American Journal of Mahemaical and Compuer Modelling 28; 3(): -2 hp://www.sciencepublishinggroup.com/j/ajmcm doi:.648/j.ajmcm.283.2 Sochasic Ineres Rae Approach of ricing aricipaing Life Insurance olicies wih Embedded Surrender Opion Musapha Abdul-Rahaman, *, Francis Oduro, Al-Hassan Issahaku 2 Deparmen of Mahemaics, Kwame Nkrumah Universiy of Science and echnology, Kumasi, Ghana 2 Deparmen of Informaics, Regen Universiy College of Science and echnology, Accra, Ghana Email address: * Corresponding auhor o cie his aricle: Musapha Abdul-Rahaman, Francis Oduro, Al-Hassan Issahaku. Sochasic Ineres Rae Approach of ricing aricipaing Life Insurance olicies wih Embedded Surrender Opion. American Journal of Mahemaical and Compuer Modelling. Vol. 3, No., 28, pp. -2. doi:.648/j.ajmcm.283.2 Received: February 5, 28; Acceped: March 9, 28; ublished: April 8, 28 Absrac: Life insurance conracs are priced and analysed using echniques from acuarial and modern financial mahemaics, which requires ha, he condiions for he risk-neural valuaion are fulfilled and ha, a specified underlying securiy and an equivalen maringale measure mus exis. his paper analysed life insurance endowmen policy, paid by sequence of periodical premiums in Ghana wih a guaraneed minimum reurn o he policyholder. Again, his paper presens wo premium deerminaion schemes for he insurance policy, he consan premium case and he periodical adjusmen case in which boh he benefi and he periodical premiums are annually adjused in relaion o he performance of a reference porfolio. I was realized ha, wih rising guaraneed ineres rae, he rae of reurn on he reference porfolio, he premiums of he whole conrac decreased boh in he consan and he periodical adjusmen cases whiles an increase in he paricipaing coefficien and age of he insured led o an increase in he whole premium boh in he consan and periodical adjusmen cases. Also, i was revealed ha, he premium of he non-surrendered bonus opion is smaller in he consan premium case han in he periodical adjusmen case and he premium of he bonus opion in he surrendered paricipaing policy looks cheap in he consan premium case han in he periodical adjusmen case. hus, i s abou.3% and 6.95% respecively of he oal premium for he consan and for he periodical adjusmen cases. Keywords: Sochasic Ineres Rae, Surrender Opion, aricipaing olicies, Life Insurance olicy, eriodical remiums. Inroducion Insurance provides a medium hrough which coningen fuure losses are exchange for fixed premium paymens [8]. he underlying principle for he acuarial deerminaion of premium is ha, here need o be adequae on average o cover fuure loses. he equivalen principle is he consequen of his raionale as a basis for pricing insurance producs such ha he presen value of premiums equals he presen value of he expeced fuure losses. ricing boh life and non-life insurance producs originaes from he equivalen principle, however, is applicaion as observed by Biener requires divergen approaches in relaion o differen properies of risk in differen line of business [6]. In life, losses are a imes ineviable. eople migh become ill and lose heir income o pay off medical bills. Individuals or heir relaives may die of illness or accidens. All hese aciviies are subjec of risk of loss from unforeseen evens. o lessen hese burdens, insurance companies are formed o provide producs, wih he common goal of pooling relaed risks and offering a cushion o he unforeseen incidens. In expanding he coverage of life insurance producs, life insurance companies have recenly begun offering complex policies wih embedded opions. Among hem is he paricipaing policies wih ineres rae guaranee and surrender possibiliy. Bacinello defined paricipaing policy as a conrac in which he policyholder is eniled o a share of he excess profi if he realized ineres rae during he insurance period is above he assumed ineres rae []. Grosen and Jorgensen claimed ineres rae guaranees, bonus disribuion sysems and surrender opions are common feaures of a sandard paricipaing life insurance
American Journal of Mahemaical and Compuer Modelling 28; 3(): -2 policy issued in he Unied Saes, Europe, Japan and mos developing counries of which Ghana is no lef ou []. Each of hese opions conribues o he insurer s liabiliy and presen a value ha consiues a poenial hazard o company s solvency. Similar sudy by Briys and De Varenne revealed ha, many life insurance companies disregard he significance of hese opions, hereby exposing hemselves o he risk of insolvency [9]. According o Bacinello, he rules for compuing premium(s) are fixed in any case, hus fair pricing is achievable by selecing he appropriae parameers ha characerized he conrac []. Conras o a sandard financial insrumen such as pu or call opions, life insurance conracs are more complex producs ha incorporae feaures like moraliy/survival, periodical premiums, guaranee ineres rae and he righ o surrender. Wih-profis life insurance policies are conracs which include an annual minimum rae of reurn guaranee as well as a bonus disribuion schemes deermined by managemen decisions of he insurance company. hey have several feaures and guaranees such as bonus opions, surrender possibiliy, ineres rae guaranees which poses liabiliy o he insurer and consequenly consiue he possible risks o he company's solvency [3]. Modelling and pricing paricipaing policy ha incorporae all hese facors is complex and challenging. he ideal approach is o include he imporan facors and a he same ime keep he model racable [6]. As argued by Briys and De Varenne, mos life insurance companies neglec he significance of hese facors and ha exposed hem o he risk of insolvency [9]. Grosen and Jorgensen examined and priced paricipaing policy wih a guaranee minimum ineres rae and argued ha paricipaing policies should provide a low-risk, sable and ye compeiive invesmen opporuniies. heir model includes he opion o surrender and o receive he surrender value implied by a surrender charge []. However, hey were unable o presen a closed-form formula of heir bonus accoun owing o is pah dependen naure hence he adopion of Mone Carlo Mehods (MCM). Bacinello analysed life insurance paricipaing policies wih a guaraneed minimum ineres rae and consider boh he cases in which he conrac is paid by an upfron premium a issuance, and in siuaions in which i is paid by a sequence of periodical premiums annually adjused in accordance wih he performance of a reference porfolio. However, her analysis did no include a surrender possibiliy []. In her laer aricle of fair valuaion of a guaraneed life insurance paricipaing conrac embedding a surrender opion, she used Cox, Ross and Rubinsein model o deermine he fair value of he policy paid by an upfron single premium a incepion of he conrac and perform sensiiviy analysis on he conracual parameers ha characerized he conrac. She obained a closed-form relaion ha makes he conrac fair under Black, Scholes and Meron framework [2]. Jorgensen sudied he American-syle conrac wih a guaranee ineres rae using binomial laice [3]. Whereas Jensen, Jorgensen and Grosen priced he embedded surrender opion and he bonus policy by means of finie difference approach [2]. aricipaing policies wih embedded surrender opion have been valued wihin he framework of consan risk-free ineres rae. Ideally, he guaranee ineres rae offered by he conrac is more likely o change hroughou he life of he policy raher han been consan. Holders of paricipaing policies wih embedded surrender possibiliy migh surrender heir conrac o ake advanage of he higher yield in he financial marke. Surrender opions have herefore become a major concern for life insurers especially during ineres rae volailiy. Owing o he long mauriy naure of life insurance producs, if he guaranee reurn is no sufficienly high enough compared o oher forms of invesmens, policy holders may erminae heir exising policies early in order o go in for he higher yields offered in he capial marke [5]. he idea of embedding sochasic ineres raes ino he modelling and pricing of life insurance policy is no new. Milersen and ersson adoped he General Healh-Jarrow- Meron approach in a sochasic ineres raes o model he minimum guaraneed rae of reurn. hey adaped he Vasicek and Cox-Ingersoll-Ross (CIR) shor rae models o derived pricing formulae for poin-o-poin and clique syle guaraneed rae of reurn on boh sock marke reurn process and shor erm ineres rae process [6]. Briys and De Varenne considered coninuous ime valuaion case o modelled life insurance liabiliy ha accouns for boh ineres rae risk and defaul risk. heir model employs he insananeous shor rae by he Ornsein Uhlenbeck (OU) and obain a closed-form formulae of cerain life insurance liabiliy [9]. Zaglauer and Bauer deermined he risk-neural value of paricipaing life insurance policy in a sochasic ineres rae environmen. hey used wo asse marke model and considered he componens of he insurance company s asses porfolio implicily by selecing adequae volailiies and correlaion beween he asse process and he ineres rae process. heir pricing model adap he framework proposed by [4]. hey used an OU and he CIR shor rae models o model he insananeous risk-free ineres rae o deermine he risk-neural price of he paricipaing life insurance policy [9]. Furhermore, Liao, Chang and Lin priced paricipaing conracs inroduced by Bacinello embedded wih surrender possibiliy in a sochasic ineres rae model. heir sudy proposed a wo-dimensional CRR model capable of deermining he value of he surrender opion embedded in he conrac [4]. he purpose of his paper is o esimae he fair-price of life insurance paricipaing policies using a sochasic ineres rae model. his will provide insigh abou he ineracion of he disinc facors ha influences he premium of life insurance conrac and he risk ha comes along wih insurers liabiliies. 2. Maerials and Mehods of Analysis 2.. Source of Daa he moraliy daa for his paper was exraced from 28 Sociey of Acuaries (SOA) life able.
2 Musapha Abdul-Rahaman e al.: Sochasic Ineres Rae Approach of ricing aricipaing Life Insurance olicies wih Embedded Surrender Opion 2.2. Mehods of Daa Analysis his paper fixed he erm o mauriy, iniial benefi, 25 and g 24.6% as guaraneed ineres which was he bank of Ghana reasury bill rae. Furhermore, a minimum of η 5% as a paricipaion level on book value earnings was credied o he policy s reserve and i g 26% as he spo rae of he zero-coupon bond. he choice of N was informed by a daily change in he uni price of he relaive reference porfolio since here are abou 25 rading days in a year. 2.2.. he olicy remium and he Reserve Consider a sandard endowmen paricipaing life insurance policy issued a ime () and maures a ime (). Under he policy arrangemen, he beneficiary receive a specified amoun of money if he insured dies wihin he conracual period or survive he mauriy dae. Assumed in he even of deah during he conrac year (.2,, ) benefi is paid a he end of he year of deah oherwise paid a mauriy. Again, if represens he enry age of he insured a ime (), as he iniial amoun insured, payable if deah occurs wihin he firs year of he policy and as he benefi due a ime ( 2,3,,). If he policy is paid by a series of periodical premiums due a he beginning of each policy year if he insured is sill alive. he iniial premium, paid a he ime of issuance of he conrac is given by; : :!: ( + ) + ( + ) / x x i q i p ( + ) x i () Wherei >, is he annual (compounded) echnical ineres rae, /qx denoes he probabiliy ha he insured dies wihin he policy year (beween imes and ) and p x is he probabiliy ha he insured survives o ime. he probabiliies inroduced are dependen on he age, of he insured and are usually exraced from a suiable risk-neural moraliy able implied on he echnical ineres rae. From equaion (), he expeced value of he iniial benefi b, discouned from he random ime of paymen o ime zero wih he echnical ineres ( ) rae b A x :, equals he expeced value a ime, of he sequence of he periodical premiums he same rae ( ɺɺ x : ) discouned also wih a. Hence makes he policy fair a issuance on he grounds of firs order echnical bases. he benefi reserve a ime, of a policy issued a age x, ha is sill in force years laer is defined according o [8] as he excess acuarial presen value a age x + of he fuure premiums including any premium payable a age x +. his excess represens a liabiliy o he insurer and are usually calculaed a he end of each policy year. If is paid by an insured, he echnical rae could be considered as he rae of reurn credied o he policy reserve a he onse of he policy year. hus, he benefi is annually adjused, hereby resuling he dependence of he periodical premium on he performance of he reference porfolio. However, he adjusmen is done in a manner ha, he policy remains fair on he grounds of he firs order echnical bases in relaion o he residual conrac period. If he insured is alive and he conrac is sill in force, and R ( R + ) as he mahemaical reserve of he policy a ime, shorly prior o he paymen of he periodical premiums, and (shorly afer) an adjusmen respecively. hen, given b and, : aɺɺ x+ x+ : R b A R is given by; b + i h qx + + i px + i hpx h ( ) h ( ) / + ( ) + ( ) +,,..., (2) where h /qx+ is he probabiliy of he insured dying wihin (+h) policy year (beween +h and +h ) condiioned on he of survivorship of he insured a ime, and h p x+ is he probabiliy ha he insured is sill alive a ime (+h) condiioned on he same even. 2.2.2. he Reserve, Benefi and remium Adjusmen Raes Following he compuaion of R from Equaion (2), R + is immediaely adjused a a rae δ such ha: where δ is given by; ( δ ) + R R +,,2,..., (3) ηg ig δ max,,,2,3,, (4) + ig where g represens he rae of reurn on he reference porfolio during he policy year, η is he paricipaing
American Journal of Mahemaical and Compuer Modelling 28; 3(): -2 3 coefficien beween and and ig is he (minimum) guaraneed ineres rae. Following he adjusmen of he mahemaical reserve, he oal reurn credied o he insured during he conrac year implied on he guaraneed rae riches is maximum beween i g and η g given by; { ig ηg} ( ig ) ( δ ) max, + +,,..., (5) assuming α and γ are he adjusmen raes of he benefi and he premium respecively. I is usual in pracice ha he benefi and he periodical premiums of a paricipaing policy be adjused in he same measure as he bonus rae δ, credied o he benefi reserve. herefore, he benefi and he periodical premiums are given by Equaion (6) and (7) respecively: and ( α ) b + b +,,2,..., (6) ( γ ) +,,2,..., (7) Wih regards o he residual conrac period, he policy is fairly priced on he basis of firs order echnical bases + + ɺɺ x+ : x+ : R b A a,,2,..., (8) R where ω b A x+ : ( ) α ω δ + γ ω (9) 2.2.3. Idenical Adjusmen Raes his paper assumed ha life insurers se he same adjusmen raes α γ δ for any ime such ha he mahemaical reserve, he benefi and he periodical premiums are all adjused in he same degree [2]. For a given b, one can ascerain an iniial premium of which he conrac is fairly priced a incepion given by Equaion () and () respecively: ( δ ) b + b +,,2,..., () ( δ ) +,,2,..., () where δ is as defined earlier. Ieraively, Equaion () and () can be expressed as ( δ ), 2,3,..., (2) b b + k ( δk ),,2,..., (3) + k 2.2.4. Consan eriodical remiums In order o uphold he idea of consan periodical premiums, i is apparen o assume γ, so ha will be consan a any given ime,. Suppose denoe he consan periodical premium, analysing Equaion (), (6) and (9) imply + ( + ) b ( ) ( ) + δ b ω b ( δ ) ( i) b b ω δ δ + (4) x+ : where x+ : A aɺɺ x+ : x+ : Equaion (4) depics ha, he benefi adjusmen rae depends on he pair( x +, ). However, i is pruden o assume in pracice ha, he adjusmen rae depends only on he duraion, and mauriy, and no on he age of he insured [2]. herefore, Equaion (4) has o be approximaed by replacing wih premium obained from Equaion () for policies belonging o he same porfolio and his gives b + b ( + δ ) b δ Applying a convenion ha, ( δ ) and (5) becomes, respecively h (5) + h, Equaion (4) δk b b ( + δk ) ( + δh ), 2,3,..., (6) k k x+ k : k h k+ k b b ( + δk ) δk ( δh ), 2,3,..., + k k h k + 2.2.5. Surrender Condiion (7) A surrender opion defined by Bacinello is an American pu opion ha eniles he policyholder o sell back he conrac o he issuer a a surrender value [2]. he inclusion of a surrender possibiliy in a policy implies ha he insured can sell back he conrac o he issuing company before mauriy. hus, policy owners have he righ o early erminaion of heir conrac o receive he surrender value implied by a surrender charge. Assume ha surrender decisions are made a he beginning of he policy year shorly afer he declaraion of he renewal benefi for he coming year, and prior o he paymen of he periodical premium. If SV denoe he surrender value a ime defined by:
4 Musapha Abdul-Rahaman e al.: Sochasic Ineres Rae Approach of ricing aricipaing Life Insurance olicies wih Embedded Surrender Opion where,,,2 SV b + A, 3,4,...,, x + : ( h ) ( ) : ( g ) ( x+ + ) h / x+ + + g ( ) x+ 3 A i q i p (8) he above equaion is consisen wih life insurance policies sold in he Ghanaian insurance marke. hus, no benefi is paid o he policy owner unless a leas hree periodical premiums are colleced. 2.2.6. he Model Dynamics Consider a life insurance endowmen paricipaing conrac wih embedded surrender opion, he insured a mauriy receive a guaranee benefi in addiion o a paricipaion bonus in reurn for he periodical premiums paid. In a riskneural world, he insurer is a subjec of moraliy and financial risk. Assumed independence beween boh risks, he price of a coningen insurance claim can be formulaed as he discouned expeced value in relaion o he risk neural moraliy and he financial elemens. As oulined by Bernard and Lemieux, he dynamic process of a zero-coupon bond in a risk neural world wih mauriy τ and a reference porfolio is given by; db(, τ) rd + σ (, τ ) dz( ) (9) B(, τ) ds( ) rd + σdz( ) (2) S( ) where Z( ) and Z ( ) are sandard Brownian moions under he risk neural measure Q and r is he insananeous riskfree ineres rae. If ρ denoe he correlaion facor beween wo Brownian movemens such ha dz( ) dz( ) ρd. Now considering anoher Brownian moion Z 2 ( ) independen from Z ( ) such ha dz( ) dz 2( ). hen, he Brownian moion Z( ) is given by; 2 ρ 2 dz( ) ρdz ( ) + dz ( ) (2) Equaion (2) does no correlae wih he ineres rae risk from he reference porfolio risk. herefore, he reference porfolio dynamics from Equaion (2) can be rewrien as; ds() 2 rd + σρdz( ) + σ ρ dz2( ) (22) S( ) From Girsanov s heorem, he exisence of such measure is guaraneed, herefore, Z ( ) defined by dz ( ) dz ( ) + σ( u, ) du is a Q Brownian moion. Furhermore, building he process, Z 2 in such a way ha Z and Z 2 are uncorrelaed and applying Ioˆ' s formula o Equaion (9) and (22), he dynamics of he reference porfolio and he zero bond under he ransformed Q measure are given by; 2 ( σρ σ( u, ) ) dz ( u) + σ ρ dz2 ( u) S( ) S() exp B(, ) B(, ) 2 2 2 (( σ ) ) ρ σ ( u, ) + σ ( ρ ) du 2 (23) ( ) B(, ) B(, ) exp ( u, ) + ( u, ) dz ( u) ( u, ) + ( u, ) du B(, ) 2 2 ( σ σ ) ( σ σ ) (24) Owing o he long mauriy naure of mos life insurance conracs and he consrains imposed on he compuaional viabiliy, we assumed σ ( u, ) σ ( u) where $ is consan as proposed by ([4]; [7]). he dynamics of relaive price, R( ) in successive years (,) of he reference porfolio and he zero-coupon bond under he ransformed Q measure are given by; 2 ( σρ σ( u) ) dz ( u) + σ ρ dz2 ( u) S() R( ) exp B(, ) B(, ) 2 2 2 (( σ ) ) ρ σ ( u) + σ ( ρ ) du 2 (25)
American Journal of Mahemaical and Compuer Modelling 28; 3(): -2 5 B(, ) 2 B(, ) exp σ ( ) + σz ( ) B(, ) 2 (26) 2.2.7. he ricing Framework Life insurance paricipaing policy is a ypical example of a coningen claim since i is affeced by boh moraliy and financial risks. Assuming independence beween hese risks and ha he financial and insurance markes are perfecly compeiive and free of arbirage opporuniies. Also, assuming policyholders are raional and non-saiaed, and o share he same informaion. As a resul, surrender decision can only be made following he comparison of he surrender value and he coninuaion value, such ha policyholder s surrender heir conrac, if and only if, he surrender value is more han he coninuaion value oherwise keeps he policy unil he end of he coming year. he financial risk ha affecs he policy under sudy comes from he sochasic evoluion of he raes of reurn on he reference porfolio and he zero-coupon bond. In such insance, assume he reference porfolio is well diversified and spli ino unis such ha any yield realized is immediaely reinvesed and shared among all unis. hus, he reinvesed yield only affecs he uni price and no he oal number of unis involve when a wihdrawal or new invesmen is made. his sugges ha he reference porfolio price is deermined by he evoluion of is relaive uni price. If R τ denoe his price a ime ( τ > ), hen R l,,2,..., (27) R o fairly price he paricipaing policy wih he embedded surrender opion under sochasic ineres rae model, we divided each conrac year ino N-equal sub-period wih equal lengh such ha N. Following binomial evoluion as proposed by [], he relaive price, R( ) a each period has wo possible values, a good one denoed by UR and a bad one represened by DR wih mahemaical relaions given by; r R U R e (28) D (29) R U R Under he risk-neural measure, he probabiliy of an even R R condiioned on all he available relevan { } τ + u τ informaion a ime τ is given by; R U DR R D And he probabiliy of he even ha { R R } given by; R (3) τ + d τ is R UR U D R R (3) o avoid any arbirage opporuniy, a volailiy parameer r is fixed for a given drif erm such haur > e > DR. his implies a sricly posiive value less han one () for boh R and R. Also, he above assumpions imply ha l,, 2,..., are i.i.d and can ake N + possible values: N i N ψi UR DR, i,,..., N B(, ) wih corresponding risk-neural probabiliies given by; N N i i R ( ) ( ),,,..., ( ) i R R i N i (32) (33) Furhermore, δ,,2,...,, are i.i.d and can ake n + possible values given by; ηψi ig Φ i, i,,..., n + i wih probabiliy R ( ) ( ) i n R i ( ) i ( ) g (34) and wih probabiliy 2.2.8. Fair ricing of he Basic Conrac he insurer s liabiliy of he basic conrac is he deerminisic benefi, b due a random ime of deah or mauriy. he marke value a ime zero, is is expeced value wih respec o he risk-neural moraliy of he benefi discouned from he random ime of paymen o ime zero implied by he risk-free ineres rae, r is given by; (, ) / (, ) : b A b B qx + B p x (35) x Also, he sequence of consan periodical premium, due a beginning of each policy year has is ime zero marke value given by; aɺɺ (, ) x : B p x (36) herefore, he periodical premium which equal hese wo is given by;
6 Musapha Abdul-Rahaman e al.: Sochasic Ineres Rae Approach of ricing aricipaing Life Insurance olicies wih Embedded Surrender Opion A x : b a ɺɺ x : x : b (37) 2.2.9. Fair ricing of he Bonus Opion Under he non-surrendered paricipaing policy, he insurer s liabiliy represens he sochasic benefi due a he random ime of deah of he insured or a mauriy. he fair value of he insurer s liabiliy is given by; Q / x Q x ( ) (, ) ( ) + (, ) ( ) π b B E b q B E b p (38) he ime zero value of he sequence of periodical premiums are also deermined in he same manner. he paper idenified a disincion beween he adjused and he consan periodical premiums. Idenical Adjusmen Rae Assuming he benefi b, defined by Equaion (6) and exploiing he sochasic dependence of δ k, for k,2,...,, implies; ( b ) b B (, ) E ( + ) π δ (39) Q K k Also, aking in o accoun he i.i.d naure ofδ k, Equaion (39) becomes where EQ [ δ k ] herefore ( b ) Φ Φ ( b ) b B ( )( + Φ) π, (4) L i i π from equaion (39) is rewrien as b x x π ( b ) B (, ) q + B (, ) p ( + Φ) A ( + Φ) + Φ b + Φ x : (4) he periodical premiums have he same form as he benefi since boh are adjused by he same measure. If denoe he iniial premium wih marke value ( ) π, hen from Equaion (37), k Also, ( ) (, ) ( ) + δ (42) π ( ) Q E B Hence, he fair value a ime zero of he sequence of periodical premiums, ( ) ( ) x x equals he fair value of he insurer s liabiliy, if and only if: k B, ( + Φ) (43) is given by; π p B, p ( + Φ) a ɺɺ ( + Φ) (44) : x b A x : ( + Φ) ( + Φ ) aɺɺ ( + Φ) x : b + Φ x : (45) Consan eriodical remiums Consider he consan periodical benefi and he i.i.d naure of δ k, for k,2,...,, he insurer s liabiliy of he benefi is given by; ( ) (, ) ( + Φ) k k b b π b bb Φ ( + Φ) b (, ) + (, ) (, ) k B B B Φ Φ (46) Consequenly, equaion (38) is rewrien as b b π ( b ) b A + A ( IA) (47) Φ : Φ : : x x x B qx B p x where, ( IA) (, ) + (, ) x : If denoe he consan periodical premium due a he beginning of each policy year if he insured is alive. As i is in he basic conrac, he ime zero marke value of he
American Journal of Mahemaical and Compuer Modelling 28; 3(): -2 7 sequence of periodical premiums, is given by a ɺɺ. herefore, he fair premium of he non-surrendered paricipaing policy is given by; where, A + b Φ( IA) x x : Φ Φaɺɺ x : is he fair premium of he basic conrac. x : (48) 2.2.. Fair ricing of he Whole Conrac he insurer s liabiliy under he whole conrac is he sochasic benefi, b due a he insured random ime of deah or a mauriy, if only policy owners does no surrender heir conrac. Likewise, policy owner s liabiliies are presened by he periodical premiums (consan or adjused) due a he beginning of each policy year unil eiher surrender, deah of he insured or a mauriy. For any given iniial premium, he periodical premiums, he whole conrac value given by he difference beween he insurer s liabiliy and ha of he policy owner s liabiliy, and he coninuaion value. If CV and W for,2,..., denoes he coninuaion and he whole conrac values respecively a he beginning of (+) policy year. A any given ime, he benefi is b +, paid a ime +, if he insured dies wihin imes, and +, else he whole policy value isw +. he coninuaion value CV a ime if he insured is alive is given by: (, ) ( / ) CV B + q x + b + + p x + E Q W + I,,2,..., (49) Special case is a ime, if he conrac is sill in force, policy owners received b, a he end of he year irrespecive of wheher he insured died wihin he year or survive he conrac. Hence he policy value a mauriy equalsb. Consequenly, Equaion (49) becomes (, ) CV b B (5) he value of he whole conrac is hen deermined as he maximum of he coninuaion value and he surrender value since policy owners are raional and non-saiaed, given by; { } W max CV, SV (5) 3. Resuls and Discussion 3.. Compuaional Basis - he iniial premium paid a he ime of issuance of he conrac - he iniial premium of he basic conrac SV - he iniial premium of he surrender opion. c - he iniial premium of he non-surrendered bonus opion in he consan premium case a - he iniial premium of he non-surrendered bonus opion in he periodical adjusmen case a - he iniial premium of he whole conrac in he periodical adjusmen case c - he iniial premium of he whole conrac in he consan premium case U - he iniial premium of he bonus opion in he a adjusable case given by c a U - he iniial premium of he bonus opion in he consan case given by 3.2. Numerical Resuls c he fair price of he whole conrac is deermined as he summaion of he individual componens of he policy as: remium of he whole conrac remium of he basic conrac + remium of non-surrendered paricipaing opion + remium of he surrender opion. Whiles he premium of he non-surrendered paricipaing policy is deermined by: remium of non-surrendered paricipaing policy remium of he basic conrac + remium of he surrender opion. able shows he whole policy and is componens versus insured age, he iniial premiums of he various componens ha makes he policy fair a incepion are: 29.9, 27.2, SV.32, c 27.48, U.28, 27.8, 28.99, U.89, c a 29.3 c I was revealed ha; he premium of he non-surrendered bonus opion is smaller in he consan premium case han in he periodical adjusmen case and he premium of he bonus opion in he surrendered paricipaing policy looks cheap in he consan premium case han in he periodical adjusmen case. hus, i s abou.3% and 6.95% respecively of he oal premium for he consan and for he periodical adjusmen cases. Furhermore, he premium defined by Equaion () is below he whole policy premium in he periodical adjusmen case ( a ) ime, above he premium in he consan case ( c ) a <, and a he same a >. he premium of he basic conrac and he premiums compued by insurer are boh increasing wih increasing age of he insured. hese have resuled an increasing premium for he non-surrendered paricipaing opion boh in he consan case and in he periodical adjusmen case. he premiums for he surrender opion also increases wih he aged of he insured, and so does he premium for he whole conrac since + U + SV in he consan case, and c c c a a a + U + SV in he periodical adjusmen case. Also, incidence of he premium on he bonus opion for he whole policy decreases from abou.4% o.2% in he consan case, and abou 6.4% o 6.7% in he periodical adjusmen case. Likewise, he incidence of he premium of
8 Musapha Abdul-Rahaman e al.: Sochasic Ineres Rae Approach of ricing aricipaing Life Insurance olicies wih Embedded Surrender Opion he surrender opion on he whole policy increases from.6% o 2.2% in he consan case and from.57% o 2.2% in he periodical adjusmen case. Finally, he premium, c is always smaller han he premium, while x able. he whole policy and i s componens versus insured age. CONSAN REMIUMS SV c U c a seems o be very close o for ages 4 o 44, and hereafer greaer han he premium. ADJUSABLE REMIUMS c a U a 4 26.67 24.76.5 25.2.26 25.7 26.39.63 26.54 4 26.82 24.92.6 25.8.26 25.34 26.56.64 26.72 42 26.99 25.9.7 25.35.26 25.52 26.74.65 26.9 43 27.8 25.27.9 25.54.27 25.73 26.94.67 27.3 44 27.38 25.48.2 25.75.27 25.95 27.6.68 27.36 45 27.6 25.7.22 25.97.27 26.9 27.4.7 27.62 46 27.85 25.95.23 26.22.27 26.45 27.66.7 27.89 47 28.2 26.22.25 26.49.27 26.74 27.95.73 28.2 48 28.42 26.52.27 26.79.27 27.6 28.26.74 28.53 49 28.74 26.84.29 27.2.28 27.4 28.6.77 28.9 5 29.9 27.2.32 27.48.28 27.8 28.99.89 29.3 5 29.48 27.59.35 27.87.28 28.22 29.4.8 29.75 52 29.9 28..37 28.3.29 28.67 29.86.85 3.23 53 3.38 28.48.4 28.78.3 29.9 3.36.88 3.77 54 3.89 28.99.44 29.3.3 29.74 3.9.9 3.34 55 3.45 29.56.48 29.86.3 3.34 3.5.94 3.98 56 32.7 3.7.52 3.49.32 3. 32.6.99 32.68 57 32.75 3.85.57 3.7.32 3.74 32.88 3.3 33.45 58 33.49 3.59.62 3.92.33 32.54 33.67 2.8 34.29 59 34.3 32.4.68 32.74.34 33.42 34.54 2.4 35.22 6 35.9 33.29.73 33.64.35 34.37 35.49 2.2 36.22 Figure shows he influence of he insured age on premium boh in he consan case and in he case in which he premium is periodically adjused according o he performance of he reference porfolio wih special references made o he premium compued by insurance companies. I shows ha, he premium payable is an increase funcion of age. a Figure. he influence of he insured age on premium.
American Journal of Mahemaical and Compuer Modelling 28; 3(): -2 9 3.3. he olicy Componens and he Rae of Reurn able 2 shows he whole premium and is componens versus he rae of reurn. All he values presened are very sensiive o he rae of reurn on he reference porfolio. he premium of he basic conrac is clearly decreasing wih raising rae of reurn, and so are he premium of he non-surrendered paricipaing policy and ha of he whole policy despie an increasing rend in he premium of he walk away opion. Bacinello produced he same rend on her paper on fair valuaion of guaranee life insurance paricipaing policy embedding a surrender opion [2]. his is for he reason ha policyholders are eniled o able 2. he whole premium and is componens versus rae of reurn. paricipae in he profi sharing of heir heavily loaded premiums. he bonus opion seems cheap in he consan premium case and is incidence on he whole premium decreases from.39% o.78%. Whereas he premiums of he surrendered bonus opion in he periodical adjusmen case appears expensive wih is incidence on he oal premium decreasing from abou 9.25% o 4.3%. Finally, here exis a value of g beween 25% and 25.5% for which c in he consan premium case, and beween 26% and 26.5% in he periodical adjusmen case for which. CONSAN REMIUMS ADJUSABLE REMIUMS g SV c U c c a U a a.2 36.52.27 37.4.52 37.3 4.27 3.75 4.54.2 34.74.28 35.2.47 35.49 38.8 3.34 38.36.22 33.5.28 33.48.43 33.76 36.6 3. 36.34.23 3.46.29 3.85.39 32.4 34. 2.64 34.39.24 29.96.3 3.3.35 3.6 32.29 2.33 32.59.25 28.54.3 28.86.32 29.7 3.59 2.5 3.9.26 27.2.32 27.48.28 27.8 28.99.79 29.3.27 25.93.33 26.9.26 26.52 27.48.55 27.8.28 24.73.33 24.96.23 25.29 26.7.34 26.4.29 23.6.34 23.8.2 24.4 24.74.4 25.8.3 22.54.35 22.72.8 23.7 23.5.96 23.85 29.9 3.4. he olicy Componens and he Guaraneed Ineres Rae From able 3, he guaranee ineres rae, i g has a lile influence on he premiums of he bonus opion as well as he surrender opion, a leas in he range of values considered. Wih increasing guaraneed ineres rae, he premium of he bonus opion also increases boh in he consan and in he periodical adjusmen cases. However, he premiums of he basic conrac, non-surrendered paricipaing policy as well as he premiums of he whole conrac decreases wih increasing guaraneed ineres rae. Also, as i g raises, he probabiliy ha policy owners finding more profiable invesmen afer surrendering heir policy decreases. Again, i g SV able 3. he whole premium and is componens versus guaranee ineres rae. CONSAN REMIUMS c U c a here exis a level of i g ha makes he premium compued by insurance companies equals ha of he whole policy premium (beween 25% and 25.5% in he consan premium case ( c ) and beween 26% and 26.5% in he periodical adjusmen case). Moreover, he incidence of he surrender opion on he oal premium increases from.73% o.5% in he consan premium case and from.72% o.38% in he periodical adjusmen case, whiles he incidence of he premium of he bonus opion on he whole policy s premium increases from.73% o.38% in he consan premium case and from.36% o 2.57% in he periodical adjusmen case. ADJUSABLE REMIUMS c a U a.2.27 36.52 36.79.27 37.6 37.49.97 37.76.25.27 35.6 35.89.28 36.6 36.7. 36.98.2.28 34.74 35.2.28 35.3 35.95.2 36.23.25.28 33.88 34.7.29 34.45 35.2.33 35.49.22.28 33.5 33.34.29 33.62 34.5.45 34.78.225.29 32.24 32.54.3 32.83 33.79.55 34.8.23.29 3.46 3.76.3 32.5 33..64 33.39.235.3 3.7 3..3 3.3 32.43.73 32.73.24.3 29.96 3.26.3 3.56 3.78.82 32.8.245.3 29.24 29.54.3 29.85 3.4.9 3.45.25.3 28.54 28.85.3 29.6 3.5.97 3.82.255.3 27.86 28.7.3 28.48 29.9 2.5 3.22.26.32 27.2 27.5.3 27.83 29.3 2. 29.63 a
2 Musapha Abdul-Rahaman e al.: Sochasic Ineres Rae Approach of ricing aricipaing Life Insurance olicies wih Embedded Surrender Opion i g SV CONSAN REMIUMS c U c ADJUSABLE REMIUMS c a U a.265.32 26.55 26.87.32 27.9 28.73 2.8 29.5.27.33 25.93 26.24.32 26.57 28.7 2.24 28.5.275.33 25.32 25.64.32 25.97 27.62 2.3 27.95.28.33 24.73 25.5.32 25.38 27.8 2.35 27.4.285.34 24.6 24.48.32 24.82 26.56 2.4 26.9.29.34 23.6 23.92.32 24.26 26.6 2.46 26.4.295.35 23.6 23.38.32 23.73 25.55 2.49 25.9.3.35 22.53 22.85.32 23.2 25.6 2.53 25.4 29.9 a 3.5. he olicy Componens and he aricipaing Co-efficien As far as he paricipaing co-efficien is concerned, he resuls as repored in able 4 revealed a very srong influence on he premiums of he surrendered paricipaing policy as well as he bonus opion, boh in he consan case and in he periodical adjusmen case. All he premiums are increasing, hough he rend in he non-surrendered paricipaing policy in he periodical adjusmen case beas ha of he consan case. he paricipaing co-efficien has no or lile influence on he premiums of he basic conrac and ha lives is value same for differen levels of he paricipaing co-efficien. he bonus opion valued less in he consan premium case han in he periodical adjusmen case, a leas in he range of values considered for all he differen levels of η. he incidence of he bonus opion on he whole premium in he consan premium case increases from abou.5% o 9.8% whiles from.94% o abou 8.53% in he periodical premium case. Finally, here exis a value of η for which he whole premium equals (beween 55% and 6% in he consan premium case, and beween 4% and 45% in he periodical adjusmen case). η CONSAN REMIUMS c U c able 4. he whole premium and is componens versus η. ADJUSABLE REMIUMS c a U a.5 27.34.4 27.66 27.46.26 27.78.2 27.4.2 27.73 27.76.56 28.8.25 27.55.35 27.89 28.6.86 28.38.3 27.7.5 28.8 28.36.6 28.68.35 27.9.7 28.22 28.67.47 28.99.4 28.8.88 28.4 28.99.79 29.3.45 28.26.6 28.58 29.32 2.2 29.64.5 28.44.24 228.76 29.65 2.45 29.97.55 28.62.42 28.94 29.99 2.79 3.3.6 28.8.6 29.3 3.34 3.4 3.66.65 29..8 29.32 3.7 3.5 3.2.7 29.2 2. 29.52 3.7 3.87 3.39.75 29.4 2.2 29.72 3.44 4.24 3.76.8 29.6 2.4 29.93 3.82 4.62 32.4.85 29.87 2.67 3.9 32.32 5.2 32.64.9 3.3 2.83 3.35 32.62 5.42 32.94.95 3.25 3.5 3.57 33.3 5.83 33.35. 3.48 3.28 3.8 33.46 6.26 33.78 29.9, 27.2, SV.32 a 4. Conclusion he purpose of his paper was o esimae he fair-price of life insurance paricipaing policy using a sochasic ineres rae model. he resuls indicae ha, for a life policy, he premium of he non-surrendered bonus opion was smaller in he consan premium case, han i is, in he periodical adjusmen case. In comparing he policy componens and he rae of reurn, i was revealed ha, he basic conrac was decreasing wih increasing rae of reurn and so were he premiums. he guaraneed ineres rae had a lile influence on he premium of he bonus opion as well as he surrender opion. Also, he paricipaing co-efficien had a posiive influence on he premiums of he surrendered paricipaing policy as well as he bonus opion. References [] Bacinello, A. R. (2). Fair pricing of life insurance paricipaing policies wih a minimum ineres rae guaraneed. Asin Bullein, 3(2): 275 297. [2] Bacinello, A. R. (23a). Fair valuaion of a guaraneed life insurance paricipaing conrac embedding a surrender opion. Journal of Risk and Insurance, 7(3): 46 487.
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