Valuation and Risk Assessment of a Portfolio of Variable Annuities: A Vector Autoregression Approach

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1 Jounal of Mahemacal Fnance, 8, 8, 49-7 hp:// ISSN Onlne: 6-44 ISSN Pn: 6-44 Valuaon and Rk Aemen of a Pofolo of Vaable Annue: A Veco Auoegeon Appoach Albna Olando, Gay Pake Iuo pe le Applcazon del Calcolo Mauo Pcone CNR, Napol, Ialy Depamen of Sac and Acuaal Scence, Smon Fae Unvey, Bunaby, Canada How o ce h pape: Olando, A. and Pake, G. (8) Valuaon and Rk Aemen of a Pofolo of Vaable Annue: A Veco Auoegeon Appoach. Jounal of Mahemacal Fnance, 8, hp://do.og/.46/jmf.8.8 Receved: Febuay, 8 Acceped: May 7, 8 Publhed: May, 8 Copygh 8 by auho and Scenfc Reeach Publhng Inc. h wok lcened unde he Ceave Common Abuon Inenaonal Lcene (CC BY 4.). hp://ceavecommon.og/lcene/by/4./ Open Acce Abac h pape focue on aeng he fnancal poon of an nue ung a pofolo of Vaable Annue (VA). wo mulvaae model fo he undelyng and he nee ae ae condeed. he f model ue a ngle oal ae of eun fo he bake of ae. he econd one, jonly model he ae of eun on he n ae n he bake. Fo mplcy, he nue aumed o be able o mplemen a ac hedgng pogamme o manage he k. A an example, a homogeneou pofolo of VA wh GMDB and GMMB guaanee offeng dffeen nvemen oppoune o he polcyholde uded. he nue can chooe o ebalance he bake of ae egulaly o no. Reul fo hee wo cae ae peened. Keywod Vaable Annue, VAR Model, Sac Hedgng, Condonal Value a Rk. Inoducon Ove he yea, many paccal and academc conbuon have been offeed decbng VA and he embedded guaanee. Vaable annue (VA) ae lfe nuance poduc wh nvemen guaanee povdng ubanal nvemen guaanee along wh ax pvlege and pudenly managed ae. VA have exed n USA nce he 95. Recenly hey have been peadng aco Euope. Some of he moe gnfcan and hgh pofle launche have been AXA n Fance, Gemany, Span, Ialy and Belgum a well a ING launche n Span, Hungay and Poland, Geneal launch (Decembe 7) n Ialy and Ego launch (Febuay 8) n Gemany. h DOI:.46/jmf.8.8 May, 8 49 Jounal of Mahemacal Fnance

2 A. Olando, G. Pake n addon o he vaou launche by Aegon, Hafod, Melfe and Lncoln n he U.K. []. VA allow he allocaon of pemum no a ange of nvemen opon whch uually conan ock, bond, money make numen o ome combnaon of he hee. he benef o he polcyholde wll depend on he pefomance of he choen nvemen opon; ypcally, he benef he hghe of he accoun value and he guaaneed amoun. he majo ype of VA guaanee ae: Guaaneed Mnmum Deah Benef (GMDB) ha guaanee a eun of he pncpal nveed upon he deah of he polcyholde; Guaaneed Mnmum Accumulaon Benef (GMAB) whee he guaanee ypcally be on pecfed polcy annveae o beween pecfed dae f he polcy ll n foce. If guaanee ae avalable a mauy hey ae called Guaaneed Mnmum Mauy Benef (GMMB); Guaaneed Income Benef (GMIB) guaanee a mnmum ncome eam (ypcally n he fom of a lfe annuy) fom a pecfed fuue pon n me; Guaaneed Mnmum Whdawal Benef (GMWB) guaanee a mnmum ncome eam hough egula whdawal fom he accoun. A dealed ovevew of vaable annue and a decpon of he VA nenaonal make ae gven n [] and []. he man empha of he cuen leaue on pcng and hedgng he opon embedded n hee lfe nuance conac. A geneal famewok n whch any degn of opon and guaanee cuenly offeed whn VA can be modeled, peened n []. In he ecen leaue we fnd ome neeng conbuon on denfcaon and aemen of k. Among ohe, [4] deal wh k aemen, moly va ochac mulaon. he man conbuon o peen an negaed appoach o k n VA poduc. Analycal mehod fo he calculaon of k meaue fo vaable annue guaaneed benef and a quanave model fo make k on he lably de and evenue k n he ae de, ae popoed n [5]. he nue wll be holdng he lably fo he VA guaanee unl lape, deah o mauy. he mpac of boh moaly k (due o andom flucuaon n he numbe of deah) and nvemen k can be ahe hgh n he peence of ome ype of guaanee and opon. he eulng economc mpac o he nue ellng he guaanee baed on how he fuue eal wold cenao play ou. Gven hee condeaon, may be ueful fo an nue o ae eal wold k aocaed wh embedded guaanee of VA pofolo, once pcng done and a k managemen aegy adoped. Mo nue manage he k of he VA pofolo by adopng a ophcaed hedgng pogamme. Some ue ac hedgng o mgae he k, a lea paally. Hee, ac hedgng efe o aege conng of buyng and ellng long-em fnancal numen, e.g. ove he coune (OC) opon, a me of ue and holdng hem unl mauy wh lle ebalancng, f any. Avalable ac hedge ofen fal o cloely mach he lable of VA poduc DOI:.46/jmf Jounal of Mahemacal Fnance

3 A. Olando, G. Pake and ae expenve o e up. Fo hee eaon, dynamc hedgng aege ae moe popula among nue. Hee, a pofolo of lqud ae e up o ha, ove a ho peod, change n value wll appoxmaely mach he change n value of he lable. he pofolo of ae hen ebalanced a a fequency ha ake no accoun adng co and effecvene of he hedge. In h pape, aumed ha he nue able o mplemen a ac hedge. h allow u o compae he mpac of he wo mulvaae pocee ued o model he ae of eun of he bake of ae on he value of he pofolo. We can alo udy whehe ebalancng he bake of ae held n he VA accoun o no affec he k. Alhough ung a dynamc hedgng aegy moe ealc, we conde a ac hedgng pogam a a f ep of ou eeach whch no pecfcally focued on he knd of aegy choen o hedge he pofolo. Dynamc hedgng wll be exploed n fuue wok. In h pape we chooe o focu on aeng he fnancal poon a ue of an nue ellng a pofolo of VA poduc offeng GMDB and GMMB guaanee. We meaue he pofolo value a ncepon a he dffeence beween he peen value of he ochac fuue nflow and ouflow n a eal wold cenao. Ou man conbuon con of condeng wo mulvaae VAR () pocee o model he undelyng and he nee ae and compae he pofolo value a ncepon. Alhough vaable annue ae wen on moe han one ae, n pacce, ngle ae unvaae ochac nvemen model ae moly ued fo pcng and hedgng pupoe. h may lead o poblem uch a ba k o a lack of flexbly wh epec o ae allocaon [6]. In lgh of hee condeaon, we compae a wo-dmenonal VAR () model, whee he ae eun on each fund ae combned n a ngle me ee, o an (n + )-dmenonal VAR () model whee he ae eun on each fund ae modelled epaaely. A pecal cae of hee model popoed o bee capue he muual dependence of he vaable nvolved. Anohe ue fo he nue he decon o ebalance o no he bake of ae conung he fund elf. Rebalancng he poce of ealgnng he weghng of a pofolo of ae. I nvolve peodcally buyng and ellng ae n a pofolo o manan a pecfc ae allocaon. In ou wok, we conde wo dffeen accumulaon funcon dependng on whehe he bake of ae ebalanced o no. Fo ake of mplcy, no anacon co ae condeed. Fnally, we aume ha he VA we offe o polcyholde dffeen pemum allocaon wh dffeen nvemen yle. hee knd of Fund chaaceze ome VA poduc old n Euope (among ohe, he Accumulao old by AXA). he dffeen cae decbed above ae evaluaed n a mulaon famewok. he eul unde ac hedgng ae compaed wh hoe obaned when he DOI:.46/jmf Jounal of Mahemacal Fnance

4 A. Olando, G. Pake nue decde no o hedge. he emande of he pape oganzed a follow. Secon noduce he poduc, Secon decbe he VAR () model employed, Secon 4 devoed o he pofolo evaluaon a ncepon. In Secon 5, he eul ae peened and dcued. In Secon 6, ome concludng emak ae gven.. he Conac Le u conde a pofolo of VA conac offeng GMDB and GMMB guaanee ued o C ndependen lve aged x, each payng a unque pemum P a ue. he nue nve he pemum, ne of nal chage fo geneal expene, no eveal nenal fund chaacezed by dffeen nvemen yle. Polcyholde may chooe n whch fund o nve. By vue of he GMDB guaanee, f he nued de n polcy yea, he nue pay a um equal o he geae of he guaaneed amoun o he fund value. On he ohe hand, by vue of he GMMB, a mauy he nue pay he geae of he guaaneed amoun o he fund value, f he nued ll alve. he benef ae pad monhly. A monhly managemen chage deduced and he money gaheed by he nue hough h chage ued o cove he guaanee and ohe expene lnked o he Fund uch a fund managemen and axe. he oblgaon he nue ha o fon fo he GMDB a me ae valued GMDB = N Max F, G () wh [,,, ]. Fo he GMMB we have: ( ) [ ] D ( ) [, ] GMMB = N Max F G () S whee ND ( ) he acual numbe of deah n [, ] and NS ( ) he acual numbe of uvvo a me. Accodng o he conacual degn, he fund value ued n () and () : f( ) F = P ( m) e () and he guaaneed value : wh [,,, ] G g = (4) P e. Hee, F he fund value obaned by nveng he ne pemum P = P c no he choen nenal fund, c he nal chage and P he ngle f( ) pemum. e he accumulaon funcon and m he monhly managemen chage deduced. G he guaaneed amoun obaned by accumulang he ne pemum a he guaaneed ae g (oll up). Dnc fund ae offeed, each made up of a mx of eveal ae. he combnaon of he ae gve e o vaou nvemen yle. DOI:.46/jmf Jounal of Mahemacal Fnance

5 A. Olando, G. Pake Moeove, wh efeence o he fund value F n (), we conde wo cae: he ebalancng aegy of he bake of ae conung he fund elf and he no ebalancng one (ee Secon ). By mean of he well known pu decompoon pncple, we can ewe he guaanee () and () a follow: wh [,,, ] and ( ) ( ) [,] GMDB = N F + G F (5) D ( ) ( ) [,] GMMB NS F G F + = +. (6) (5) and (6) mply ha upon he deah of he polcyholde o expaon of he conac, he nuance Company pay he accumulaed fund value F o he nveo plu an addonal payoff of a pu opon wh ke pce equal o G wh [,,, ] and G epecvely. heefoe f he fund pefom o pooly ha value, when he conac maue o when deah occu, below he coepondng guaaneed value, he nue pay he dffeence. Fo he GMDB, he nue monhly paymen ae: wh [,,, ] ( ) [,] L = N G F. (7) GMDB D. On he ohe hand fo he GMMB we have: ( ) [,] GMMB L NS G F = (8) he andom lable evaluaed a me zeo ae hen gven by: whee y ( ) GMDB e [,,, ] D( ) (,) + y ( ) (9) = L = N G F e ( ) ( ) y ( ) GMMB L = N G F, e () S a andom dcounng faco. +. Modelng he Fnancal Vaable: he Veco Auoegeve Model We chooe a mulvaae famewok modellng jonly all fnancal vaable, namely ae eun and nee ae, by mean of a Veco Auoegeve model of ode one, VAR (). he VAR model one of he mo flexble and eay o ue model fo analyzng mulvaae me ee. I a naual exenon of he unvaae auoegeve model o udy mulvaae me ee. he VAR model ha poven o be epecally ueful fo decbng he dynamc behavou of economc and fnancal me ee and fo foecang [7]. An n-dmenonal me ee andom vaable X ad o follow a VAR () model f DOI:.46/jmf Jounal of Mahemacal Fnance

6 A. Olando, G. Pake ( ) X µ =Φ X µ + a () whee: Z, µ an ( n ) max of he long em mean value of X, Φ an ( n n) max of auoegeve coeffcen, a an n-dmenonal whe noe em whch follow a mulvaae nomal dbuon wh zeo mean and covaance max Σ a. A uffcen condon of aonay ha all egenvalue of Φ ae le han n abolue value. When condonng on he nal value X, he mean of E X X =Φ ( X µ ) + µ he covaance beween X and k <, alo condonal on X : X k k +, k a = X (), whee k a non negave nege, Cov X X X = Φ Σ Φ () We conde wo model: a wo-dmenonal VAR () model fo he compoe ae of eun of he n ae and he nee ae and a (n + )-dmenonal VAR () fo he ae of eun and he nee ae... wo-dmenonal VAR () Model In h cae we model he compoe ae of eun of he n ae conung each fund and he nee ae by mean of a wo-dmenonal VAR () model. Refeng o () we can we: X Φ Φ= Φ = Φ Φ,,,, a a = a he compoe ae of eun can be compued n wo way, dependng on whehe he bake of ae ebalanced o no. When ebalancng he bake of ae we have: n = ω = (4) (5) (6) (7) wh no ebalancng, he compoe ae of eun : n = ( + k) * (8) = k= n whee = ω = and,,,. In (7) and (8) wh,,, n ae he monhly connuou ae of eun of he n ae and ω,,, n ae he popoon of he fund DOI:.46/jmf Jounal of Mahemacal Fnance

7 A. Olando, G. Pake nveed n each of he n ae. If we model he fnancal vaable by a wo-dmenonal VAR () model, n () can be ewen a: F wh [,,, ]. In (9), ebalanced o no. nvemen aegy. he dcounng faco gven by: wh [,,, ] = ( ) F = P m e (9) gven by (7) o (8) dependng on whehe he fund. F aume dffeen value dependng on he eleced y ( ) = e e.. (n + )-Dmenonal VAR () Model = () Hee we conde a (n + )-dmenonal VAR (), modellng jonly he ae of eun of he n ae n whch he nue nve ( wh,,, n) and he dcounng nee ae ( ). Refeng o () we can we: X (,,,, n, ) = () Φ, Φ, Φ, n Φ, n+ Φ, Φ, Φ, n Φ, n+ Φ= Φn, Φn, Φnn, Φnn, + Φn+, Φn+, Φn+, n Φn+, n+ (,,,, n, ) () a = a a a a a. () Baed on ome obevaon on he emaed paamee, we alo popoe a modfed veon of he (n + )-VAR () whch nclude a Geomec Bownan Moon (GBM) o model he ock ndexe (ee Secon 5). he ue of he (n + )-dmenonal model mple ha we have dffeen value of F n (), n cae of ebalancng o no he bake of ae ) Rebalancng cae: ) No ebalancng cae: n ( ) = = ω F = P m e (4) n ω = ( ) = F = P m e (5) whee,,, n ae he pecenage of capal nveed by he nue n each of he n ae. In boh cae, accodng o he value of ω, we oban dffeen accumulaon faco dependng on he choen nvemen yle. he dcounng faco gven by: ϖ [ ] DOI:.46/jmf Jounal of Mahemacal Fnance

8 A. Olando, G. Pake wh [,,, ]. y ( ) = e e = (6) 4. Hedgng he Guaanee Beang n mnd ha he VA conac embed equy pu opon, a cucal poblem o hedge hee opon. Hedgng he pu opon mean holdng a hedge pofolo eplcang he guaanee payoff. A poble oluon dynamc hedgng whch eque a connuou ebalancng of he hedge pofolo. In h famewok, we conde he cae of ac hedgng; fo example, he nue buy equy pu on he make place. 4.. he Co of he Hedgng Pofolo In cae of ac hedgng, he nue buy a pofolo of pu opon, a me zeo, o hedge he guaanee payoff. We know ha, n h cae, he opon pce may be oo hgh becaue hee opon dffe n naue and em fom he ypcal ove he coune opon cuenly avalable. Moeove, he coune pay k may be a faco, gven he long em nvolved. Sac hedgng no alway poble, nevehle h a f ep of ou udy. Fuue eeach wll conde dynamc hedgng, oo. We pce he eplcang pofolo by adapng he Black-Schole eul fo he guaanee embedded n he VA conac lke n [8] whle condeng he dffeen aumpon we make abou he fnancal vaable. he em o mauy of each opon n he VA conac andom, dependng on he emanng lfeme of he polcyholde. Fo he GMDB guaanee depend on he monhly deah pobable and fo he GMMB depend on he uvval pobably a he mauy of he conac. heefoe, he expeced co of he hedge pofolo a me zeo : ( ) ( ) ( ) ( ) PP = E BSP + E BSP (7) D S = whee ED ( ) he expeced numbe of deah a he end of he monh and ES ( ) he expeced numbe of uvvo a he mauy dae of he conac. We aume ha { ND( ) } { N ( )} S mulnomal wh paamee = ( Cq ; x,, qx, npx) whee C he numbe of polce ued a me zeo, q x he pobably ha a lfe aged x a ue de n [, ] and p x he pobably ha a lfe aged x a ue alve a me. So we have ha ED( ) = C q and x ES ( ) = C px A afoemenoned, we calculae he quany BSP n (7) by an adapaon of he andad Black-Schole fomula. We need o modfy he andad fomula fo he pu opon o eflec he fac ha he undelyng ae no he ock pce elf, bu he fund value, and h dffe fom he ock pce hough he deducon of he managemen chage. DOI:.46/jmf Jounal of Mahemacal Fnance

9 A. Olando, G. Pake Ne of he monhly managemen chage he fund : S F = F ( m) (8) S whee S he ock ndex. Le F = S, hen he opon pce wh mauy : BSP ( ) = e E Q G S ( m ) ( ) Ung h appoach, we eplace S m n he andad Black-Schole fomula. hen he pu opon pce compued a follow: + S by ( ) ( ) ( ) ( ) ( ) (9) BSP = G e Φ d S m Φ d () whee he k-fee (foce) of nee, Φ ( ) denoe he cumulave nomal g deny funcon, = P e he nceang ke pce, m he monhly chage and G ( ( ) ) ( σ S ) log S m G + + d =, σ d d = σ, σ he emaed volaly of he ae eun. S If we model he fnancal vaable by a wo-dmenonal VAR model, σ S he emaed volaly of he weghed connuouly compounded ae of eun. I aume dffeen value whehe he bake of ae ebalanced o no (Secon.) Modellng he ae by a ( n + ) -dmenonal VAR () model, σ he weghed volaly of he n eun on ae, ha σ S n n S ωω jσ = j=, j =. Le u now conde he cae n whch he nal allocaon doe no change and he fnancal vaable ae modelled by a ( n + ) -dmenonal VAR () model. In h cae, a me zeo, he nue buy a pofolo of bake opon. A well known, a bake opon an opon on a bake of ae, ypcally ock. In h cae, he Fund value n (8) can be modfed a follow: n ω = S () () S F = F ( m) () whee S he -h ock ndex, ω he popoon of he fund nveed n ae. Leng C F = S, we can we: n ω = ( ) F = S m. (4) Ou nee n he dynamc of he weghed ahmec um of he undelyng S ( =,,,, n). Smla o a plan vanlla opon, he value of a bake opon equal o expeced payoff dcouned a he k fee ae. he DOI:.46/jmf Jounal of Mahemacal Fnance

10 A. Olando, G. Pake expecaon compued wh epec o he ae-pce deny funcon (SPD) alo known a he k neual pobably deny funcon. he SPD, n h cae, no known and cloed fom oluon o he poblem ae no avalable. A common appoach o emae he ae deny funcon by a lognomal funconal fom, ung dffeen machng echnque. One way o compue he f wo momen of he bake payoff ucue a mauy and hen mach hem o hoe of he lognomal dbuon. Followng [9], we defne he peudo-fowad me of he bake: whee ( ) n ωa = A = (5) A = S m e, beng he k fee ae. * Dvdng F by A and denong he ao by F, we nomalze o have a mean of (.e. M = ). * he econd momen of F : M n n ρ, jσσ j ωω jaae j A = j= = (6) whee ρ he coelaon coeffcen beween, j A and A. j * Aumng ha he nomalzed bake F lognomal wh mean M =, follow ha he vaance of he bake v= ln ( M ) (7) he momen-mached lognomal ae-pce deny funcon gve he pce of he pu opon, by mean of he uual Black-Schole fomula: ( ) ( ) ( ) LNBSP = G e Φ d A Φ d (8) whee Φ ( ) he cumulave nomal deny funcon and Equaon (7) become ( ( ) ) log A m G + v d = (9) v d = d qv (4) ( ) ( ) ( ) ( ) PP = E LNBSP + E LNBSP (4) D S = 4.. he Pofolo Value a Incepon he pofolo value a me zeo gven by he dffeence beween he andom nflow and ouflow ang fom he conac. Le u now conde he nflow conneced o he conac. hey ae gven by he expene chage pad a ue by each nued a a pecenage c of he pemum P plu he peen value of he colleced monhly chage m o he nued accoun. Denong by IF he nflow evaluaed a me zeo, n he cae of a wo-dmenonal VAR () model and efeng o (9) DOI:.46/jmf Jounal of Mahemacal Fnance

11 A. Olando, G. Pake and () we have: S = = ( D ( ) ( ( ) ) ) IF = C P c + N m P e e = ( ) ( ) ( ) = = + N m P e e (4) whee he value depend on ebalancng o no and on he nvemen aegy (ee Secon.). In he cae of he ( n + ) dmenonal VAR () model, ung Equaon (4)-(6), we ge: ) Rebalancng cae: S n = = ω = ( D ( ) ( ( ) ) ) IF = C P c + N m P e e = ( ) ( ) ( ) n = = ω = + N m P e e (4) ) No Rebalancng cae: S n = = ( D( ) ( ( ) ) ω ) IF = C P c + N m P e e = = ( ) ( ) ( ) = = ω. + N m P e e (44) In (4) and (44) ω, =,,, n, ae he popoon nveed n each of he n ae and he value change accodng o he choen nvemen aegy. Le u now conde he cah flow fo each, conneced o he conac. hey ae gven by he dffeence beween he oblgaon ang fom he guaanee (lnked o he acual numbe of deah each monh (GMDB) and o he acual numbe of uvvo a me (GMMB)) and he nflow ang fom he pu payoff. A me zeo we ge: + (45) = ( ) = ( D( ) D( )) (,) CF GMDB N E G F e = ( ) ( S ( ) S ( )) ( ) = CF GMMB = N E G F, e. (46) heefoe, he pofolo value a ncepon : VP = IF PP CF ( GMDB) CF ( GMMB) (47) whee he value of each quany depend on he pecfc model and nvemen aegy, a pevouly explaned. We ae neeed n capung he dbuon of he poenal fuue make and moaly ae. o h am we eo o a Mone Calo mulaon, aeng VP aco a eal wold cenao. 5. Illuaon In h econ we udy an lluave pofolo of VA conac offeng GMDB and GMMB guaanee (ee Secon ) ued o ndependen lve aged 55 and payng a ngle pemum P = 6 a ue. + DOI:.46/jmf Jounal of Mahemacal Fnance

12 A. Olando, G. Pake able ummaze he pofolo chaacec. he pemum P, ne of he nal chage (% of P), nveed no he fund. he benef ae pad monhly and a monhly chage m =.5 deduced. he guaaneed amoun obaned by accumulang he ne pemum a he guaaneed ae g. In ou example we conde hee dffeen guaanee ae:.5,. and.5. We aume ha he VA we offe wo ae allocaon o he polcyholde. he f one, called modeae, chaacezed by a lage popoon of ae nveed n bond. he econd one, mxed, chaacezed by equal popoon n bond and ock. hee ae allocaon chaaceze ome fund avalable n VA poduc old n Euope (among ohe he Accumulao old by AXA) Accodng o he modeae allocaon (ay Allocaon A) we aume ha he Company nve 7% n he Euo Bg ndex, 8% n he Euooxx ndex and % n he S & P ndex. Fo he mxed allocaon (ay Allocaon B) he fund allocaon : 5% n he Euo Bg ndex, he % n he Euooxx ndex and he % n he S & P ndex. Moeove we ue wo dffeen accumulaon funcon o calculae he expeced eun on nvemen n each fund, dependng on whehe he bake of ae conung he fund elf ebalanced o no. he Euo Bg ndex (Boad nvemen Euo BIG Bond ndex) povde a well known benchmak fo Euo Bond fxed-ncome pofolo. I cove all eco of he nvemen gade fxed ncome make ha ae acceble o nuonal nveo and accuaely meaue he pefomance and chaacec. he Euooxx ndex a boad, ye lqud, ube of he Soxx Euope 6 Index. Wh a vaable numbe of componen, he ndex epeen lage, md and mall capalzaon compane of Euozone coune: Aua, Belgum, Fnland, Fance, Gemany, Geece, Ieland, Ialy, Luxemboug, he Neheland, Pougal and Span. he S & P5 a fee-capalzaon-weghed ndex of he able. he lluave pofolo. Pemum P 6 Monhly chage m.5 Inal chage c %.5% Guaaneed ae Modeae allocaon Mxed allocaon g Allocaon A Allocaon B %.5% 7% EuoBIG, 8% Euooxx, % S & P5 5% EuoBIG, Euooxx, % S & P5 DOI:.46/jmf Jounal of Mahemacal Fnance

13 A. Olando, G. Pake pce of 5 lage-cap common ock acvely aded n he Uned Sae. he dcounng ae modelled ung he Ialan Rendao ha a weghed aveage yeld on a bake of Ialan govenmen ecue. Fo each ndex, we ue he daa fom June o Apl 6. he daa ouce ae: he yeld book of C fxed ncome ndce fo he EuoBIG ndex, Yahoo fnance fo he Euooxx and he S & P5 ndce and he hocal ee publhed by he Bank of Ialy, fo he Ialan Rendao. Fgue lluae he behavo of he hee condeed ndexe a well a he Ialan Rendao ee. Fo he moaly ae, we efe o a non paamec moaly able: IPS Emaon of he VAR Model Paamee he paamee emaon eul ae obaned ung he VAR package of he ofwae R. All he paamee ae emaed on a monhly ba. Befoe emang he paamee, we ubac he long em mean veco µ fom he daa. We conde he ample mean of each ee a an emao fo he long em mean veco µ. Le u f conde he cae of he wo-dmenonal VAR () model decbed n Secon.. he long em mean of he dcounng ae.8. able how he long em mean of he compoe ae of eun on ae n each cae: ebalancng fo allocaon A and B (column ) and no ebalancng fo allocaon A and B (column ). able and able 4 how he paamee emaon eul fo Allocaon A. We have a poce ha able nce he egenvalue of he Φ max ae le han one. hey ae(.999,.977) fo he ebalancng cae and (.999,.957) Fgue. EuoBIG, Euooxx, S & P 5 and Ialan Rendao ee. June -Apl 6. able. μ emaon. wo-dmenonal VAR (). Rebalancng No Rebalancng Allocaon A Allocaon B Allocaon A Allocaon B DOI:.46/jmf Jounal of Mahemacal Fnance

14 A. Olando, G. Pake n cae of no ebalancng. able 5 and able 6 how he paamee emaon eul fo Allocaon B. We have a poce ha aonay nce he egenvalue of he Φ max ae le han one. In h cae hey ae (.999,.9) fo he ebalancng cae and (.999,.9) n cae of no ebalancng. Le u now conde he ( n + ) -dmenonal VAR () model: n he pecfc example we ae condeng n =, heefoe we wll have a fou-dmenonal VAR () model. We ndcae by he Euobg ndex, by he Euooxx5 ndex, by he S & P ndex and by he Ialan Rendao (Fgue ). able. Φ max emaon. Allocaon A (modeae). wo-dmenonal VAR () model fo he connuou weghed aveage ae of eun n ebalancng and no ebalancng cae and Ialan Rendao ee. June -Apl 6. Rebalancng No Rebalancng able 4. Σ max emaon. Allocaon A (modeae). wo-dmenonal VAR () a model fo he connuou weghed aveage ae of eun n ebalancng and no ebalancng cae and Ialan Rendao ee. June -Apl 6. Rebalancng No Rebalancng.e e 8.78e 4.69e e 8 4.8e 8.69e 8 4.e 8 able 5. Φ max emaon. Allocaon B (mxed). wo-dmenonal VAR () model fo he connuou weghed aveage ae of eun n ebalancng and no ebalancng cae and Ialan Rendao ee. June -Apl 6. Rebalancng No Rebalancng able 6. Σ max emaon. Allocaon B (mxed). wo-dmenonal VAR () model a fo he connuou weghed aveage ae of eun n ebalancng and no ebalancng cae and Ialan Rendao ee. June -Apl 6. Rebalancng No Rebalancng 5.876e 4 5.9e e e 8 4.7e e 9 4.9e 8 DOI:.46/jmf Jounal of Mahemacal Fnance

15 A. Olando, G. Pake he long em mean on he hee ae ae: µ =.4, µ =., µ =.44. he emaon eul fo he Φ max and he covaance max of he edual ae hown n able 7 and able 8. he poce aonay nce he abolue value of he egenvalue of he Φ max, whch ae.999,.95,.5 and.964, ae le han one. Nevehele le u look a able 7. Each ow how he egeon paamee of he equaon decbng he evoluon n me of each vaable. Lookng a he f and econd ow, f of all we obeve a ong auoegeve dependence of he wo bond ndexe, and. Indeed he paamee explanng he evoluon baed on he own lag ae hgh:.9887 and.958, epecvely. he egeon paamee explanng he dependence on he ohe vaable how a weak dependence on ock ndexe. he hd and fouh ow of able 7 gve he emaed paamee of ock ndexe pocee. he egeon paamee explanng he dependence on he lag of bond ndexe ae:.7956 and 9.69 fo equaon and and fo he equaon. All hee obevaon ell u ha hee no a clea dependence of bond ndexe ( and ) on ock ndexe ( and ). Le u now focu on ock ndexe. he wo ndexe how a weak auoegeve dependence:.8769 and.58, epecvely. In ode o bee capue he muual dependence, we modelled he wo ndexe by a wo-dmenonal VAR () model. he emaed paamee ae hown n able 9. he auoegeve dependence of each ndex ll weak. Moeove he egenvalue ae low:. and.645. Baed on he afoemenoned obevaon, we conclude ha a VAR () queonable when neeed n decbng he behavo of ock ndexe. We decde o model he ock ndexe by a Geomec Bownan Moon model (GBM), he mo wdely ued model fo ock pce. A well known, he ock ndexe S follow a GBM f: ds = µ Sd+ σsdw (48) whee W a Bownan moon and μ and σ ae pove conan. h mple ha he nananeou ae of eun a Bownan Moon: ds = = µ + σd W. (49) S able 7. Φ max emaon. Fou-dmenonal VAR () model. Euobg, Euooxx 5, SP and Ialan Rendao ee. June -Apl DOI:.46/jmf Jounal of Mahemacal Fnance

16 A. Olando, G. Pake able 8. Σ max emaon. Fou-dmenonal VAR () model. Euobg, Euooxx5, a SP and Ialan Rendao ee. June -Apl 6..88e 8.89e 8.4e 7.7e 9.89e 8.99e 8.689e 7.9e 7 9.9e 8.459e 7.54e.85e.7e 9.9e 7.98e.75e able 9. Φ max emaon. wo-dmenonal VAR () model. Euooxx 5 and S & P ee. June -Apl he oluon we popoe n ode o nclude he GBM n he model, a modfed veon of he fou-dmenonal VAR () model. F of all, we emae agan he paamee of a wo-dmenonal VAR () model condeng only he bond ndexe, Euobg and Rendao. he Φ max emaon eul ae gven n able. Agan he poce aonay becaue he egenvalue of he Φ max ae.99 and.95. Le u now come back o he fou-dmenonal VAR () model. We eplace * he emaed Φ max n able 7, wh he Φ max of able. he * Φ max obaned by eng all he paamee conneced o he Euooxx and he S & P equal o zeo and by eng he ohe paamee equal o he one gven n able. Wh efeence o () he edual max obaned a follow: ( µ ) ( µ ) * * a = X Φ X (5) whee X gven by (). * I han poble o emae he covaance max of he edual Σ (ee a able ). In ode o compae he VAR () model and he popoed modfed veon, we an, mulaon fo value of,, and value wh aumng ha he evoluon n me decbed by he full [,,,] fou-dmenonal VAR () model (able 7 and able 8) and by he modfed veon of he ame model (able and able ). Fgue and Fgue how he evoluon ove me of he expeced value of he fou vaable n boh cae and fo dffeen eleced nal value. Fo each vaable and fo each, he expeced value ae compued a he mean of he, mulaed value. We obeve ha he wo pocee have he ame long em dynamc wh dffeen behavo only fo a few monh. DOI:.46/jmf Jounal of Mahemacal Fnance

17 A. Olando, G. Pake able. Φ max emaon. wo-dmenonal VAR () model. Euobg and Ialan Rendao ee. June -Apl able. Φ max emaon. Fou-dmenonal VAR () model. Euobg and Ialan Rendao ee. June -Apl * able. Σ a max emaon. Modfed fou-dmenonal VAR () model. Euobg, Euooxx5, SP and Ialan Rendao ee. June -Apl 6..7e 8.6e e e 8.6e 8.94e 8.667e 8.94e e 8.667e e 8.596e Fgue. Fou-dmenonal VAR () model. Expeced value wh dffeen nal value. Numbe of mulaon =,. DOI:.46/jmf Jounal of Mahemacal Fnance

18 A. Olando, G. Pake Fgue. Modfed fou-dmenonal VAR () model. Expeced value wh dffeen nal value. Numbe of mulaon =,. 5.. Some Reul In h econ, we efe o he lluave pofolo decbed n Secon 5.. We ae neeed n he expeced value and he condonal value a k a a choen confdence level of 95% of he pofolo a ncepon, fo a ac hedgng aegy (Secon 4.). he ofwae ued fo he followng analy Malab. We eo o a Mone Calo mulaon, aeng he quane of nee aco a eal-wold cenao e fo fnancal and moaly vaable. Hee ae he ep of he mulaon pocedue: ) Randomly geneae he monhly ae of eun and he monhly dcounng ae ung he eul of he emaed paamee; ) Randomly geneae he monhly deah pobable. Fo faconal age a Unfom dbuon of deah aumpon made: fo example he pobably of dyng n any monh of a gven yea he ame. ) Ae VP value (Equaon (47)); 4) Repea he pocedue N =, me; 5) Emae E [ VP ] by akng he ample mean of he N mulaed value of VP ; 6) Emae he condonal value a k a a 95% confdence level fom he empcal dbuon of VP. A well known, he CVaR ( α )% he aveage of he ( α )% wo cenao. In ou example we chooe α = 95% and he CVaR 95% he aveage of he 5% lowe value of VP. he mulaon pocedue epeaed fo each aumpon abou he fnancal vaable pevouly decbed, he wo and fou-dmenonal VAR () model, boh wh ebalancng of he bake of ae and whou ebalancng. DOI:.46/jmf Jounal of Mahemacal Fnance

19 A. Olando, G. Pake We hen compae he eul wh he choce of no hedge, ha he cae n whch he nue decde no o buy he hedge pofolo of pu opon. able how he co of he hedgng pofolo n each cae. he pofolo pce ha been deemned n a k neual famewok on he ba of he eul obaned n Secon 4. and 5.. F of all we obeve ha n each cae he co lowe han he nal chage pad by he nue, ha C P c, ha n ou lluave pofolo amoun o 8, euo. A he guaaneed ae ge hghe, he pu opon pce nceae and he co of he ene pofolo nceae, oo. In geneal, he co of he hedgng pofolo hghe fo he ebalancng aegy han he no ebalancng one, a expeced. Indeed, he emaed volale of he undelyng ae hghe n he ebalancng cae, fo each condeed opon. he ame happen gong fom Allocaon A o Allocaon B. he econd aegy moe ky becaue chaacezed by a hghe popoon of ae nveed n ock and heefoe a hghe volaly. able 4-7 compae he eul fo ac hedgng and hoe fo when he Company decde no o hedge a all. F of all, we obeve ha n each condeed cae he expeced value of he pofolo E [ VP ] hghe f we hf fom no hedgng o ac hedgng. h expeced becaue he no hedgng cae doe no adap o moaly expeence ove me, a ac hedgng doe o ome exen. Moeove we obeve ha Allocaon B gve lowe value of E [ VP ] and ha able. Co of he hedgng Pofolo. g Allocaon A Allocaon B D VAR () Rebalancng.4%,95 4,.5% 7,4 54,97 % 5,4,5 g Allocaon A Allocaon B D VAR () No Rebalancng.4% 8, 6,8.5%, 45,48 % 4,8 9,78 g Allocaon A Allocaon B 4D VAR () Rebalancng.4%,99 4,8.5% 6,9 5,9 % 5,56 7,4 g Allocaon A Allocaon B 4D VAR ()No Rebalancng.4% 7,5 4,.5% 9,88 4,48 % 9, 87, DOI:.46/jmf Jounal of Mahemacal Fnance

20 A. Olando, G. Pake able 4. Smulaon eul fo VP : DVAR (), Rebalancng cae, N =,, α = 5%. NO HEDGE E[ VP ] CVaRVP ( α ) Allocaon A 6,8 99,4 g =.4% g =.5% g = % Allocaon B 6,78,8, SAIC HEDGING Allocaon A 6,57 8,9 Allocaon B 55,5 4,55 NO HEDGE Allocaon A 9,5,7,9 Allocaon B 5,6,867, SAIC HEDGING Allocaon A 57,9 77,8 Allocaon B 47, 6,86 NO HEDGE Allocaon A 9,65,7,4 Allocaon B 89,,7, SAIC HEDGING Allocaon A,78 5,6 Allocaon B 9,76 6,55 able 5. Smulaon eul fo VP : DVAR (), No Rebalancng cae, N =,, α = 5%. NO HEDGE E[ VP ] CVaRVP ( α ) Allocaon A 4,6 87,6 g =.4% Allocaon B 7,8,65, SAIC HEDGING Allocaon A 7,96,8 Allocaon B 67, 46,94 NO HEDGE Allocaon A 86, 856,6 g =.5% Allocaon B 9,58,689,8 SAIC HEDGING Allocaon A 7,59 98,99 Allocaon B 56,9 7, NO HEDGE Allocaon A,7,76, g = % Allocaon B,49,94,8 SAIC HEDGING Allocaon A 5,7 78,5 Allocaon B,7 9,46 DOI:.46/jmf Jounal of Mahemacal Fnance

21 A. Olando, G. Pake able 6. Smulaon eul fo VP : 4DVAR (), Rebalancng cae, N =,, α = 5%. NO HEDGE E[ VP ] CVaRVP ( α ) Allocaon A 9,,,9 g =.4% g =.5% g = % Allocaon B 5,6,8,4 SAIC HEDGING Allocaon A 48,6 8,8 Allocaon B 45,9 6,7 NO HEDGE Allocaon A 7,4,4, Allocaon B 59,75,885, SAIC HEDGING Allocaon A 44,54 77, Allocaon B 5,6 5,9 NO HEDGE Allocaon A 68,5,46,7 Allocaon B 56,,9,6 SAIC HEDGING Allocaon A 9,7 5, Allocaon B 8, 7,8 able 7. Smulaon eul fo VP : 4DVAR (), No Rebalancng cae, N =,, α = 5%. NO HEDGE E[ VP ] CVaRVP ( α ) Saegy A 69, 78,4 g =.4% SaegyB 88,76,94,8 SAIC HEDGING Saegy A 6,8 6,9 Saegy B 7,86 7,99 NO HEDGE Saegy A,7 8,5 g =.5% g = % Saegy B 4,89,46,4 SAIC HEDGING Saegy A 6,86 4,6 Saegy B 6, 6, NO HEDGE Saegy A 6,68,9, Saegy B 84,7,69,4 SAIC HEDGING Saegy A 4,74 84, Saegy B 7,66 6,9 DOI:.46/jmf Jounal of Mahemacal Fnance

22 A. Olando, G. Pake hold fo each condeed cae. Indeed Allocaon B moe ky, heefoe we ge hghe expeced lable. In pacula, n he no hedge cae, Allocaon B alway lead o a lo. Regadng Allocaon A, even f we ge ome pove value, hey ae lowe han he nal um of 8, euo. Lookng a ac hedgng, E [ VP ] alway lowe f allocaon B choen, bu we neve ge negave value. h woenng due o he afoemenoned hghe co of he hedgng pofolo (able ). Movng fom he cae of ebalancng he bake of ae o no ebalancng, he eul mpove n each cae. Refeng o ac hedgng aegy, h a conequence of he hghe co of he hedgng pofolo (able ) due o a hghe level of he undelyng volaly n he ebalancng cae. Moeove he hghe volaly of he undelyng allow fo hghe exeme eun and conequenly hghe lable. h effec obevable fo he no hedgng cae a well. Fnally le u look a he eul obaned by mean of wo-dmenonal VAR () and fou-dmenonal VAR () model (D VAR () and 4D VAR (), epecvely). Fo he ac hedgng cae, D VAR () gve lghly hghe value han 4D VAR boh n cae of ebalancng o no. Even f he co of he hedgng pofolo lowe n he 4D VAR () cae, we can obeve ha n D he ae volale ae aveaged ou n he emao bu n 4D hey ae emaed epaaely. h allow fo moe exeme eun n he 4D model and conequenly hghe lable. h effec onge when no hedgng. Lookng a Condonal Value a Rk value, we obeve he ame behavou a he expeced value, of coue. In pacula, n he no hedge cae he dffeence beween he condonal value a k and he expeced value vey hgh. h due o he hghe k aocaed wh h choce. Moeove, fo boh he no hedge and he ac hedge cae, we obeve a woenng n em of condonal value a k, when changng fom allocaon A o he moe ky allocaon B. 6. Concludng Remak h pape ude, fom he nue pon of vew, a VA poduc offeng GMDB and GMMB guaanee. In pacula, he fnancal poon of an nue ung a homogeneou pofolo of VA offeng hoe guaanee, aeed. o h am, he expeced value and he condonal value a k of h pofolo ae quanfed n a eal wold cenao. he eul unde a ac hedgng aegy ae compaed wh hoe obaned when no hedgng. Sac hedgng, a expeced, gve bee eul. Boh n ac hedgng and no hedgng cae, he pofolo value depend on many faco and we nvegae ome of hem. Regadng he choce offeed o he polcyholde, we look a wo dffeen ae allocaon and, a expeced, he pofolo value deceae a he ae allocaon become moe ky. DOI:.46/jmf Jounal of Mahemacal Fnance

23 A. Olando, G. Pake A cucal apec he choce of a model fo he ae eun and he dcounng ae. h choce mpac he valuaon of he pofolo fo k managemen pupoe. VA guaanee ae lnked o moe ha one ae and he ue of ngle-ae (unvaae) model fo VA pcng and k managemen pupoe, may lead o eul ha ae no elable. We how he eul obaned by modellng jonly he fnancal vaable nvolved by mean of wo knd of VAR () model. We obeve ha ung a fou-dmenonal VAR () model gve lowe value of he pofolo han a wo-dmenonal VAR () model. In he f cae, he ae eun on each fund ae combned n a ngle me ee whle n he fou-dmenonal VAR () model, he ae eun on each fund ae modelled epaaely. h ucue add moe nfomaon n modellng he fnancal vaable and he eul could be moe elable. h povde ueful nfomaon o he nue concened abou k managemen. We alo conde wo dffeen accumulaon funcon dependng on whehe he bake of ae ebalanced o no. he ebalancng cae gve lowe value of he pofolo. Followng he analy pefomed n h pape, fuhe wok could be done. h may nclude, fo nance, he mplemenaon of a dynamc hedgng aegy and he ue of ochac moaly model. Refeence [] Ledle, M.C., Coy, D.P., Fnkelen, G.S., Rche, A.J., Su, K. and Wlon, D.C.E. (8) Vaable Annue. Bh Acuaal Jounal, 4, hp://do.og/.7/s [] Kalbee,. and Ravndan, K. (9) Vaable Annue. A Global Pepecve. Rk Book. [] Baue, D., Klng, A. and Ru, J. (8) A Unveal Pcng Famewok fo Guaaneed Mnmum Benef n Vaable Annue. An Bullen, 8, hp://do.og/.7/s5565 [4] Bacnello, A.R., Mlloovch, P., Olve, A. and Pacco, E. () Vaable Annue: Rk Idenfcaon and Rk Aemen. CAREFIN Reeach Pape 4/, -48. hp://n.com/abac=79966 [5] Feng, R. and Volkme, H.W. () Analycal Calculaon of Rk Meaue fo Vaable Annuy Guaaneed Benef. Inuance: Mahemac and Economc, 5, hp://do.og/.6/j.nmaheco..9.7 [6] L, J.S.-H. and Ng, A.C.-Y. () Pcng and Hedgng Vaable annuy guaanee wh mulae ochac nvemen model. Noh Amecan Acuaal Jounal, 7, 4-6. hp://do.og/.8/ [7] Zvo, E. and Wang, J. (6) Veco Auoegeve Model fo Mulvaae me See. In: Modellng Fnancal me See wh S-PLUS, Spnge Scence, [8] Hady, M. () Invemen Guaanee: Modelng and Rk Managemen fo Equy Lnked Lfe Inuance. John Wley & Son, Hoboken, NJ. [9] Mlevky, M.A. and Pone, S.E. (998) A Cloed-Fom Appoxmaon fo Valung Bake Opon. he jounal of Devave, 5, hp://do.og/.95/jod DOI:.46/jmf Jounal of Mahemacal Fnance