Valuing Surrender Options in Korean Interest Indexed Annuities

Transcription

1 Valung Surrender Opons n Korean Ineres Indexed Annues Chang Km * Absrac We presen surrender rae models wh explanaory varables such as he dfference beween reference raes and credng raes, polcy age snce ssue, unemploymen raes, and economy growh raes usng he log model. We calculae he values of he surrender opons n Korean neres ndexed annues. I s neresng o noe ha he values of he surrender opons wh surrender charges are negave numbers even hough he surrender opons are he rghs gven o he polcy holders. Key words: Surrender Rae Models, Surrender Opons, Amercan Type Opon Prcng. Inroducon Surrender raes are one of he mporan facors of he neres rae sensve cash flow movemens. Km (005 b) shows ha surrender rae flucuaons really affec he cash flows of he neres ndexed annues and nvesgaes he surrender rae mpacs on he value, he duraon, and he convexy of neres ndexed annues. Kuo, Tsa, and Chen (00) uses he conegraon approach o reexamne he conendng lapse rae hypoheses: he emergency fund hypohess and he neres rae hypohess. The paper shows ha he neres rae hypohess s favored agans he emergency fund hypohess n he sense ha he neres rae s more economcally sgnfcan han he unemploymen rae n explanng he lapse rae dynamcs. Ineres-ndexed annuy (IIA) s one of he popular sngle premum deferred annues (SPDA) sold n Korea. The dsncve feaures of IIA n Korea are he surrender opons and annuzaon opons. The surrender opons are Amercan ype pu opons gven o he polcy holders. In hs paper we ry o nvesgae he characerscs of polcy holder surrender behavors and calculae he value of he surrender opons n Korean neres-ndexed annues. There are a few examples on he surrender opon prcng. Carrere (996) calculaes he early-exercse prce for opons usng smulaons and nonparamerc regresson. Albzza and Geman (994) calculaes he value of he surrender opon n lfe nsurance polces. They derve a closed-form soluon n he case of a snglepremum polcy. The prce of he surrender opon s compued n he case of French * Dr. Chang Km s Lecurer a Acuaral Sudes, Faculy of Commerce & Economcs, The Unversy of New Souh Wales, Sydney NSW 05 Ausrala. Tel: , Emal: c.m@unsw.edu.au

2 conracs usng boh he closed-form expresson and Mone Carlo smulaons. Bacnello (00) calculaes he prce of a guaraneed lfe nsurance parcpang polcy, sold n he Ialan mare, whch embeds a surrender opon. One of he man dfferences beween our approach and he prevous mehods n calculang surrender opons s he surrender rae model. We ry o model he surrender raes wh a few explanaory economc varables such as he dfference beween reference raes and credng raes, polcy age snce ssue, unemploymen raes, and economy growh raes. We use a sasc model, called he log model, n modelng surrender raes. The log model enables us o use many explanaory varables for he surrender raes. And hs log model can be a useful ool for praccal purposes. Km (005 a) shows ha he log model and he complemenary log-log model are generally beer han he exsng surrender rae models such as he arcangen model. The paper also shows ha he surrender rae models should be dfferen accordng o nsurance polcy ypes. We calculae he values of he surrender opons usng wo neres rae models, Blac, Derman, and Toy (990) model and Ho and Lee (986) model o chec f he values are ndependen on neres rae models. One of he neresng hngs we fnd s ha he values of he surrender opon wh surrender charges are negave numbers. These negave surrender opon values may be some profs o he nsurance companes bu no o he polcy holders who have he opon (or he rgh!). One of he reasons for he negave surrender opon value s ha he polcy holders do no use he surrender opons properly for her profs. The rraonal or un-opmal surrender behavors of he polcy holders may be due o bad economy condons such as low economy growh raes and hgh unemploymen raes. The surrender opon s a rgh gven o he polcy holders and we may expec ha he value of he surrender opons be posve. If he polcy holders use he surrender opons for her profs (.e. raonally or even opmally) hen he value of surrender opons can be posve and nsurance companes may no have posve gans from surrender. Anoher reason for he negave surrender opon value s he hgh surrender charges. We noce ha surrender charges really have an effec on he value of he surrender opons of IIA. The nsurance companes should fnd far surrender charges o he polcy holders consderng he values of he surrender opons. The Srucure of Korean Ineres Indexed Annues In Korea, he annuy mare s young and growng slowly. Many nsurance companes are sellng sngle premum deferred annues (SPDA). Bu SPDA are sold wh he prmary focus on accumulaon. Only a few of he polcy holders purchase SPDA for he purpose of annuzaon. Approxmaely less han % of deferred annuy values are annuzed each year n Korea. The dsncve feaures of IIA n Korea are he surrender opons and annuzaon opons. The purchasers of SPDA can surrender a any me before annuzaon f he new money raes move o her advanage wh reasonable surrender charges. A he dae of annuzaon, hey can also selec one ype of annuy ou of four choces: lump sum of her accoun value, whole lfe annuy, fxed erm annuy, or

3 nherance annuy. The dscusson ha follows wll prmarly address he basc annual guaraneed credng neres producs, so called, neres ndexed annues (IIA).. Credng Ineres Raes of IIA There are several nds of neres ndexed annues (IIA) sold n Korea. Almos all conracs guaranee a mnmum neres rae below whch he renewal credng neres raes wll no fall. Currenly he mnmum guaraneed neres rae s % n mos cases. Hsorcally, has been hgher n hgh neres envronmens. A maory of companes announce her credng neres raes based on he mare raes, nvesmen ncome gan raes, and expeced fuure nvesmen ncome gan raes. Under hs approach, he neres/gan raes currenly avalable on new nvesmens wll be refleced n he nal credng neres rae. The credng neres raes are announced every monh based on curren mare raes, curren nvesmen gan raes, and he expeced fuure porfolo ncome gan raes. Curren mare raes are usually based on he average of hree year governmen bond raes, hree year maor company bond raes, and one year maor ban depos raes. The curren nvesmen gan raes are calculaed by he average of company porfolo gans durng he prevous sx monhs. The expeced fuure porfolo ncome gan raes are esmaed reflecng he rend of he mare condons. Also mnmum level of credng raes s guaraneed.. Surrender Charges of IIA Many conracs cred he full premum o he accoun value and assess surrender charges when he polcy holder surrenders. The amoun of surrender charges are usually from 7% o 0% of he accoun value and decreased o zero over a 6-0 year perod. The range of surrender charges of dfferen companes may be hgher or lower and he penaly perods may run for shorer or longer. The amoun of new conrac acquson coss or commssons wll be recovered by he amoun of surrender charges and fuure nvesmen ncome. The surrender charges wll preven he cusomers from ermnang he conrac early and cover he unamorzed acquson coss or commssons gven o he agens a he me of ssue.. Free Paral Whdrawals/ Provsonal Loans of IIA A poron of he accoun value can be whdrawn a any me whou surrender charges o provde lqudy o he conrac owner. The maxmum level s 90% of he accoun value a he me of paral whdrawal, bu a few companes mgh lm he maxmum level much lower han 90% of he accoun value. The neres rae for hs paral whdrawal s usually credng rae plus.5%. Ofen he polcy holders can ae advanage of hs paral whdrawal opon several mes a year. For example, when he For varous characerscs and valuaon of SPDA, we may refer Socey of Acuares (99), Cox, Lapore, Lnney, and Lombard (99), and Asay, Bouyoucos, and Marcano (99).

4 soc mares show sgns of an upward ump, he polcy holders can draw ou her savngs from he accoun whou any surrender charges and nves hs amoun of money n he soc mares. Afer enoyng he profs from he soc mare, hey can reurn o her nsurance conracs payng relavely low neres. So hs characersc of hgh maxmum level of paral whdrawal whou surrender charges s a source ha one mgh overuse he paral whdrawal opon. Moreover he deah benef amoun s sll guaraneed durng he paral whdrawal perod..4 Deah Benefs n IIA Usually he deah benef s he accoun value. A few varaons of deah benefs are consdered accordng o he companes, for example, he accoun value plus 0% of premum, and anoher 0% of premum n he case of accdenal deah. Some conracs allow he spouse o ae over ownershp of he conrac a he me of deah of he owner f he spouse was a benefcary..5 Annuzaon and Annuy Opons n IIA The polcy holder can choose he nal annuzaon dae. The owner may change before he chosen nal annuzaon dae. The range of he nal annuzaon dae s from age 45 o age 70 and usually 0 years afer ssue. Guaraneed annuzaon raes may be announced by he company, bu hese raes are really conservave. The credng raes reflec he curren mare raes and porfolo ncome gan raes wh mnmum guaraneed rae of %. Bu he guaraneed annuzaon raes may be based on he mnmum guaraneed rae of % plus very conservave bonus. Some polcy holders prefer mnmum rae of reurn guaraneed producs. The moraly may be mldly conservave reflecng annual mprovemen facors, n recognon of ancpaed fuure moraly reducons. There are several annuy opons provded by he conrac. The ypcal ypes of annuy opons are (a) he lump sum whdrawal of he accoun value a he dae of annuzaon, (b) ceran perods from 5 o 0 years, (c) lfe ncome wh a guaraneed perod of 0 years, and (d) nherance annuy. The annuan of nherance annuy receves only he neres of he accoun value whle he/she s alve and he accoun value wll be gven o he her/heress when he annuan des. A he me of annuzaon, he polcy holder can choose he ype of annuy consderng he healh condon of he annuan and he neres rae level. So here may be poenal moraly and neres rae rss o he companes wh hs annuy opon. Modelng Surrender Raes for Korean Ineres Indexed Annues As dscussed n Km(005 a), he surrender opon s no a funcon of neres rae only. I depends on he polcy age snce he conrac was ssued. I also reflecs he unemploymen rae and he economy growh rae. For old examples of lapse sudes, we may refer Rchardson and Harwell (95), Buc (960), and Brzezns (975). We wan o model he polcy holder surrender behavor sascally. The varables consdered are (a) he dfference beween reference new money raes and 4

5 produc credng raes wh surrender charges, (b) he polcy age snce he conrac was ssued, (c) unemploymen raes, (d) economy growh raes, and (e) seasonal effecs. We wll use a cascade mehod o generae he varables raher han use a full mulvarae analyss, even hough he explanaory varables are based on he mehods of me-seres analyss. Fgure shows he cascade srucure for he surrender rae modelng. Fgure. Cascade Srucure for he Surrender Rae Modelng Term Srucure Shor Raes Announced Raes Credng Raes Surrender Charges Reference Mare Raes Economy Growh Raes Economy Cycle Polcy Age Unemploymen Raes Seasonal Effecs Surrender Raes. Smulaon of Basc Raes We need a few basc raes o be used n modelng surrender raes and valung he opons n IIA conracs. The basc raes are shor raes {(,), = 0,,, T-, and =0,,,, for each } for dscounng he fuure cash flows, new money raes { m (, ), = 0,,, T-} for reference raes, economy growh raes { EG (, ), = 0,, For cascade mehods, refer Wle (987 and 995). 5

6 , T-}, unemploymen raes { UE (, ), = 0,,, T-}, announced raes { a (, ), = 0,,, T-}, IIA credng raes { c (, ), = 0,,, T-}, mnmum guaraneed raes g, and surrender charges Sc. For he deals and usfcaons on he smulaon mehods of he basc raes, refer Km (005 b). We resae he smulaon resuls from Km (005 b) for our uses n hs paper... Shor Raes For a gven erm srucure, we calculae he se { P (0,)} of zero coupon bonds from a gven neres rae model. P (0,) s he me 0 prce of a zero coupon bond payng a maury. The se of shor raes { (,)}, for = 0,, and for =0,,,, for each, should be smulaed sasfyng he followng equaon, P (0,) = P ˆ(0, ), () where P ˆ(0, ) s he me 0 prce of a zero coupon bond payng a maury from he mare. We use Blac-Derman-Toy (BDT) model and Ho-Lee (HL) model o generae shor raes,{, =0,,T-}. In valung a sream of pah-dependen cash flows, we may face praccal lmaons snce he oal number of pahs s really huge. For example, n valung he surrender rae opon prces of neres rae ndexed annues (IIA) wh 0 year of surrender opon maury, we have 0 pahs, whch s compuaonally nfeasble. So we need o pc a subse of all neres rae pahs wh a compuaonally reasonable number, K, of sample pahs. Fgure. Smulaed Shor Rae from a Bnomal Lace (, ) = (5,) For an example of he procedure for selecng sample pahs, see Ho (99). =5 6

7 Le us denoe he se of all neres rae pahs as Ω and he number of elemens of 0 Ω as Ω. Then Ω =. We denoe he subse of sample pahs as Ωˆ wh Ω ˆ = K, Ωˆ = {,,..., K }. Here we denoe he -h pah of Ωˆ as funcon,, from me o sae such ha s he sae, s, a me on pah. For a pah, we can defne a, = s, () where =,,, K, = 0,,, T-, and s =0,,,. For noaonal smplcy, le us denoe (, ) o be he node a me on pah, nsead of (, ), on he bnomal lace. We show an example n Fgure. Snce he sae s a me =5 on he pah, (, ) s (5,) on he bnomal lace... New Money Raes We propose o use he maxmum of he year yeld raes (under he ncreasng erm srucure) and he shor raes (under he decreasng erm srucure) as he reference new money raes. We wll use hese reference raes o calculae he value from surrender of surrendered conracs under he assumpon ha he surrendered cash value s nvesed wh he reference new money raes4. A node (,), o calculae he year yeld raes,y(,,+year), for he nex en years, we can use neres rae models such as Blac-Derman-Toy (BDT) or Ho-Lee (HL) models wh he followng formula, u P(,,u) = A(,, u, ) = 0 =, () u { + Y, u)} where P(,,u) s he prce a node (,) of he zero-coupon bond maurng a me u>, and A(,,u,) s he Arrow-Debreu prce. 4 As one of examples, we choose year raes for he reference new money raes. We may choose oher reference raes such as 5 year or 0 year raes ec. We denoe m o be reference raes, reference mare raes, and new money raes a me wh he same meanng. 7

8 For a gven erm srucure, we can generae shor raes by an neres rae model, { (, ), =0,,,,T, and =,,,K} n he fuure. The resulng formula for new money raes, m (, ), s m (, ) = max{ (, ), Y(,,6)}. (4) Noe ha as he reference raes, we use he shor raes when he erm srucure s decreasng or he year yeld raes when he erm srucure s ncreasng a (, )... Economy Growh Raes We collec pas daa on economy growh raes { EG } and pas reference raes { m } of IIA, and nduce a funcon h EG such ha EG = h EG ( m ) + e EG, (5) where e EG s an error erm. We observe ha here are paerns on economy crculaons wh perods of 0 monhs. We consder hs n modelng economy growh raes. We show he formula for he economy growh raes below, where π EG = * m *sn( ) 0 π * cos( ) + ξ, (6) 0 ξ = ε - ξ, - φ ξ, (7) ε follows he normal dsrbuon, N(0, φ wh φ = , φ = 0.50 and ). We use hs funcon o smulae fuure economy growh raes, EG (, ) = h EG ( m (, )) + e EG, (8) for he gven new money rae m (, )...4 Unemploymen Raes We collec pas daa on unemploymen raes { UE } and observe ha he unemploymen raes are gven by he followng formula, UE = UE (-) * { EG DV DV DV DV DV DV DV DV 0 } + ε, (9) where EG s he economy growh rae a me, DV s a monh and 0 oherwse, and ε follows he normal dsrbuon, N(0, 0.005)...5 Announced Raes 8

9 The ssung companes usually announce credng neres raes every monh based on curren mare raes, curren nvesmen gan raes, and he expeced fuure porfolo ncome gan raes. Consderng smplcy and hgh correlaon beween reference mare raes and announced raes, we collec pas reference raes { m }, and nduce a funcon h a such ha a = h a ( m ) + e a, (0) where e a s an error erm. Now we have he formula for he announced raes, a = * m ( ) + ε, () where ε follows he normal dsrbuon, N(0, ). We use hs funcon o smulae fuure announced raes, a (, ) = h a ( m (, )) + e a, () for he gven reference rae m (, )...6 Credng Raes of IIA The credng rae, c (, ), s he maxmum beween he raes announced by he company and he guaraneed neres rae, c (, ) = max{ a (, ), g }, () where he announced rae a (, ) s gven, and he guaraneed annual neres rae g s, g = % annually. (4) In pracce, consderng he surrender behavors of he polcy holders, he credng rae c (u, ) a me u on pah s dependen on he laes surrender me before me u, * c (u, ) = c (u,, u ), (5) where * u = 0, f here s no surrender before me u = he laes surrender me before me u. * And we may use he noaon c (u, ) and c (u,, u ) ogeher hrough ou he whole chapers. For compuaonal purposes, we use he formula for he announced raes * a (u, ) = a (u,, u ) = * 0.m ( u, ) where ε u follows he normal dsrbuon, N(0, ). m ( u, ) + ε u, (6)..7 Surrender Charges 9

10 The surrender charges durng he year, Sc, are Sc = 7%, 0 < Sc = 6%, < Sc = 5%, < Sc = 4%, < 4 Sc = %, 4 < 5 Sc = %, 5 < 6 Sc = %, 6 < 7 Sc = 0%, 7 < 0. (7)..8 Surrender Raes I s neresng o noe ha he probably of surrender depends on so many varables such as he dfference beween new money raes and credng raes, economy growh raes, unemploymen raes, polcy-age snce IIA polcy was ssued, and seasonal effecs. The exsng surrender rae models such as arcangen model or exponenal model are usng he dfference beween reference mare raes and credng raes only. Bu we have observed ha surrender raes are dependen on dverse varables5. For explanaory purposes we consder only year of polcy-age snce IIA polcy was ssued. The seasonal effecs are consdered wh he dummy-varable, on h monh DV =, (8) 0 oherwse where -s monh s January, -nd monh s February, -rd monh s March, and so on. So DV = on January, DV = on February and so on. The probably of surrender beween me and + and on pah, qs ), s gven by d s ) qs ) = l(, ) = f( m (-, )- c (-, ), UE (, ), EG (, ), DV ), (9) where l (, ) s he number of conracs survved, d s (, ) s he number of conracs surrendered, and DV s he seasonal effec dummy-varable. Here, we use he Log Model, 5 In Cox, Lapore, Lnney, and Lombard (99), hey examne he perssency of sngle-premum deferred annues wh varous varables such as produc characerscs, produc dsrbuon mehods, creded neres raes, and age of he annuan. Even hough hey do no nclude oher varables such as federal ax penaly for early whdrawals, changes n famly ncome or wealh, healh of he annuan, dvorce, marrage, employmen saus, hey menon ha hese varables may also be relaed o whdrawal acvy. They quanfy he relaonshp beween whdrawal acvy and varables assocaed wh whdrawal acves. 0

11 qs ) ln qs ) = β 0 + β = 0,,4,6,8,0, *( m (-, )- c (-, + β UE * UE (, ) + β EG * EG (, ) + )) = β *DV. (0) monh The parameer esmaes are shown n Table. I s neresng o noe ha he parameer β UE for he unemploymen raes s very large, I means ha he surrender raes change very grealy accordng o he unemploymen rae movemens. Bu, consderng he unemploymen rae change rao s no so radcal as ha of he reference mare raes (new money raes), s no srange for us o have a large β UE. I seems also reasonable ha he parameer β EG for he economy growh raes s a negave number, We can guess ha when he economy condon s good he polcy holders may no surrender her IIA polces. Now our fnal model for he IIA surrender raes, { qs ), =0,,,, T and =,,,K}, s gven by he followng formula, where α = β + β * ) + β qs ) =, () + exp( α ) * ) 0 UE UE EG EG + β *{ (, ) (, )} + β DV. () m c = 0,,4,6,8,0, = monh_ * We show he graph of he real and predced (usng Log model) surrender raes of IIA polces below. Table. Parameer Esmaes wh Log Model (IIA) Analyss of Maxmum Lelhood Esmaes Sandard Parameer DF Esmae Error Ch-Square Pr > ChSq Inercep <.000 DIFFLAG <.000 DIFFLAG <.000 DIFFLAG <.000 DIFFLAG <.000 DIFFLAG <.000 DIFFLAG <.000

12 DIFFLAG <.000 Unemploy <.000 Eco GROWTH <.000 MONTH <.000 MONTH <.000 MONTH <.000 MONTH <.000 MONTH <.000 MONTH <.000 MONTH <.000 MONTH <.000 MONTH <.000 MONTH <.000 MONTH <.000 Fgure. Real and Predced Surrender Raes of IIA Expeced Real 0.00% 9.00% 8.00% 7.00% 6.00% 5.00% 4.00%.00%.00%.00% 0.00% Tme 4 Valung Surrender Opons n Ineres Indexed Annues As dscussed above, here are several mplc opons n neres ndexed annues (IIA) such as surrender opon, mnmum neres rae guaranees, mnmum rae of reurn guaranees, and annuy selecon opon. Also here are deah benefs before annuzaon wh he amoun of he accoun value plus 0% of nal premum whou he applcaon of surrender charge. The man facor of rss among he mplc opons

13 n IIA conracs s he surrender opon gven o he polcy holders. The new acquson coss are pad o he agens or broers as commssons a he me of ssue. If a cusomer surrenders early before he company maes enough prof o recover he amoun of he new acquson coss, here may be losses o he company. Also early surrender mgh cause he msmach of asse and lably cash flows resulng n he duraon msmach. In hs secon, we wan o examne he surrender opon of IIA and calculae he value of hs opon. The surrender opon may no be exercsed opmally. Even when he compeve mare raes are hgh, some cusomers may no surrender her polces. As we see above, he polcy holder surrender behavor depends no only on he dfference beween he reference mare raes (new money raes) and credng raes, bu also on he polcy age snce he conrac was ssued, he unemploymen raes, surrender charges, seasonal effecs, and economc growh raes. A me and on pah, he probably of surrender, q, ), s gven by where s ( qs ) =, () + exp( α ) α = β + β * ) + β * ) 0 UE UE EG EG + β *{ (, ) (, )} + β DV. (4) m c = 0,,4,6,8,0, = * monh_ We consder he same IIA conrac for a represenave male clen of age 40 wh nal sngle premum of A as he one n he prevous secon. We already defned he accoun values for a pah, { AV (, )}, as follows. The accoun value a me 0 and on pah, AV (0, ), s he nal sngle premum, AV (0, ) = A. (5) The accoun value a me and on pah s gven by AV (, ) = AV (-, ) [+ c (-, )], (6) where c (-, ) s he credng rae a me - and on pah. A me and on pah, he polcy holder has he accoun value of AV (, ). And he/she also has he opon o surrender a me. If he/she hns would be advanageous o surrender IIA polcy and accumulae wh hgh yeldng new money raes, he/she mgh surrender he IIA conrac a me. A me on pah, he polcy holder has he nformaon on he new money rae (reference rae) m (, ), he credng rae c (, ), and he surrender charges Sc. A me and on pah, he polcy holder has he accoun value AV (, ) of IIA conrac and can consder alernave value from surrender, SV (, ), he/she may ge when he/she surrenders he conrac wh he new money rae (reference rae) m (, ) and surrender charge. The polcy holder may also consder he eepng value, KV (, ), he value ha he polcy holder may ge when he/she does no surrender he conrac. The exercse value for he polcy holder who decdes o surrender he IIA conrac s calculaed by he dfference beween he value from surrender, SV (, ), and he eepng value, KV (, ). To fnd he value

14 from surrender and eepng value a he node (, ) a me on a pah, we have o consder he fuure pahs. Le us denoe Ω (, ) o be he whole pahs from he node (, ) and Ωˆ (, ) o be a subse of sample pahs of Ω (, ). Then he number of T elemens or pahs n Ω (, ), denoed by Ω (, ) s, whch may be a large number we need o smulae a subse Ωˆ (, ) of sample pahs of Ω (, ), wh a compuaonally reasonable number of pahs, Ωˆ (, ). We show a sample pah of Ωˆ (, ),.e. Ωˆ (, ) below6. Le us denoe (s, (s)) o be he node a me s> on pah on he bnomal lace. We show an example n Fgure 4. Snce he sae s 5 a me s=9 on he pah, (s, (s)) s (9,5) on he bnomal lace. For smplcy, we denoe he node as (s, ) nsead of (s, (s)). Fgure 4. A Sample Pah from (, ) (s, ) = (9,5) (, ) = (5,) =5 s=9 6 We use Blac-Derman-Toy model and Ho-Lee model for he bnomal shor rae lace. The sample pahs can be seleced usng he lnear pah space model. 4

15 Generally, we descrbe a mahemacal srucure for he mulperod model of a fne se of sample pahs, Ωˆ = {,,, K }, as follows. We assume ha each Ωˆ has a posve probably of occurrence. We assume ha a fne sequence of parons, { P 0, P,, P, P T }, of Ωˆ s defned sasfyng P 0 P v P P T. We also assume ha P 0 = { Ωˆ } = {{,,, K }}, and P T = {{ }, { },, { K }}. The par ( Ωˆ, { P }) s called as a flered space. The number of ses of s comprsng he paron P s denoed by v so ha v = P and P = ( ) ( ) { H, H,, H }. The elemens { H } of P are called o he me- hsores. The elemen of paron P whch conans a pah s denoed by v o be he number of elemens of conssng of one represenave from each of he ses H H ( ). We also denoe ( ) ( ) H, H ˆ ( ) ˆ ( ) ( ). We denoe ˆ ( ) Ω o be a se,, H comprsng P. Then he number of elemens of Ω s v, Ω = v. Noe ha he choce of ( ) represenave ses { Ω ˆ (0) (), =0,,,, T} s no unque. We assume ha ˆΩ ˆΩ ( ) Ωˆ ( ) Ω ˆ T. v Fgure 5. A Se of Sample Pahs Tme 0 Tme Tme Tme We show an example below. Suppose T=. Then we have Ω =. We assume a se of sample pahs, Ωˆ = {,,, 7 }, wh K=7. From Fgure 5 we have 5

16 P 0 = { Ωˆ }, K=7 () () P = { H, H }, v =, wh () () H = {,,, 4 }, v = 4, () H = { 5, 6, 7 }, () v =, () () () () P = { H, H, H, H 4 }, v = 4, wh () () H = {, }, v =, () () H = {, 4 }, v =, () H = { 5 }, () H 4 = { 6, 7 }, () v =, () v 4 =, P = {{ }, { },, { 7 }}, v = 7 = K. (0) () We may have a parcular choce of represenave ses, ˆΩ, ˆΩ, (0) () () () me such ha ˆΩ ˆΩ ˆΩ ˆΩ. For example (0) ˆΩ = { }, () ˆΩ = {, 5 }, () ˆΩ = {,, 5, 6 }, () ˆΩ = {,,, 7 } = Ωˆ. () ˆΩ, and () ˆΩ a each The polcy holder has he opon o surrender a any me on any pah or o eep hs/her polcy. If he/she surrenders he polcy, hen he/she can decde o surrender agan or eep hs/her polcy afer one perod. If he/she eeps he polcy hen he/she can also decde o surrender or eep hs/her polcy afer one perod. Fgure 6. Decson Trees surrender surrender eep eep surrender eep 6

17 We use he reference raes (new money raes) o calculae he value from surrender of surrendered conracs under he assumpon ha he surrendered cash value s accumulaed by he reference raes (new money raes) on a pah. And we use he credng raes o calculae he accumulaed value when he polcy holder does no surrender. Noe ha he credng raes are he funcon of curren reference rae and he reference rae of laes surrender me. If here s no surrender before, hen he credng rae s he funcon of curren reference rae and he reference rae of nal me. The credng rae s greaer han or equal o he mnmum guaraneed rae. If he polcy holder surrenders an IIA conrac ssued a me 0 a me and on pah, he/she would have he accoun value reduced by surrender charges a me, AV (, ) (- Sc ). (7) Afer me on a pah H ( ), he polcy holder may surrender or eep he polcy unl he end of opon perod. We denoe C(,,T, ) o be he se of surrender behavors of he polcy holder from me o me T-, gven surrender a me, on a pah H ( ), C(,,T, ) = { e, T, ) e l, T, ), e, T, ), e, T, ),, e, T, ) T }, (8) T where each, l =,,,, s a surrender behavor of he polcy holder from me o me T-, gven surrender a me, on a pah H ( ). Here we assume ha he polcy holders always renves. We show an example of he se C(T-,,T, ) of surrender behavors gven surrender a T-,, T, ) C(T-,,T, ) = { e, e wh each surrender behavor s as follows, T, e ), T, ), T, ) e, T, ), e = (s, s, s), = (s,, s),, T, ), e, T, ) 4 }, (9) e = (s, s, ), and, T, ) e4 = (s,, ), (0) where s denoes surrenderng and denoes eepng. 7

18 If he polcy holder nvess he surrendered amoun of money, AV (, )(-, T, ) Sc ), unl he opon maury7 T under he surrender behavor e l, hen he accumulaed value from surrender8 a me T and on a pah H ( ) s denoed by, T, ) T SV ( e l ), l =,,,. Ths value depends on he polcy holder surrender behavors such as he number of surrenders and surrender mes. As an example, f we assume ha he surrendered cash value s nvesed wh new (,,, ) money raes for wo consecuve perods only,.e. under e T = (s,s,,,., ), hen we have (,,, ) SV ( e T ) = AV (, ) (-Sc) (+ m (, )) (-Sc()) (+ m (+, )) * ( + * c ( u,, u )), u= + () where * u = 0, f here s no surrender before me u = he laes surrender me before me u, () * and c (u,, u ) s he credng rae a me u, on pah H ( ). Snce we assume ha surrender occurs a he end of perod we apply he surrender charge Sc() n he above formula. We also assume ha here s no surrender charge a me T. A me T, he average of he accumulaed value from surrender, SV (,,T, ), gven, T }, where Pr( e H ( SV (,,T, ) = ( T,,, ) C(,,T, )9. ), s he condonal expeced value of { SV ( Q, T, ) ( ) E [ SV ( e ) H ( )], T, ) T, T, ) e l ), l =,, = Pr( e ) SV ( e ), () (,, T, ) e C, T, ), T, ) ) s a rs neural probably of he surrender behavor e ( T,,, ) 7 Surrender opon maury T s he annuzaon dae of IIA conracs. 8 Here we assume ha he surrendered cash values are accumulaed by he reference new money raes on a gven se of pahs of new money raes under he smulaon based valuaon mehod. We may choose anoher reference raes and accumulaon mehods. We also assume ha he polcy holders always renves. In pracce, hey may ermnae her conracs wh lump-sum whdrawals before me T. (00). 9 Noe ha we use a rs neural probably of he surrender behavor. As an example see Moller 8

19 A curren me, he value from surrender, SV (, ), s he expeced dscouned value of { SV (,,T, ), H ( )} dscouned by he shor raes, {(u, ), u=,,t-, H ( )}, SV (, ) = SV Q, T, ) ( ) E H ( ) T ( + ( u, )) u= = SV, T, ) Pr( ), T (4) ( ) H ( ) ( + ( u, )) where Pr( ) s a rs neural probably. If he polcy holder does no surrender he IIA conrac a me and on pah, he/she would have he accoun value AV (, ). The accoun value wll be ncreased wh credng rae c (,, u= * ). Afer one perod, a me u> and on a pah ( ) H ( ), he polcy holder may decde o surrender or no o surrender (.e. eep) he polcy. And he accoun value wll be accumulaed accordng o he surrender behavor of he polcy holder. We denoe F(,,T, ) o be he se of surrender behavors of he polcy holder from me o me T-, gven eepng a me, on a pah H ( ),, T, ), T, ), T, ), T, ) T F(,,T, ) = { f, f, f,, f }, (5), T, ) T where each f l, l =,,,, s a surrender behavor of he polcy holder from me o me T-, gven eepng a me, on a pah H ( ). We show an example of he se F(T-,,T, ) of surrender behavors gven eepng a T-,, T, ), T, ), T, ), T, ) F(T-,,T, ) = { f, f, wh each surrender behavor s as follows, T, ) f = (, s, s), f = (,, s), f, T, ), f, T, ) 4 }, (6) f = (, s, ), and, T, ) f 4 = (,, ), (7) where s denoes surrenderng and denoes eepng. 9

20 If he polcy holder nvess he amoun of money, AV (, ), unl he opon, T, ) maury T under he surrender behavor f l, hen he accumulaed value a me T, T, ) T and on a pah H ( ) s denoed by KV ( f l ), l =,,,. Ths value depends on he polcy holder surrender behavors such as he number of surrenders and surrender mes. As an example, f we assume ha he accoun value s no surrendered for wo consecuve perods (me and +), surrendered for he nex wo consecuve perods (,,, ) (me + and +), and here s no surrender afer me u>+,.e. under f T = (,,s,s,,,,, ), hen we have KV ( f (,, T, ) ) = AV (, )*(+ c (,, * * ))*(+ c (+,, ( +) )) * (-Sc(+))* (+ m (+, ))*(-Sc())*(+ m (+, )) T * ( + * c ( u,, u )), (8) = + u 4 * where H ( ), and c (u,, A me T, he average of he accumulaed value, KV (, u ) 0 s he credng rae a me u, on pah. H ( ), s he condonal expeced value of { KV ( T }, KV (,,T, ) = (,, T, ) Q, T, ) ( ) E [ KV ( f ) H ( )], T, ) f l, T, ),T, ), gven ), l =,,, = Pr( f ) KV ( f ), (9) (,, T, ) f F, T, ), T, ) (,, T, ) where Pr( f ) s a rs neural probably of he surrender behavor f F(,,T, ). A curren me, he eepng value, KV (, ), s he expeced dscouned value of { KV (,,T, ), H ( )} dscouned by he shor raes, {(u, ), u=,,t-, H ( )}, KV (, ) = Q E KV ( ), T, ) ( ) H T ( + ( u, )) u= 0 Noe ha he credng raes for he value from surrender and for he eepng value may be * dfferen a me u >, snce hey are dependen on pas new money raes appled. u s defned o be he laes surrender me before u or 0 f here s no surrender before me u. 0

21 = ( ) H ( ) KV, T, ) Pr( ), (40) T ( + ( u, )) where Pr( ) s a rs neural probably. A me on a pah, some polcy holders surrender her IIA polces wh posve probably q, ). Here we wan o calculae he dfference beween he s ( value from surrender SV (, ) and he eepng value KV (, ). Ths can be he exercse value, EV (, ), for he polcy holder who surrenders her IIA conrac a me on a pah even hough hey do no now he fuure. Ths exercse value can be posve or negave. EV (, ) = p ) * qs )*{ SV (, ) - KV (, )}. (4) Noe ha, a he annuzaon me T, he polcy holder has he opon o surrender IIA polcy (ae ou he lump sum) or use he annuy opon. Bu, rgh now, we do no have enough experence daa on he use of he annuy opon, and we do no have relable sascs on he rao of surrender a he me of annuzaon T, because much of he busness s sll young. Acually annuy opon s anoher source of rss n IIA. Here we us consder he surrender cases before he annuzaon me T and assume ha he ermnal exercse values are gven by EV (T, ) = 0. (4) Now we calculae he value of surrender opon (VSO) for he oal exercse values a me 0 as below, u= K T EV ) VSO = Pr( ), (4) = = ( + ( u, )) where K denoes he number of pahs a me 0. We presen he value of surrender opon (VSO) n Table. Case s for surrender charges Sc(0) = 7% and decreased by % each year unl 0%. Case s for surrender charges Sc = 0% for all me. Here he nal sngle premum s 0,000. u= 0 Table. Value of Surrender Opon (VSO) Surrender VSO (BDT) VSO (HL) Charge Case 7% - 0% Case 0% We noce ha surrender charges really have an effec on he value of he surrender opon of IIA.

22 We calculae he values of he surrender opon usng wo neres rae models, Blac-Derman-Toy (BDT) model, and Ho-Lee (HL) model o chec f he values are ndependen on neres rae models. And he wo values of he surrender opon of IIA wh BDT model and HL model are almos he same, bu no exacly he same. So he values may be dependen on he parcular choce of neres rae model. I s also neresng o noe ha he values of he surrender opon wh surrender charges are negave numbers. These negave values may be some profs o he nsurance companes and no o he polcy holders who have he opon (or he rgh!). The surrender opon s a rgh gven o he polcy holders and we may expec ha he value of he surrender opons be posve. I may no be really surprsng for someone who noce ha some nsurance companes ge posve gans from surrender. If he polcy holders can use he surrender opons for her profs (.e. raonally or even opmally) hen he value of surrender opons can be posve and nsurance companes may no have posve gans from surrender. We may consder, for fuure research opc, he rraonal surrender behavors of he polcy holders probably caused by oher economy condons such as low economy growh raes, hgh unemploymen raes, ec and he raonal and opmal surrender behavors. 5 Far Surrender Charges As dscussed above he values of he surrender opon wh surrender charges are negave numbers. The negave values of he surrender opon are some profs o he nsurance companes and no o he polcy holders who have he opon. Bu he polcy holders may expec ha he values of he surrender opon should be posve. Even hough he reasons for he negave values of he surrender opon may be he rraonal polcy holder surrender behavors due o unemploymen or oher facors, we sll noce ha he curren surrender charges are very hgh. Here we calculae he values of surrender opon under four surrender charge scenaros: from 7% o 0% decreased by % each year for 7 years, from 5% o 0% For morgage-baced secures, Green and LaCour-Lle (999) show ha borrower prepaymen behavor appears o be hghly rraonal somemes, n he sense ha many borrowers prepay her morgages when s no opmal o do so and fal o prepay her morgages when he prepaymen opon s subsanally advanageous. Rchard (99) menons several reasons why people do no prepay when he pay off s posve. One reason s ha he borrower has los hs ob. Anoher reason s ha he value of he house s lower han he morgage value. There s also a group of people who are no raonal and hold hgh-premum loans whou prepayng hem. We may nvesgae he raonal surrender behavors,.e. he polcy holders surrender her conracs only when he exercse values are posve. For raonal prepaymen of morgage-baced secures, see Sanon (995), and McConnell and Sngh (994).

23 decreased by % each year for 5 years, only % for 5 years, and no surrender charges. Usng varous surrender charge scenaros, he nsurance company may fnd far surrender charges no only for he company bu also for he polcy holders consderng he economc facors n he surrender rae models. Table. Fndng Far Surrender Charges Surrender VSO (BDT) VSO (HL) Charges 7% - 0% % - 0% % % Concluson In Korea, many nsurance companes are sellng sngle premum deferred annues (SPDA). Especally neres-ndexed annues (IIA) are one of he mos popular SPDA producs. There are several opons n IIA such as surrender opon, mnmum neres rae guaranees, mnmum rae of reurn guaranees, and annuy selecon opon. The surrender behavors of IIA polcy holders are complcaed and somemes rraonal. I s really dffcul o nvesgae and quanfy he polcy holder surrender behavors wh numercal mehods. We have consdered a model on he polcy holder surrender behavors sascally. The basc explanaory raes are modeled and he surrender raes are calculaed usng he Log model. The varables consdered are (a) he dfference beween reference raes (new money raes) and produc credng raes wh surrender charges, (b) he polcy age snce he conrac was ssued, (c) unemploymen raes, (d) economy growh raes, and (e) seasonal effecs. We use a cascade mehod o generae he varables raher han use a full mulvarae analyss. We use Blac-Derman-Toy model and Ho-Lee model o generae shor raes. We noce ha he choce of neres rae model s a consderaon n valung neres rae conngen cash flows. The values of he surrender opon of IIA may depend on he choce of neres rae models. We also noce ha he surrender charges have an effec on he value of he surrender opon of IIA. Fndng far surrender charges no only for he nsurance companes bu also for he polcy holders should be consdered. I s neresng o noe ha he values of he surrender opon wh surrender charges are negave numbers, whch can be some profs o he nsurance companes no o he polcy holders who have he opon (or he rgh!). For he fuure research opc we may nvesgae he raonal/rraonal surrender behavors of he polcy holders. References

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