sill in force (hence no puped) in year i+x: rx = j= 1 ( 1 q x+ j 1 sx+ j 1) In he lifeie of he policy he value of he benefis as calculaed in year a ie (jus before paying he h preiu) VB, is n 1. n px+ + + i. irx+. n ipx+ + i = i= 0 VB Dn. n 1 + i+ 1. β + i+ 1. i rx +. sx+ + i. n ( x+ + i+ 1) px+ + i= 1 The value of he fuure preius VP, jus before he h preiu payen is VP n = 1 P. i,. i i= 0 ( D r ) x+ The echnical reserves of he in force policy a ie jus before he h preiu payen TR, IF, is i IF TR = VB, VP If he policy is puped jus before he h preiuhe value of he paid up policy a ie is =. n Ex+ TR, The iniial echnical reserves a ie afer he signing of he conrac, before he firs preiu is received, TR 0, IF is: IF TR 0, = VB0, VP0 If he ariff is profiable, TR 0, IF is negaive. In his way he expeced profi of he policy is released in one ie. Noe ha a profiable policy iniially leads o a negaive liabiliyhus an asse. This is ofen considered o be undesirable, because an asse is a clai ha is enforceable. The fuure preius are no enforceable by he copany, clearly conradicing he characerisics of an asse. The value of he fuure, no ye insured, benefis inus he preius is VF, VF = VB. n Ex+ VP The value of he echnical reserves a ie + for an in-force policy can now be rewrien as IF = TR, + TR VF, If he ariff would be profiable for he copany a ie + he value of VF, is posiive, oherwise i is negaive. If he policy is puped when he ariff is profiable, a loss o he

9 copany arises equal o VF,. In his case he echnical reserves of he in-force policy does no cover he value of he paid up policy, so creaing a dangerous siuaion in case of disconinuiy of he copany. Dynaic behavior In he previous secion we assued ha he policyholder behavior was known in advance and independen fro he econoic environen. In his secion we will ake a look a a odel ha assues ha he policyholder acs dynaically in his own bes ineres in response o he econoic environen. If we assue ha he policyholder behaves in an econoic raional wayhe value of he policy us include he righ of he policyholder o ake decisions dependen on he sae of he econoy. This is fully consisen wih he valuaion principles as used in he valuaion of (ebedded) derivaives, raher han he classical acuarial approach assuing a fixed behavior like we discussed in he previous secion. The analysis of dynaic behavior of he policyholder assues econoic raionaliy of he policyholder wih regard o he decision o coninue his policy or o pup i. Therefore he valuaion is spli in wo pars: he value of he paid up aoun due o previous preiu payens and he value of he curren and fuure preiu payen. The laer exiss of he presen value (a arke raes) of he addiional paid up insured aoun and he value o ake he sae decisions wih laer preiu payens. The single preiu in his par of he analysis uses he real expeced oraliy ha off course can differ fro he oraliy in he ariff. The direc value of he addiional insured aoun can be se of o he price of i, and ha is he classical preiu P as previously calculaed. The value of fuure preius and guaranees is excluded fro his variable, so he value of he h preiu payen a ie PP, is: PP = SP,. P The value of he decision o pay preius in he fuure is defined using a recursive relaion, as in year +1 he sae decision us be ade like in year. I would be wrong o assue ha he decision o coninue he preius a ie would iply ha all he furher payens will be perfored. The forula also akes ino accoun one-year discouning and one year oraliy. FV = 1 p x+ + 1 r( s) ds Q. E ( e VP ) This forula uses he risk neural probabiliy easure Q. The funcion r(s) has a double role, as i boh is he discouning of he fuure cash flows and defines hese cash flows heselves. The value of he h preiu payen includes he righ o pup he policy. If he preiu is paid hen here is an addiion o he insured aoun ha is se of agains he preiu ha us be paid. Only when he preiu is paid you have he righ o coninue aking he sae decision a year laer. If he preiu is no paid he coss will be he reducion in he paid up insured aoun wih he facor β. The odel saes ha he policyholder will choose he axiu of boh values. The choice is ade jus before every preiu payen, so his opion is a Berudan syle opion. VP = ax( PP + FV ; RV, )

10 This forula inroduces he pup-behavior in he odel. The policyholder only will pup his policy if ha is in his econoic advanage. When he policy is in forcehe value of he echnical reserves a ie, RT, IF, is he su of he value of he paid up insured aouns and he value of he nex preiu payen IF = TR, + RT VP If here is no reducion in paid up insured aoun when he policy is puped (β=0)hen VP, can no becoe negaive. The echnical reserves RT, IF will no be lower hen he paid up value of he policy. Fuure profiabiliy of he preius is excluded fro he echnical reserves, as i would lead o a reducion in he reserves. Fuure losses however are included and lead o an increase in he reserves. In his way he value of old guaranees will gradually increase he reserves, no due o a board- decision, bu raher due o he arke price of hese guaranees. The value of he guaranees is recognized even if hese guaranees are ou of he oney. If β>0 hen i is possible ha VP, <0: whaever he policyholder decides he will lose oney in cerain circusances. If he sops paying preius he value of his paid up insured aoun will be reduced, leaving hi wih a loss. If he coninues o pay preius he ariffrae can lead o a lower value for oney hen if he invess in he arke. Only in his case he value of he echnical reserves are lower hen he value of he paid up policy (before reducion). Also i becoes apparen ha i is no possible ha a new policy direcly will lead o a profi (becoing an asse on he insurers balance shee). The payen of he firs preiu can yield a profi, bu only he profi fro ha payen will be aken ino accoun. If he ariff however is loss aking o he insurer he whole loss will be aken direcly when he policy is sold. Nuerical exaple Consider a pure endowen policy wih insured aoun IA= he policy expires afer five years (n=5). In he even he insured dies before ha dae here will be no payen fro he insurance copany o he insured person or his survivors. The policy is financed wih equal annual preiu payens as long as he policy is in force. The ariff of he policy is based on a echnical ineres of 5% and an annual oraliy rae (equal for all ages) of 1%. This leads o an annual preiu of ,72. The developen of he paid up insured aoun can be seen in able 1. Table 1: Developen of he paid up insured aoun M The iniial (=0) er srucure of ineres is depiced as a zero coupon curve using he Nelson- Siegel v inerpolaion wih paraeers ã 0 =0,062538; ã 1 =-0,013053; ã 2 =-0,034068; τ=2,5.

11 Fixed behavior The policy cash flows are fully known in advance, as he behavior is independen fro he fuure sae of he econoy. The echnical reserves for he in-force policy are dependen on he acual ineres raes ha are observed during he lifeie of he policy. Using he sandard assupions (α =100%; β =0; s x =2%) and using he aforeenioned iniial er srucure of ineres and he ariff as enioned above, we perfor he reserve calculaions for a siple policy where he preius are paid unil he end of he policy. In he odel i is assued ha par of he policy is lapsed, i is clear ha his is no possible for his siple policy. Tables 2.1, 2.2 and 2.3 show for hree differen ineres rae pahs in he fuure he value of he echnical reserves based on he odel wih a fixed pup-behavior. The ineres rae is depiced as "reszero"ha is he zerorae in he er srucure wih he sae auriy as he policy. Table 2.1: Technical reserves for a pah (1) (fixed) z n-+1,+-1 TR,+-1 VF,+-1 IF TR, ,68% 0,00 317,14 317,14 2 6,80% , , ,88 3 7,46% , , ,78 4 6,39% ,54-733, ,92 5 8,05% ,10-521, ,02 Table 2.2: Technical reserves for a pah (2) (fixed) z n-+1,+-1 TR,+-1 VF,+-1 IF TR, ,68% 0,00 317,14 317,14 2 3,09% , , ,39 3 3,14% , , ,61 4 3,67% ,20 568, ,82 5 3,29% ,77 267, ,38 Table 2.3: Technical reserves for a pah (3) (fixed) z n-+1,+-1 TR,+-1 VF,+-1 IF TR, ,68% 0,00 317,14 317,14 2 5,56% , , ,79 3 4,87% ,01-61, ,11 4 3,67% ,20 568, ,82 5 3,29% ,77 267, ,38 The hree pahs are chosen in order o show a high (1), low (2) and ixed (3) fuure ineres developen. In all he ree pahs he policy shows an iniial loss of 317,14, because he ariff ineres rae is higher hen he ineres raes in he iniial er srucure. The high pah (1) shows ha he echnical reserves ge low copared o he oher pahs because of wo reasons. Firshe high ineres raes gives a lower value o he already insured paid up value of he policy in colun TR,+ -1. Secondly, colun VF,+ -1 shows ha he value of he fuure expeced preius is high for he copany, as he ariff conains a lower reurn o he cusoer. Noe ha puping of he policy becoes dangerous for he copany, because he echnical reserves in colun TR,+ -1 IF are no sufficien o cover he paid-up policy. The low pah (2) gives exacly he opposie picure of he high pah, as he low pah has higher echnical reserves and because he losses due o he fuure preius wih a relaively high

12 guaranee are direcly reserved for. The ixed (3) pah shows ha i is possible ha during he lifeie of he policy he value aribued o he fuure preius changes of sign. Dynaic behavior The odel for he fuure ineres raes is based on he Hull Whie vi rinoial ree. This odel was chosen as i is arbirage free, widely known, easy o ipleen in a Microsof Excelspreadshee and relaively easy o undersand. I is as well very pracical for he ipleenaion of decision rees, as one node describes a full zero-curve. A ajor disadvaage of he Hull Whie odel is he possibiliy of negaive ineres raes in he fuure. In he curren lieraure on ineres rae derivaives ore coplex ineres rae odels are widely used. This paper however focuses no on he exac price of an ineres rae derivaive bu raher is abou he inclusion of fuure possible ineres raes in he price of an insurance conrac. The ineres rae ree is buil using seps of one onh; his leads o soe discreizaion errors ha see o be accepable. Furher paraeers used in he odel are a=0,1 and σ=0,01. The echnical reserves RT 0, for his policy (=iniial loss o he copany) is 780,48. For he hree previously analyzed pahshe value of he echnical reserves and he value of he curren preiu payen a ha oen is shown in ables 3.1, 3.2 and 3.3. The siuaion is fully coparable wih he previous analyzed siuaion, as he policyholder here also coninues o pay his preius, even if ha is no raional. The sae paraeers for he policy are used as in he previous analysis. The variable RT, can be calculaed as RT, =TR, +ax(pp, +FV, ;0). Table 3.1: Technical reserves for a pah (1) (dynaic) z n-+1,+-1 TR,+-1 PP,+ -1 FV,+-1 RT, ,68% 0 169,04 611,94 780,48 2 6,80% , ,90 3, ,56 3 7,46% , ,46 0, ,63 4 6,39% ,54-495,95 2, ,54 5 8,05% ,10-521, ,10 Table 3.2: Technical reserves for a pah (2) (dynaic) z n-+1,+-1 TR,+-1 PP,+ -1 FV,+-1 RT, ,68% 0 169,04 611,94 780,48 2 3,09% , , , ,52 3 3,14% ,27 895,94 706, ,70 4 3,67% ,20 409,96 168, ,86 5 3,29% ,77 267, ,38 Table 3.3: Technical reserves for a pah (3) (dynaic) Z n-+1,+ -1 TR,+-1 PP,+ -1 FV,+-1 RT, ,68% 0 169,04 611,94 780,48 2 5,56% ,50-448,51 83, ,50 3 4,87% ,01 6,77 131, ,72 4 3,67% ,20 409,96 168, ,86 5 3,29% ,77 167, ,38

13 In he high pah (1) only he firs preiu is raionalhe oher preius are loss aking for he policyholder. This can be seen as ax (PP+FV;0)=0 for all he preius excep for he firs one. Pah 2 is in a low ineres rae environen and he ariff ineres guaranee is in he oney as can be seen fro he posiive PP+FV. This is he reason ha all he preius are profiable o he clien. Jus before he second preiu payen he value of he echnical reserves are for a large par (12%) due o he possibiliy o pay he following preiu. Table 3.3 deonsraes ha for a ixed pah (3) i is possible ha irraional behavior (he second preiu payen is irraional) can be becoe profiable o he policyholder, if he is lucky. There is a clear relaion beween VF, in he fixed behavior odel and VP, (=ax(pp, +FV, ;0)) in he dynaic behavior odel. If he value of he fuure preius in he fixed odel is negaivehan he value of he coing preiu in he dynaic odel will (in general) be nil, as he dynaic odel assues ha he policyholder will no pay he preiu if i is loss aking for hi. The fixed odel assues ha he policyholder is no aware of he loss he is aking and will coninue o pay. Moraliy selecion The ariff of he policy assues a cerain oraliy paern for he insured. In general he insured will know ore abou his healh han he insurance copany, creaing an asyeric siuaion. Ofen insurance copanies will ry o encouner his proble wih a procedure o ask for healh checks when he policy is sold. However, during he policy he healh of he insured can change and ha akes i for he policyholder possible o decide wha o do wih he policy dependen on he own healh siuaion. The insurer can no reac o his, insurance is ean o give his proecion. Fixed behavior The fixed odel does no allow for he possibiliy ha he knowledge of he own oraliy probabiliies influences he behavior of he policyholder. Though he relaion beween he oraliy facor and he iniial reserves will no be linearhere will be a seady increase in he iniial reserves when he oraliy facor goes down. In able 4 he relaion beween he oraliy facor and he iniial reserves are given for he fixed odel. The oher assupions are he sandard assupions (β =0, s x =2%). Table 4: Iniial echnical reserves for differen oraliy levels (fixed) α TR 0, 0% 2.692,13 50% 1.489,01 100% 317,14 200% ,95 400% ,89 The relaion beween á and TR 0, is clear: a higher oraliy (higher á) for a pure endowen is in he ineres of he copany, as he probabiliy ha he policy us be paid ou decreases. The negaive TR 0, indicaes a high iniial profi of he policy. Dynaic behavior In he policy considered here i is clear ha he policyholder will pup he policy if his healh siuaion deerioraes very uch. On he oher side: if he policyholder feels very healhy he will be ore inclined o pay he preius, as he hopes he will "ouperfor" he oraliy rae in he ariff.

14 The iniial echnical reserves of he policy are clearly influenced by he real oraliy versus he ariff oraliy. In able 5 he iniial value of he policy o he policyholder (before he firs preiu payen) is given for differen α. The oher paraeers are he sandard ones. Table 5: Iniial echnical reserves for differen oraliy levels (dynaic) α RT 0, 0% 2 809,73 50% 1 725,37 100% 780,48 200% 0,00 400% 0,00 The able akes clear ha if he insured has a oraliy rae ha is double he rae fro he ariff (α=200%) hen he policy is no profiable for hi. However, if he is very healhy (α=50%) hen he policy is good for hi. If he is fully sure ha he will survive he whole period (α=0%) hen he policy even generaes a higher profi. The relaion beween he fixed and he dynaic odel is clear. The posiive values of he echnical reserves are in he sae agniude for á=0%; 50% and 100%. For higher á he raional behavior of he policyholder is differen fro he expeced behavior, because he will no pay he preius when his is loss aking for hi, leading o a nil echnical reserve for he dynaic behavior when he reserves becoes negaive in he fixed odel. The opial behavior fro he policyholder can be drawn in a graph wih on he horizonal axis he preius and on he verical axis he ineres rae level. For differen oraliy levels α a line is drawn ha is known as he early exercise boundary. Under he line i is raional o coninue paying preius, above he line i is beer o pup he policy. Graph 1 for insance akes clear ha wih α=200% i is for he hird preiu raional o pay he preiu only when he ineres rae for he zero for he res of he policy-period is below 4%. Graph 1: Early exercise boundary and oraliy á Early exercise boundary and oraliy 6.00% reszero 5.00% 4.00% 3.00% 2.00% 1.00% 50% 100% 200% 400% 0.00% preiu nuber

15 The graph illusraes ha he ineres dependency of he policyholder behavior changes wih oher facors, i.e. oraliy. Though boh facors are independen fro each oher (in he odel as well as in realiy) he behavior is influenced by boh joinly. In a low ineres environen he ineres guaranee is ore valuable copared o a high ineres environen. In order o keep he value of he guaranee he raional policyholder is willing o accep soe oraliy losses, as long as he oal value of boh resuls is posiive. The decision o pay he preiu is ade on he basis of he oal value of he policy, no jus he ineres or oraliy par. Reducion in insured aoun The odel allows for a reducion in he paid up insured aoun when he policy is puped. Ofen his reducion reflecs he par of he acquisiion coss no ye earned back. In ha case he reducion faco β declines when (he nuber of preius acually paid) increases. For a new policy he deferred acquisiion coss end o be largehis is parly recovered by applying a high β. Typical values for β are in he 2%-4% range. For reasons of sipliciy all calculaions in his aricle are perfored using a fla β =β. The facor β is iporan for he odeled behavior of he policyholder. If a nuber of preius are paidhe paid up insured aoun is relaively high copared o he preius, so he policyholder will be ore inclined o coninue paying his preius as he value of he reducion increases. In he iniial echnical reserves a higher β will resul in lower echnical reserves in boh odels. The facor β is a kind of erinal bonus when all he preius are paid, i can be regarded as a very siplified way of including a spefic for of profi-sharing in he odel. Fixed behavior The iniial value of he policy is influenced by boh β and he esiaed pup-probabiliy s x. In able 6 for differen β and s x he iniial value of he echnical reserves using he fixed behavior is presened. There are wo "absurd" siuaions added: a β of 100% eaning ha he policy is canceled when one preiu is no paid. Anoher absurd siuaion is a bonus when he policy is puped (β=-10%), so giving a serious siulaion no o pay he preiu (especially he las preiu). Table 6: Reducion facor â and TR 0, (fixed) s x  0% 2,5% 5% 100% -10% 0% 319,92 319,92 319,92 319,92 319,92 2% 317,14 242,06 166, ,08 617,46 5% 312,86 136,17-40, , ,64 10% 305,48-13,43-332, , ,13 The able clearly indicaes ha he pup-probabiliy and he reducion facor boh influence he iniial echnical reserves (=iniial loss o he copany). The colun wih â=0 depics a slighly decreasing iniial valueha can only be aribued o less preius received. As he ariff is iniially loss aking, fewer preius lead o a lower iniial loss. Wih higher reducions in he puped value his is clearly o he advanage of he copany. The colun wih he 100% shows ha he policy becoes pup suppored: a higher pup-rae leads o a higher value for he copany. The siuaion wih â=-10 (i.e.: he policyholder receives a bonus when he policy is puped) shows ha in his siuaion he copany will be afraid of a higher pup-percenage.

16 Dynaic behavior Wih â i is possible ha he echnical reserves of he policy will be lower han he paid up value. Consider for insance he siuaion where a he second preiu payen he zerorae is 6,80%, β=2,5%, 4 p x+1 =96,06%. The value of he preiu now can be calculaed in hree subpars (here can be rounding differences in he calculaions): PP 1,+1 = ,86 * 96,06% * e -4*6,80% -/ ,72 = ,90 FV 1,+1 = 764,71 RV 1,+1 = 2,5% * ,01 * 96,06% * e -4*6,80% = 410,19 VP 1,+1 = ax( , ,71; -410,19) = - 410,19 RT 1,+1 = ,01 * 96,06% * e -4*6,80% -/- 410,19 = ,41 In his exaple he loss of he pup-reducion â is saller for he policyholder hen he loss ha would be ade by paying he preius plus he righ for he fuure FV. The odel assues hus ha he policy will be puped. Table 7 shows he influence on he iniial value of he policy for differen β. Table 7: Reducion facor β and value for he policyholder (dynaic) β RT 0, 0% 780,48 2,5% 493,31 5% 372,71 100% 229,41-10% 6 432,78 I is enlighening o draw he relaion beween he raional behavior a a cerain poinhe hen prevailing ineres raes and he value of β. In graph 2 he early exercise boundary for he five preiu payens are drawn for differen β. Graph 2: Early exercise boundary and paid up reducion ß Early exercise boundary and paid up reducion 12.00% reszero 10.00% 8.00% 6.00% 4.00% 2.00% 0% 2.50% 5% 100% -10% 0.00% preiunuber

17 The early exercise boundary for he hree siuaions wih β=0%, 2,5% and 5% indicaes ha a higher β leads o a higher early exercise boundary, so decreasing he probabiliy of policyholders leaving he policy. For β=0% he boundary does no differ uch fro he ariffineres (5%), indicaing ha he loss of he righ o pay in he fuure preius is relaively low in ha area. Wih a very high β i is unlikely ha he raional policyholder will pup his policy. In he case where he odel indicaes ha he will never pup he policy he value of he policy equals he value of a policy wihou he righ o pup; he opion is iaerial. This consiues an ineresing check on he resuls. In he odel ha was buil here reained soe differences even when here was no raional oen o leave he policy. These differences are aribued o discreizaion errors ha arise in he Hull-Whie odel. In he graph he siuaion wih β=100% deonsraes ha he early exercise boundary sars a he lowes poin fro all he lineshis can be aribued o a low iniial value of he policy. If he firs payen is done hen he policyholder will pracically always coninue wih he policy, because he loss of all he paid up value will be larger hen a loss due o a difference beween he ariff and he acual ineres rae. The siuaion wih a negaive β gives an enirely differen picure. The early exercise boundary is relaively high, indicaing ha he payen of he preius is in general profiable for he policyholder. Only a he las preiu he siuaion changes: now i is very profiable o pup he policy. This also influences previous preius: he addiional paid up insured aoun fro a preiu payen will be increased by 10% if he policy is puped jus before he las preiu payen. Policyholder oives In he odel wih he fixed behavior he policyholder is assued o be consan in his behavior. On he basis of pas experience fro he copany i is possible o esiae he pupparaeer. I is possible ha his value will change in he fuure, because due o a differen environen policyholders can change. For insance, a ore aggressive salesforce fro oher copanies can lead o an increase in he nuber of cliens shopping around. This will be ore pronounced when copeiors offer policies wih a higher guaraneed inereshis will be he case when he ineres increases. A general econoic slowdown will lead o a saller nuber of policyholders being able o pay he preius. Dependen on he circusances, i can be possible ha a higher pup-rae will lead o financial probles for he copany. In cases when he echnical reserves of he puped policy is higher hen he echnical reserves for he in force policy, an increase in pup-rae will lead o an increase in echnical reserves ha can ge dangerous. The posiive values of he fuure preius ha are expeced o coe do no aerialize. The odel wih he dynaic behavior assues econoic raional behavior fro he policyholder. In general he policyholder will have oher oives o pay his preius, like he desire for insurance proecion and fiscal siuli. He also can have anoher percepion of he sae of he econoy. The dynaic odel does no include ha kind of oives and does no ry o odel he acual behavior fro he policyholder. The dynaic odel akes i ipossible ha acual behavior fro he policyholder will have negaive repercussions for he insurer, as he os disadvanageous behavior already is included.

18 If he policyholder pays his preius when his is expeced in he odel no addiional charge or profi arises. If in ha siuaion he preiu is no paid hen a profi arises for he insurer. The opposie also applies: if he odel does expec ha he policy will be puped and he policyholder sops paying his preiushen no profi or loss coes up. If he preiu is paidhen he insurer akes a profi RI,. This profi can be hough of as caused by irraional behavior fro he policyholder. The value of he evenual resul is a every node RI = ax( PP + FV ; RV ) / in( PP + FV ; RV, ) Hedging he policy Boh odels have a differen hedge sraegy. A hedge fully replicaes he exposure of he liabiliies wih a porfolio of asses. For he fixed behavior odel he hedge can be consruced using long and shor posiions in zero bonds. The fuure preius ha are "cerain" are replicaed wih shor posiions in zero bonds. All he fuure benefis o he policyholders are replicaed wih a long posiion in zero bondshere is no difference beween he replicaion of he already paid up fuure benefis and he benefis relaed o he fuure preius. When a policy is pupedhe replicaing porfolio is changed, buying back he shor posiion in zero bonds and selling he zero bonds ha are relaed o he benefis relaed o fuure preius. In general his liquidaion will no be cos neural because he fuure ineres raes will be differen fro he ineres raes iplicaed in he previous er srucure of ineres. I is possible o define a replicaing invesen sraegy for his policy when i is valued assuing dynaic behavior. I is spli in wo pars: he value of he paid up policy is invesed in zero bondshe value of he fuure preiu can be replicaed using a dynaic hedge. This posiion is self-financing: he value of he hedge porfolio auoaically equals he value of he fuure preiu. When a policy is puped when he odel does no expec ihe unwinding of he hedge will lead o a cash flow o he copany. Final rearks I is possible o use echniques fro he financial econoics o analyze siuaions ha are ypical for he life-insurance indusry. These siuaions ay include legal opions for policyholders. So i is possible o beer value hese opions and o inegrae he in he risk anageen. An invesen policy based on he uncerainy of fuure preius can boh proec he paid up insured aouns and ake ino accoun he hedging of guaranees given on fuure preius. The valuaion of fuure preius is a echnical acuarial proble bu i can have big repercussions on he eergence of profi in a Fair Value accouning environen. The eergence of profi ore closely follows he acual reasons of he profis raher hen leading o a large one off profi. The price ha us be paid lies in he reducion of ransparency of he ehod. In earlier ies acuaries were ofen hough of as being overpruden, bu in recen ies he collapse of he Equiable in he UK and he probles for pension funds all over he world

19 gave ha iage of he acuary a blow. The proposed ehod using econoic raionaliy diinishes he possibiliy for he acuary o be overopiisic; i leads o inherenly safe reserves and a ehod ha does no allow for oo uch professional judgeen. This can significanly increase he uniforiy of he calculaions. References 1 Bouwkneg, P. and Pelsser, A. (2002) Marke Value of Insurance Conracs wih Profi Sharing Journal of Risk Finance 3, Gerber, H. (1990) Life Insurance Maheaics Springer-Verlag Heidelberg 3 Grosen, A. and Jorgensen, P. (2000) Fair valuaion of life insurance liabiliies: The ipac of ineres rae guaranees, surrender opions, and bonus policies Insurance: Maheaics and Econoics 26, Gueran, S. (2002) The Coing Revoluion in Insurance Accouning Norh Aerican Acuarial Journal 6, Hull, J. (2000) Opions, Fuures, & Oher Derivaives Prenice-Hall Inernaional, New Jersey. 6 Inernaional Accouning Sandards Board (2002) Draf Saeens of Principles 7 Milevsky, M. and Salisbury, T. (2001) The Real Opion o Lapse a Variable Annuiy: Can Surrender Charges Coplee he Marke? Proceedings of Xi h AFIR Colloquiu, Pelsser, A. (2003) Pricing and Hedging Guaraneed Annuiy Opions via Saic Opion Replicaion Working paper, Erasus Universiy of Roerda 9 Sid C., and Wolhuis, H. (2001) Bedrijfsanalyse en Ebedded Value Insiuu voor Acuariaa en Econoerie, Aserda 10 Wallace, M. (2002) Perforance Reporing under Fair Value Accouning Norh Aerican Acuarial Journal 6, Waers, H. Wilkie, A. and Yang, S. (2003) Reserving, Pricing and Hedging for Policies wih Guaraneed Annuiy Opions Briish Acuarial Journal, forhcoing i The auhor wishes o hank Wies de Boer, Angela van Heerwaarden and Anoon Pelsser for heir valuable and criical suppor for earlier versions of his paper. ii Also known as Coninuaion Opion or Pay As You Go Opion, iii In classical acuarial accouning he righ o pup a policy is ofen no seen as a danger, because he preiu reserve jus before and afer he uaion reain he sae (ofen even leaving he copany wih a sall profi due o penalies). Manageen however ofen knows ha hey are loosing soe fuure profiabiliy due o he acion. The classical accouning hus has difficulies o correcly assess he influence of he acion, as i can show a profi even when fro an econoic perspecive a loss would be ore appropriae. iv I is ofen expeced ha under fair value accouning he issue of a policy will lead o an iniial profi, in he subsequen years only followed by he expeced release of he MVMhus showing only a sall run-off profi. v The general for of he Nelson-Siegel zerocurve is: r( ) = γ vi The ree is buil on he basis of chaper in Hull (2000) 0 + ( γ 1 (1 e + γ2). / τ / τ ) γ 2 e / τ