A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *


 Margaret Carol Chase
 2 years ago
 Views:
Transcription
1 A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies * Daniel Bauer Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, Alana, GA 333, USA Phone: ( , Fax: ( Alexander Kling Insiu für Finanz und Akuarwissenschafen Helmholzsraße 22, 898 Ulm, Germany Phone: 49 ( , Fax: 49 ( Jochen Russ Insiu für Finanz und Akuarwissenschafen Helmholzsraße 22, 898 Ulm, Germany Phone: 49 ( , Fax: 49 ( Absrac Variable Annuiies wih embedded guaranees are very popular in he US marke. here exiss a grea variey of producs wih boh, guaraneed minimum deah benefis (GMDB and guaraneed minimum living benefis (GMLB. Alhough several approaches for pricing some of he corresponding guaranees have been proposed in he academic lieraure, here is no general framework in which he exising variey of such guaranees can be priced consisenly. he presen paper fills his gap by inroducing a model, which permis a consisen and exensive analysis of all ypes of guaranees currenly offered wihin Variable Annuiy conracs. Besides a valuaion assuming ha he policyholder follows a given sraegy wih respec o surrender and wihdrawals, we are able o price he conrac under opimal policyholder behavior. Using boh, MoneCarlo mehods and a generalizaion of a finie mesh discreizaion approach, we find ha some guaranees are overpriced, whereas ohers, e.g. guaraneed annuiies wihin guaraneed minimum income benefis (GMIB, are offered significanly below heir riskneural value. * he auhors hank HansJoachim Zwiesler for useful insighs and commens. Corresponding auhor  
2 Inroducion Variable Annuiies, i.e. deferred annuiies ha are fundlinked during he defermen period were inroduced in he 97s in he Unied Saes (see Sloane (97. Saring in he 99s, insurers included cerain guaranees in such policies, namely guaraneed minimum deah benefis (GMDB as well as guaraneed minimum living benefis (GMLB. he GMLB opions can be caegorized in hree main groups: Guaraneed minimum accumulaion benefis (GMAB provide a guaraneed minimum survival benefi a some specified poin in he fuure o proec policyholders agains decreasing sock markes. Producs wih guaraneed minimum income benefis (GMIB come wih a similar guaraneed value G a some poin in ime. However, he guaranee only applies if his guaraneed value is convered ino an annuiy using given annuiizaion raes. hus, besides he sandard possibiliies o ake he marke value of he fund unis (wihou guaranee or conver he marke value of he fund unis ino a lifelong annuiy using he curren annuiy conversion raes a ime, he GMIB opion gives he policyholder a hird choice, namely convering some guaraneed amoun G ino an annuiy using annuiizaion raes ha are fixed a incepion of he conrac (. he hird kind of guaraneed minimum living benefis are socalled guaraneed minimum wihdrawal benefis (GMWB. Here, a specified amoun is guaraneed for wihdrawals during he life of he conrac as long as boh he amoun ha is wihdrawn wihin each policy year and he oal amoun ha is wihdrawn over he erm of he policy say wihin cerain limis. Commonly, guaraneed annual wihdrawals of up o 7% of he (single upfron premium are guaraneed under he condiion ha he sum of he wihdrawals does no exceed he single premium. hus, i may happen ha he insured can wihdraw money from he policy, even if he value of he accoun is zero. Such guaranees are raher complex since he insured has a broad variey of choices. Mos of he earlier lieraure on Variable Annuiies, e.g., Renz Jr. (972 or Greene (973 is empirical work dealing wih produc comparisons raher han pricing issues. I was no unil recenly, ha he special ypes of guaranees were discussed by praciioners (cf. JPMorgan (24, Lehman Brohers (25, or analyzed in he academic lieraure. Milevsky und Posner (2 price various ypes of guaraneed minimum deah benefis. hey presen closed form soluions for his ianic Opion 3 in case of an exponenial moraliy law and numerical resuls for he more realisic GomperzMakeham law. hey find ha in general hese guaranees are overpriced in he marke. In Milevsky und Salisbury (22, a model for he valuaion of cerain GMLB and GMDB opions is presened in a framework where he insured has he possibiliy o parially surrender he policy. he auhors call his a Real Opion o Lapse 4. hey presen closed 3 he auhors denoe his opion as ianic Opion since he paymen srucure falls beween European and American Opions and he paymen is riggered by he decease of he insured. 4 heir Real Opion is a financial raher han a real opion in he classical sense (cf. Myers (
3 form soluions in he case of an exponenial moraliy law, consan surrender fees and no mauriy benefis. I is shown ha boh, he value and he opimal surrender sraegy, are highly dependen on he amoun of he guaranee and of he surrender fee. Ulm (26 addiionally considers he real opion o ransfer funds beween fixed and variable accouns, and analyzes he impac of his opion on he GMDB rider and conrac as a whole, respecively. In Milevsky und Salisbury (26, he same auhors price GMWB opions. Besides a saic approach, where deerminisic wihdrawal sraegies are assumed, hey calculae he value of he opion in a dynamic approach. Here, he opion is valuaed under opimal policyholder behavior. hey show ha under realisic parameer assumpions opimally a leas he annually guaraneed wihdrawal amoun should be wihdrawn. Furhermore, hey find ha such opions are usually underpriced in he marke. In spie of hese approaches for he pricing of several opions offered in Variable Annuiies, here is no general framework in which he exising variey of such opions can be priced consisenly and simulaneously. he presen paper fills his gap. In paricular, we presen a general framework in which any design of opions and guaranees currenly offered wihin Variable Annuiies can be modeled. Asides from he valuaion of a conrac assuming ha he policyholder follows a given sraegy wih respec o surrender and wihdrawals, we are also able o deermine an opimal wihdrawal and surrender sraegy, and price conracs under his raional sraegy. he res of he paper is organized as follows. In Secion 2, we give a brief overview over he exising forms of guaranees in Variable Annuiies. Secion 3 inroduces he general pricing framework for such guaranees. We show how any paricular conrac can be modeled wihin his framework. Furhermore, we explain how a given conrac can be priced assuming boh, deerminisic wihdrawal sraegies and opimal sraegies. he laer is referred o as he case of raional policyholders. Due o he complexiy of he producs, in general here are no closed form soluions for he valuaion problem. herefore, we have o rely on numerical mehods. In Secion 4, we presen a Mone Carlo algorihm as well as a discreizaion approach based on generalizaions of he ideas of anskanen und Lukkarinen (24. he laer enables us o price he conracs under he assumpion of raional policyholders. Our resuls are presened in Secion 5. We presen he values for a variey of conracs, analyze he influence of several parameers and give economic inerpreaions. Secion 6 closes wih a summary of he main resuls and an oulook for fuure research. 2 Guaraneed Minimum Benefis his Secion inroduces and caegorizes predominan guaranees offered wihin Variable Annuiy conracs. Afer a brief inroducion of Variable Annuiies in general in Secion 2., we dwell on he offered Guaraneed Minimum Deah Benefis (Secion 2.2 and Guaraneed Minimum Living Benefis (Secion 2.3. We explain he guaranees from he cusomer s poin of view and give an overview over fees ha are usually charged
4 2. Variable Annuiies Variable Annuiies are deferred, fundlinked annuiy conracs, usually wih a single premium paymen upfron. herefore, in wha follows we resric ourselves o single premium policies. When concluding he conrac, he insured are frequenly offered opional guaranees, which are paid for by addiional fees. he single premium P is invesed in one or several muual funds. We call he value A of he insured s individual porfolio he insured s accoun value. Cusomers can usually influence he riskreurn profile of heir invesmen by choosing from a selecion of differen muual funds. All fees are aken ou of he accoun by cancellaion of fund unis. Furhermore, he insured has he possibiliy o surrender he conrac, o wihdraw a porion of he accoun value (parial surrender, or o annuiize he accoun value afer a minimum erm. he following echnical erms are needed o describe he considered guaranees: he rache benefi base a a cerain poin in ime is he maximum of he insured s accoun value a cerain previous poins in ime. Usually, i denoes he maximum value of he accoun on all pas policy anniversary daes. his special case is also referred o as annual rache benefi base. In order o simplify noaion, in wha follows, we only consider producs wih annual rache guaranees. Furhermore, he rollup benefi base is he heoreical value ha resuls from compounding he single premium P wih a consan ineres rae of i % p.a. We call his ineres rae he rollup rae. 2.2 Guaraneed Minimum Deah Benefis If he insured dies during he defermen period, he dependans obain a deah benefi. When Variable Annuiies were inroduced, a very simple form of deah benefi was predominan in he marke. However, since he mid 99s, insurers sared o offer a broad variey of deah benefi designs (cf. Lehmann Brohers (25. he basic form of a deah benefi is he socalled Reurn of Premium Deah Benefi. Here, he maximum of he curren accoun value a ime of deah and he single premium is paid. he price for his kind of benefi usually is already included in he charges of he conrac, i.e. his opion is available wihou addiional charges. Anoher varian is he Annual RollUp Deah Benefi. Here, he deah benefi is he maximum of he rollup benefi base (ofen wih a rollup rae of 5% or 6% and he accoun value. A ypical fee for ha deah benefi wih a rollup rae of 6% is approximaely.25% p.a. of he accoun value (see, e.g., JPMorgan (24. If he conrac conains an Annual Rache Deah Benefi, he deah benefi consiss of he greaer of he annual rache benefi base and he curren accoun value. he charges for his ype of deah benefi are similar
5 Furhermore, he varian Greaer of Annual Rache or Annual RollUp Deah Benefi is offered. Wih his kind of opion, he greaer of he rollup benefi base and he annual rache benefi base, bu a leas he curren accoun value is paid ou as he deah benefi. Wih a rollup rae of i6%, insurers ypically charge abou.6% p.a. for his guaranee (see, e.g., JPMorgan ( Guaraneed Minimum Living Benefis I was no unil he lae 99s ha Guaraneed Minimum Living Benefis have been offered in he marke. oday, GMLB are very popular. he wo earlies forms, Guaraneed Minimum Accumulaion Benefis (GMAB and Guaraneed Minimum Income Benefis (GMIB originaed almos a he same ime. Boh guaranees offer he insured a guaraneed mauriy benefi, i.e. a minimum benefi a he mauriy of he conrac. However, wih he GMIB, his guaranee only applies if he accoun value is annuiized. Since 22, a new form of GMLB is offered, he socalled Guaraneed Minimum Wihdrawal Benefi (GMWB. Here, he insured is eniled o wihdraw a prespecified amoun annually, even if he accoun value has fallen below his amoun. hese guaranees are exremely popular. In 24, 69% of all Variable Annuiy conracs sold included a GMWB opion. Each of he 5 larges Variable Annuiy providers offered his kind of guaranee a his ime (cf. Lehmann Brohers ( Guaraneed Minimum Accumulaion Benefis (GMAB Guaraneed Minimum Accumulaion Benefis are he simples form of guaraneed living A benefis. Here, he cusomer is eniled o a minimal accoun value G a mauriy of he conrac. Usually, G A is he single premium P, someimes a rollup benefi base. he corresponding fees vary beween.25% and.75% p.a. of he accoun value (cf. Mueller ( Guaraneed Minimum Income Benefis (GMIB A mauriy of a Variable Annuiy wih a GMIB, he policyholder can as usual choose o obain he accoun value (wihou guaranee or annuiize he accoun value a curren marke condiions (also wihou any guaranee. However, he GMIB opion offers an addiional I choice: he policyholder may annuiize some guaraneed amoun G a annuiizaion raes ha have been specified upfron. herefore, his opion can also be inerpreed as a guaraneed annuiy, saring a, where he annuiy paymens have already been specified a. Noe ha if he accoun value a mauriy is below he guaraneed value G, he cusomer I canno ake ou he guaraneed capial G as a lump sum bu only in he form of an annuiy a he prespecified annuiizaion raes. hus, he opion is in he money a ime if he I  5 
6 resuling annuiy paymens exceed he annuiy paymens resuling from convering he acual accoun value a curren annuiy raes. I he guaraneed amoun G usually is a rollup benefi base wih, e.g., i 5% or 6%, or a rache benefi base. Someimes here is no one specified mauriy, bu he policyholder can annuiize wihin a cerain (ofen raher long ime period. he offered rollup raes frequenly exceed he riskfree rae of ineres, whereas he prespecified annuiizaion facors are usually raher conservaive. hus, a mauriy he opion migh no be in he money, even if he guaraneed amoun exceeds he accoun value. Furhermore, he pricing of hese guaranees is ofen based on cerain assumpions abou he cusomers behavior raher han assuming ha everybody exercises he opion when i is in he money. Such assumpions reduce he opion value. 5 Depending on he specific form of he guaranee, he curren fees for GMIB conracs ypically vary beween.5% and.75% p.a. of he accoun value Guaraneed Minimum Wihdrawal Benefis (GMWB Producs wih a GMWB opion give he policyholder he possibiliy o wihdraw a specified W amoun G (usually he single premium in small porions. ypically, he insured is eniled W o annually wihdraw a cerain proporion x W of his amoun G, even if he accoun value has fallen o zero. A mauriy, he policyholder can ake ou or annuiize any remaining funds if he accoun value did no vanish due o such wihdrawals. Recenly, several forms of socalled Sepup GMWB opions have been inroduced: Wih one popular version, he oal guaraneed amoun which can be wihdrawn is increased by a predefined raio a cerain poins in ime, if no wihdrawals have been made so far. In wha follows, we will only analyze his form of Sepup GMWB. Alernaively, here are producs in he marke, where a cerain poins in ime, he remaining oal guaraneed amoun which can be wihdrawn is increased o he maximum of he old remaining guaraneed amoun and he curren accoun value. he laes developmen in his area are socalled GMWB for life opions, where only some maximum amoun o be wihdrawn each year is specified bu no oal wihdrawal amoun. W his feaure can be analyzed wihin our model by leing G and. From a financial poin of view, GMWB opions are highly complex, since he insured can decide a any poin in ime wheher and, if so, how much o wihdraw. hey are currenly offered for beween.4% and.65% p.a. of he accoun value. However, Milevsky and Salisbury (26 find ha hese guaranees are subsanially underpriced. hey conclude ha insurers eiher assume a subopimal cusomer behavior or use charges from oher (overpriced guaranees o crosssubsidize hese guaranees. 5 Cf. Milevsky and Salisbury (
7 While his summary of GMDB and GMLB opions covers all he basic designs, a complee descripion of all possible varians would be beyond he scope of his paper. hus, some producs offered in he marke may have feaures ha differ from he descripions above. For curren informaion regarding Variable Annuiy producs, ypes of guaranees, and curren fees, we refer, e.g., o Our model and noaion presened in he following Secion is designed o cover all he guaranees described in his Secion as special cases. Of course, he underlying general framework allows for any specific variaions of he guaranees ha migh deviae from he producs described above. 3 A General Valuaion Framework for Guaraneed Minimum Benefis 3. he Financial Marke As usual in his conex, we assume ha here exiss a probabiliy space (Ω,F,Q equipped wih a filraion F ( [, ] I, where Q is a riskneural measure under which, according o he RiskNeural valuaion formula (cf. Bingham and Kiesel (24, paymen sreams can be valued as expeced discouned values. Exisence of his measure also implies ha he financial marke is arbirage free and ha here exiss some selffinancing invesmen B, sraegy which allows he insurer o hedge his liabiliies. We use a bank accoun ( [ ] he numéraire process, which evolves according o as db B r d, B >. ( Here, r denoes he shor rae of ineres a ime. We furher assume ha he underlying muual fund S of he Variable Annuiy is modeled as a righconinuous F adaped sochasic process wih finie lef limis (RCLL. 6 In paricular, he discouned asse process S B. S B [ ], is a Qmaringale. For convenience, we assume 3.2 A Model for he Insurance Conrac In wha follows, we presen a model suiable for he descripion and valuaion of variable annuiy conracs. Wihin his framework, any combinaion of guaranees inroduced in 6 For our numerical calculaions, we assume ha S evolves according o a geomeric Brownian moion wih consan coefficiens
8 Secion 2 can be represened. In our numerical analysis however, we resric ourselves o conracs wih a mos one GMDB and one GMLB opion. We consider a Variable Annuiy conrac wih a finie ineger mauriy, which is aken ou a ime for a single premium P. Alhough he model generally allows for flexible expiraion opions, in order o simplify he noaion, we only consider a fixed mauriy. We denoe he accoun value by A and ignore any upfron charges. herefore, we have A P. During he erm of he conrac, we only consider he charges which are relevan for he guaranees, i.e. coninuously deduced charges for he guaranees and a surrender fee. he surrender fee is charged for any wihdrawal of funds from he conrac excep for guaraneed wihdrawals wihin a GMWB opion. he coninuously deduced guaranee fee ϕ is proporional o he accoun value and he surrender fee s is proporional o he respecive amoun wihdrawn. In order o valuae he benefis of he conrac, we sar by defining wo virual accouns: W denoes he value of he cumulaive wihdrawals up o ime. We will refer o i as he wihdrawal accoun. Every wihdrawal is credied o his accoun and compounded wih he riskfree rae of ineres up o mauriy. A ime zero, we have W. Similarly, by D we denoe he value of he deah benefis paid up o ime. Analogously o he wihdrawals, we credi deah benefi paymens o his deah benefi accoun and compound he value of his accoun wih he riskfree rae unil ime. Since we assume he insured o be alive a ime zero, we obviously have D. In order o describe he evoluion of he conrac and he embedded guaranees, we also need he following processes: he guaraneed minimum deah benefi a ime is denoed by G D a ime is given by { A ; } D. hus, he deah benefi max G. We le G D A if he conrac conains one of he D D described GMDB opions (cf. Secion 2.2, oherwise we le G. he evoluion of G over ime depends on he ype of he GMDB opion included in he conrac. I will be described in deail in Secion 3.3. he guaraneed mauriy benefi of he GMAB opion is denoed by G. In order o accoun for possible changes of he guaranee over he erm of he conrac, we le ( [ ] A A G, represen he evoluion of his guaranee (see Secion 2.3. for deails. We have G A A A for conracs wih one of he described GMAB opions and G for conracs wihou a GMAB opion
9 I Analogously, we le G denoe he guaraneed mauriy benefi ha can be annuiized in G, I he case of a GMIB opion and model is developmen by ( [ ] G I I and G for conracs wih and wihou a GMIB opion, respecively. A. Also, we have Finally, o be able o represen GMWB opions, we inroduce he processes ( G [, ] E ( G [, ]. W and W G denoes he remaining oal amoun ha can be wihdrawn afer ime, and E G is he maximum amoun ha can be wihdrawn annually due o he GMWB opion. If he conrac conains a GMWB, we le G W A and G E xw A, where x W is he porion of he premium ha can be wihdrawn annually. For conracs wihou GMWB, we le W E G G. he evoluion over ime of hese processes is also explained in deail in Secion 3.3. Due o he Markovpropery 7 of he underlying processes, all informaion available a ime is A I D W compleely conained in he socalled sae variables A, W, D, G, G, G, G and E G. o simplify noaion, we inroduce he following sae vecor A I D W E ( A, W, D, G, G, G, G G y,. 3.3 Evoluion of he Insurance Conrac During he erm of he conrac here are four possible ypes of evens: he insured can wihdraw funds as a guaraneed wihdrawal of a GMWB opion, perform a parial surrender, i.e. wihdraw more han he guaraneed wihdrawal amoun, compleely surrender he conrac, or pass away. For he sake of simpliciy, we assume ha all hese evens can only occur a a policy anniversary dae. herefore, a ineger ime poins,2,...,, for all sae variables we disinguish beween ( and (, i.e. he value immediaely before and afer he occurrence of such evens, respecively. he saring values a of all accouns and processes describing he conrac were given in Secion 3.2. Now, we will describe heir evoluion in wo seps: Firs, for,,2,...,, he developmen wihin a policy year, i.e. from o (  is specified. Subsequenly, we will 7 See Secion in Bingham and Kiesel (
10 describe he ransiion from (  o (, which depends on he ype of guaranees included in he conrac and he occurrence of he described evens. Finally, we describe he mauriy benefis of he conrac Developmen beween and (  As indicaed in Secion 3., he price of he underlying muual fund evolves sochasically over ime. hus, aking ino accoun coninuous guaranee fees ϕ, for he accoun value we have S ϕ A e. (2 A S he accouns W and D are compounded wih he riskfree rae of ineres, i.e. r s ds r s ds W W e and D D e. D A I he developmen of he processes G, G and G depends on he specificaion of he corresponding GMDB, GMAB and GMIB opion: if he corresponding guaraneed benefi is he D / A / I D / A / I single premium or if he opion is no included, we le G G. If he guaraneed D / A / I D / A / I benefi is a rollup base wih rollup rae i, we se G G ( i D / A / I D / A / I. For rache guaranees, we have G G, since he rache base is adjused afer possible wihdrawals, and herefore considered in he ransiion from (  o ( (cf. Secion W E he processes G and G do no change during he year, i.e ransiion from (  o ( A he policy anniversary dae, we disinguish four cases: a he insured dies wihin he period (,] W / E W / E. G G Since our model only allows for deah a he end of he year, dying wihin he period (,] is equivalen o a deah a ime. he deah benefi is credied o he deah benefi accoun and will hen be compounded wih he riskfree rae unil mauriy : D D D max{ G ; A }. Since afer deah, no fuure benefis are possible, we le A / I / W / D / E A as well as G. he wihdrawal accoun, where possible prior wihdrawals have been colleced, will no be changed, i.e. W compounded unil mauriy. W. his accoun will be  
11 b he insured survives he year (,] and does no ake any acion (wihdrawal, surrender a ime Here, neiher he accoun D nor W is changed. hus, we have A A, W W A / I / D D D and. For he GMAB, GMIB, and GMDB, wihou a rache ype guaranee, we also A / I / D have G G. If, however, one or more of hese guaranees are of rache ype, A / I / D A / I / D we adjus he corresponding guaranee accoun by max{ G A } G. ; If he conrac includes a GMWB opion wih sepup and is a sepup poin, he GMWB processes are adjused according o he sepup feaure, bu only if here were no pas wihdrawals: If iw denoes he facor, by which he oal amoun o be wihdrawn is ( W W increased (cf. Secion 2.3.3, we ge G G Ι { } i W W and W / E W / E any oher case, we have G G. E W G x W G. In c he insured survives he year (,] and wihdraws an amoun wihin he limis of he GMWB opion A wihdrawal wihin he limis of he GMWB is a wihdrawal of E W { G G } E min ; wihdrawal amoun G an amoun, since he wihdrawn amoun may neiher exceed he maximal annual E W nor he remaining oal wihdrawal amoun G. he accoun value is reduced by he wihdrawn amoun. In case he wihdrawn amoun exceeds he accoun value, he accoun value is reduced o. hus, we have A { A E } max ;. Also, he remaining oal wihdrawal amoun is reduced by he W W wihdrawn amoun, i.e. G G E. Furhermore, he wihdrawn amoun is credied o he wihdrawal accoun: W W E. he maximal annual wihdrawal amoun as well E E as he deah benefi accoun remain unchanged: G G and D D. Usually, living benefi guaranees (GMAB and GMIB and, in order o avoid adverse selecion effecs, also he guaraneed deah benefis are reduced in case of a wihdrawal. We will resric our consideraions o a socalled pro raa adjusmen. Here, guaranees which are no of rache ype are reduced a he same rae as he accoun value, i.e.  
12 A G A / I / D A / I / D G. If one or more of he guaranees are of rache ype, for he A respecive guaranees, we le G A A / I / D max A G. 8 A A / I / D ; d he insured survives he year (,] and wihdraws an amoun exceeding he limis of he GMWB opion A firs, noe ha his case includes he following cases as special cases: d he conrac does no comprise a GMWB opion and an amoun wihdrawn. < < A E is d2 A GMWB opion is included in he conrac, bu he insured wihdraws an amoun < < A E W E wih { G G } E > ; min. d3 he insured surrenders by wihdrawing he amoun E A 9. We le E W E, where min { G G } 2 E E E ;. Consequenly, E is he porion of he wihdrawal wihin he limis of he GMWB opion. If he conrac does no include a GMWB opion, we obviously have E. As in case c, he accoun value is reduced by he amoun wihdrawn, i.e. A A E, and he wihdrawn amoun is credied o he wihdrawal accoun. However, he insured has o pay a surrender fee for he second componen which leads o ( s 2 W W E E. he deah benefi accoun remains unchanged, i.e. D D. 8 Besides pro raa adjusmens, here are also reducions by he socalled dollar mehod. Here, all he respecive A / I / D A / I / D processes are reduced by he wihdrawn amoun, i.e. G max[ G E, ]. In order o model and evaluae producs where he dollar mehod or any oher reducion scheme applies, he respecive formulas can be adjused. 9 E W If he conrac comprises a GMWB opion and if { G G } A < W G A min ; as well as, hen a wihdrawal of E A is wihin he limis of he GMWB and does no lead o a surrender of he conrac. However, his case is covered by case c
13 Again, he fuure guaranees are modified by he wihdrawal: For he guaranees which are no of rache ype, we have A / I / D A A / I / D G G, whereas for he rache ype A guaranees, we le G A A / I / D max A G. A A / I / D ; E W For conracs wih a GMWB, wihdrawing an amoun { G G } E > min ; also changes fuure guaraneed wihdrawals. We consider a common kind of GMWB opion, where he W W W A guaraneed fuure wihdrawals are reduced according o G min G E ; G, A i.e. he wihdrawal amoun is reduced by he higher of a pro raa reducion and a reducion according o he dollar mehod. For fuure annual guaraneed amouns, we use E E A G G. A Mauriy Benefis a If he conrac neiher comprises a GMIB nor a GMAB opion, he mauriy benefi L is simply he accoun value, i.e. L A. In conracs wih a GMAB opion, he survival benefi A a mauriy is a leas he GMAB, hus { } A L A ; G guar max. Insured holding a GMIB opion can decide wheher hey wan a lump sum paymen of he accoun value A or annuiize his amoun a curren annuiizaion raes. Alernaively, hey can annuiize he guaraneed annuiizaion amoun a prespecified condiions. If we denoe by ä curren and ä guar he annuiy facors 2 when annuiizing a he curren and he guaraneed, prespecified condiions, respecively, he value of he guaraneed benefi a I ä curren mauriy is given by G. hus, a financially raional acing cusomer will chose he ä annuiy, whenever we have G I ä ä curren guar I I ä curren is given by L max A ; G. ä guar > A. herefore, he value of he benefi a ime Cf. Pioneer (25, pp. 36. Cf. Pioneer (25, page 36f. Also, a reducion of he form G E E G W W G is frequenly offered. G 2 Here, an annuiy facor is he price of an annuiy paying one dollar each year
14 If he conrac conains boh, a GMAB and a GMIB opion, he mauriy value of he conrac A I is L max{ L ; L }. 3.4 Conrac Valuaion We make he common assumpion ha financial markes and biomeric evens are independen. Furhermore, we assume riskneuraliy of he insurer wih respec o biomeric risks (cf. Aase and Persson (994. hus, he riskneural measure for he combined marke (insurance and financial marke is he produc measure of Q and he usual measure for biomeric risks. In order o keep he noaion simple, in wha follows, we will also denoe his produc measure by Q. Even if riskneuraliy of he insurer wih respec o biomeric risk is no assumed, here are sill reasons o employ his measure for valuaion purposes as i is he socalled variance opimal maringale measure (see Møller (2 for he case wihou sysemaic moraliy risk and Dahl and Møller (26 in he presence of sysemaic moraliy risk. Le x be he insured s age a he sar of he conrac and a x year old o survive years. By q x p x denoe he probabiliy for, we denoe he probabiliy for a ( x year old o die wihin he nex year. he probabiliy ha he insured passes away in he year (,] is hus given by p x q x. he limiing age is denoed by ω, i.e. survival beyond age ω is no possible Valuaion under Deerminisic Policyholder Behavior A firs, we assume ha he policyholder s decisions (wihdrawal/surrender are deerminisic, i.e. we assume here exiss a deerminisic sraegy which can be described by IR. 3 Here, ξ denoes he amoun o be wihdrawn a he end of year, if he insured is sill alive and if his amoun is admissible. If he amoun a wihdrawal vecor ξ ( ξ ;...; ξ ( ξ is no admissible, he larges admissible amoun E < ξ is wihdrawn. In paricular, if he conrac does no conain a GMWB opion, he larges admissible amoun is E { ; A } A full surrender a ime is represened by ξ. By ( min ξ. Ψ Ψ Ψ IR... we denoe he se of all possible deerminisic sraegies. In paricular, every deerminisic sraegy is F measurable. If a paricular conrac and a deerminisic sraegy are given, hen, under he assumpion ha he insured dies in year {,2,..., x } ω, he mauriyvalues L ;ξ, W ;ξ and 3 Here, IR denoes he non negaive real numbers (including zero; furhermore we le IR IR { }
15  5  ;ξ D are specified for each pah of he sock price S. hus, he ime zero value including all opions is given by:. ; ; ; ; ; ; ; ; ; ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ω D W L e E p D W L e E q p D W L e E q p V ds r Q x ds r Q x x x ds r Q x x s s s ( Valuaion under Probabilisic Policyholder Behavior By probabilisic policyholder behavior, we denoe he case when he policyholders follow cerain deerminisic sraegies wih cerain probabiliies. If hese deerminisic sraegies ( ( j j j IR ( ( ( ;...;ξ ξ ξ, n j,...,,2 and he respecive probabiliies ( j p ξ are known ( n j j p ( ξ, he value of he conrac under probabilisic policyholder behavior is given by ( ( j n j j V p V ξ ξ. (4 his value also admis anoher inerpreaion: if he insurer has derived cerain forecass for he policyholders fuure behavior wih respec o wihdrawals and surrenders, and assigns he respecive relaive frequencies as probabiliies o each conrac, hen he sum of he probabilisic conrac values consiues exacly he value of he insurer s whole porfolio given ha he forecas is correc. hus, his cumulaive value equals he coss for a perfec hedge of all liabiliies, if policyholders behave as forecased. However, in his case he risk ha he acual clien behavior deviaes from he forecas is no hedged Valuaion under Sochasic Policyholder Behavior Assuming a deerminisic or probabilisic cusomer behavior implies ha he wihdrawal and surrender behavior of he policyholders does no depend on he evoluion of he capial marke or, equivalenly, on he evoluion of he conrac over ime. A sochasic sraegy on he oher hand, is a sraegy where he decision wheher and how much money should be wihdrawn is based upon he informaion available a ime. hus, an admissible sochasic sraegy is a discree F measurable process (X, which deermines he amoun o be wihdrawn depending on he sae vecor y. hus, we ge: ( y Ε, X,,...,,2.
16 For each sochasic sraegy (X and under he hypohesis, ha he insured deceases in year {,2,..., ω x }, he values L ( ;(X, W ( ;(X and D ( ;(X are specified for any given pah of he process S. herefore, he value of he conrac is given by: V ω x r ds x x Q, s ( (X p q E e ( L (,(X W (,(X D ( (X. (5 We le Ξ denoe he se of all possible sochasic sraegies. hen he value V of a conrac assuming a raional policyholder is given by (X Ξ ((X V supv. (6 4 Numerical Valuaion of Guaraneed Minimum Benefis For our numerical evaluaions, we assume ha he underlying muual fund evolves according o a geomeric Brownian moion wih consan coefficiens under Q, i.e. ds S rd σ dz, S, (7 where r denoes he (consan shor rae of ineres. hus, for he bank accoun we have r B e. Since he considered guaranees are pahdependen and raher complex, i is no possible o find closed form soluions for heir riskneural value. herefore, we have o rely on numerical mehods. We presen wo differen valuaion approaches: in Secion 4., we presen a simple Mone Carlo algorihm. his algorihm quickly produces accurae resuls for a deerminisic, probabilisic or a given F measurable sraegy. However, Mone Carlo mehods are no preferable o deermine he price for a raional policyholder. hus, in Secion 4.2, we inroduce a discreizaion approach, which addiionally enables us o deermine prices under opimal policyholder behavior. 4. MoneCarlo Simulaion 8 Le (X : IR IR IR a F measurable wihdrawal sraegy. By Iô s formula (see, e.g. Bingham and Kiesel (24, we obain he ieraion A 2 S ϕ σ A e A exp r z ; z ~ S ϕ 2 σ N (, iid,  6 
17 which can be convenienly used o produce realizaions of sample pahs ( j a of he underlying muual fund using Mone Carlo Simulaion. 4 For any conrac conaining Guaraneed Minimum Benefis, for any sample pah, and for any ime of deah, we obain he evoluion of all accouns and processes, employing he rules of Secion 3. Hence, ( j ( j ( j realizaions of he benefis l (,(X w (,(X d (,(X a ime, given ha he insured dies a ime are uniquely defined in his sample pah. hus, he ime zero value of hese benefis in his sample pah is given by v [ ] ω x ( j x x, r ( j ( j ( j ((X e p q l (,(X w (,(X d ( (X J ( i Hence, V ((X v ((X is a MoneCarlo esimae for he value of he conrac, J j where J denoes he number of simulaions. However, for he evaluaion of a conrac under he assumpion of raional policyholders following an opimal wihdrawal sraegy, MoneCarlo simulaions are no preferable. 4.2 A Mulidimensional Discreizaion Approach anskanen and Lukkarinen (24 presen a valuaion approach for paricipaing life insurance conracs including a surrender opion, which is based on discreizaion via a finie mesh. We exend and generalize heir approach in several regards: we have a mulidimensional sae space, and, hus, need a mulidimensional inerpolaion scheme. In addiion, heir model does no include fees. herefore, we modify he model, such ha he guaranee fee ϕ and he surrender fee s can be included. Finally, wihin our approach a sraegy does no only consis of he decision wheher or no o surrender. We raher have an infinie number of possible wihdrawal amouns in every period. Even hough we are no able o include all possible sraegies in a finie algorihm, we sill need o consider numerous possible wihdrawal sraegies. We sar his Secion by presening a quasianalyic inegral soluion o he valuaion problem of Variable Annuiies conaining Guaraneed Minimum Benefis. Subsequenly, we show how in each sep he inegrals can be approximaed by a discreizaion scheme which leads o an algorihm for he numerical evaluaion of he conrac value. We resric he presenaion o he case of a raional policyholder, i.e. we assume an opimal wihdrawal sraegy. However, for deerminisic, probabilisic or sochasic wihdrawal sraegies he approach works analogously afer a sligh modificaion of he funcion F ~ in Secion For an inroducion o Mone Carlo mehods see, e.g., Glasserman (
18 4.2. A quasianalyic soluion he ime value V of a conrac depends solely on he sae variables a ime A I D W E ( A, W, D, G, G, G, G G y,. Since besides A, he sae variables change deerminisically beween wo policy anniversaries, he value process V is a funcion of, A and he sae vecor a he las policy anniversary, i.e. V V (, A ; y. A he discree poins in ime benefi paymens and wihdrawals V V (, A ; y V V (, A, y. If he insured does no die in he period (, ],2,...,, we disinguish he value righ before deah and he value righ afer hese evens, he knowledge of he wihdrawal amoun E and he accoun value o (. We denoe he corresponding ransiion funcion by f E ( A (,, y A y Similarly, by f ( A, y ( A, y wihin (, ]. A deermine he developmen of he sae variables from we denoe he ransiion funcion in case of deah By simple arbirage argumens (cf. anskanen and Lukkarinen (24, we can conclude ha V is a coninuous process. Furhermore, wih Iô s formula (see, e.g. Bingham and Kiesel (24 one can show ha he value funcion V τ for all τ [, saisfies a BlackScholes parial differenial equaion (PDE, which is slighly modified due o he exisence of he fees ϕ. Hence, here exiss a funcion v : IR IR IR wih V ( τ, a, y v ( τ, a τ [,, a IR and v saisfies he PDE. dv d v dv rv dτ da da σ a 2 ( r ϕ a (8 wih he boundary condiion v (, a ( q V (, f ( a, y q V (, f ( a, y x E x, a IR, which, in paricular, is dependen on he insured s survival. For a derivaion and inerpreaion of he PDE (8 and he boundary condiion, see Ulm (26. hus, we can deermine he imezero value of he conrac V by he following backward ieraion: : A mauriy, we have V ( A, y L W D,
19 k: V, k k a ime (k be known for all possible values of he sae vecor. hen, he ime (k value of he conrac is given by he soluion v k, a of he PDE (8 wih boundary condiion Le ( k, A y ( v ( k, a ( q sup V ( k, f ( a, y q V ( k, f ( a, y. x k E k k x k E k IR r ϕ 2 2 A soluion of he PDE (8 can be obained by defining υ :, ρ : σ υ r and σ x x g x e σ υ ρτ σ x x ( τ, v τ, e. hen, g x e σ υ ρ σ lim ( τ, v, e and g saisfies a one ( dimensional hea equaion, τ ( ( 2 d g dg, (9 2 2 dx d a soluion of which is given by 5 2 ( x u ( τ g ( τ, x exp g (, u du. ( 2π (( τ 2 ( hus, we have v 2π (( τ exp 2 σ 2 ( 2 ( logλ ( τ ρ (( τ (, a e υ λ v 2 σ ( λa dλ k. ( By subsiuing 2 λ( u exp σ u r ϕ σ, we obain 2 V ( k, A k, y ( ( k sup (, ( (, q x V k f u A y k E k k k r E k ir e u Φ q V ( k, f ( λ( u A, y x k k λ (2 du, k where Φ denoes he cumulaive disribuion funcion of he sandard normal disribuion. 5 Cf. heorem 3.6 of chaper 4, Karazas and Shreve (
20 4.2.2 Discreizaion via a Finie Mesh In general, he inegral (2 canno be evaluaed analyically. herefore, we have o rely on numerical mehods o find an approximaion of he value funcion on a finie mesh. Here, a finie mesh is defined as follows: Le ( 8 Y IR be he se of al possible sae vecor values. We denoe a finie se of possible values for any of he eigh sae variables as a se of mesh basis values. Le a se of mesh basis values for each of he eigh sae variables be given. Provided ha he Caresian produc of hese eigh ses is a subse of Y, we denoe i by Grid Y and call i a Y mesh or simply a mesh or a grid. An elemen of Grid is called a grid poin. For a given grid Grid, we ierae he evaluaion backwards saring a. A mauriy, he value funcion is given by: V ( A, y L W D, y Grid,. We repea he ieraion sep described above imes and hereby obain he value of he conrac a every ineger ime poin for every grid poin. In paricular, we obain he ime zero value of he conrac V. Wihin each ime period, we have o approximae he inegral ( wih he help of numerical mehods. his will be described in he following Secion Approximaion of he Inegral Following anskanen and Lukkarinen (24, for a IR and a given sae vecor define he funcion y k, we ~ F k ( a, y k ( q sup V ( k, f ( a, y q V k, f ( a, y x k E k E k IR k x k (. k (3 hus, (2 is equivalen o V r ~ ( k A, y e Φ( u F ( ( u A, y du k k, λ for y k Grid k, k 2 where λ( u exp σ u r ϕ σ as above. In order o evaluae he inegral, we evaluae he funcion F 2 ~ k ( a, y k for each y k Grid k and for a selecion of possible values of he variables a. In beween, we inerpolae linearly. hus, le y k k Grid k and A max >, a maximal value for a, be given. We spli he inerval Amax ~ [, A ] in M subinervals via α : m, m {,,2 M }. Le F (, y max hen, for any IR ~ a, F ( a, y k m,..., M k k can be approximaed by m k α m k γ
21 ~ F k M a α m ( a, y k γ m ( γ m γ m Ι [ αm, αm m γ ( γ M γ M Ι [ A, ( a ( a M [ bm, a bm, ] Ι [ α, ( a [ b, a b, ] [, ( a, m α m M M Ι Amax m M where b γ m ( γ γ m a α α α M α α M m M m, m m m,,..., M m ; M, b M, m,..., M ; b M, b M, and I denoes he indicaor funcion. hus, we have V max b and b ( γ γ M m, m m, Amax M ϕ r ( k, a, y [ ( Φ( Φ( ( Φ( k a e bm, u m σ u m σ bm,e u m Φ( u m ] m Amax m r ϕ σ where u, u m log and u M. σ M a σ σ 2 Defining b b, we obain V,, ( k, A k, y k M ϕ r A k e ( bm, bm, ( Φ( u m σ e ( bm, bm, Φ( u m m [ ( ]. Hence, i suffices o deermine he values γ F ( α, y, m {,,2 M } ~ m k m k,...,,. When deermining he γ m, heoreically he funcion f E has o be evaluaed for any possible k wihdrawal amoun E k. For our implemenaion, we resric he evaluaion o a finie ~ amoun of relevan values E k. Furhermore, due o he definiion of F k (see (3, i is necessary o evaluae V afer he ransiion of he sae vecor from ( k o ( k. Since he sae vecor and, hus, he argumens of he funcion are no necessarily elemens of Grid k, V ( k, A k, y k has o be deermined by inerpolaion from he surrounding mesh poins. We inerpolae linearly in every dimension. Due o he high dimensionaliy of he problem, he compuaion ime highly depends on he inerpolaion scheme. In order o reduce calculaion ime and he required memory capaciy, we reduced he dimensionaliy by only considering he relevan accouns for he considered conracs. In paricular, when he deah benefi accoun D is sricly posiive, i.e. if he insured has died before ime, he accoun value A will be zero. Conversely, as long as A is greaer han zero, D remains zero, i.e. he insured is sill alive a ime. hus, he dimensionaliy can always be reduced by one
22 Furhermore, in our numerical analyses, we only consider conracs wih a mos one GMDBopion and a mos one GMLBopion. herefore, by only considering he relevan sae variables, we can furher reduce he dimensionaliy o a maximum of 4. However, for a conrac wih erm o mauriy of 25 years, using abou 4, o 65, laice poins, 6 seps for he numerical calculaion of he inegral, and a discreizaion of he opimal sraegy o 52 poins, he calculaion of one conrac value under opimal policyholder sraegy on a single CPU (Inel Penium IV 2.8 GHz,. GB RAM sill akes beween 5 and 4 hours. 5 Resuls We use he numerical mehods presened in Secion 4 o calculae he riskneural value of Variable Annuiies including Guaraneed Minimum Benefis for a given guaranee fee ϕ. We call a conrac, and also he corresponding guaranee fee, fair if he conrac s riskneural value equals he single premium paid, i.e. if he equilibrium condiion P V V ( ϕ holds. Unless saed oherwise, we fix he riskfree rae of ineres r 4%, he volailiy σ 5%, he conrac erm 25 years, he single premium amoun P,, he age of he insured x 4, he sex of he insured male, he surrender fee s 5%, and use bes esimae moraliy ables of he German sociey of acuaries (DAV 24 R. For conracs wihou GMWB, we analyze wo possible policyholder sraegies: Sraegy assumes ha cliens neiher surrender nor wihdraw money from heir accoun. Sraegy 2 assumes deerminisic surrender probabiliies which are given by 5% in he firs policy year, 3% in he second and hird policy year, and % hereafer. In addiion, we calculae he riskneural value of some policies assuming raional policyholders. For conracs wih GMWB, we assume differen sraegies which are described in Secion Deermining he fair Guaranee Fee In a firs sep, we analyze he influence of he annual guaranee fee on he value of conracs including hree differen kinds of GMAB opions. For conrac, he guaraneed mauriy value is he single premium (moneyback guaranee, conrac 2 guaranees an annual rache base, whereas a rollup base a a rollup rae of i 6% is considered for conrac 3. Figure shows he corresponding conrac values as a funcion of he annual guaranee fee assuming neiher surrenders nor wihdrawals
23 V 8, 6, 4, 2,, 8, 6, 4,.%.5%.%.5% 2.% 2.5% 3.% 3.5% 4.% 4.5% 5.% ϕ 6% RollUp annual rache moneyback guaranee premium Figure : Conrac value as a funcion of he annual guaranee fee For conrac, a guaranee fee of ϕ.7% leads o a fair conrac. he fair guaranee fee increases o.76% in he rache case. he riskneural value of conrac 3 exceeds, for all values of ϕ. hus, under he given assumpions here exiss no fair guaranee fee for a conrac including a 6% rollup GMAB. As a consequence, such guaranees can only be offered if he guaranee coss are subsidized by oher charges or if irraional policyholder behavior is assumed in he pricing of he conrac. 5.2 Fair Guaranee Fees for Differen Conracs 5.2. Conracs wih a GMDB Opion We analyze hree differen conracs wih a minimum deah benefi guaranee. Conrac provides a moneyback guaranee in case of deah, conrac 2 an annual rache deah benefi and conrac 3 a 6% rollup benefi. able shows fair guaranee fees for hese conracs under he wo policyholder sraegies described above. conrac sraegy : no wihdrawals or surrenders 2: deerminisic surrender probabiliy Moneyback guaranee Rache benefi base 6% rollup benefi base.%.4%.4% < % < %.5% able : Fair guaranee fee for conracs wih GMDB under differen consumer behavior
24 Assuming ha cusomers neiher surrender heir conracs nor wihdraw any money before mauriy, he fair guaranee fee for all hese conracs is raher low. However, he guaraneed deah benefi included in conrac 3 is significanly more expensive han he oher guaranees. If policyholders surrender heir conracs a he surrender raes assumed in sraegy 2, he fair guaranee fee srongly decreases for wo reasons: Policyholders pay fees before surrendering bu will no receive any benefis from he corresponding opions. Secondly, surrender fees can be used o subsidize he guaranees of he cliens who do no surrender. For conracs and 2, surrender fees exceed he value of he remaining cliens opions. hus, he riskneural value of he conrac exceeds he single premium even if no fee is charged for he opion. hus, our resuls are consisen wih Milevsky and Posner (2, who find ha GMDB opions are generally overpriced in he marke. Overall, he guaranee fees are raher low, since a benefi paymen is only riggered in he even of deah. here is no possibiliy for raional consumer behavior in erms of exercising he opion when i is in he money. he only way of raional policyholder behavior is surrendering a conrac when he opion is far ou of he money: I is opimal o surrender he conrac if he expeced presen value of fuure guaranee fees exceeds he value of he opion plus he surrender fee. However, for he considered surrender charge of 5%, surrendering a conrac is almos never opimal. hus, he conrac value for a raional policyholder hardly differs from he value under sraegy. However, for lower surrender charges, policyholder behavior would be more imporan Conracs wih a GMAB Opion We analyze hree differen conracs wih a minimum accumulaion benefi guaranee. Again, conrac provides a moneyback guaranee a he end of he accumulaion phase, conrac 2 an annual rache guaranee and conrac 3 a 6% rollup benefi base. he value of hese conracs under policyholder sraegy has been displayed as a funcion of ϕ in Figure above. able 2 shows he fair guaranee fee for hese hree conracs under he wo given policyholder sraegies. In addiion, we show he fair guaranee fee if an addiional 6% rollup deah benefi is included (columns wih DB. conrac Moneyback guaranee Rache benefi base 6% rollup benefi base sraegy w/o DB wih DB w/o DB wih DB w/o DB wih DB : no wihdrawals or.7%.23%.76% surrenders 2: deerminisic surrender probabiliy < %.2%.57%.74% able 2: Fair guaranee fee for conracs wih GMAB under differen consumer behavior
25 he fair guaranee fees for he conracs differ significanly. For he moneyback guaranee, he fair guaranee fee is below.25%, even if he GMDB opion is included. he fee for he rache guaranee is significanly higher. Even under sraegy 2 and wihou addiional deah benefi i exceeds.5%. In any case, he fair guaranee fee of he rache guaranee is a leas four imes as high as he corresponding fair guaranee fee of he moneyback guaranee. For a rollup rae of 6%, he value of he pure mauriy guaranee wihou fund paricipaion (i.e. ϕ % exceeds, under boh surrender scenarios. hus, even under he assumed surrender paern, a 6% rollup GMAB canno be offered a a fair price. he addiional fee for deah benefi (difference beween columns wih DB and w/o DB always exceeds he fair guaranee fee of he pure deah benefi guaranee shown in able, and is hardly reduced by he assumed surrenders. Furher analyses showed ha raional policyholder behavior hardly influences he riskneural value of he conracs: he values under opimal policyholder behavior are very close o he values under sraegy (no surrender or wihdrawal. his is no surprising since for he moneyback guaranee, surrender is rarely opimal due o he raher high surrender charges. In he case of a rache guaranee, he acual guaranee level is annually adjused o a poenially increasing fund value. hus, he guaranee is always a or in he money a a policy anniversary dae. However, as explained above, surrendering is usually only opimal if he opion is ou of he money Conracs wih a GMIB Opion A GMIB opion gives he policyholder he possibiliy o annuiize he minimum benefi base a an annuiy facor ha is fixed a. Wheher or no he opion is in he money depends on boh, he fund value and he raio of he guaraneed annuiy facor and he curren annuiy facor a annuiizaion. Usually, he guaraneed annuiy facor is calculaed based on conservaive assumpions which are supposed o lead o a raio ä ä ä curren : <. However, increasing longeviy and decreasing ineres raes may change his raio during he erm of he conrac and make he guaranee exremely valuable a annuiizaion. We analyze hree differen GMIBconracs for differen values of ä. Again, he minimum benefi base for conrac is he single premium, conrac 2 includes an annual rache guaranee whereas conrac 3 comes wih a 6% rollup benefi base. he hree conracs are analyzed wih and wihou he addiional GMDB opion from he previous Secion. he respecive fair guaranee fees are shown in able 3. Obviously, for ä, he fair guaranee fees are he same as for he corresponding GMAB opions. he value of he guaranee highly depends on he value of ä. Since bes esimaes abou fuure moraliy raes are subjec o high uncerainy, his assumpion bears a significan risk for he insurer ha canno be hedged wih exising financial insrumens. guar
GMWB For Life An Analysis of Lifelong Withdrawal Guarantees
GMWB For Life An Analysis of Lifelong Wihdrawal Guaranees Daniela Holz Ulm Universiy, Germany daniela.holz@gmx.de Alexander Kling *) Insiu für Finanz und Akuarwissenschafen Helmholzsr. 22, 8981 Ulm, Germany
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees
1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.
The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationA Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *
Universal Pricing Fraewor for Guaraneed Miniu Benefis in Variable nnuiies * Daniel Bauer Research raining Group Ul Universi Helholzsraße 8 8969 Ul Geran Phone: 49 (73 5388 Fax: 49 (73 53239 Daniel.Bauer@uniul.de
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationPricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates
Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear
More informationThe Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies
1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationAnalyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective
Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive Alexander Bohner, Nadine Gazer Working Paper Chair for Insurance Economics FriedrichAlexanderUniversiy
More informationA TwoAccount Life Insurance Model for ScenarioBased Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul
universiy of copenhagen Universiy of Copenhagen A TwoAccoun Life Insurance Model for ScenarioBased Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationIMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **
IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include socalled implici or embedded opions.
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationOn the Management of Life Insurance Company Risk by Strategic Choice of Product Mix, Investment Strategy and Surplus Appropriation Schemes
On he Managemen of Life Insurance Company Risk by raegic Choice of Produc Mix, Invesmen raegy and urplus Appropriaion chemes Alexander Bohner, Nadine Gazer, Peer Løche Jørgensen Working Paper Deparmen
More informationThe Grantor Retained Annuity Trust (GRAT)
WEALTH ADVISORY Esae Planning Sraegies for closelyheld, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More informationPRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES
PRICING AND PERFORMANCE OF MUUAL FUNDS: LOOKBACK VERSUS INERES RAE GUARANEES NADINE GAZER HAO SCHMEISER WORKING PAPERS ON RISK MANAGEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAGEMEN
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationTHE IMPACT OF THE SECONDARY MARKET ON LIFE INSURERS SURRENDER PROFITS
THE IPACT OF THE ECONDARY ARKET ON LIFE INURER URRENDER PROFIT Nadine Gazer, Gudrun Hoermann, Hao chmeiser Insiue of Insurance Economics, Universiy of. Gallen (wizerland), Email: nadine.gazer@unisg.ch,
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationINVESTMENT GUARANTEES IN UNITLINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE
INVESMEN UARANEES IN UNILINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN
More informationAnnuity Decisions with Systematic Longevity Risk
Annuiy Decisions wih Sysemaic Longeviy Risk Ralph Sevens This draf: November, 2009 ABSTRACT In his paper we invesigae he effec of sysemaic longeviy risk, i.e., he risk arising from uncerain fuure survival
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationFair Valuation and Risk Assessment of Dynamic Hybrid Products in Life Insurance: A Portfolio Consideration
Fair Valuaion and Risk ssessmen of Dynamic Hybrid Producs in ife Insurance: Porfolio Consideraion lexander Bohner, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen FriedrichlexanderUniversiy
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationThe fair price of Guaranteed Lifelong Withdrawal Benefit option in Variable Annuity
Problems and Persecives in Managemen, olume 7, Issue 4, 9 Gabriella Piscoo (Ialy) he fair rice of Guaraneed Lifelong Wihdrawal Benefi oion in ariable Annuiy Absrac In his aer we use he No Arbirage ricing
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationLEASING VERSUSBUYING
LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss
More informationFifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance
Fifh Quaniaive Impac Sudy of Solvency II (QIS 5) Naional guidance on valuaion of echnical provisions for German SLT healh insurance Conens 1 Inroducion... 2 2 Calculaion of besesimae provisions... 3 2.1
More informationSPEC model selection algorithm for ARCH models: an options pricing evaluation framework
Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationII.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal
Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationDynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint
Dynamic Hybrid Producs in Life Insurance: Assessing he Policyholders Viewpoin Alexander Bohner, Paricia Born, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen FriedrichAlexanderUniversiy
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationTHE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS
VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely
More informationPREMIUM INDEXING IN LIFELONG HEALTH INSURANCE
Far Eas Journal of Mahemaical Sciences (FJMS 203 Pushpa Publishing House, Allahabad, India Published Online: Sepember 203 Available online a hp://pphm.com/ournals/fms.hm Special Volume 203, Par IV, Pages
More informationDynamic Option Adjusted Spread and the Value of Mortgage Backed Securities
Dynamic Opion Adjused Spread and he Value of Morgage Backed Securiies Mario Cerrao, Abdelmadjid Djennad Universiy of Glasgow Deparmen of Economics 27 January 2008 Absrac We exend a reduced form model for
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationCLASSIFICATION OF REINSURANCE IN LIFE INSURANCE
CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationTHE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS
HE PERFORMANE OF OPION PRIING MODEL ON HEDGING EXOI OPION Firs Draf: May 5 003 his Version Oc. 30 003 ommens are welcome Absrac his paper examines he empirical performance of various opion pricing models
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationValuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate
Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his
More informationSupplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect RiskTaking?
Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec RiskTaking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationRationales of Mortgage Insurance Premium Structures
JOURNAL OF REAL ESTATE RESEARCH Raionales of Morgage Insurance Premium Srucures Barry Dennis* Chionglong Kuo* Tyler T. Yang* Absrac. This sudy examines he raionales for he design of morgage insurance premium
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationARTICLE IN PRESS Journal of Computational and Applied Mathematics ( )
Journal of Compuaional and Applied Mahemaics ( ) Conens liss available a ScienceDirec Journal of Compuaional and Applied Mahemaics journal homepage: www.elsevier.com/locae/cam Pricing life insurance conracs
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationThe Uncertain Mortality Intensity Framework: Pricing and Hedging UnitLinked Life Insurance Contracts
The Uncerain Moraliy Inensiy Framework: Pricing and Hedging UniLinked Life Insurance Conracs Jing Li Alexander Szimayer Bonn Graduae School of Economics School of Economics Universiy of Bonn Universiy
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationRISKSHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction
RISKSHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION AN CHEN AND PETER HIEBER Absrac. In a ypical paricipaing life insurance conrac, he insurance company is eniled o a
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationSome Quantitative Aspects of Life Annuities in Czech Republic
Some Quaniaive Aspecs of Life Annuiies in Czech Republic Tomas Cipra The conribuion deals wih some quaniaive aspecs of life annuiies when applied in he Czech Republic. In paricular, he generaion Life Tables
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationVARIABLE STRIKE OPTIONS IN LIFE INSURANCE GUARANTEES
Opions in life insurance guaranees VARIABLE SRIKE OPIONS IN LIFE INSURANCE GUARANEES Piera MAZZOLENI Caholic Universiy Largo Gemelli,, (03) Milan, Ialy piera.mazzoleni(a)unica.i Absrac Variable srike opions
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationTask is a schedulable entity, i.e., a thread
RealTime Scheduling Sysem Model Task is a schedulable eniy, i.e., a hread Time consrains of periodic ask T:  s: saring poin  e: processing ime of T  d: deadline of T  p: period of T Periodic ask T
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More informationA General Pricing Framework for NoNegativeEquity. Guarantees with Equityrelease Products: A Theoretical and
A General Pricing Framework for NoNegaiveEquiy Guaranees wih Equiyrelease Producs: A Theoreical and Empirical Sudy JrWei Huang 1 ChuangChang Chang 2 Sharon S. Yang 3 ABSTRACT We invesigae sochasic
More informationOptions and Volatility
Opions and Volailiy Peer A. Abken and Saika Nandi Abken and Nandi are senior economiss in he financial secion of he Alana Fed s research deparmen. V olailiy is a measure of he dispersion of an asse price
More informationTime Consistency in Portfolio Management
1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen
More informationThe naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces imeseries smoohing forecasing mehods. Various models are discussed,
More informationCALCULATION OF OMX TALLINN
CALCULATION OF OMX TALLINN CALCULATION OF OMX TALLINN 1. OMX Tallinn index...3 2. Terms in use...3 3. Comuaion rules of OMX Tallinn...3 3.1. Oening, realime and closing value of he Index...3 3.2. Index
More information