Pricing and Hedging Guaranteed Annuity Options via Static Option Replication 1

Transcription

1 Picing and Hedging Guaaneed Annuiy Opions via Saic Opion Replicaion Anoon Pelsse Head of ALM Dep Pofesso of Mahemaical Finance Naionale-Nedelanden Easmus Univesiy Roedam Acuaial Dep Economeic Insiue PO Bo 796 PO Bo A Roedam 3 DR Roedam he Nehelands he Nehelands el: el: Fa: Fa: [email protected] [email protected] Fis vesion: Januay 22 his vesion: 2 Febuay 23 his aicle epesses he pesonal views and opinions of he auho. Please noe ha ING Goup o Naionale-Nedelanden neihe advocae no endose he use of he valuaion echniques pesened hee fo is eenal epoing. he auho would like o hank Piee Bouwkneg, Pee Ca, Eduado Schwaz, Andew Cains, Phelim Boyle, an anonymous efeee, paicipans a he Deivaives Day 22 in Amsedam and paicipans a he Insuance: Mahemaics and Economics 22 confeence in Lisbon fo valuable insighs and commens.

2 Picing and Hedging Guaaneed Annuiy Opions via Saic Opion Replicaion Absac In his pape we deive a make value fo wih-pofis Guaaneed Annuiy Opions using maingale modelling echniques. Fuhemoe, we show how o consuc a saic eplicaing pofolio of vanilla inees ae swapions ha eplicaes he wih-pofis Guaaneed Annuiy Opion. Finally, we illusae wih hisoical UK inees ae daa fom he peiod 98 unil 2 ha he saic eplicaing pofolio would have been eemely effecive as a hedge agains he inees ae isk involved in he GAO, ha he saic eplicaing pofolio would have been consideably cheape han up-fon eseving and also ha he eplicaing pofolio would have povided a much bee level of poecion han an up-fon eseve. JEL Codes: G3, G22 2

3 . Inoducion Recenly, consideable publiciy is dawn o wih-pofis life-insuance policies wih Guaaneed Annuiy Opions GAO s. Equiable, a lage Biish insuance office, had o close fo new business as a pofolio of old insuance policies wih GAO s became an unconollable liabiliy. In his pape we wan o popose a hedging mehodology ha can help insuance companies o avoid such poblems in he fuue. Duing he las few yeas, many auhos have applied no-abiage picing heoy fom financial economics o calculae he value of embedded opions in life-insuance conacs. Iniially, he wok was focussed on valuing eun guaanees embedded in equiy-linked insuance policies, see fo eample Bennan and Schwaz 976, Boyle and Schwaz 977, Aase and Pesson 994, Boyle and Hady 997 and Bacinello and Pesson 22. In equiy-linked conacs, he minimum eun guaanee can be idenified as an equiy pu opion, and hence he classical Black-Scholes 973 opion picing fomula can be used o deemine he value of he guaanee. Many life-insuance policies ae no eplicily linked o he value of a efeence equiy fund. adiionally, life-insuance policies pomise o pay a nominal amoun of money o he policyholde a epiaion of he conac. In ode o compensae he policyholde fo he elaively low base-aes which ae used fo pemium calculaion, vaious pofi-shaing schemes have been employed by insuance companies. hough a pofi-shaing scheme, pa of he ecess eun i.e. eun on invesmens above he base ae ha he insuance company makes is being euned o he policyholdes. Howeve, since only he ecess eun is being shaed wih he policyholdes and no he shofall, having a pofi-shaing scheme in place is equivalen o giving a minimum eun guaanee a he level of he base ae o he policyholdes. his ype of embedded eun guaanees has only ecenly been analysed in he lieaue, see fo eample Aase and Pesson 997, Gosen and Jøgensen 997, 2a and 22, Milesen and Pesson 999 and 2 and Bouwkneg and Pelsse 22. Guaaneed Annuiy Opions ae anohe eample of minimum eun guaanees, bu in he case of GAO s he guaanee akes he fom of he igh o conve an assued sum ino a life annuiy a he bee of he make ae pevailing a he ime of convesion and a guaaneed ae. Many lifeinsuance companies in he UK issued pension-ype policies wih GAO s in he 97 s and 98 s. Duing his ime UK inees aes wee vey high, above % beween 975 and

4 Hence, adding GAO s wih implici guaaneed aes aound 8% was consideed hamless a ha ime due o he fac ha hese opion wee so fa ou-of-he-money. Due o he fall of UK inees aes fa below 8% cuenly UK inees aes ae a a level of 5%, he GAO s have become an unconollable liabiliy which caused he downfall of Equiable in 2. he issue of deemining he value of GAO s has been addessed in ecen yeas by Bolon e al. 997, Lee 2, Cains 22, Balloa and Habeman 22, Wilkie, Waes and Yang 23 and Boyle and Hady 23. As is eviden fom he lieaue oveview povided hee, he main focus has been given o deemining he value of embedded opions. Wih he downfall of Equiable i has, in ou view, become appaen ha no only he valuaion should be addessed, bu also he hedging of embedded opions. Alhough he hedging issue seems ivial a fis sigh: any deivaive can be eplicaed by eecuing a dela-hedging saegy. Howeve, he opions wien by insuance companies have such long mauiies and he insued amouns ae so high ha eecuing a delahedging saegy can have disasous consequences. ypically, an insuance company has sold pu opions o is policy holdes. o ceae a delaneual posiion he insuance company has o sell he undelying asse of he pu opion. If makes fall, he insuance company has o sell off moe of is asse posiion o emain delaneual. his will ceae moe downside pessue on he asse pices, especially if he insuance company is ying o ebalance a lage posiion. Hence, eecuing a dela-hedging saegy fo a sho pu posiion can ceae dangeous feedback loops in financial makes which can have disasous consequences. Simila feedback loops wee pesen in Pofolio Insuance saegies which used dela-hedging o ceae synheic pu opions and wee vey popula duing he 98 s. Auomaed selling odes geneaed by compues ying o follow blindly he dela-hedging saegy have been blamed fo iggeing he Ocobe 987 cash. Afe he 987 cash, Pofolio Insuance saegies vey quickly los hei appeal. A second complicaion wih eecuing a delahedging saegy is ha dela hedging equied fequen ebalancing of he hedging asses in ode o emain dela-neual. Especially fo long mauiy opions, his can be quie epensive because of he ansacions coss involved. We wan o popose he use of saic opion eplicaion as a viable alenaive fo insuance companies o hedge hei embedded opions. A saic opion eplicaion can be se up if a pofolio of acively aded opions can be found ha appoimaely eplicaes he payoff of he deivaive 4

5 unde consideaion. Once he payoff of he deivaive has been eplicaed, he no-abiage condiion implies ha also fo all pio imes he value of he deivaive is eplicaed by he saic pofolio. Saic eplicaion hedging echniques fo eoic equiy opions have been inoduced by Bowie and Ca 994, Deman, Egene and Kani 995 and Ca, Ellis and Gupa 998. he advanages of saic eplicaion ae obvious: once he iniial saic hedge has been se up, no ebalancing is needed in ode o keep he deivaive hedged. In pacice, i is no always possible o find a se of acively aded opions ha pefecly eplicaes he payoff of a given deivaive. Howeve, if he appoimaion is close enough he saic eplicaion pofolio will ack he value of he deivaive unde a wide ange of make condiions. In his pape we wan o show how Guaaneed Annuiy Opions can be saically eplicaed using a pofolio of vanilla inees ae swapions. Inees ae swapions ae acively aded fo a wide vaiey of mauiies and single ades can be eecued fo lage noional amouns. Using he hisoy of UK inees aes, we demonsae ha a judiciously chosen saic pofolio of swapions can hedge GAO s ove a long ime hoizon and unde a wide ange of make condiions. Hence, we illusae ha saic eplicaion offes a ealisic possibiliy fo insuance companies o hedge hei eposue o embedded opions in hei pofolios. he emainde of his pape is oganised as follows. In Secion 2 we descibe he payoff of Guaaneed Annuiy Opions and we deive a picing fomula using maingale modelling. In Secion 3 we consuc he saic eplicaion pofolio consising of vanilla swapions. In Secion 4 we illusae he effeciveness of he saic pofolio wih a hypoheical back es using UK inees ae daa fom 98 unil 2. Finally, we conclude in Secion Guaaneed Annuiy Opions Le us conside he make value of annuiies a he momen when hey ae bough. An annuiy is financed by a single pemium, in ou case his single pemium equals he lump sum paymen of he capial policy. Suppose he annuiy is bough a ime by a peson of age. Condiional on he suvival pobabiliies n p fom he moaliy able we can wie he make value of he annuiy ä wih an annual paymen of as ω n a p D, 2. + n 5

6 whee n p denoes he pobabiliy ha an yea old peson suvives n yeas and D +n denoes he make value a ime of a discoun faco wih mauiy +n. Also noe ha, he sum is uncaed a age ω, he maimum age in he moaliy able. In his pape we will make he assumpion ha he suvival pobabiliies n p evolve deeminisically ove ime. his allows fo ends in he suvival pobabiliies, which ae impoan o ake ino consideaion given he long ime hoizons fo his ype of poduc. Alhough in pacice we know ha he suvival pobabiliies ae sochasic, he volailiy of he suvival pobabiliy pocess is much smalle han volailiy of he discoun bond pocesses. Hence, he main isk faco diving he unceainy in he value of annuiies is he make isk, which we analyse in his pape. Given he make value ä, he make annuiy payou ae ove an iniial single pemium of is given by /ä. 2.2 Noe, ha we assume ha he lump sum paymen L a ime is a deeminisic quaniy. his may seem inconsisen wih he fac ha GAO s have been issued on uni-linked and wih-pofis conacs, because in hese ypes of conacs he value of he capial policy a ime is unknown. he papes by Balloa and Habeman 22, Wilkie, Waes and Yang 23 and Boyle and Hady 23 eplicily model he unceainy of he capial policy a ime by eaing he policies as uni-linked conacs. In his pape we ake a diffeen appoach. Ou appoach eplois he fac ha mos of he policies offeed, especially he policies of Equiable, ae wih-pofis policies. Bolon e al. 997, Appendi 2 epo ha wih-pofis policies accoun fo 8% of he oal liabiliies fo conacs which include GAO s. In he case of wih-pofis policies, he capial paymen L o be paid ou a ime depends on he bonuses declaed. Unde a adiional UK wih-pofis conac pofis ae assigned using evesionay and eminal bonuses. Revesionay bonuses ae assigned on a egula basis as guaaneed addiions o he basic mauiy value L and ae no disibued unil he mauiy dae. he eminal bonuses ae no guaaneed. Via he pofi-shaing mechanism, he amoun L can heefoe only incease and neve decease. In each yea he evesionay bonus will add an 6

7 addiional laye L o he conac wih an addiional GAO. Fo he emainde of he conac his laye L is fied. Hence, he analysis we offe in his pape is valid fo wih-pofis policies, since each laye L of pofi-shaing can be valued and hedged a ime when he evesionay bonus is declaed. Suppose ha an yea old policyholde has an amoun of money L a his disposal a ime which is he payou of his capial policy. he GAO opion gives he policyholde he igh o choose eihe an annual paymen of L based on he cuen make aes see fomula 2.2 o an annual paymen L G using he Guaaneed Annuiy G. A aional policyholde will selec he highes annuiy payou given he cuen em sucue of inees aes. heefoe, we can ewie he value of he GAO a he eecise dae as L ma G, Σ n p D +n L Σ n p D +n + L ma G, Σ n p D +n L + L ma G, ä 2.3 Hence, he make value of he GAO policy a he eecise dae is equal o he lump sum paymen L plus L imes he value of he GAO pu-opion. In he emainde of his pape we will focus only on he value V G of he GAO pu-opion V G ma G, ä 2.4 o calculae he make value V G of he GAO pu-opion oday a ime, we can poceed along seveal pahs. he unceainy abou he value of he opion is due o he fac ha he discoun facos D S a ime ae unknown quaniies a ime. One possible appoach heefoe, is o model he complee em-sucue of inees aes wih a em-sucue model, like he Heah- Jaow-Moon 992 model HJM model, o obain an opion value. he disadvanage of such an appoach is ha he opion pice canno be deemined analyically. Resuls have o be obained hough numeical appoimaions which povide us wih elaively lile insigh in he behaviou of he GAO. 7

8 o obain a bee handle on he behaviou of he GAO, we daw an analogy beween he GAO and a swapion. A swapion gives he holde of he opion he igh, bu no he obligaion, o ene ino he undelying swap conac fo a given fied ae. As he value of he swap depends on he em-sucue of inees aes, we could use a em-sucue model o deemine he value of he bond opion. In he case of a swap, all unceainy abou he em-sucue of inees aes is efleced in a single quaniy: he pa swap ae. Hence, he value of a swapion can be deemined moe diec by modelling he bond-pice iself as a sochasic pocess. his is eacly he appoach ha financial makes adop o calculae he pices of swapions wih he Black 976 fomula. In he case of he GAO pu-opion, all he unceainy abou he em-sucue of inees aes is efleced in he make annuiy payou ae. Hence, if we model he make annuiy payou diecly as a sochasic pocess, we have sufficien infomaion o pice he GAO opion. he appoach of using make aes, such as LIBOR aes and swap aes, has been applied in ecen yeas wih gea success o em-sucue models. his ype of models, which have become known as make models, was inoduced independenly by Milesen, Sandmann and Sondemann 997, Bace, Gaaek and Musiela 997 and Jamshidian 998. he main mahemaical esul on which his modelling echnique is based is he maingale picing heoem which saes ha, given a numeaie i.e. a efeence asse ha is used as a new basis o epess all pices in he economy in ems of his asse, an economy is abiage-fee and complee if and only if hee eiss a unique equivalen pobabiliy measue such ha all numeaie ebased pice pocesses ae maingales unde his measue. Fo a poof of he maingale picing heoem we efe o he oiginal pape by Geman e al Fo a geneal inoducion ino he mahemaics involved and he applicaion of maingale mehods o financial modelling we efe o Musiela and Rukowski 997. he books by Hun and Kennedy 2 and Pelsse 2 focus moe eplicily on inees ae deivaives. In he economy we ae consideing, he aded asses ae he discoun bonds D S fo he diffeen mauiies S. Any abiage-fee inees model can be embedded in he HJM famewok. Unde he isk-neual measue Q * which is he pobabiliy measue associaed wih he money-make accoun as he numeaie he pocess fo D S in he HMJ famewok is given by d b dw * dds DS + S, 2.5 8

9 9 whee denoes he spo inees ae, W * denoes Bownian Moion unde he measue Q * and b S denoes he volailiy of he discoun bond. Noe ha in he HJM famewok b S is allowed o be sochasic. Diffeen specificaions of b S lead o diffeen inees ae models. Fo eample, he choice b S / S e κ κ σ leads o he well-known Vasicek-Hull-Whie model ha is used in he papes by Balloa and Habeman 22, Wilkie, Waes and Yang 23 and Boyle and Hady 23 o deemine pices of GAO s. o illusae he change of numeaie appoach, we will also conside he pocesses of discoun bond pocess unde he -fowad measue Q. his is he pobabiliy measue associaed wih he mauiy discoun bond D as he numeaie, see Geman e al Fo a poof of he esuls we deive below, we efe o Musiela and Rukowski 997, Secion he Radon-Nikodym deivaive ρ fo he change of measue is given by he aio of numeaies + def s dw s b ds s b ds s D D d d * 2 2 * ep ep / Q Q ρ. 2.6 Hence, he Radon-Nikodym kenel b κ and we have ha unde he -fowad measue he pocess dw dw * -b d is a sandad Bownian Moion. his implies ha unde he - fowad measue he pocess fo a discoun bond D S wih mauiy S> is given by. dw b d b b D d b dw b d D dd S S S S S S An applicaion of Iô s Lemma confims ha he -fowad discoun bond pice D S /D is indeed a maingale unde he -fowad measue: dw b b D D D D d S S S. 2.8

10 A paicula convenien choice of he numeaie fo he GAO pu-opion is he annuiy ä Σ n p D +n. Noe, ha unde he assumpion ha he suvival pobabiliies n p ae deeminisic, his is a pofolio of aded asses he discoun bonds and hence a pemissible choice as numeaie. 2 he annuiy payou ae fo ime was defined in 2.2. A imes pio o we can conside he value of he pofolio of discoun bonds ha eplicaes he cash flows of an annuiy saing fom. A peson ha will be yeas old a ime, has a ime an age of --. Hence, he make value a ime of a fowad annuiy saing a is given by ω ω n+ p D + n p n pd + n p a, 2.9 whee we have used he acuaial ideniy n+m p m p n p +m see, e.g., Bowes e al. 997, Chape 3. A ime, an insuance company can finance he fowad annuiy by boowing money fom ime unil ime. Only in he cases he insued suvives unil ime, will he insuance company have o epay he loan. Hence, he make value a ime of his loan is given by p D. 2. Combining equaions 2.5 and 2.6, we can define he fowad annuiy ae as D / ä, 2. whee we see ha he suvival pobabiliy faco - p -- in he numeao and he denominao has cancelled. Noe, ha if his definiion coincides wih 2.2 since D. Also noe ha he fowad annuiy ae is he numeaie ebased pice of he discoun bond D using he numeaie ä. 2 Alhough his is a dividend paying numeaie, no dividends ae paid befoe he mauiy dae of he GAO, and his is heefoe a valid choice of numeaie o analyse he pice of he GAO. Noe,

11 he change of numeaie heoem saes ha unde he maingale pobabiliy measue Q A associaed wih he numeaie ä, all ä -ebased pice pocesses ae maingales. Hence, also he pice pocess fo he fowad annuiy is a maingale unde he measue Q A. he Radon-Nikodym deivaive ρ A fo he change of measue o Q A is given by he aio of numeaies: def ρ A dq dq A * ω n p D a ep + n. 2.2 s ds By an applicaion of Iô s Lemma we obain ha ρ A follows he pocess ω n p D + n * dρ A b dw + n. 2.3 a ep s ds he Radon-Nikodym kenel κ A is he volailiy of ρ A. Hence, we can idenify κ A 2.3 as fom p D κ A 2.4 ω n + n wnb + n wih wn ω m pd + m m and we have ha unde Q A he pocess dw A dw * -κ d is a sandad Bownian Moion. We can now deive ha he fowad annuiy ae is a maingale unde he measue Q A and follows he pocess A ω A d wn b + n b dw. 2.5 σ ha simila numeaies ae used in Swap Make Models o analyse he pice of swapions. See, e.g., Jamshidian 998.

12 Fom his epession we see ha he fowad annuiy ae volailiy σ is a weighed aveage of he fowad discoun bond volailiies 2.8. Fuhemoe, he numeaie ebased make value V G /ä of he GAO pu-opion is also a maingale pocess unde he pobabiliy measue Q A. Using equaion 2.4 which gives he value of he GAO pu-opion a ime, he value of he GAO opion fo any ime can be epessed as G G G V AV Ama, a E E a a a 2.6 A G E [ ma,], whee E A [] denoes an epecaion unde he pobabiliy measue Q A. Muliplying boh sides of equaion 2.6 by ä leads o he following epession fo he make pice of he GAO: V G G [ ma,] A a E. 2.7 Given he pocess 2.5 fo unde he measue Q A, we can use epession 2.7 o calculae he value of he GAO opion eplicily. Howeve, since he weighs w n ae sochasic, i is quie complicaed o evaluae 2.7 analyically. A An alenaive appoach is o appoimae he pocess 2.5 as d σ dw wih deeminisic volailiy σ. his implies ha we appoimae he pobabiliy disibuion of by a lognomal disibuion. Given such an appoimaion, we can infe σ fom 2.5 by feezing he sochasic weighs a hei cuen values w n. If he discoun bond volailiies b S ae deeminisic funcions like in he Vasicek-Hull-Whie model, we can hen appoimae 2 σ by he quadaic vaiaion of ln as ω wn b + n s b s 2 σ ds

13 Insead of pesuming a paicula funcional fom fo he discoun bond volailiies b S, we can also esimae σ diecly fom hisoical obsevaions of he fowad annuiy ae. Given a value fo σ, we can appoimae he pice fo he GAO pu-opion via he Black 976 fomula as: V G a d,2 ln G N d N d G 2 ± 2 σ. σ We have adoped he lae appoach in Secion 4 of his pape. 3. Saic Replicaing Pofolio he GAO pu-opion we have discussed in he pevious secion, is no a sandad inees ae opion. o hedge he isk of such a non-sandad opion, an insuance company can eecue a dynamic eplicaion saegy dela hedging. his eplicaion saegy equies coninuous ebalancing of a pofolio of discoun bonds. Discussions on how o se up dela hedging saegies can be found in Boyle and Hady 23 and Wilkie, Waes and Yang 23. Eecuing such a ading saegy in pacice can be cosly due o ansacion coss o even unsuccessful due o inconsisencies in he model assumpions and he acual behaviou of he make. Especially he long ime hoizons ha ae ypically involved in life-insuance poducs make he implemenaion of a dela hedging saegy a challenging ask. We heefoe wan o popose a saic opions eplicaion saegy ha can be used o hedge he isk of GAO s. In a saic opions eplicaion saegy one ses up a pofolio of acively aded opions such ha he payoff of he GAO a mauiy is eacly eplicaed. Due o he fac ha his pofolio maches he payoff of he GAO a mauiy, he pofolio will also accuaely ack a all pevious imes he value of he GAO. Wee his no he case, an abiage oppouniy would aise. Hence, once he iniial pofolio of opions is bough, is composiion neve needs o be adjused unil he ime ha he GAO epies. Even when he acual behaviou of he make is inconsisen wih he model assumpions of he undelying opions, his has sill no impac on he hedge effeciveness of he saic eplicaing pofolio. In ohe wods, no only he make isk bu also he model isk is eliminaed by a saic hedge pofolio. 3

14 In he emainde of his secion we show how a saic eplicaion pofolio of vanilla inees ae swapions can be se up fo wih-pofis GAO s. In inees ae makes, inees ae swapions ae he mos acively aded opions conacs and can be aded in lage quaniies fo a wide vaiey of mauiies and eecise pices. he consucion we popose fo GAO s is inspied by he saic eplicaion saegy poposed by Hun and Kennedy 2, Ch. 5 fo iegula swapions. Noe ha he use of swapions as a hedging saegy has been poposed peviously by Bolon e al. 997, Lee 2 and Wilkie, Waes and Yang 23. Howeve, none of he menioned conibuions uses he idea of saic hedging. Bolon a al. 997 popose a paicula simple appoach, whee hey buy eceive swapions wih a sike equal o he ae of inees undelying he GAO. Howeve since he seam of cash flows associaed wih an inees ae swap has a adically diffeen sucue fom he cash flows of an annuiy, such a hedging saegy will no be vey effecive in pacice. A he eecise dae, he GAO pu-opion gives he holde he igh, bu no he obligaion, o ene ino an annuiy a he guaaneed ae G : V G ω G G, a ma p D, ma n + n, 3. whee we have subsiued he definiion ä given in equaion 2.. Hence, he GAO gives he igh o obain a seies of cash paymens n p G a he diffeen daes +n fo he pice of a ime. Noe ha, due o he fac ha he annuiy paymens ae made a he beginning of each yea, a ime one has o pay bu one eceives G immediaely so ha he ne cash flow a ime is equal o - G. A vanilla inees ae swapion gives he igh, bu no he obligaion, o ene a ime ino an inees ae swap in which duing N yeas he floaing LIBOR inees ae is echanged fo a fied inees ae K N. I is well known ha he make value S N of a eceive swap in which he fied ae is eceived annually is given by see, e.g. Hull 2, Ch. 5 S N N K N D + n + + K N D + N

15 Hence, he make value V N of a eceive swapion ha gives he igh o ene ino an N-yea eceive swap a ime can be epessed as V N N N S, ma K N D + n + + K N D N,. ma Fom epession 3.3 we see ha, simila o he GAO, a swapion also gives he igh o obain a seies of cash paymens fo a pice of. Howeve, he paen of he cash paymens is vey diffeen in he wo opions. he cash flows np G associaed wih he guaaneed annuiy ae gadually deceasing ove ime due o he gadually deceasing suvival pobabiliies n p. he cash flows associaed wih an N-yea swap follow a vey diffeen paen: he fis N- yeas one eceives an amoun of K N, wheeas in he Nh yea, a cash amoun of +K N is eceived. By combining posiions in eceive swap conacs all saing a dae wih diffeen mauiies N, i is possible o eplicae he cash flow paen n p G of he guaaneed annuiy fo all daes +n. o find he igh amouns ha has o be invesed in each swap, we poceed backwads fom ime +ω- o ime +. o eplicae he cash flow ω- p G we have o ene ino he ω--yea eceive swap S ω- wih fied ae K ω-. A ime +ω- his swap has a cash flow of +K ω-. Hence, if we inves an amoun L ω- ω- p G / +K ω- in swap S ω- we eplicae he cash flow of he guaaneed annuiy a ime +ω-. One yea ealie, a ime +ω--, he guaaneed annuiy pays ou a cash flow of ω-- p G. Fom he posiion L ω- in swap S ω- we aleady eceive a cash flow of K ω- L ω- ω- p G - L ω-. Hence, if we inves an amoun L ω-- L ω- + G ω-- p - ω- p / +K ω-- in swap S ω-- we eplicae he cash flow of he guaaneed annuiy a ime +ω--. Coninuing his backwad consucion, we find ha we can eplicae he cash flow of he guaaneed annuiy a a geneal dae +n by invesing an amoun L n L n+ + G n p n+ p / +K n in swap S n. Poceeding backwads in his fashion, we coninue o mach all he cash paymens of he guaaneed annuiy up unil ime +. 5

16 Howeve, hee is a cach. Fom equaion 3.2 we see ha a he sa dae of he swap conac we equie an iniial cash paymen of. Hence, he oal pofolio of eceive swaps consuced above o eplicae he cash flows of he guaaneed annuiy equies an iniial cash paymen of ω L n. Bu in equaion 3. we deived ha he GAO pu-opion gives he igh o ene he guaaneed annuiy fo an iniial ne cash paymen of - G. Founaely, we can adjus he amouns L n by consideing eceive swaps wih diffeen fied aes K n. his implies ha we have o choose a se of fied aes K n * fo all he swaps S n such ha he invesed amouns L n * saisfy ω L * n G. Wih he pofolio of swaps Σ L n * S n we have eplicaed all he cash flows of he guaaneed annuiy wih ae G. Hence, he GAO which gives he igh, bu no he obligaion, a ime o ene ino he guaaneed annuiy is equivalen o he opion o ene ino he pofolio Σ L n * S n. his implies ha he value V G a ime of he GAO can be epessed in ems of swapions V n as: V G ma ω L * n S n, ω L * n ma ω n * n S, Ln V, 3.4 whee he inequaliy sems fom he fac ha he value an opion on a pofolio of swaps is less han o equal o he value of he pofolio of he coesponding swapions. An inuiive eplanaion fo his fac is ha in he opion on he pofolio you have only an all-o-nohing choice o obain all undelying swaps a once o none a all, wheeas in he pofolio of swapions you can chey pick he individual swaps ha have posiive make values a ime. If all he inees aes in he economy ae pefecly coelaed, i.e. all inees aes move all he ime in pefec locksep, hen hee eiss only one single se of make swap aes K n * fo which he swaps S n eacly eplicae he cash flow seam of he guaaneed annuiy. Due o he pefec coelaion of he inees aes, all make swap aes will eihe be simulaneously above he aes 6

17 K n * o simulaneously below. Hence, in he case of pefecly coelaed inees aes, he inequaliy in equaion 3.4 becomes an equaliy fo he se of swapions wih sikes K n *. 3 Bu his implies ha in he case of pefecly coelaed inees aes, we have eplicaed he payoff of he GAO via a pofolio of vanilla inees ae swapions and, a foioi, ha we have idenified a saic opions eplicaion fo he GAO. In pacice we know ha he inees aes in he economy ae no pefecly coelaed, and heefoe ha he pofolio of swapions has a highe pice han he GAO due o he inequaliy in equaion 3.4. Howeve, GAO s ypically ae poducs wih a vey long mauiy. heefoe, hei value depends mainly on he behaviou of inees aes wih long mauiies and hese inees aes ae vey highly coelaed. We heefoe conjecue ha he pice of he saic hedge eplicaion will be vey close o he ue pice of he GAO. 4. Hisoical es o es he pefomance of he saic eplicaion saegy we have poposed in Secion 3, we have conduced a hypoheical hisoical es using UK inees ae daa. his is only a hypoheical es, because in 98 he swap make in he UK was no as fa developed as i is oday. his means ha he swaps and swapions needed o eecue he saic hedge wee no available in 98. Howeve, since he hisoical peiod fom 98 unil 2 does povide a vey ineesing sesses fo ou saic hedge appoach, we eso o a hypoheical es wee we impue swap and swapion pices on he basis of UK Govenmen Bond yield daa. We downloaded fom Daaseam UK Govenmen Bond yields wih mauiies 2, 3, 5, 7,, 5, 2 and 3 yeas. We used he daa a he las ading day of each yea fom 98 unil 2. On he basis of he UK Govenmen Bond yields we consuced hypoheical swap aes by aking he bond yields as poies fo he pa swap aes wih he same mauiies. In each yea we used a Nelson-Siegel 987 paameeisaion o obain a complee em sucue of zeo-aes. In each yea he Nelson-Siegel paamees wee obained by a leas squaes fi of he swap aes implied by he zeo-cuve o he obseved Govenmen Bond yields. he esuls of he paamee fis ae 3 his emakable esul was deived fo he fis ime by Jamshidian 989 whee he showed ha in a one-faco inees ae model an opion on a coupon beaing bond can be epessed as a pofolio of opions on zeo coupon bonds. Noe also, ha in he case of pefecly coelaed aes he appaen ambiguiy in choosing he aes K n * is esolved. 7

18 epoed in able. able can be found a he end of his pape. Noe ha, in ode o sess-es ou saic hedge, we have also allowed he ime-scale paamee au o vay ove ime, o obain as much as possible vaiaion in he inees aes wih long mauiies. Paciiones usually keep he value of au consan o sabilise he long end of he yield cuve. Given he Nelson-Siegel paameeisaion, we have zeo-aes available fo all possible mauiies. Using he PMA92 moaliy able 4, we deemined he fowad annuiy aes using fomula 2.. In Figue below, we have ploed he fowad annuiy aes fo a male ha was 45 yeas old in 98 and ha would eie a age 65 in 2. Iniially, he fowad annuiy ae was above he guaaneed level of.%. Howeve, due o he falling inees aes we see ha he fowad annuiy ae dopped below he guaaneed level vey quickly afe 98. Fowad Annuiy annual payoff pe capial Dec-8 Dec-82 Dec-84 Dec-86 Dec-88 Dec-9 Dec-92 Dec-94 Dec-96 Dec-98 Dec- Figue : Fowad annuiy ae fo UK daa and PMA92 moaliy able Fom he moaliy able, we calculaed ha he minimum annuiy ae 65 * is equal o 4.56%. Fom he ime-seies of he fowad annuiy aes, we esimaed he volailiy of he fowad annuiy ae pocess a.3%. o accoun fo he fac ha implied volailiies ae highe han hisoical volailiies, we muliplied he hisoical volailiy wih a faco of.25. On he basis of a volailiy of 4.2% in fomula 2.9, we calculaed he make value of he GAO pu-opion. 4 he auho would like o hank Andew Cains fo supplying he PMA92 ables. 8

19 GAO pu-opion value pe capial Dec-8 Dec-82 Dec-84 Dec-86 Dec-88 Dec-9 Dec-92 Dec-94 Dec-96 Dec-98 Dec- Figue 2: Make value of GAO pu-opion. he calculaed make values of he GAO pu-opion have been ploed in Figue 2. Again, we see ha he value of he GAO pu-opion inceased damaically in value wih he falling inees aes duing he lae 99 s. In fac, he value of he GAO inceased almos a faco 3: fom.56% in 98 o 5.24% in Decembe 2. his aleady indicaes wha he disadvanages ae of only eseving fo mauiy guaanees insead of eplicaion: eseving is vey epensive and does no give complee poecion. See, fo eample, he esuls epoed by Wilkie, Waes and Yang 23, able hey calculae, on he basis of he 984 Wilkie model, ha he eseve a a 99% level ha would have o be se aside in 98 fo he policy wih em 2 was equal o 5,36%. As we see hee, he acual value of he GAO a he end of he 2-yea peiod 5.24% was much highe han his 99% eseve. Hence, even eseving a a 99% pobabiliy-level would no have povided sufficien poecion agains he eplosive gowh in value of he GAO pu-opion duing he 2 yea peiod fom 98 unil 2. 9

20 Fowad Swap Rae Saic Hedge Fied Rae 4.% 3.% 2.%.%.% 9.% 8.% Fowad Swap Mauiy Figue 3a: Fowad swap aes and saic hedge fied aes in Dec-98 Seing up he saic eplicaion pofolio of vanilla swapions would have been consideably cheape han only eseving, and would have povided supeio poecion. In 98, he insuance company should have foecased he annuiy paymens fo a hen 45 yea old peson which would each he eiemen age 65 in he yea 2. In Figue 3a we have ploed he hypoheical fowad swap aes of Dec-98. All swap aes ae 2 yea fowad aes, wih vaious swap mauiies. We see ha he fowad swap aes slowly deceased fom 2.79% fo he 2-yea fowad -yea swap ae, unil.25% fo he 2-yea fowad 45-yea swap ae. As was eplained in Secion 3, o se up he saic eplicaing pofolio, we have o selec a se of fied aes K n *. If all he inees aes ae coelaed pefecly, his will be he swap aes fo which he GAO will be eacly a-he-money. o consuc he saic hedge pofolio, we have made he assumpion ha all inees aes ae pefecly coelaed and also ha all inees aes move eacly paallel. 5 Hence, we have shifed all he aes by he same amoun unil he invesed ω 65 * amouns L n * saisfied L We found ha his was achieved fo a downwad n shif of.3%-poin. he se of fied aes K n * obained by his paallel shif of he swap aes has also been depiced in Figue 3a. 2

21 Cash Flow Saic Hedge Swap Fowad Swap Mauiy Figue 3b: Saic Replicaion Pofolio of Annuiy cash flows In Figue 3b, we have ploed he pojeced cash flows fo he annuiy fo he yeas 2 unil 245. Also, we have ploed he weighs L n * ha would have o be invesed in all he swaps wih fied aes K n * fo o 45. Hence, wih he weighs L n * he insuance company could have bough he pofolio of vanilla swapions Σ n L n * V n. his pofolio of swapions would have cosed 6.87 pe capial in 98, which is only.3 pe capial moe epensive han he ue make value of he GAO pu-opion. Once his pofolio of swapions would have been aained, no fuhe buying o selling would have been necessay unil Decembe 2, when he pofolio would have been unwound o cove he cos of he GAO pu-opion. 5 A moe sophisicaed appoach would be o selec a one-faco inees ae model o model he possible changes in he em sucue moe accuaely. Such an appoach would lead o an even lowe pice fo he saic hedge. Howeve, fo ease of eposiion we ae using jus a paallel shif. 6 We have calculaed he hisoical volailiy of each fowad swap-ae. o calculae he pice of each swapion we used an implied volailiy which was.25 imes highe han he hisoical volailiy. 2

22 Opion Value pe capial ime o Epiaion y 3 2 Saic Hedge Pofolio GAO pu-opion Figue 4: Pefomance of Saic Hedge Pofolio vs. GAO pu opion In Figue 4 we have ploed he value of he saic eplicaing pofolio agains he make value of he GAO pu opion fo he peiod Dec-98 unil Dec-2. he lines wih diamonds and squaes depic he make value pe capial of he saic eplicaing pofolio and he make value of he GAO pu-opion especively. We see ha he value of he saic eplicaing pofolio acks he make value of he GAO eemely closely duing he whole peiod of 2 yeas. 5. Summay and Conclusion In his pape we have deived a make value fo wih-pofis Guaaneed Annuiy Opions using maingale modelling echniques. Fuhemoe, we have shown how o consuc a saic eplicaing pofolio of vanilla swapions ha eplicaes he wih-pofis Guaaneed Annuiy Opion. Finally, we have shown in a hypoheical back es using hisoical UK inees ae daa fom 98 unil 2 ha he saic eplicaing pofolio would have been eemely effecive as a hedge agains he inees ae isk involved in he GAO, and ha he saic eplicaing pofolio would have been consideably cheape han up-fon eseving and also ha he eplicaing pofolio would have povided a much bee level of poecion han a fied eseve. 22

23 able : Nelson-Siegel zeo-cuves bea bea bea2 au 2/3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /29/ /3/ /3/ /3/ /3/ /3/ /29/ /3/ /3/ /3/ /3/ /29/

24 Refeences K. Aase and S. Pesson, 994, Picing of Uni-linked Life Insuance Policies, Scandianavian Acuaial Jounal, Vol., K. Aase and S. Pesson, 997, Valuaion of he Minimum Guaaneed Reun Embedded in Life Insuance Poducs, Jounal of Risk and Insuance, Vol. 644, A. Bacinello and S. Pesson, 22, Design and Picing of Equiy-Linked Life Insuance unde Sochasic Inees Raes, Jounal of Risk Finance, Vol. 32, 6-2. L. Balloa and S. Habeman, 22, Valuaion of Guaaneed Annuiy Convesion Opions, Woking pape, Ciy Univesiy, London. F. Black, 976, he picing of Commodiy Conacs, Jounal of Financial Economics, Vol. 3, F. Black and M. Scholes, 973, he picing of Opions and Copoae Liabiliies, Jounal of Poliical Economy, Vol. 83, M.J. Bolon, D.H. Ca, P.A. Collins, C.M. Geoge, V.P. Knowles and A.J. Whiehouse 997, Reseving fo Mauiy Guaanees, he Repo of he Annuiy Guaanees Woking Pay, Insiue of Acuaies, London, UK. P. Bouwkneg and A. Pelsse, 22, Make Value of Pofi-shaing, Jounal of Risk Finance, Vol. 33, N. Bowes, H. Gebe, J. Hickman, D. Jones and C. Nesbi, 997, Acuaial Mahemaics 2 nd Ed., he Sociey of Acuaies, Illinois. J. Bowie and P. Ca, 994, Saic Simpliciy, Risk, Augus 994, P. Boyle and M. Hady, 997, Reseving fo Mauiy Guaanees: wo Appoaches, Insuance: Mahemaics and Economics, Vol. 2,

25 P. Boyle and M. Hady, 23, Guaaneed Annuiy Opions, Woking Pape, Univesiy of Waeloo, Canada. P. Boyle and E. Schwaz, 977, Equilibium pices of equiy linked insuance policies wih an asse value guaanee, Jounal of Risk and Insuance, Vol. 44, A. Bace, D. Gaaek and M. Musiela, 997, he Make Model of Inees Rae Dynamics, Mahemaical Finance, Vol. 7, M. Bennan and E. Schwaz, 976, he picing of equiy-linked life insuance policies wih an asse value guaanee, Jounal of Financial Economics, Vol. 3, A. Cains, 22, A Family of Models fo he em-sucue of Inees Raes: An Applicaion o Guaaneed Annuiy Opions, Pesenaion a he IME22 confeence, Lisbon. P. Ca, K. Ellis and V. Gupa, 998, Saic Hedging of Eoic Opions, Jounal of Finance, Vol. 533, E. Deman, D. Egene and I. Kani, 995, Saic Opions Replicaion, he Jounal of Deivaives, Vol. 24, H. Geman, N. El Kaoui and J.C. Roche 995, Changes of Numeaie, changes of pobabiliy measues and picing of opions, Jounal of Applied Pobabiliy, Vol. 32, A. Gosen and P. Jøgensen, 997, Valuaion of Ealy Eecisable Inees Rae Guaanees, Jounal of Risk and Insuance, Vol. 64 3, A. Gosen and P. Jøgensen, 2a, Fai valuaion of life insuance liabiliies: he impac of inees ae guaanees, suende opions, and bonus policies, Insuance: Mahemaics and Economics, Vol. 26, A. Gosen and P. Jøgensen, 22, Life Insuance Liabiliies a Make Value, Jounal of Risk and Insuance, Vol 69,

26 D. Heah, R. Jaow and A. Moon, 992, Bond Picing and he em Sucue of Inees Raes: A New Mehodology fo Coningen Claims Valuaion, Economeica, Vol. 6, J. Hull, 2, Opions, Fuues, and Ohe Deivaives 4 h Ed., Penice-Hall, New Jesey. P. Hun and J. Kennedy, 2, Financial Deivaives in heoy and Pacice, Wiley and Sons, Chichese. F. Jamshidian, 989, An Eac Bond Opion Fomula, Jounal of Finance, Vol. 44, F. Jamshidian, 998, LIBOR and Swap Make Models and Measues, Finance and Sochasics, Vol. 4, P. Lee, 2, Picing and Reseving fo Guaaneed Annuiy Opions, pesenaion fo Den Danske Akuafoening, downloadable fom hp:// K. Milesen and S. Pesson, 999, Picing Rae of Reun Guaanees in a Heah-Jaow- Moon Famewok, Insuance: Mahemaics and Economics, Vol. 253, K. Milesen and S. Pesson, 22, Guaaneed Invesmen Conacs: Disibued and Undisibued Ecess Reun, Scandinavian Acuaial Jounal, fohcoming. K. Milesen, K. Sandmann and D. Sondemann, 997, Closed Fom Soluions fo em Sucue Deivaives wih Lognomal Inees Raes, Jounal of Finance, Vol. 52, M. Musiela and M. Rukowski, 997, Maingale Mehods in Financial Modelling, Spinge Velag, Belin. C. Nelson and A. Siegel, 987, Pasimonious Modelling of Yield Cuves, Jounal of Business, Vol. 64, A. Pelsse, 2, Efficien Mehods fo Valuing Inees Rae Deivaives, Spinge Velag, Belin. 26

27 D. Wilkie, H. Waes and S. Yang, 23, Reseving, Picing and Hedging fo policies wih Guaaneed Annuiy Opions Biish Acuaial Jounal, fohcoming. 27