FAIR VALUAION OF VARIOUS PARIIPAION SHEMES IN LIFE INSURANE PIERRE DEVOLDER AND INMAULADA DOMÍNGUEZ-FABIÁN BY Insttut des Scences Actuaelles, Unvesté atholque de Louvan, 6 ue des Wallons, 348 Louvan la Neuve, Belgum <devolde@fn.ucl.ac.be> Deatment of Fnancal Economy, Unvesty of Etemadua, San domngu@guadana.une.es (second evsed veson Novembe 4 ABSRA Fa valuaton s becomng a mao concen fo actuaes, esecally n the esectve of IAS noms. One of the key asects n ths contet s the smultaneous analyss of assets and labltes n any sound actuaal valuaton. he am of ths ae s to llustate these concets, by comang thee common ways of gvng bonus n lfe nsuance wth oft: evesonay, cash o temnal. Fo each atcaton scheme, we comute the fa value of the contact takng nto account lablty aametes (guaanteed nteest ate and atcaton level as well as asset aametes (maket condtons and nvestment stategy. We fnd some equlbum condtons between all those coeffcents and comae, fom an analytcal and numecal ont of vew, the systems of bonus. Develoments ae made fst n the classcal bnomal model and then etended n a Black and Scholes economy. KEYWORDS Fa value, atcaton scheme, asset and lablty management
FAIR VALUAION OF VARIOUS PARIIPAION SHEMES IN LIFE INSURANE. INRODUION If fo a long tme lfe nsuance could have been consdeed as a sleeng beauty, thngs have changed damatcally as well fom a theoetcal ont of vew as fom ndustal concens. Nowadays, the fnancal sks nvolved n lfe nsuance oducts ae suely amongst the most motant challenges fo actuaes. he need to udate ou actuaal backgound takng nto account the eal fnancal wold has been ecently emhaszed by Hans Buhlman n a ecent edtoal n ASIN BULLEIN (Buhlman (. he classcal way of handlng fnancal evenues n lfe nsuance was chaactezed by two assumtons: statonaty of the maket (no tem stuctue of nteest ate and absence of uncetanty (detemnstc aoach; all ths leadng to the famous actuaal aadgm of the techncal guaanteed ate: all the futue was summazed n one magc numbe. lealy thngs ae not so smle and lfe nsuance s a efect eamle of stochastc ocess (even moe than non lfe; the two dmensons of tme and uncetanty ae comletely nvolved n the oducts: long tem asect and fnancal sk (Nobeg (. he uose of ths ae s to focus on the stochastc asect of the etun and to show how to calbate techncal condtons of a lfe nsuance oduct, usng classcal fnancal models. In ths contet fa valuaton s the cental toc. he comng ntoducton of accountng standads fo nsuance oducts wll undoubtedly ncease the motance of fa valuaton of lfe contngences. Even f
motalty sk can deely nfluence the oftablty of a lfe nsue (fo nstance n the annuty maket, t seems clea that the fnancal sk s the heat of the matte. Pcng and valuaton of lfe nsuance oducts n a stochastc fnancal envonment stated fst fo equty lnked olces wth a matuty guaantee (Bennan and Schwatz (976, Delbaen (986, Aase and Pesson (994, Nelsen and Sandmann (995.he newness of these oducts elans obably why hstocally they wee the fst studed wth stochastc fnancal models. But moe classcal oducts, lke nsuance wth oft, ae at least as motant fo the nsuance ndusty and nduce also clealy mao concens n tems of fnancal sk. Dffeent models of valuaton of these contacts have been oosed based on the classcal neutal aoach n fnance (Bys and De Vaenne (997, Bacnello (, 3b. he motance to develo, n ths feld, good models and to analyse all the embedded otons, wll suely ncease wth the IAS wold (Gosen and Jogensen (. Even f fo comettve o legal easons the cng of these oducts has vey often lmted degee of feedom, the valuaton eques a comlete undestandng of vaous sks nvolved n each oduct. Amongst them, the elaton between the guaanteed ate and the atcaton level s of fst motance, n close connecton wth the asset sde. he am of ths ae s to focus on ths asect of atcaton, fo the one hand by ntoducng some lnks between the lablty condtons and the asset stategy and, fo the othe hand, by comang thee classcal bonus systems. wo oblems ae esented: fst f all the aametes of the contact ae fed (actuaal valuaton of an estng oduct, fomulas of fa value ae gven. Secondly, n ode to desgn new aametes of a oduct, equlbum condtons ae develoed. he ae s ogansed as follows. Secton esents the man assumtons used n tems of assets and labltes n a fnancal bnomal envonment and ntoduces 3
thee dffeent atcaton systems: evesonay, cash and temnal bonus. In secton 3, a fst model on a sngle eod s develoed, showng how to comute fa values and equlbum condtons n functon of the chosen nvestment stategy. In ths case, t aeas clealy that the thee atcaton schemes ae dentcal. hen secton 4 etends the fa valuaton comutaton n a multle eod model; each atcaton system havng then dffeent valuaton. Secton 5 s devoted to the equlbum condton n the multle eod; n atcula, we show that the condtons ae dentcal fo the evesonay and the cash bonus even f n geneal fa values ae dffeent as seen n secton 3. hs motvates to take a deee look on the elaton between fa values n these two systems. Secton 6 shows that cash systems have a bgge valuaton when the contact s unbalanced n favou of the nsue and vce vesa. Secton 7 llustates numecally the esults and secton 8 etends the esults n a contnuous tme famewok, usng the classcal Black and Scholes economy. In ths case, we show that elct fomulas fo the fa values and the equlbum value of the atcaton ate ae stll avalable; but mlct elatons ae only ossble fo the equlbum value of the guaanteed ate.. NOAIONS AND DEFINIIONS We consde a lfe nsuance contact wth oft: n etun fo the ntal ayment of a emum, the olcyholde obtans, at matuty, a guaanteed beneft lus a atcaton bonus, based on the eventual fnancal sulus geneated by the undelyng nvestments. Insue s suosed to be sk neutal wth esect to motalty. Futhemoe, we assume ndeendence between motalty and fnancal elements. he asset and the lablty sdes of the oduct can be chaactezed as follows:.. Lablty sde 4
We consde a ue endowment olcy wth sngle emum ssued at tme t and matuatng at tme. he beneft s ad at matuty at tme t f and only f the nsued s stll alve. No suende oton ests befoe matuty. We denote by the guaanteed techncal nteest ate and by the ntal age. he suvval obablty wll be denoted by. If we assume wthout loss of genealty that the ntal sngle emum s equal to, the guaanteed beneft ad at matuty (dsegadng loadngs and taes s smly gven by: ( G ( π ( We ntoduce then the atcaton lablty; we denote by B the atcaton ate on the fnancal sulus ( < B. hee dffeent atcaton schemes wll be comaed: - Revesonay bonus : a bonus s comuted yealy and used as a emum n ode to buy addtonal nsuance (ue endowment nsuance nceasng beneft only at matuty. - ash bonus : the bonus s also comuted yealy but s not ntegated n the contact. It can be ad back dectly to the olcyholde o tansfeed to anothe contact wthout guaantee. - emnal bonus : the bonus s only comuted at the end of the contact, takng nto account the fnal sulus. We denote by B, B and B the coesondng atcaton ates... Asset sde We assume efectly comettve and fctonless makets, n a dscete tme famewok wth one sky asset and one sk less asset. 5
he annual comounded sk fee ate s suosed to be constant and wll be denoted by. he sky asset s suosed to follow a bnomal evoluton (o-ross- Rubnsten (979; two etuns ae ossble at each eod: a good one denoted by u and a bad one denoted by d. In ode to avod any abtage ootunty, we suose as usual: d < < u hese values can be also eessed altenatvely n tems of sk emum and volatlty: whee u d λ λ [.] λ : sk emum ( λ > : volatlty ( > λ On ths maket, the nsue s suosed to nvest a at of the emum n the sky asset and the othe at n the sk less asset. A stategy s then defned by a coeffcent (wth constant < gvng the sky at of the nvestment. he etun geneated by a stategy s a andom vaable, takng one of two ossble values denoted by u and d and defned as follows: u u ( ( ( λ [.] d d ( ( ( λ [3.] he case wll be not beng taken nto account. 3. A FIRS ONE PERIOD MODEL We stat wth the vey secal case. hen by defnton the thee ways of gvng bonus, as defned n secton., ae dentcal. 6
A contact s chaactezed by ts vecto of techncal and fnancal aametes: he othe aametes ( u, d 3.. Fa valuaton v (, B, [4.], can be seen as constants of the maket. Usng the standad sk neutal aoach n fnance, the fa value of the contact can be eessed as the dscounted eectaton of ts futue cash flows, unde the sk neutal measue, and takng nto account the suvval obablty. he sk neutal obabltes, assocated esectvely wth the good etun and the bad etun, ae gven by: d u d λ u ( λ u d he coesondng labltes ae esectvely: L L [ ( ] B[ ( λ ] B u [ ( ] B[ ( λ ] B d [5.] [6.] he fa value of a contact v s then defned by: ( v ( v wth ( v defned as the fnancal fa value of the beneft and gven by : ( v λ λ B L λ [ B[ ( λ ] ] B[ ( λ ] [ ] λ [ ( λ ] [ ( λ ] [7.] 7
he fnancal fa value conssts of two ats: wth ( v G ( v P ( v G ( v : fa value of the guaantee [8.] wth P ( v : fa value of the atcaton coesondng to a call oton and gven by: B λ λ P ( v P P [9.] P P [ ( λ ] [ ( λ ] [.] o go futhe, we have to make some assumtons on the values of P and P. Of couse, we have P P. he followng stuatons can haen fo a gven contact: ase : P P ase : P < P ase 3: < P < P ase : P P In ths case, the techncal guaanteed ate s so bg that no atcaton can be gven. In atcula, P mles: ( λ ( λ > [.] 8
he contact becomes uely detemnstc, and the fa value s then: ( v [.] ase : P < P hs case can be consdeed as the ealstc assumton: - If the sky asset s u, thee s a sulus and atcaton. - But f the sky asset s down, the guaantee s layng and thee s no atcaton. he condton can be wtten as follows: ( λ < ( λ [3.] In ths stuaton, the fa value becomes: λ ( v B [4.] ( ( λ ase 3: < P < P In ths case, the techncal guaanteed ate s so low that even n the down stuaton of the maket, thee s a sulus: he fa value becomes then: ( λ < [5.] λ ( v B [ B( ] λ ( ( λ ( ( λ [6.] he fa value of the atcaton s ust based on the dffeence between the sk fee ate and the guaanteed ate. 3.. Equlbum 9
A vecto v of aametes s sad to be equlbated f the coesondng ntal fa value s equal to the sngle emum ad at tme t: ( v π [7.] that s equvalent to ( v. Snce elaton [7.] mles : ( v we have a fst geneal equlbum condton gven by : [8.] So, n ode to obtan equlbum, the techncal guaanteed ate must always be lowe o equal to the sk fee ate. secton 3.. We ty now to look at equlbum stuatons n the thee cases esented n ase : P P In ths case, elaton [.] shows that >, and no equlbum s ossble. ase : P < P akng nto account smultaneously elaton [8.] and [3.], we must have: ( λ < [9.] he elaton can be wtten n tems of equlbum values of each of the aametes of the contact: - Guaanteed ate (functon of the atcaton ate and the stategy coeffcent: λ B B ( λ [.] wth condtons: < B and <. he condton [9.] s well esected, takng nto account these two condtons. - Patcaton ate (functon of the guaanteed ate and the stategy coeffcent:
B λ ( λ [.] wth condtons: < and ( λ <. - Stategy coeffcent (functon of the guaanteed ate and the atcaton ate: he uose hee s to see f, fo a coule of coheent values of the techncal aametes and B, thee ests an asset stategy geneatng an equlbum stuaton: B B λ ( λ [.] wth condtons: < B he condton on s obtaned by the condton < : a > fo < and B( λ f b ase 3: < P < P B < B (.e. fo all B snce > λ λ ( λ he equlbum condton on the fa value becomes then, by [6.]: [ B( ] o B( Because n ths case <, ths mles B All ths develoment shows that the eal nteestng stuaton wth a non-tval soluton s the case. 4. FAIR VALUAION IN MULIPLE PERIOD MODELS We etend hee the comutaton of fa values fo a geneal matuty. he thee atcaton schemes defned n secton.. must now be studed seaately.
4.. Revesonay bonus akng nto account the bnomal stuctue of the etuns, the total beneft to be ad at matuty s a andom vaable gven by B(, whee s the numbe of L L u cases of the sky asset amongst the yeas and L and L ae the coesondng total etuns. he fa value s then gven by: ( v LL [ L L ] [ ( v ] [3.] whee denotes the bnomal coeffcent and ( v s the fnancal fa value on one yea (cf [7.]. 4.. ash bonus In ths case, each yea, the ate of bonus s aled only to the eseve accumulated at ate and takng nto account the suvval obabltes. Fo an ntal sngle emum equal to π, the eseve V(t to use at tme t s gven by: V( t π( t / t ( t / t he at of the labltes to be ad at tme t ( t,,..., as cash bonus s a andom vaable gven by: t - n u case: ( t B P ( ( / c t - n down case: ( t B P ( ( / c t t n case of suvval he fa value can be eessed as dscounted eected value of all futues cash flows unde the sk neutal measue and takng nto account the suvval obabltes:
t t ( v ( ( t ( t t ( v B t ( ( t P P t ( whee ( v s the fnancal fa value and s gven by : ( v B [4.], and P, P beng defned n [5.] and [.] ( ( ( P P ( In the atcula case P ; P, ths gves: > ( λ ( ( λ ( ( ( v ( B [5.] ( he coesondng value fo the evesonay bonus s (cf [3.]: ( v ( B ( λ ( ( λ [6.] As eected, fo these two values ae ndentcal. A deee comason between [5.] and [6.] wll be develoed n secton 6, afte calculaton of equlbum values n secton 5. 4.3. emnal bonus he bonus s only comuted at the end of the contact, comang the fnal techncal lablty ( wth the asset value at matuty. If we denote by A ( the temnal assets at tme t, usng stategy, the fa value can be wtten as follows: 3
( v ( [( B ΕQ [ A ( ( ] ] ( v [7.] whee Q s the sk neutal measue and E Q denotes eectaton unde Q. Altenatvely, we can eess ths fa value n tems of oton ce: whee A ; ; ( ( ( ( v B ( A ; ; ( ( matuty and stke ce ( s the ce of a call oton on the asset A, wth akng nto account the bnomal stuctue of the model, the geneal fom of ths ce s gven by: ( ( A ; ;( [ u d ( ] ] he fa value s then gven elctly by : ( ( B ( [ u d ( ] ( v [8.] a nf As eamle, let us look at the model on two eods of tme (. If we denote by a the mnmal numbe of ums n ode to gve atcaton: { ( ( ( ( ( } Ν : λ λ > stuatons can haen: [( u ] Fst stuaton: a >, the followng he techncal guaanteed ate s so bg that no atcaton can be gven. he fa value s ust: 4
Second stuaton: a ( v he only case whee bonus can be gven s the stuaton of two u ums of the sky asset: he fa value becomes then: ( ( u u d < [9.] λ [ ] ( v ( B u ( hd stuaton: a As soon as thee s one u um on the two eods, bonus s gven: he fa value becomes then : ( u d d < [3.] ( λ ( ( ( u d ( λ ( v ( B u ( Fouth stuaton: a A bonus s gven each yea whateve ae the etuns of the asset: he fa value s smly: ( v ( [3.] d > [3.] ( ( B [33.] 5
he fst and the last stuatons can be consdeed as degeneate. If we want to avod these lmt stuatons, we must have: ( u d < [34.] We ae then n stuaton o stuaton 3. Remak: It s easy to see that we wll be automatcally n stuaton 3 (a f: and ( < λ λ [35.] <. he fst condton [35.] s ndeendent of the chaactestcs of the contact; t ust means that the volatlty has not to be too motant.. he second condton [ < ], elated to the contact, seems to be qute easonable (cf [8.] on one eod. 5. EQUILIBRIUM RELAION IN MULIPLE PERIOD MODELS he am of ths secton s to genealze on eods the equlbum elatons obtaned n secton 3.. fo one eod, usng the elct fomulas of fa value obtaned n secton 4. 5.. Revesonay bonus Fomula [3.] shows clealy that: ( v ( v Equlbum esults obtaned n secton 3.. ae unchanged n a multle eod model wth evesonay bonus. 5.. ash bonus 6
Fomulas [5.] and [6.], fo nstance, show that the fa values ae nomally dffeent usng the evesonay bonus o the cash bonus. Nevetheless, we wll see that the equlbum values of the aametes of the contact ae the same. Usng [4.], the contact wll be equlbated n the cash bonus scheme f: P and P. [ ] ( ( ( ( ( B P P ( [36.] Lke n secton 3.., we can consde dffeent cases, deendng on the values of ase : P P No equlbum s ossble lke n the evesonay scheme. ase : P < P (cental assumton Usng the values [5.] and [.] of P and P, fomula [36.] becomes: [ ] ( ( ( ( ( ( λ B λ ( λ o B [ ( λ ] o B λ B c ( λ [37.] dentcal to elaton [.] obtaned fo the evesonay bonus. ase 3: < P < P Fomula [36.] becomes then: [ ] ( ( B ( ( whch mles lke n the evesonay case : B ONLUSION: he equlbum condtons on the aametes of the contact ae the same fo the evesonay bonus and fo the cash bonus. 7
8 5.3. emnal bonus Fomula [8.] gves as equlbum condton: ( ( [ ] ( d u ( B [38.] Fom ths elaton, t s aleady ossble to obtan an equlbum value fo the atcaton ate, as a functon of the guaanteed ate and the stategy coeffcent (to be comaed wth fomula [.] fo the evesonay bonus: ( ( [ ] ( d u B [39.] On the othe hand, f the atcaton ate s known, t s ossble to eess the equlbum value of the guaanteed ate as follows: ( λ λ λ λ a a a a B d u B ( B d u B ( [4.] whee a s defned as usual by : } ( d N : u nf{ k a k k > Unfotunately, ths elaton s not elct because the coeffcent a deends on the level of. he elaton can be comuted, assumng a cetan value fo a and then check aftewads f the condton on a s fulflled. As eamle, let us see agan what haens on two eods.
9 We fst concentate on the standad stuaton a ; ths means that constant [3.] has to be fulflled. Statng fom [4.], we wll get an equlbum canddate fo and check aftewads ths constant. Fo and a, we get: ( ( ( ( λ λ λ λ B d u u B [4.] wth the followng condton to check: < d u ( d Smlay, we can ty to fnd a canddate fo the stuaton a ; now constant [9.] has to be checked. Usng the same aoach, the equlbum value s gven by: ( ( λ λ B u B [4.] wth the followng condton to check : ( u d u < 6. ANALYIAL OMPARISON BEWEEN REVERSIONARY AND ASH SYSEMS Revesonay and cash systems seem not so dffeent; the only dffeence s the ntegaton of the bonus nsde the contact. Moeove, we saw n secton 5 that the equlbum condtons ae the same, even f the fa value fomulas look qute dffeent. Fom now on, and wthout loss of genealty, we wll wok wthout motalty effect. he am of ths secton s to ove the followng elaton:
( ( cash evesonay > f and only f the atcaton ate s geate than ts equlbum value. In ode to get ths elaton, we have to comae the valuatons n the two atcaton schemes usng the same aametes: - Fo the evesonay bonus (cf [3.]: ( [ ] [ ] B K L L v ( wth [ ] P P K - Fo the cash bonus(cf [4.]: ( ( ( K B v ( Develong the evesonay fomula gves: BK v ( ( ( ( ( he condton to have a bgge fa value fo the evesonay system becomes then: BK BK > ( ( ( ( ( ( ( ( O: BK > ( ( ( ( O assumng K (.e. P > : BK BK BK BK > ( ( ( ( ( ( ( [43.] Let us consde the functon:
( ( G( wth Fo ( (, we have: G( On the othe hand, t s easy to show that fo >,the functon G s stctly nceasng. So when condton BK > s fulflled, the fa value fo the evesonay bonus s bgge than the fa value fo the cash bonus and vce vesa. hs last condton can be wtten as follows: becomes: B [ P P ] > [44.] akng nto account the dffeent cases studed n secton 3., condton [44.] ase : P P : not elevant hee ( K. ase : P < P : λ B P B [ ( λ ] > o B > λ ( λ whch means that the atcaton ate s bgge than ts equlbum value (cf [.]. ase 3: < P < P : B( P P B( > O B > that s the equlbum value n that case. 7. NUMERIAL ILLUSRAION In ths secton, we wll comae the thee atcaton schemes n tems of fa values and equlbum values of the aametes n a two eods model and wthout motalty effect.
In tems of fnancal maket, we wll use a cental scenao based on the followng values: skfee ate.3 λ sk emum. volatlty.6 In tems of nvestment stategy undelyng the oduct, we wll manly comae two choces:. : consevatve stategy.6 : aggessve stategy 7.. Fa values fo dffeent guaantees and atcaton levels Fgue shows fo each chosen stategy ( consevatve o aggessve the elaton between the guaanteed ate, the atcaton level and the fa value of the contact. he fa value s clealy an nceasng functon of the atcaton level, whateve s the atcaton scheme. Fgue. FAIR VALUE Investment stocks % FAIR VALUE Investment stocks 6%,3,3,,,99,99,97,95..3 Guaanteed ate.5 % 8% 6% 4% % Patcaton ate,97,95.5.5 Guaanteed ate.5 % 8% 6% 4% % Patcaton ate ash Revesonay emnal
7.. Equlbum values Lke seen n secton 5., the equlbum condtons on the aametes of the contact ae the same fo the evesonay bonus and fo the cash bonus. Fgue comaes, fo the aggessve nvestment stategy, the equlbum values of the atcaton ate and of the guaanteed ate between evesonay and temnal bonus. Fgue. Equlbum value of atcaton ate -6% nvestment n stocks- Equlbum value of guaanteed ate -6% nvestment n stocks-,8,6,4,,667,933,,467,733,3,5,,5,,5,,4,6,8 Guaanteed ate Patcaton ate emnal bonus Revesonay o cash bonus emnal bonus Revesonay o cash bonus 7.3. Fa value and equlbum value able comaes fo vaous values of the techncal aametes, chosen n elaton wth the equlbum values, the fa values n the thee atcaton schemes fo the aggessve nvestment stategy (6% n sky asset. Fgues n bold and talc coesond to stuatons whee the ntal fa value s bgge than the ad emum. able 6%.5.5.6 Equlbum values of B ERMINAL BONUS nteest ate,76 %,995485,978396,96357,45 4%,6553,9856995,973645,4 6%,5856,99363,985763,47 8%,9958,33,99788,6 %,666,7699 3
REVERSIONARY BONUS,657 %,99754,97939,9654,6 4%,469,98755,97858,8 6%,933,995345,986767,54 8%,783,3498,99348,6 %,47467,6844 ASH BONUS,657 %,99759,97988,966363,6 4%,43,987777,977,8 6%,8473,9953737,9888,54 8%,7695,34696,9949,6 %,45357,5656 lealy, the dffeence between the dffeent bonus schemes would be much moe onounced n models on moe than two eods. 8. GENERALIZAION IN ONINUOUS IME MARKE he ncles of comason between the dffeent atcaton schemes can be easly adated n a contnuous tme fnancal maket. We develo hee the model, usng the Black and Scholes envonment. he classcal assumtons on the maket ae suosed to be fulflled. wo knds of assets ae suosed to est: - the sk less asset X, lnked to the sk fee ate: dx ( t X ( t dt whee ln ( s the nstantaneous sk fee ate. - the sky fund X, modelled by a geometc Bownan moton: dx ( t ηx ( t dt σ X ( t dw( t whee w s a Wene ocess. Once agan, the sk neutal obablty measue wll be denoted by Q. he efeence otfolo of the nsue conssts of a constant ooton nvested n the sky fund and a ooton (- nvested n the sk fee asset. he evoluton equaton of ths otfolo denoted by X s then gven by: 4
dx ( t ( η ( X ( t dt σ X ( t dw( t [45.]. 8.. he one eod model We etend hee the esults of secton 3 obtaned n a bnomal envonment. On one eod, the thee knds of atcaton schemes ae dentcal. We comute the fa values fo a gven contact v (, B, and obtan equlbum condtons on the coeffcents n ode to have a fa contact. he fa value s gven now by: ( ν ( B c(,, [46.] whee c(,, eesents the ce of a call oton on the efeence otfolo X fo one eod and fo a stke ce equal to the guaantee (. In the Black and Scholes envonment ths ce s gven by: c(,, EQ(( X ( ( Φ( d(,, Φ( d (,, whee : d(,, (ln( σ d (,, d (,, σ / σ s the cumulatve dstbuton functon of a standad nomal vaable. Fnally the fa value can be wtten as follows: ( ν X ( B( Φ( d(,, Φ( d (,, [47.] he equlbum condton gven by [7.] can be eessed, n ths model, as an elct equlbum value B of the atcaton ate, fo a gven guaanteed ate and a gven 5
stategy coeffcent (equvalent of fomula [.] n the bnomal model: B [48.] ( Φ( d (,, ( Φ( d (,, Imlct elatons can only be obtaned n ths model f we want to solve t fo the two othe aametes (equlbum value esectvely fo the guaanteed ate and fo the stategy coeffcent. 8.. Fa value n multle eod models 8... Revesonay bonus: Eactly as n the dscete case and takng nto account the stuctue of the etun ocess, the fa value fo a contact of eods usng a evesonay bonus s gven by: ( ν X ( B( Φ( d(,, Φ( d (,, [49.] 8... ash Bonus he fa value s eessed as the dscounted eected value of all futue cash flows unde the sk neutal measue Q and the suvval obabltes: ( ν ( E ( B( t Q t t ( t whee B(t s the cash bonus ad at tme t : t EQ( B( t Bc (( ( c(,, Fnally, the fa value s gven by : t ( ν (( Bc( c(,, ( ( ( [5.] 8..3. emnal Bonus: he fa value wll have the same stuctue as n the one eod model: 6
( ν (( B c(,, [5.] wth: c(,, ( EQ(( X ( ( Φ( d(,, ( Φ( d (,, whee : d (,, (ln( d (,, d (,, σ σ / σ 8.3. Equlbum elaton n multle eod models 8.3.. Revesonay Bonus Accodng to fomula [49.] and lke n the bnomal model, the equlbum condton s the same as n the one eod model. 8.3. ash Bonus Usng fomula [5.], the equlbum condton becomes fo the cash bonus: ( ( B ( c(,, ( ( ( o: B ( c(,, ( Φ( d (,, ( Φ( d (,, whch s agan equal to the equlbum value on one eod ( cf [48.]. 8.3.3 emnal Bonus Usng fomula [5.], the equlbum condton fo the temnal bonus becomes: B ( ( ( Φ( d (,, ( Φ( d (,, [5.] 8.4 omason between evesonay and cash systems A same methodology as n secton 6 can be used n ode to obtan a ankng between 7
fa values fo evesonay and cash bonus when the aametes ae not n equlbum. Indeed usng esectvely fomulas [49.] and [5.], the fa values can be wtten as follows: - n the evesonay case : ( ν ( wth : K* ( c(,, -n the cash case : ( B K* (( ν (( B K * ( ( ( ( whch have eactly the same fom as n the bnomal case. So the same concluson can be dawn. 9. ONLUSION In ths ae, we have develoed vaous fomulatons n ode to comae the fa value fo lfe nsuance oducts based on thee atcaton schemes: evesonay, cash and temnal bonus, takng nto account smultaneously the asset sde and the lablty sde n a multle eod model. We have shown that the fa value deends on the nvestment stategy (and on the assocated sk, on the atcaton level and on the guaanteed ate but also on the bonus system chosen. We have found some elct equlbum condtons between all these aametes. A dee comason has been made between the thee atcaton schemes, as well n tems of comutaton of the fa value as n the equlbum condtons. Usng fst a bnomal model, we have obtaned closed foms and gven clea nteetatons on the lnk between the maket condtons, the volatlty of the assets and the aametes of the oduct. A same aoach, leadng to smla conclusons, has been oosed n a 8
tme contnuous model. he model could be also etended n ode to take nto account othe asects lke suende otons, eodcal emums o the longevty sk.. REFERENES AASE, K.K. and PERSSON, S.A. (994: Pcng of unt-lnked lfe nsuance olces, Scandnavan Actuaal Jounal, 6-5 BAINELLO, A.R. (: Fa cng of Lfe Insuance atcatng olces wth a mnmum nteest ate guaanteed, ASIN Bulletn 3(, 75-98 BAINELLO, A.R. (3a: Fa valuaton of a guaanteed lfe nsuance atcatng contact embeddng a suende oton, Jounal of Rsk and Insuance 7(3, 46-487 BAINELLO, A.R. ( 3 b: Pcng guaanteed lfe nsuance atcatng olces wth annual emums and suende oton, Noth Amecan Actuaal Jounal 7(3, -7 BRIYS, E and DE VARENNE, F.(997: On the sk of nsuance labltes: debunkng some common tfalls, Jounal of Rsk an Insuance 64(4, 673-694 BUHLMAN, H. (: New Math fo Lfe Actuaes, ASIN Bulletn 3(, 9- BRENNAN, M.J. and SHWARZ, E.S.(976: he cng of equty lnked lfe nsuance olces wth an asset value guaantee, Jounal of Fnancal Economcs 3, 95-3 OX, J., ROSS, S. and RUBINSEIN, M.(979:Oton Pcng: a smlfed aoach, Jounal of Fnancal Economcs 7, 9-63 DELBAEN, F. (986: Equty lnked olces, BARAB 8, 33-5 DEVOLDER, P. and DOMÍNGUEZ-FABIÁN, I.(4: Deflatos, actuaal dscountng and fa value, to aea n Fnance GROSEN, A. and JORGENSEN, P. L. (: Fa valuaton of lfe nsuance labltes: the mact of nteest ate guaantees, suende otons and bonus olces, Insuance: Mathematcs and Economcs 6(, 37-57 HABERMAN, S., BALLOA, L. and WANG, N.(3: Modellng and valuaton of guaantees n wth-oft and untsed wth oft lfe nsuance contacts, 7 th Insuance: Mathematcs and Economcs congess, Lyon 9
HANSEN, M. and MILERSEN, K. R. (: Mnmum ate of etun guaantees: the Dansh case, Scandnavan Actuaal Jounal 4(4, 8-38 HARDY,M.(3: Investement guaantees: modellng and sk management fo equty-lnked nsuance, John Wley& Sons MOLLER,. (998: Rsk mnmzng hedgng stateges fo unt-lnked lfe nsuance contacts, Astn Bulletn 8, 7-47 NIELSEN, J. A. and SANDMANN, K. (995: Equty-lnked lfe nsuance: a model wth stochastc nteets ates, Insuance:Mathematcs and Economcs 6, 5-53 NORBERG,R. (: Lfe Insuance Mathematcs, 6 th Insuance: Mathematcs and Economcs congess, Lsbon 3