FAIR VALUATION OF VARIOUS PARTICIPATION SCHEMES IN LIFE INSURANCE ABSTRACT

Size: px
Start display at page:

Download "FAIR VALUATION OF VARIOUS PARTICIPATION SCHEMES IN LIFE INSURANCE ABSTRACT"

Transcription

1 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 Deatment of Fnancal Economy, Unvesty of Etemadua, San (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

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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:

11 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.

12 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:

13 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 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

14 ( 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

15 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

16 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 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

17 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

18 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.

19 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:

20 ( ( 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:

21 ( ( 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.

22 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, Guaanteed ate.5 % 8% 6% 4% % Patcaton ate ash Revesonay emnal

23 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% Equlbum values of B ERMINAL BONUS nteest ate,76 %,995485,978396,96357,45 4%,6553, ,973645,4 6%,5856,99363,985763,47 8%,9958,33,99788,6 %,666,7699 3

24 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, ,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

25 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

26 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.] emnal Bonus: he fa value wll have the same stuctue as n the one eod model: 6

27 ( 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 Revesonay Bonus Accodng to fomula [49.] and lke n the bnomal model, the equlbum condton s the same as n the one eod model 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.] 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

28 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

29 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(, BAINELLO, A.R. (3a: Fa valuaton of a guaanteed lfe nsuance atcatng contact embeddng a suende oton, Jounal of Rsk and Insuance 7(3, 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, 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(, 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

30 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

Order-Degree Curves for Hypergeometric Creative Telescoping

Order-Degree Curves for Hypergeometric Creative Telescoping Ode-Degee Cuves fo Hyegeometc Ceatve Telescong ABSTRACT Shaosh Chen Deatment of Mathematcs NCSU Ralegh, NC 7695, USA schen@ncsuedu Ceatve telescong aled to a bvaate oe hyegeometc tem oduces lnea ecuence

More information

Impact on inventory costs with consolidation of distribution centers

Impact on inventory costs with consolidation of distribution centers IIE Tansactons (2) 33, 99± Imact on nventoy costs wth consoldaton of dstbuton centes CHUNG PIAW TEO, JIHONG OU and MARK GOH Deatment of Decson Scences, Faculty of Busness Admnstaton, Natonal Unvesty of

More information

Additional File 1 - A model-based circular binary segmentation algorithm for the analysis of array CGH data

Additional File 1 - A model-based circular binary segmentation algorithm for the analysis of array CGH data 1 Addtonal Fle 1 - A model-based ccula bnay segmentaton algothm fo the analyss of aay CGH data Fang-Han Hsu 1, Hung-I H Chen, Mong-Hsun Tsa, Lang-Chuan La 5, Ch-Cheng Huang 1,6, Shh-Hsn Tu 6, Ec Y Chuang*

More information

A New replenishment Policy in a Two-echelon Inventory System with Stochastic Demand

A New replenishment Policy in a Two-echelon Inventory System with Stochastic Demand A ew eplenshment Polcy n a wo-echelon Inventoy System wth Stochastc Demand Rasoul Haj, Mohammadal Payesh eghab 2, Amand Babol 3,2 Industal Engneeng Dept, Shaf Unvesty of echnology, ehan, Ian (haj@shaf.edu,

More information

Perturbation Theory and Celestial Mechanics

Perturbation Theory and Celestial Mechanics Copyght 004 9 Petubaton Theoy and Celestal Mechancs In ths last chapte we shall sketch some aspects of petubaton theoy and descbe a few of ts applcatons to celestal mechancs. Petubaton theoy s a vey boad

More information

A Coverage Gap Filling Algorithm in Hybrid Sensor Network

A Coverage Gap Filling Algorithm in Hybrid Sensor Network A Coveage Ga Fllng Algothm n Hybd Senso Netwok Tan L, Yang Mnghua, Yu Chongchong, L Xuanya, Cheng Bn A Coveage Ga Fllng Algothm n Hybd Senso Netwok 1 Tan L, 2 Yang Mnghua, 3 Yu Chongchong, 4 L Xuanya,

More information

AREA COVERAGE SIMULATIONS FOR MILLIMETER POINT-TO-MULTIPOINT SYSTEMS USING STATISTICAL MODEL OF BUILDING BLOCKAGE

AREA COVERAGE SIMULATIONS FOR MILLIMETER POINT-TO-MULTIPOINT SYSTEMS USING STATISTICAL MODEL OF BUILDING BLOCKAGE Radoengneeng Aea Coveage Smulatons fo Mllmete Pont-to-Multpont Systems Usng Buldng Blockage 43 Vol. 11, No. 4, Decembe AREA COVERAGE SIMULATIONS FOR MILLIMETER POINT-TO-MULTIPOINT SYSTEMS USING STATISTICAL

More information

An Algorithm For Factoring Integers

An Algorithm For Factoring Integers An Algothm Fo Factong Integes Yngpu Deng and Yanbn Pan Key Laboatoy of Mathematcs Mechanzaton, Academy of Mathematcs and Systems Scence, Chnese Academy of Scences, Bejng 100190, People s Republc of Chna

More information

I = Prt. = P(1+i) n. A = Pe rt

I = Prt. = P(1+i) n. A = Pe rt 11 Chapte 6 Matheatcs of Fnance We wll look at the atheatcs of fnance. 6.1 Sple and Copound Inteest We wll look at two ways nteest calculated on oney. If pncpal pesent value) aount P nvested at nteest

More information

Gravitation. Definition of Weight Revisited. Newton s Law of Universal Gravitation. Newton s Law of Universal Gravitation. Gravitational Field

Gravitation. Definition of Weight Revisited. Newton s Law of Universal Gravitation. Newton s Law of Universal Gravitation. Gravitational Field Defnton of Weght evsted Gavtaton The weght of an object on o above the eath s the gavtatonal foce that the eath exets on the object. The weght always ponts towad the cente of mass of the eath. On o above

More information

On the Efficiency of Equilibria in Generalized Second Price Auctions

On the Efficiency of Equilibria in Generalized Second Price Auctions On the Effcency of Equlba n Genealzed Second Pce Auctons Ioanns Caaganns Panagots Kanellopoulos Chstos Kaklamans Maa Kyopoulou Depatment of Compute Engneeng and Infomatcs Unvesty of Patas and RACTI, Geece

More information

Electric Potential. otherwise to move the object from initial point i to final point f

Electric Potential. otherwise to move the object from initial point i to final point f PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Electc Potental Dsclame: These lectue notes ae not meant to eplace the couse textbook. The content may be ncomplete. Some topcs may be unclea. These

More information

Decomposing General Equilibrium Effects of Policy Intervention in Multi-Regional Trade Models

Decomposing General Equilibrium Effects of Policy Intervention in Multi-Regional Trade Models Decomosng Geneal Equlbum Effects of Polcy Inteventon n ult-regonal Tade odels ethod and Samle Alcaton by hstoh Böhnge ente fo Euoean Economc Reseach (ZEW) boehnge@zew.de and Thomas F. Ruthefod Deatment

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

Keywords: Transportation network, Hazardous materials, Risk index, Routing, Network optimization.

Keywords: Transportation network, Hazardous materials, Risk index, Routing, Network optimization. IUST Intenatonal Jounal of Engneeng Scence, Vol. 19, No.3, 2008, Page 57-65 Chemcal & Cvl Engneeng, Specal Issue A ROUTING METHODOLOGY FOR HAARDOUS MATIALS TRANSPORTATION TO REDUCE THE RISK OF ROAD NETWORK

More information

Orbit dynamics and kinematics with full quaternions

Orbit dynamics and kinematics with full quaternions bt dynamcs and knematcs wth full quatenons Davde Andes and Enco S. Canuto, Membe, IEEE Abstact Full quatenons consttute a compact notaton fo descbng the genec moton of a body n the space. ne of the most

More information

Degrees of freedom in HLM models

Degrees of freedom in HLM models Degees o eedom n HLM models The vaous degees o eedom n a HLM2/HLM3 model can be calculated accodng to Table 1 and Table 2. Table 1: Degees o Feedom o HLM2 Models Paamete/Test Statstc Degees o Feedom Gammas

More information

Joint Virtual Machine and Bandwidth Allocation in Software Defined Network (SDN) and Cloud Computing Environments

Joint Virtual Machine and Bandwidth Allocation in Software Defined Network (SDN) and Cloud Computing Environments IEEE ICC 2014 - Next-Geneaton Netwokng Symposum 1 Jont Vtual Machne and Bandwdth Allocaton n Softwae Defned Netwok (SDN) and Cloud Computng Envonments Jonathan Chase, Rakpong Kaewpuang, Wen Yonggang, and

More information

AN EQUILIBRIUM ANALYSIS OF THE INSURANCE MARKET WITH VERTICAL DIFFERENTIATION

AN EQUILIBRIUM ANALYSIS OF THE INSURANCE MARKET WITH VERTICAL DIFFERENTIATION QUIIRIUM YI OF T IUR MRKT WIT VRTI IFFRTITIO Mahto Okua Faculty of conomcs, agasak Unvesty, 4-- Katafuch, agasak, 8508506, Japan okua@net.nagasak-u.ac.p TRT ach nsuance poduct pe se s dentcal but the nsuance

More information

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important

More information

TRUCK ROUTE PLANNING IN NON- STATIONARY STOCHASTIC NETWORKS WITH TIME-WINDOWS AT CUSTOMER LOCATIONS

TRUCK ROUTE PLANNING IN NON- STATIONARY STOCHASTIC NETWORKS WITH TIME-WINDOWS AT CUSTOMER LOCATIONS TRUCK ROUTE PLANNING IN NON- STATIONARY STOCHASTIC NETWORKS WITH TIME-WINDOWS AT CUSTOMER LOCATIONS Hossen Jula α, Maged Dessouky β, and Petos Ioannou γ α School of Scence, Engneeng and Technology, Pennsylvana

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

Statistical modelling of gambling probabilities

Statistical modelling of gambling probabilities Ttle Statstcal modellng of gamblng pobabltes Autho(s) Lo, Su-yan, Vcto.; 老 瑞 欣 Ctaton Issued Date 992 URL http://hdl.handle.net/0722/3525 Rghts The autho etans all popetay ghts, (such as patent ghts) and

More information

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly

More information

PCA vs. Varimax rotation

PCA vs. Varimax rotation PCA vs. Vamax otaton The goal of the otaton/tansfomaton n PCA s to maxmze the vaance of the new SNP (egensnp), whle mnmzng the vaance aound the egensnp. Theefoe the dffeence between the vaances captued

More information

(Semi)Parametric Models vs Nonparametric Models

(Semi)Parametric Models vs Nonparametric Models buay, 2003 Pobablty Models (Sem)Paametc Models vs Nonpaametc Models I defne paametc, sempaametc, and nonpaametc models n the two sample settng My defnton of sempaametc models s a lttle stonge than some

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

Institut für Halle Institute for Economic Research Wirtschaftsforschung Halle

Institut für Halle Institute for Economic Research Wirtschaftsforschung Halle Insttut fü Halle Insttute fo Economc Reseach Wtschaftsfoschung Halle A Smple Repesentaton of the Bea-Jaque-Lee Test fo Pobt Models Joachm Wlde Dezembe 2007 No. 13 IWH-Dskussonspapee IWH-Dscusson Papes

More information

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant

More information

Chapter 4: Matrix Norms

Chapter 4: Matrix Norms EE448/58 Vesion.0 John Stensby Chate 4: Matix Noms The analysis of matix-based algoithms often equies use of matix noms. These algoithms need a way to quantify the "size" of a matix o the "distance" between

More information

FAIR VALUATION OF VARIOUS PARTICIPATION SCHEMES IN LIFE INSURANCE ABSTRACT KEYWORDS. Fair value, participation scheme, asset and liability management

FAIR VALUATION OF VARIOUS PARTICIPATION SCHEMES IN LIFE INSURANCE ABSTRACT KEYWORDS. Fair value, participation scheme, asset and liability management FAIR VALUAION OF VARIOUS PARICIPAION SCHEMES IN LIFE INSURANCE BY PIERRE DEVOLDER AND INMACULADA DOMÍNGUEZ-FABIÁN ABSRAC Fair valuation is becomin a maor concern for actuaries, especially in the perspective

More information

A Resource Scheduling Algorithms Based on the Minimum Relative Degree of Load Imbalance

A Resource Scheduling Algorithms Based on the Minimum Relative Degree of Load Imbalance Jounal of Communcatons Vol. 10, No. 10, Octobe 2015 A Resouce Schedulng Algothms Based on the Mnmum Relatve Degee of Load Imbalance Tao Xue and Zhe Fan Depatment of Compute Scence, X an Polytechnc Unvesty,

More information

Drag force acting on a bubble in a cloud of compressible spherical bubbles at large Reynolds numbers

Drag force acting on a bubble in a cloud of compressible spherical bubbles at large Reynolds numbers Euopean Jounal of Mechancs B/Fluds 24 2005 468 477 Dag foce actng on a bubble n a cloud of compessble sphecal bubbles at lage Reynolds numbes S.L. Gavlyuk a,b,,v.m.teshukov c a Laboatoe de Modélsaton en

More information

Efficient Evolutionary Data Mining Algorithms Applied to the Insurance Fraud Prediction

Efficient Evolutionary Data Mining Algorithms Applied to the Insurance Fraud Prediction Intenatonal Jounal of Machne Leanng and Computng, Vol. 2, No. 3, June 202 Effcent Evolutonay Data Mnng Algothms Appled to the Insuance Faud Pedcton Jenn-Long Lu, Chen-Lang Chen, and Hsng-Hu Yang Abstact

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

Prejudice and the Economics of Discrimination

Prejudice and the Economics of Discrimination Pelmnay Pejudce and the Economcs of Dscmnaton Kewn Kof Chales Unvesty of Chcago and NB Jonathan Guyan Unvesty of Chcago GSB and NB Novembe 17, 2006 Abstact Ths pape e-examnes the ole of employe pejudce

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

PREVENTIVE AND CORRECTIVE SECURITY MARKET MODEL

PREVENTIVE AND CORRECTIVE SECURITY MARKET MODEL REVENTIVE AND CORRECTIVE SECURITY MARKET MODEL Al Ahmad-hat Rachd Cheaou and Omd Alzadeh Mousav Ecole olytechnque Fédéale de Lausanne Lausanne Swzeland al.hat@epfl.ch achd.cheaou@epfl.ch omd.alzadeh@epfl.ch

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Top-Down versus Bottom-Up Approaches in Risk Management

Top-Down versus Bottom-Up Approaches in Risk Management To-Down vesus Bottom-U Aoaches in isk Management PETE GUNDKE 1 Univesity of Osnabück, Chai of Banking and Finance Kathainenstaße 7, 49069 Osnabück, Gemany hone: ++49 (0)541 969 4721 fax: ++49 (0)541 969

More information

CONSTRUCTION PROJECT SCHEDULING WITH IMPRECISELY DEFINED CONSTRAINTS

CONSTRUCTION PROJECT SCHEDULING WITH IMPRECISELY DEFINED CONSTRAINTS Management an Innovaton fo a Sustanable Bult Envonment ISBN: 9789052693958 20 23 June 2011, Amsteam, The Nethelans CONSTRUCTION PROJECT SCHEDULING WITH IMPRECISELY DEFINED CONSTRAINTS JANUSZ KULEJEWSKI

More information

Time Series Perturbation by Genetic Programming

Time Series Perturbation by Genetic Programming Tme Sees Petbaton by Genetc Pogammng G. Y. Lee Assstant Pofess Deatment of Comte and Infomaton Engneeng Yong-San Unvesty San 5 -Nam-R Ung-Sang-E Yang-San-Sh Kyng-Nam Soth Koea sky@java-tech.com Abstact-

More information

LINES ON BRIESKORN-PHAM SURFACES

LINES ON BRIESKORN-PHAM SURFACES LIN ON BRIKORN-PHAM URFAC GUANGFNG JIANG, MUTUO OKA, DUC TAI PHO, AND DIRK IRMA Abstact By usng toc modfcatons and a esult of Gonzalez-pnbeg and Lejeune- Jalabet, we answe the followng questons completely

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

Charging the Internet Without Bandwidth Reservation: An Overview and Bibliography of Mathematical Approaches

Charging the Internet Without Bandwidth Reservation: An Overview and Bibliography of Mathematical Approaches JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 9, -xxx (2003) Chagng the Intenet Wthout Bandwdth Resevaton: An Ovevew and Bblogaphy of Mathematcal Appoaches IRISA-INRIA Campus Unvestae de Beauleu 35042

More information

International Business Cycles and Exchange Rates

International Business Cycles and Exchange Rates Revew of Intenatonal Economcs, 7(4), 682 698, 999 Intenatonal Busness Cycles and Exchange Rates Chstan Zmmemann* Abstact Models of ntenatonal eal busness cycles ae not able to account fo the hgh volatlty

More information

Time Value of Money Module

Time Value of Money Module Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the

More information

AMB111F Financial Maths Notes

AMB111F Financial Maths Notes AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed

More information

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error Intra-year Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor

More information

Green's function integral equation methods for plasmonic nanostructures

Green's function integral equation methods for plasmonic nanostructures Geens functon ntegal equaton methods fo plasmonc nanostuctues (Ph Couse: Optcal at the Nanoscale) Thomas Søndegaad epatment of Phscs and Nanotechnolog, Aalbog Unvest, Senve 4A, K-9 Aalbog Øst, enma. Intoducton

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value 8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Valuation of Floating Rate Bonds 1

Valuation of Floating Rate Bonds 1 Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

More information

Competitive Targeted Advertising with Price Discrimination

Competitive Targeted Advertising with Price Discrimination Compette Tageted Adetsng wth Pce Dscmnaton Rosa Banca Estees Unesdade do Mnho and NIPE banca@eeg.umnho.pt Joana Resende Faculdade de Economa, Unesdade do Poto and CEF.UP jesende@fep.up.pt Septembe 8, 205

More information

An Overview of Financial Mathematics

An Overview of Financial Mathematics An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

More information

FAST SIMULATION OF EQUITY-LINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION. Carole Bernard Christiane Lemieux

FAST SIMULATION OF EQUITY-LINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION. Carole Bernard Christiane Lemieux Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. FAST SIMULATION OF EQUITY-LINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

A Novel Lightweight Algorithm for Secure Network Coding

A Novel Lightweight Algorithm for Secure Network Coding A Novel Lghtweght Algothm fo Secue Netwok Codng A Novel Lghtweght Algothm fo Secue Netwok Codng State Key Laboatoy of Integated Sevce Netwoks, Xdan Unvesty, X an, Chna, E-mal: {wangxaoxao,wangmeguo}@mal.xdan.edu.cn

More information

Level Annuities with Payments Less Frequent than Each Interest Period

Level Annuities with Payments Less Frequent than Each Interest Period Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

More information

The Mathematical Derivation of Least Squares

The Mathematical Derivation of Least Squares Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell

More information

Security of Full-State Keyed Sponge and Duplex: Applications to Authenticated Encryption

Security of Full-State Keyed Sponge and Duplex: Applications to Authenticated Encryption Secuty of Full-State Keyed Sponge and uplex: Applcatons to Authentcated Encypton Bat Mennnk 1 Reza Reyhantaba 2 aman Vzá 2 1 ept. Electcal Engneeng, ESAT/COSIC, KU Leuven, and Mnds, Belgum bat.mennnk@esat.kuleuven.be

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

FI3300 Corporate Finance

FI3300 Corporate Finance Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multi-peiod multi-cf time-value-of-money poblems: Geneal case Pepetuity

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

A PARTICLE-BASED LAGRANGIAN CFD TOOL FOR FREE-SURFACE SIMULATION

A PARTICLE-BASED LAGRANGIAN CFD TOOL FOR FREE-SURFACE SIMULATION C A N A L D E E X P E R I E N C I A S H I D R O D I N Á M I C A S, E L P A R D O Publcacón núm. 194 A PARTICLE-BASED LAGRANGIAN CFD TOOL FOR FREE-SURFACE SIMULATION POR D. MUÑOZ V. GONZÁLEZ M. BLAIN J.

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

Determinants of Borrowing Limits on Credit Cards Shubhasis Dey and Gene Mumy

Determinants of Borrowing Limits on Credit Cards Shubhasis Dey and Gene Mumy Bank of Canada Banque du Canada Wokng Pape 2005-7 / Document de taval 2005-7 Detemnants of Boowng mts on Cedt Cads by Shubhass Dey and Gene Mumy ISSN 1192-5434 Pnted n Canada on ecycled pape Bank of Canada

More information

The Analysis of Outliers in Statistical Data

The Analysis of Outliers in Statistical Data THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

Mixed Task Scheduling and Resource Allocation Problems

Mixed Task Scheduling and Resource Allocation Problems Task schedulng and esouce allocaton 1 Mxed Task Schedulng and Resouce Allocaton Poblems Mae-José Huguet 1,2 and Pee Lopez 1 1 LAAS-CNRS, 7 av. du Colonel Roche F-31077 Toulouse cedex 4, Fance {huguet,lopez}@laas.f

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

REAL INTERPOLATION OF SOBOLEV SPACES

REAL INTERPOLATION OF SOBOLEV SPACES REAL INTERPOLATION OF SOBOLEV SPACES NADINE BADR Abstact We pove that W p s a eal ntepolaton space between W p and W p 2 fo p > and p < p < p 2 on some classes of manfolds and geneal metc spaces, whee

More information

Statistical Discrimination or Prejudice? A Large Sample Field Experiment. Michael Ewens, Bryan Tomlin, and Liang Choon Wang.

Statistical Discrimination or Prejudice? A Large Sample Field Experiment. Michael Ewens, Bryan Tomlin, and Liang Choon Wang. Statstcal Dscmnaton o Pejudce? A Lage Sample Feld Expement Mchael Ewens, yan Tomln, and Lang Choon ang Abstact A model of acal dscmnaton povdes testable mplcatons fo two featues of statstcal dscmnatos:

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

Real-Time Scheduling in MapReduce Clusters

Real-Time Scheduling in MapReduce Clusters Unvesty of Nebaska - Lncoln DgtalCoons@Unvesty of Nebaska - Lncoln CSE Confeence and Woksho Paes Coute Scence and Engneeng, Deatent of 2013 Real-Te Schedulng n MaReduce Clustes Chen He Unvesty of Nebaska-Lncoln,

More information

1. Math 210 Finite Mathematics

1. Math 210 Finite Mathematics 1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

Bending Stresses for Simple Shapes

Bending Stresses for Simple Shapes -6 Bendng Stesses fo Smple Sapes In bendng, te maxmum stess and amount of deflecton can be calculated n eac of te followng stuatons. Addtonal examples ae avalable n an engneeng andbook. Secton Modulus

More information

Continuous Compounding and Annualization

Continuous Compounding and Annualization Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

More information

SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.

SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976-76-10-00

More information

A) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution.

A) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution. ACTS 408 Instructor: Natala A. Humphreys SOLUTION TO HOMEWOR 4 Secton 7: Annutes whose payments follow a geometrc progresson. Secton 8: Annutes whose payments follow an arthmetc progresson. Problem Suppose

More information

10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest

10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest 1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual e ectve

More information

Ilona V. Tregub, ScD., Professor

Ilona V. Tregub, ScD., Professor Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

More information

The Can-Order Policy for One-Warehouse N-Retailer Inventory System: A Heuristic Approach

The Can-Order Policy for One-Warehouse N-Retailer Inventory System: A Heuristic Approach Atcle Te Can-Ode Polcy fo One-Waeouse N-Retale Inventoy ystem: A Heustc Appoac Vaapon Pukcanon, Paveena Caovaltongse, and Naagan Pumcus Depatment of Industal Engneeng, Faculty of Engneeng, Culalongkon

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

More information

Hedge accounting within IAS39

Hedge accounting within IAS39 Economc and Fnancal Report 2002/02 Hedge accountng wthn IAS39 Alessandro Ross, Gudo Bchsao and Francesca Campolongo Economc and Fnancal Studes European Investment Bank 00, boulevard Konrad Adenauer L-2950

More information

LIFETIME INCOME OPTIONS

LIFETIME INCOME OPTIONS LIFETIME INCOME OPTIONS May 2011 by: Marca S. Wagner, Esq. The Wagner Law Group A Professonal Corporaton 99 Summer Street, 13 th Floor Boston, MA 02110 Tel: (617) 357-5200 Fax: (617) 357-5250 www.ersa-lawyers.com

More information

Multiple discount and forward curves

Multiple discount and forward curves Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

More information

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143 1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

10.2 Future Value and Present Value of an Ordinary Simple Annuity

10.2 Future Value and Present Value of an Ordinary Simple Annuity 348 Chapter 10 Annutes 10.2 Future Value and Present Value of an Ordnary Smple Annuty In compound nterest, 'n' s the number of compoundng perods durng the term. In an ordnary smple annuty, payments are

More information

How a Global Inter-Country Input-Output Table with Processing Trade Account. Can be constructed from GTAP Database

How a Global Inter-Country Input-Output Table with Processing Trade Account. Can be constructed from GTAP Database How a lobal Inte-County Input-Output Table wth Pocessng Tade Account Can be constucted fom TAP Database Manos Tsgas and Zh Wang U.S. Intenatonal Tade Commsson* Mak ehlha U.S. Depatment of Inteo* (Pelmnay

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

RESOURCE SCHEDULING STRATEGIES FOR A NETWORK-BASED AUTONOMOUS MOBILE ROBOT. Hongryeol Kim, Joomin Kim, and Daewon Kim

RESOURCE SCHEDULING STRATEGIES FOR A NETWORK-BASED AUTONOMOUS MOBILE ROBOT. Hongryeol Kim, Joomin Kim, and Daewon Kim RESOURCE SCHEDULING SRAEGIES FOR A NEWORK-BASED AUONOMOUS MOBILE ROBO Hongyeol K, Joon K, and Daewon K Depatent of Infoaton Contol Engneeng, Myong Unvesty Abstact: In ths pape, effcent esouce schedulng

More information

Section 2.3 Present Value of an Annuity; Amortization

Section 2.3 Present Value of an Annuity; Amortization Secton 2.3 Present Value of an Annuty; Amortzaton Prncpal Intal Value PV s the present value or present sum of the payments. PMT s the perodc payments. Gven r = 6% semannually, n order to wthdraw $1,000.00

More information

Discussion Papers. Thure Traber Claudia Kemfert

Discussion Papers. Thure Traber Claudia Kemfert Dscusson Papes Thue Tabe Clauda Kemfet Impacts of the Geman Suppot fo Renewable Enegy on Electcty Pces, Emssons and Pofts: An Analyss Based on a Euopean Electcty Maket Model Beln, July 2007 Opnons expessed

More information

Mathematics of Finance

Mathematics of Finance Mathematcs of Fnance 5 C H A P T E R CHAPTER OUTLINE 5.1 Smple Interest and Dscount 5.2 Compound Interest 5.3 Annutes, Future Value, and Snkng Funds 5.4 Annutes, Present Value, and Amortzaton CASE STUDY

More information