RISK PROFILES OF LIFE INSURANCE PARTICIPATING POLICIES: MEASUREMENT AND APPLICATION PERSPECTIVES

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1 122 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 2007 RIK PROFILE OF LIFE INURANCE PARTICIPATING POLICIE: MEAUREMENT AND APPLICATION PERPECTIVE Albina Olando *, Massimiliano Poliano ** Absac The pape deals wih he calculaion of suiable isk indicaos fo Life Insuance policies in a Fai Value conex. In paicula, aim of his wok is o deemine he quanile eseve fo Life insuance Paicipaing Policies. This goal poses boh mehodological and numeical poblems: fo his eason he pape discusses boh he choice of he mahemaical models and he calculaion ecnique. Numeical applicaion illusaes he esuls. Key wods: Paicipaing policies, Fai Value, Quanile Reseve, Mahemaical Reseve. JEL classificaion: G22, G28, G Inoducion A he end of Mach 2004, The Inenaional Accouning andad Boads (IAB) issued he Inenaional Financial Repoing andad 4 Insuance Conacs (IFR 4) (e.g. [8]), poviding, fo he fis ime, guidance on accouning fo insuance conacs, and making he fis sep in he IAB s pojec o achieve he convegence of widely vaying insuance accouning pacices aound he wold. In paicula, on he one hand, he IFR 4 pemis an insue o change is accouning policies fo insuance conacs only if, as a esul, is financial saemens pesen infomaion ha is moe elevan and no less eliable, o moe eliable and no less elevan ; on he ohe hand i pemis he inoducion of an accouning policy ha involves emeasuing designaed insuance liabiliies consisenly o eflec cuen inees aes and, if insue so elecs, ohe cuen esimaes and assumpions. Thus IFR 4 give ise o a poenial eclassificaion of some o all financial asses a fai value hough pofi and loss when an insue changes accouning policies fo insuance liabiliies. The IAB defines he Fai Value an esimae of he pice an eniy have ealized if i had sold an asse o paid if i had been elieved a liabiliy on he epoing dae in an am s lengh exchange moivaed by nomal business consideaions. In paicula, he IAB allows fo using sochasic models in ode o esimae fuue cash flows. In he acuaial pespecive, he inoducion of an accouning policy and of a fai valuaion sysem implies ha he Fai Value of he mahemaical eseve could be defined as he ne pesen value of he deb owads he policyholdes evaluaed a cuen inees aes and, evenually, a cuen moaliy aes. In acuaial lieaue, many papes deal wih models fo he Fai valuaion of insuance liabiliies; in paicula Milevsky and Pomislow (e.g. [10]) popose a sochasic appoach o model he fuue moaliy hazad ae in insuance conac wih opion o annuiise, Bacinello (e.g. [1]) deals wih he poblem of picing a guaaneed life insuance paicipaing policy, Balloa and Habeman (e.g. [2]) give a hoeical model fo evaluaing guaaneed annuiy convesion opion. In his field, ou pape aims a giving a conibuion o he quesion of calculaion of suiable isk indicaos fo he mahemaical eseve of a guaaneed life insuance paicipaing policy in a fai value conex. The pape is oganised as follows: secion 2 gives a suvey abou he applicaion of he quanile eseve o acuaial liabiliies. In secion 3 he mahemaical fomalizaion is inoduced. Finally, in secion 4, a numeical evidence is offeed. Albina Olando, Massimiliano Poliano, * Isiuo pe le Applicazione del Calcolo Mauo Picone, Ialy. ** Univesià of Napoli Fedeico II, Ialy.

2 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, The quanile eseve and he acuaial liabiliies The quanificaion of a liabiliy fai value can be appoached inoducing he eplicaing pofolio, ha is a pofolio of financial insumens giving oigin o a cash flow maching ha one undelying he liabiliy iself. Of couse, in he case of liabiliy aded in an exising make, he fai value coincide wih he make value iself. The fai valuaion of insuance liabiliies, since consideing cash flows depending on he human life, and so no ading in an exising make, can be consideed exising in he economic ealiy, as saed in Buhlmann (e.g. [6]), and can be measued by means of a financial insumens pofolio. Wihin his scenaio, i is possible o inoduce quaniaive ools, such as he quanile eseve. Indicaing by W() he financial posiion a ime, ha is he sochasic mahemaical eseve of a life insuance conac, o a pofolio of conacs, he quanile eseve a confidence level,, is expessed by he value W P W W in he following equaion As one can see, he quanile eseve is a heshold value in he sense ha in (1 - )100%, W() is smalle o equal o he quanile eseve. This epesenaion gives ise o a make consisen value of he insuance liabiliy. 3. The mahemaical model Le us conside an endowmen policy issued a ime 0 and mauing a ime, wih iniial sum insued C 0. Moeove, le us define ; 1,..., and x ; 1,..., he andom spo ae pocess and he moaliy pocess especively, boh of hem measuable wih espec o he filaions F and F. The above menioned pocesses ae defined on an unique pobabiliy space F,,,, P such ha F F F (e.g. [10] and [2]). Fo he paicipaing policy, we assume ha, in case of single pemium, a he end of he -h yea, if he conac is sill in foce, he mahemaical eseve is adjused a a ae defined as follows (e.g. [1]) i max, 0 1,...,. (1) 1 i The paamee, 0 1, denoes he consan paicipaing level, and indicaes he annual eun of he efeence pofolio. The elaion (1) explains he fac ha he oal inees ae cedied o he mahemaical eseve duing he -h yea, is he maximum beween and i, whee i is he minimum ae guaaneed o he policyholde. ince we ae dealing wih a single pemium conac, he bonus cedied o he mahemaical eseve implies a popoional adjusmen a he ae also of he sum insued. Accoding o Bacinello (e.g. [1]), i is assumed ha if he insued dies wihin he em of he conac, he benefi incease of an addiional las adjusmen a he end of he yea of deah. Denoing by C, 1,...,, he benefi paid a ime if he insued dies beween ages x+-1, x+ o, in case of suvival, fo, he following ecusive elaion holds fo benefis of successive yeas C C 11 1,...,. The ieaive expession fo hem is insead C C0 1 1,...,, j1 whee we have indicaed by he eadjusmen faco

3 124 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, ,...,. j1 j In his conex, as he eliminaion of he policyholde can happen in case of deah in he yea 0, o in case of suvival, he liabiliy bone ou by he insuance company can be expessed in his manne L W 0 C 1 1Y x C J x, (2) whee Y 1 1 x 0 e if -1 Tx 0 if 0 Tx, J x. 0 ohewise e T x In he pevious expession Tx is a andom vaiable which epesens he emaining life- 0 udu is he accumulaion funcion of he spo ae. ime of an insued aged x, 3.1. Financial and moaliy scenaio The valuaion of he financial insumens involving he policy will be made by assuming a wo faco diffusion pocess obained by joining Cox-Ingesoll-Ross (CIR) model fo he inees ae isk and a Black-choles (B) model fo he sock make isk; he wo souces of unceainy ae coelaed. The inees ae dynamics ; 1,2,... is descibed by means of he diffusion pocess d f, d l, dz whee f, is he dif of he pocess, l, (3), is he diffusion coefficien Z is a andad Bownian Moion; in paicula, in he CIR model, he dif funcion and he diffusion coefficien ae defined especively as (e.g.[7]) f k l,,,, whee k is he mean eveing coefficien, is he long em peiod nomal ae, is he spo ae volailiy. I mus be poined ou ha fo picing inees ae deivaives, he Vasicek model is widely used. Neveheless, his model assigns posiive pobabiliy o negaive values of he spo ae; fo long mauiies his can have a elevan effec and heefoe he Vasicek (e.g. [12]) model appeas o be inadequae o value life insuance policies. Clealy, on he fai picing of ou policy, he specificaion of he efeence pofolio dynamics is vey impoan. The diffusion pocess fo his dynamics is given by he sochasic diffeenial equaion d f, d g, dz, (4) whee denoes he pice a ime of he efeence pofolio, is a andad Bownian Moion wih he popey CovdZ, dz d R. ince we assume a B ype model (e.g.[3]), we have f,, g,, whee is he coninously compounded make ae, assumed o be deeminisic and consan and is he consan volailiy paamee. Z

4 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, Fo he dynamics of he pocess x: ; 1,2,..., we popose o choose a model based on he Lee Cae mehodology. A widely used acuaial model fo pojecing moaliy aes is he educion faco model fo which y : y:0 RFy, subjec o RF( y,0) 1 y, whee y: 0 is he moaliy inensiy of a peson aged y in he base yea 0, y: is he moaliy inensiy fo a peson aaining age y in he fuue yea, and he educion faco is he aio of he moaliy inensiy. I is possible o age RF, in a Lee Cae appoach, y: 0 being compleely specified (e.g.[11]). Thus, y: 0 is esimaed as ˆ e, y:0 d y: y: e y : indicaes he maching pe- whee d y : denoes he numbe of deahs a age y and ime, and son yeas of exposue o he isk of deah. Taking he logaihm of equaion (3) we have log y : log y: 0 log RFy, s.c. log RF y,0 0. Defining y log y:0 logrfy, yk he Lee Cae sucue is epoduced (e.g.[9]). In fac he Lee Cae model fo deah aes is given by ln m k, y y y y whee my denoes he cenal moaliy aes fo age y a ime, y descibes he shape of he age pofile aveaged ove ime, k is an index of he geneal level of moaliy while y descibes he endency of moaliy a age y o change when he geneal level of moaliy k changes. y denoes he eo. In his famewok, fo ou puposes, wih y=x+, one can use he following model fo he ime evoluion of he hazad ae x k x: x: 0e. 4. Numeical poxies fo he quanile eseve via simulaion pocedues 4.1. The poblem backgound In his secion we pesen a simulaion pocedue o calculae he quanile eseve, poviding a pacical applicaion of he mahemaical and accouning ools pesened peviously. In paicula ou objecive is o quanify he wo ciical values of he quanile eseve W *() and W *(). The compuaion of he quanile eseve values equies he knowledge of he disibuion of W(). To his aim we use a Mone Calo simulaion pocedue which, as well known, is ipically employed o model andom pocesses ha ae oo complex o be solved by analyical mehods. Moeove he use of simulaion echniques allows o es in an easie way he effecs of changes in he inpu vaiables o in he oupu funcion. As a fis sep, as usually done in simulaion pocedues, we develop he saemen of he poblem giving he mahemaical elaion beween he inpu and oupu vaiables. The mahemaical model should be ealisic and pacically solvable. On he basis of he model pesened in secion 3, he oupu is given by he financial posiion of he insue a ime, W(), and he inpu vaiables ae given by he ime of valuaion, he suvival pobabiliies and he em sucue of ine-

5 126 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 2007 es aes, while he efeence pofolio dynamics, as peviously saed, is consideed deeminisic and consan. In his ode of ideas, we assume ha he bes pedicion fo he ime evoluion of he suviving phenomenon is epesened by a fixed se of suvival pobabiliies, oppounely esimaed aking ino accoun he impoving end of moaliy aes. As a consequence, in ou applicaion he fis wo inpus ae deeminisic while he andom inpu is epesened by he model descibing inees aes disibuion. In ou case, he esimaion of he isk-adjused mean eveing paamee is no needed, since is value has no effec on he eseve deeminaion. The example of applicaion we popose is efeed o a life insuance paicipaing conac. In paicula we quanify a he beginning of he conac he wo ciical values W *() and W *() of he eseve disibuion. The oupu of he simulaion pocedue is a sample which gives N values fo W(), being N he numbe of simulaions. In ode o pefom he simulaion pocedue i is necessay o ge he discee ime equaion fo he chosen DE descibing he evoluion in ime of he inees aes (3). We choose he fis ode Eule s appoximaion scheme, obaining he following sample pah simulaion equaion: ( k=1,2,..,t, (5) k ( k1) ( k1) ) ( k1) whee k. N(0,1) This appoximaion scheme is chaaceized by an easy implemenaion and a simple inepeaion of he esuls. The disceized pocess we conside can be epesened by he sequence, 2,..., k, whee k is he numbe of ime seps, is a consan and T is he ime hoizon. The following simulaion pocedue is caied ou in ode o gain a sample of N values of W(): a) geneaion of T pseudo-andom values k N(0,1) ; using he T b) compuaion of one simulaed pah fo he sochasic inees ae values obained in sep (a); c) compuaion of one value of he eseve on he basis of he pevious esuls. The simulaion pocedue will be epeaed N imes o gain N values fo W(). A his poin ou pupose is o quanify he wo ciical values of he eseve disibuion W *() and W *(). ince he eseve is a liabiliy, we ae ineesed in he igh hand ail of he disibuion. In he following, we popose a numeical applicaion consideing wo diffeen values of N. Being he disceized CIR model composed by a deeminisic pa and by a sochasic one k N(0,1), accoding o he Glivenko-Canelli heoem, we expec ha he empiical disibuion of W() asympoically ends o a nomal one Numeical esuls The numeical example we popose efes o a paicipaing conac issued on a peson aged 40 wih ime o mauiy 20 yeas. We assume fo he CIR pocess = , = and he iniial value 0 =0.0279, esimaed on he 3-monh T-Bill Januay 1996-Januay 2006, 0. 03, fo he ime evoluion of he efeence fund. Fo he coelaion coefficien we adop a slighly negaive value 0.06 coheenly wih he lieaue fo he Ialian ock make. Fo he suvival pobabiliies we use he moaliy Ialian daa fo he peiod of o evaluae he pojecion of he moaliy faco in a Lee Cae conex. k k

6 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, We epo he esuls obained by means of he pocedue poposed in secion 4.1 consideing N=1000, We show ha, inceasing he numbe of simulaions N, we obain a moe significan sample of W() values, geing moe exac infomaion abou is disibuion. In Table1 he chaaceisic values of he simulaed disibuion of he eseve ae epoed, coesponding o he numbe of simulaion pahs indicaed in he column. Reseve disibuion esuls obained fo N=1000 and N=10000 simulaion pahs Table 1 N Mean Median Maximum Minimum Kuosis kewness Table s 1 conens confim he asympoic behaviou of he empiical disibuion of he andom vaiable W(). Now, as aleady ecalled, he Glivenko-Canelli heoem is veified, in he sense ha, as we can easily obseve, as N inceases W() appoximaes a nomal disibuion. In paicula, in he case of N=10000, kuosis akes he value and skewness akes he value I is well known ha a nomal vaiable has a kuosis of 3 and a skewness equals zeo, heefoe he obained values in he case of N=10000 can be consideed accepable. Moeove, we ge he following esuls: Jaque-Bea es fo N=10000 Table 2 J-B es Pobabiliy As well known, he J-B (e.g. [3]) is a saisic fo esing whehe he seies is nomally disibued. The JB es is known o have vey good popeies in esing fo nomaliy; i is easy o compue and i is commonly used in he egession conex in economeics (e.g. [5]). The es saisic measues he diffeence of he skewness and kuosis of he seies wih hose fom he nomal disibuion. Unde he null hypohesis of a nomal disibuion, he J-B saisic is disibued as a chi-squae wih wo degees of feedom ( 2 (2) ). The epoed pobabiliy is he pobabiliy ha he J-B saisic exceeds (in absolue value) he value unde he null hypohesis. A small pobabiliy value leads o he ejecion of he null hypohesis of a nomal disibuion. In ou case, being he pobabiliy equal o , we can accep he hypohesis of nomal disibuion of W(). The asympoic behaviou of he empiical disibuion is shown gaphically oo, by means of he hisogams and he Quanile-Quanile plos shown below fo each value of N. As we can obseve looking a Figue 2. R() well appoximaes a nomal disibuion when N=10000.

7 128 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 2007 Fig.1. Hisogams N=1000, N=10000 Fig.2. Quanile Quanile plos N=1000, N=10000 Quanile eseve fo N=1000 and N=10000 Table 3 N W*(99%) W*(95%) Finally Table 3 shows he wo ciical values of he quanile eseve calculaed fo N=1000 and aking ino accoun ha he mahemaical povision is a liabiliy and ha he ciical values lie in he igh-hand ail of he disibuion. The diffeence beween he W* values and he M[W], he mean value of he mahemaical eseve, can be inepeed as an absolue index of he iskiness bone ou by he insue due o he unceainy abou inees and moaliy aes. Obviously, he ciical values of he quanile eseve obained by means of simulaion pahs ae moe eliable.

8 Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, Refeences 1. Bacinello A.R. (2001). Fai Picing of Life Insuance Paicipaing policies wih a minimum ines ae guaaneed. Asin Bullein, pp Balloa L. & Habeman. (2006). The Fai Valuaion poblem of guaaneed annuiy opion: he sochasic moaliy case. Insuance: Mahemaics and Economics, Vol 38, N. 1, pp Bea A., Jaque C. (1980). Efficien Tes fo nomaliy, heeskedasiciy and seial indipendence of egession esiduals: Mone Calo evidence. Economic Lee, Vol. 6, pp Black F. & choles M. (1973). The picing of opion and copoae liabiliies. Jounal of Poliical Economy, pp Bin Dog L. & Gils D.E.A. (2004). An Empiical likelihood aio es fo nomaliy. Economeics Woking Papes EWP0401Univesiy of Vicoia. 6. Buhlmann H. (2002). New Mah fo Life Acuaies. Asin Bullein, pp Cox J., Ingesoll J., Ross. (1985). A heoy of he em sucue of inees aes. Economeica, pp IAI, Januay. Inenaional Accouning andad Boad. (2004). Inenaional Financial Repoing andad 4 Insuance Conacs. 9. Lee R. (2000). The Lee Cae mehod of foecasing moaliy wih vaious exension and applicaions. Noh Ameican Acuaial Jounal, pp Milevsky M.A. and Pomislow.D. (2001). Moaliy Deivaives and he opion o annuiise. Insuance:Mahemaics and Economics, pp Renshaw A.E. & Habeman. (2003). On he foecasing of moaliy educion faco. Insuance:Mahemaics and Economics, pp Vasicek O. (1977). An equilibium chaaceizaion of he em sucue. Jounal of Financial Economics, pp