ON A FAIR VALUE MODEL FOR PARTICIPATING LIFE INSURANCE POLICIES

Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 05 ON A FAIR VALUE MODEL FOR PARTICIPATING LIFE INSURANCE POLICIES Fabio Baione, Paolo De Angeli, Andrea Forunai Abrac The aim of hi aer i o analyze boh he erm rucure of inere and moraliy rae role for evaluaing a fair value of a life inurance buine. In aricular, a fair value accouning imac on reerve evaluaion i dicued comaring a radiional deerminiic model baed on local rule for an Ialian balance hee calculaion and a ochaic one baed on a diffuion roce for boh moraliy and financial rik. A rooed by IAS Board we will earae he embedded derivaive from heir ho conrac, o he fair value of a radiional life inurance conrac would be ereed a he value of four comonen: he baic conrac, he ariciaion oion, he oion o annuiie and he urrender oion. A numerical alicaion o a radiional Ialian life inurance olicy i dicued. Key word: Fair ricing, ariciaion oion, urrender oion, guaraneed annuiy oion, Black&Schole-CIR framework, Longaff-Schwarz Lea-Square Aroach. JEL Claificaion: C5, G3, G22.. Inroducion Lieraure on Inernaional Accouning Sandard in he la hree year ha been imroved, no only in financial area, by auhoriaive cienific and rofeional conribuion. The eecaion for a final verion of an accouning andard guide line for inurance comanie by IASB organiaion ha given rie o an inene mehodological debae focued on echnical oluion for accouning rule alicaion wihin inurance ecor. The meaning of Fair Value eended o inurance conrac involve, by a ide, an adoion of new raegic choice for managing reource and rucure and, on he oher ide, rereen a way in which acuarial heory could aume a rimarily role in mehodological ah definiion. More in general, when a liquid marke of inurance conrac i available, Fair Value of an inurance conrac i equal o i marke value; oherwie i i neceary o obain an eimae of he marke value uing a heoreical model conien wih economic oeraor behaviour in a aricular rik condiion. The forhcoming IAS guide line rooe ha inurance liabiliie hould be valued a if hey are raded among well-informed, indeenden inveor in a liquid markelace. IAS guide line allow uing of ochaic mehod o eimae fuure cah flow (liabiliie including embedded oion) ariing from he conrac. In hi cone, inurance comanie hould adequae heir inernal rocedure o characerize and eimae all he olicie comonen a fair value. A an eamle, radiional Ialian ariciaing life inurance conrac, alo known a wih-rofi, enile he olicyholder of a cerain ar of he rofi generaed by he ae aociaed wih he conrac; hee rofi are credied o he mahemaical reerve increaing he inurer liabiliie. According o he recen lieraure (Groen and Jørgenen (2000), Bacinello (200)), he value of he reviou conrac could be li ino wo comonen: a baic conrac and a ariciaion or bonu oion; a bonu oion i a ariciaing Euroean-yle oion where he benefi i annually adjued according o he erformance of a reference fund. Furhermore, if a conrac rovide ha he olicyholder can urrender he conrac before mauriy, hi comonen i called urrender oion. According o he recen inurance lieraure on hi argumen, a in he regular oion lieraure, a conrac wih a urrender oion i called American, a conrac wihou a urrender oion i called Euroean (Bacinello (2003), Andreaa and Corradin (2003)). A la, ome olicie enable he olicyholder o conver cah benefi a mauriy ino a guaraneed annuiy ayable hroughou he remaining lifeime, calculaed a a guaraneed rae. A guaraneed annuiy oion i a conrac ha rovide he olicy-

06 Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 holder wih he righ o receive a mauriy eiher a cah aymen or an annuiy, deending on which ha he greaer value (Milevky and Promilow (200), Balloa and Haberman (2003)). In reference o embedded derivaive, IFRS 4 clarifie ha an inurer need no accoun for an embedded derivaive earaely a fair value if embedded derivaive mee he definiion of an inurance conrac. Moreover inurer will no need o earae urrender oion wihin dicreionary ariciaory feaure (DPF) conrac, irreecive of wheher hey ranfer ignifican inurance rik or no. Then, he above direcion eablih ha inurance comanie have o evaluae he fair value of he embedded derivaive bu no necearily accoun hem earaely from he ho conrac. Reference on he alicaion of accouning rule baed on Fair Value are eaily founded in lieraure; De Angeli (200) review inernaional guide line for Fair Valuaion of inurance comanie; De Felice and Moriconi (200) inroduce a mark o marke model for a fair ricing of life inurance ariciaing olicie wih a minimum inere rae guaraneed. In acuarial lieraure mo aer deal wih model for Fair Value of life inurance liabiliie wih embedded oion; in aricular Milevky and Promilow (200) rooe a ochaic aroach o model he fuure moraliy hazard rae in inurance conrac wih oion o annuiie; Bacinello (2003) deal wih he roblem of ricing a guaraneed life inurance ariciaing olicy which embed a urrender oion; Andreaa and Corradin (2003) rooe a imilar aroach o rice he embedded oion via Mone Carlo imulaion; Balloa and Haberman (2003) rooe a heoreical model for evaluaing guaraneed annuiy converion oion. The aim of hi aer i o analyze an acuarial model o comare reerve evaluaed on he bai of local rule wih reerve calculaed on a Fair Value bai, conidering a ochaic aroach for boh moraliy and financial rik. We focu, in aricular, on he fair valuaion of a urrendable ariciaing conrac wih minimum reurn guaraneed and oion o annuiie via Mone Carlo imulaion. We imlemen he recen conribuion of Bacinello (2003), Andreaa and Corradin (2003), conidering he cae of a guaraneed annuiy oion a in Balloa and Haberman (2003). Moreover, our aroach decribe eeced cah flow of he fair value liabiliie and embedded derivae beween inceion and erm. Secion 2 decribe heoreical model ued for he comarion decribed above. Secion 3 reen an alicaion of he model dicued in Secion 2 for an evaluaion of a urrendable ariciaing olicy wih minimum reurn guaraneed and oion o annuiie. 2. A Fair Value of he embedded oion in a guaraneed life inurance ariciaing olicy 2.. I doe no ei a unique definiion of Fair Value for whole inurance conrac. However definiion rooed by Financial Accouning Sandard Board (FASB) for financial ranacion declare Fair Value i an eimae of he rice an eniy would have realized if i had old an ae or aid if i had been relieved of a liabiliy on he reoring dae in an arm -lengh echange moivaed by normal buine conideraion. Tha i, i i an eimae of an ei rice deermined by marke ineracion. A imilar definiion of Fair Value i rooed by Inernaional Accouning Sandard Commiee (IASC): The amoun for which an ae could be echanged or liabiliy eled, beween knowledgeable, willing arie in an arm lengh ranacion ; herefore, in he radiional condiion of marke efficiency, Fair Value of an inurance olicy could be equalized o i equilibrium-rice. In abence of an efficien marke, Fair Value could be eimaed hrough a conien heoreical bid/ak model joined wih imilar ae and liabiliie. Valuaion echnique include uing recen arm lengh marke ranacion beween knowledgeable, willing arie, if available, referring o he curren fair value of anoher inrumen ha i ubanially he ame, dicouned cah flow analyi and oion ricing model. The choen valuaion echnique make maimum ue of marke inu and relie a lile a oible on eniy-ecific inu. I incororae all facor ha marke arician would conider in eing a rice and i conien wih acceed economic mehodologie for ricing financial inrumen. The fair value i baed on:

Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 07 obervable curren marke ranacion in he ame inrumen; a valuaion echnique whoe variable include rimarily obervable marke daa and ha i calibraed eriodically o obervable curren marke ranacion in he ame inrumen or o oher obervable curren marke daa; a valuaion echnique ha i commonly ued by marke arician o rice he inrumen and ha been demonraed o rovide realiic eimae of rice obained in acual marke ranacion (ee IASB (2004)). In order o reen an acuarial model for a fair value eimaion of a radiional life inurance conrac, we uoe o oerae in a radiional efficien marke. We aume, in fac, ha financial and inurance marke are erfecly comeiive, fricionle, and free of arbirage ooruniie. Moreover, all he agen are uoed o be raional and non-aiaed, and o hare he ame informaion. Conider r ;,2,... and ;,2,... a wo diffuion rocee driven he inananeou inere rae and he ineniy of moraliy (referred o an inured aged a iue), by r he filraion F and F reecively; wih reference o a generic inurance conrac ay-ou, r, he wo ochaic rocee are defined on a robabiliy ace, F, P uch ha r, r F F F. 0, i ereed a follow: Fair Value of a generic life inurance conrac in FV Ê v, r, V CFL v, CFA F, (2.),, where ˆ denoe he uual eecaion under he rik-neural robabiliy meaure; v, i he ochaic dicoun facor deenden on he o-rae dynamic beween and ; CFL and CFA, are he annual random cah flow of inurance comany and inured/olicyholder reecively, joinly deenden on he o rae and ineniy of moraliy dynamic; F r r,, F and F are he -algebra aociaed wih he above defined filraion. 2.2. A aed in Secion, o comue he fair value of a urrendable ariciaing endowmen olicy wih oion o annuiie, we earae he whole conrac in i comonen conidering a baic conrac, a ariciaion oion, an oion o annuiie and a urrender oion. The baic conrac i a andard endowmen olicy wih benefi C 0, ne conan annual remium P, echnical rae i. The Fair Value i ereed a FV B r, Ê 0, /,, V C v q v P v F. (2.2) In order o value he ariciaion oion i i neceary o comue he fair value of he non-urrendable ariciaing conrac. In accordance wih hi conrac he inurer ay a benefi C if he inured die wihin and ; oherwie C if he inured i alive a mauriy. Then, he fair value of he non-urrendable ariciaing conrac i given by: FV V (2.3) P r, Ê, /,, C v q C v P v F, where he ariciaion rule i The annual remium i comued uing fir order echnical bai.

08 Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 C C C0 (2.4) and I i ma, imin. (2.5) i 0, i he ariciaion coefficien, I rereen he rae of reurn of he reference orfolio during -h anniverary of olicy (ee Secion 3), i min i he minimum readjumen meaure. The fair value of he ariciaion oion could be eaily comued a he difference beween FV V P B and FV V. Some radiional Ialian olicie enable he olicyholder o conver cah benefi a mauriy ino a guaraneed annuiy ayable hroughou he remaining lifeime, calculaed a a guaraneed rae G. A guaraneed annuiy oion i a conrac ha rovide he olicyholder wih he righ o receive a mauriy eiher a cah aymen or an annuiy, deending on which ha he greaer value. The guaraneed annuiy oion ay-ou a mauriy i ereed a (Balloa and Haberman (2003)) ma G Ca, C C G C ma0, a K C OTA, (2.6) where K / G, OTA G C ma 0, a K and a Ê ( ) v. (2.7) r,, F In ha cae we have o ere he fair value of a non-urrendable ariciaing conrac wih oion o annuiie a OTA P r, FV V FV V ÊOTA v, F. (2.8) OTA The fair value of OTA could be eaily comued a he difference beween FV V P and FV V. A la, our aim i he comuaion of he urrender oion fair value a he difference beween he fair value of he whole conrac and he fair value of he non-urrendable ariciaing conrac wih oion o annuiie. According o Groen and Jørgenen (2000) and Bacinello (2003) he whole conrac i an American-yle conrac ha embed a urrender oion. A urrender oion i an American-yle oion ha enable he olicyholder o urrender he olicy and receive he o called urrender value. Tyically, he urrender value of a conan eriodical remium olicy i given by R C C 0 C0 iur, (2.9) where ur i i an annually comounded dicoun rae. The mechanim underlying a urrender oion i he following: a any ime =,2,..,-, he olicyholder comare he urrender value wih he eeced ayoff from he coninuaion value, and eercie he oion if he urrender value i higher.

Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 09 In order o rice he urrender oion, Groen and Jørgenen (2000) and Bacinello (2003) rooe a binomial ree model à la Co, Ro and Rubinein (979): he fair rice of he whole conrac and he coninuaion rice can be comued by mean of a backward recurive rocedure oeraing from ime - o ime 0. Inead of a binomial ree model Andreaa and Corradin (2003) ue he Lea Square Mone Carlo Aroach following Longaff and Schwarz (200). In hee aricle auhor value he urrender oion for a urrendable ariciaing conrac wihou oion o annuiie. So he coninuaion value a ime - for a given ah j i ereed a (ee Andreaa and Corradin (2003). 8): j j j W v, C P. (2.0) To diinguih our model from hoe decribed above, we have inroduced in he coninuaion value of a urrendable ariciaing olicy a guaraneed annuiy oion, remembering Ialian olicy feaure. If we uoe o be a ime -, we have o comue a differen coninuaion value. To coninue mean receive, a ime, he benefi C, if he inured die wihin - and, or o be eniled of a conrac whoe oal random value equal mag Ca, C, if he inured i alive. Therefore, he coninuaion value a ime - for a given ah j i ereed by: j j j j j j W v, C q mag C a, C P T j The fair value of he whole conrac. (2.) FV V F i herefore he maimum beween he coninuaion value and he urrender value j R : j j j ma W, R. (2.2) F Aume now o be a ime <. A in he Fackler-Fourer recurive formula, o coninue mean o immediaely ay he remium P and o receive, a ime +, he benefi C, if he inured die wihin one year, or o be eniled of a conrac whoe oal random value (including he oion of urrendering i in he fuure), equal F if he inured i alive. Therefore he coninuaion value a ime i given by he following ereion: W C q v, Eˆ v, F P. (2.3) Longaff and Schwarz (200) rooe ha he condiional eeced value of he fuure oion value E ˆv, F can be eimaed from he cro-ecional informaion in he imulaion by uing lea quare, ha i by regreing he dicouned realized ayoff from coninuaion on funcion of he value of he ae variable. Finally, he fair value of he urrender oion could be evaluaed a he difference be- T OTA ween FV V and FV V. 3. An alicaion of he fair value acuarial model 3.. The demograhic ochaic roce ;,2,... i decribed by a Mean- Revering Brownian Gomerz (MRBG) model; in aricular, we ake ino accoun a radiional acuarial aroach, where T i a random variable rereening he remaining lifeime of a olicy- Dealing wih Andreaa and Corradin (2003) we chooe o bae he regreion on wo ae variable, he cah benefi and he reference fund and regre according o a hird-order olynomial model ˆ j 2 3 2 3 2 E v, F a a C a C a C a S a S a S a C S a C S a C S. 2 2 3 4 5 6 7 8 9 0

0 Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 holder aged ; a a conequence he robabiliy of urvival o ime, condiional on being alive a age, i equal o Prob T F. (3.) We define : o be he hazard rae for an individual aged, a calendar year, i follow ha 0 can be arranged a : d Eˆ 0 e F. (3.2) The ime evoluion of he hazard rae i ereed by an eonenial form a follow g, :0e Y : :, wih g,,, : 0 0, where g, reume on ime he deerminiic correcion due o age and he effec of longeviy rik; Y i a ochaic roce inroduced o model random variaion in he foreca rend; rereen he andard deviaion of he roce : ;,2,... mean revering diffuion roce ; in aricular he ochaic roce Y i decribed by a dy by d dw, Y0 0, b 0, (3.3) where b i he mean reverion coefficien and W i a andard Brownian moion. The ime dynamic of he inananeou inere rae (o rae) r ;,2,... i modelled by a mean revering quare roo diffuion equaion a in Co, Ingeroll and Ro model (CIR); herefore, we aume he following ochaic equaion: dr r r d r dz k, (3.4) r where k i he mean reverion coefficien, i he long erm rae, r i he volailiy arameer and Z r i a andard Brownian moion. Fair ricing of an inurance ariciaing olicy alo deend on reference orfolio dynamic; in aricular we aume o work in a Black-Schole economy where he reference orfolio i comounded mainly by a bond inde and a minoriy by a ock inde. The wo comonen are decribed by he following equaion ds i i i i i r S d S dz, :ock inde i (3.5) 2 :bond inde where i ( i) i S, e Z, are, for each reference orfolio comonen, marke rice, volailiy arameer and a Wiener roce. A la, he hree ource of financial uncerainy are correlaed: k j dz dz d k, j,2, r, (3.6) k, j hence, reference orfolio could be ereed a a combinaion of he random variable inroduced above S S S 2. (3.7)

Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 A a reul, he annual rae of reurn of he reference fund a ime i defined a: S I. (3.8) S 3.2. Since a cloed form oluion of ereion 0 i no available, a olicy Fair Value can be obained via Mone Carlo imulaion. 0 reor ome conrac feaure ued for numerical analyi. Conrac Feaure Table Se Age 55 Duraion 0 Male Technical rae.00% Moraliy Table SIM 92 Sum Aured 00 Pariciaion coefficien 87.5% Minimum Adjumen Meaure 0.00% Guaraneed rae for annuiy 5.78% Annually comounded urrender dicoun rae 3.00% Reference orfolio ariciaion coefficien: Sock inde Bond Inde 0 how arameer eimaed on marke daa and ued in moraliy and financial rik diffuion rocee. In aricular, wih reference o CIR model, rik-adjued arameer are calibraed on marke value of euro wa inere rae hrough Brown and Dybvig (986) framework. Parameer for MRGB model are calibraed on moraliy hazard rae derived from an Ialian rojeced life able called RG48. A la, reference fund arameer are eimaed on daily marke value of Emu-Bond Inde and MSCI World Inde oberved beween 200 and 2003. 0% 90% Eimaed Parameer e Table 2 r r0 0.05268 b 0.5 k 0.245439 0.8 S S 23.57 0 2 S 240.22 0.058359 g 0.0 0.0 0.053524 2 r 0 0. Wih reference o an inured aged 55 and a conracual erm of en year, 0 reor, he annual eeced cah flow of mahemaical reerve comued wih local rule, fair value of baic conrac, ariciaing oion, oion o annuiie, urrender oion and whole conrac reecively; we imulae 0,000 ah for he moraliy hazard rae, he o rae and he reference fund, decribed in 0, 0 and 0.

2 Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 Local Reerve and Fair Value of a conrac wih embedded oion Table 3 Year Local Reerve Baic Conrac Pariciaion Oion Pariciaing conrac / Local Reerve Oion o annuiie Surrender Oion Whole Conrac 0 - -7.37 6.0-0.70-0.66 9.9-7.89 6.44 92.99% - 0.72 9.27 2 8.97.8 7.00 99.8% - 0.75 9.57 3 29.40.87 7.69 00.56% - 0.79 30.35 4 40.54 22.39 8.49 00.83% - 0.84 4.72 5 52.46 33.46 9.40 00.78% - 0.89 53.76 6 65.7 45.7 20.42 00.64% - 0.96 66.55 7 78.78 57.59 2.55 00.45% -.03 80.6 8 93.37 70.8 22.78 00.23% -.0 94.68 9 08.93 84.9 24.0 00.07% -.7 0.8 0 25.5 00.00 25.5 00.00% - - 25.5 The inroducion of a fair value accouning yem roduce a reducion of inurance liabiliie and o a differen rofi diribuion over ime unil he olicy mauriy. In aricular, for a 0-year conrac analyzed, 0 how ha: he liabiliy increae i an average abou of.9%; he ariciaing oion i he mo relevan embedded conrac comonen, erforming in average abou of 34.76% over he whole conrac value; a high erformance of he urrender oion comonen, in average abou of.44% over he whole conrac value, i elained by he imlici oion annuiy comonen value equal zero, in reference o a guaraneed annuiy coefficien ued in he radiional Ialian life inurance olicie. 0 how econd momen, variaion coefficien and kewne of conrac value comonen. Thee arameer offer a meaure of rikne in reference o each conrac comonen, ueful o calculae adequae margin for rik under a Fair Value accoun yem. Second Momen, Variaion Coefficien and Skewne Table 4 Year Fair Value Whole conrac 3 3 Fair Value Surrender Oion 3 3 0 5.04.07 2.8 7.57 2.67 -.03 5.20 0.36 2.76 7.74 2.55 -.02 2 5.43 0.22 2.64 7.92 2.29 -.00 3 5.76 0.6 2.45 8.07.87-0.97 4 6.20 0.3 2.20 8.7.30-0.96 5 6.80 0.2.92 8.5 0.54-0.93 6 7.59 0..64 7.99 9.60-0.9 7 8.62 0.0.42 7.58 8.48-0.89 8 9.89 0.0.27 6.75 7.03-0.87 9.49 0.0.6.59.55.79 0 2.8 0.0.4 - - -

Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 3 To analye value eniiviy of each comonen, we have derived region of Fair Value in correondence o he mo relevan arameer: K, i ur and. In aricular in 0 : Figure reen he whole conrac Fair Value behaviour in relaion o he wo arameer K and i ur, howing a non increaing monoone rend, concave in increaing he wo arameer; Figure 2 how he urrender oion Fair Value behaviour in relaion o he wo arameer K and i ur ; i highligh a non decreaing monoone rend, conve in increaing he K arameer, while i i concave in increaing he i ur arameer. Fig.. Fair Value of he Whole conrac Fig. 2. Fair Value of he Surrender Oion

4 Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 A la, Figure 3 reen he annuiy oion Fair Value behaviour in relaion o he wo coordinae and K ; i can be oberved ha reul found in Balloa and Haberman (2003) are confirmed, a he annuiy oion Fair Value how a non increaing monoone rend, concave in increaing he wo arameer. Fig. 3. Fair Value of he Oion o Annuiie Reference. ANDREATTA G., CORRADIN S. (2003), Valuing he Surrender Oion Embedded in a Porfolio of Ialian Life Guaraneed Pariciaing Policie: a Lea Square Mone Carlo Aroach, Real oion heory mee racice, 8h Annual Inernaional Conference, Monrèal Canada June 7-9, 2004. 2. BACINELLO A.R. (200), Fair Pricing of Life Inurance Pariciaing Policie wih a Minimum Inere Rae Guaraneed, Ain Bullein, 3, 275-297. 3. BACINELLO A.R. (2003), Fair Valuaion of a he Surrender Oion Embedded in a Guaraneed Life Inurance Pariciaing Policy, The Journal of Rik and Inurance 70 (3), 46-487. 4. BALLOTTA L., HABERMAN S. (2003), Guaraneed annuiy converion oion and heir valuaion, www.caac.org. 5. BRENNAN, M.J., SCHWARZ, E.S. (976), The Pricing of Equiy-Linked Life Inurance Policie wih an Ae Value Guaranee, Journal of Financial Economic n. 3, 3. 6. BROWN J.S., DYBVIG P.H. (986), The emirical imlicaion of he Co, Ingeroll, Ro heory of he erm rucure of inere rae, Journal of Finance, 4:-32. 7. COX J.C., INGERSOLL J.E., ROSS S.A. (985), A heory of he erm rucure of inere rae, Economerica, 53:-408. 8. DE ANGELIS P. (200), Gli Sandard Inernazionali er la valuazione del Fair Value delle comagnie di Aicurazione: imlicazioni auariali, Seminario enuo reo l Iiuo Ialiano degli Auari, Roma, 29 Marzo, 200.

Invemen Managemen and Financial Innovaion, Volume 3, Iue 2, 2006 5 9. DE FELICE M., MORICONI F. (200), Finanza dell aicurazione ulla via, Gruo di ricerca u Modelli er la finanza maemaica, Working aer n. 40. 0. GROSEN A., JØRGENSEN P.L. (200), Life inurance liabiliie a marke value, Working Paer, Serie no. 95, Cenre for Analyical Finance.. HARRISON J.M., KREPS D.M. (989), Maringale and Arbirage in Mulieriod Securiie Marke, Journal of Economic Theory n. 20. 2. IASB (2003), Eoure Draf:ED5 Inurance Conrac, www.iab.org. 3. IASB (2004), Eoure Draf of Prooed Amendmen o IAS 39 Financial Inrumen: Recogniion and Meauremen, The Fair Value Oion, www.iab.org. 4. JØRGENSEN P.L. (200), Life inurance conrac wih embedded oion, Working Paer Serie no. 96, Cenre for Analyical Finance. 5. LONGSTAFF F.A., SCHWARTZ E.S. (200), Valuing American oion by imulaion: a imle lea-quare aroach, The Review of Financial Sudie 4,,. 3-47. 6. MILEVSKY M.A., PROMISLOW S.D. (200), Moraliy derivaive and he oion o annuiie, Inurance: Mahemaic and Economic, (29): 23-245.