Closed-form solutions for Guaranteed Minimum Accumulation Benefits

Transcription

1 Closd-form solutions for Guarantd Minimum Accumulation Bnfits Mikhail Krayzlr, Rudi Zagst and Brnhard Brunnr Abstract Guarantd Minimum Accumulation Bnfit GMAB is on of th variabl annuity products, i.. nw typ of insuranc and rtirmnt products offring participation at th financial markts and th guarants at th sam tim. In particular, GMAB offrs at maturity th maximum of account valu and som guarantd bnfit. This papr considrs four most popular options for this guarantd amount: capital protction, minimum intrst rturn, ratcht and th maximum of thm. As ths products ar xposd to diffrnt risk factors, a multi-factor modl is rquird. Howvr, thr is oftn a tradoff btwn alistic modl and analytical tractability. Thus, som authors simplify th contract stup or / and th modling framwork to driv closd-form solutions, othrs us diffrnt numrical mthods to pric th guarants. This work fills this gap by providing analytical formulas for GMAB with common ridrs in a hybrid modl for actuarial and financial risks. Ky Words: Variabl Annuitis, Guarantd Minimum Accumulation Bnfit, Hybrid Modl. Chair of Mathmatical Financ, Tchnisch Univrsität Münchn, Parkring 11, Garching Hochbrück, Grmany risklab GmbH, Sidlstr 24, 8335 Munich, Grmany Corrsponding author: Mikhail Krayzlr, krayzlr@tum.d

2 1 INTRODUCTION 1 Introduction Variabl Annuitis VA can b dfind as dfrrd, fund-linkd annuity products that ar frquntly offrd with additional guarants including diffrnt living and dath bnfits. Ovr th yars, th guarants providd on VA products hav volvd as th markt has adaptd to mt customr nds. Dpnding on th bnfit typ diffrnt GMxBs can now b sn on th markt. Som of th most common xampls includ: Guarantd Minimum Dath Bnfit GMDB, Guarantd Minimum Accumulation Bnfit GMAB, Guarantd Minimum Withdrawal Bnfit GMWB, Guarantd Minimum Incom Bnfit GMIB. All of ths products might also diffr in th way th guarantd amount is dtrmind. In som products only initial prmiums ar guarantd, othrs guarant all prmiums paid plus accumulatd intrst roll-up or includ th so-calld ratcht options that allow to rais th guarantd amount dpnding on th undrlying fund prformanc. W rfr an intrstd radr to Brunnr t al. 29] for mor information on diffrnt typs of guarants. Ovr th last dcad, variabl annuitis hav bn a major succss story in th North Amrican insuranc markt. Thy hav vn ovrtakn traditional fixd annuitis to bcom th primary form of protctd invstmnt 1. This succss has bn rpatd in Japan whr sinc 21 th VA markt has grown from $1.3 billion up to $216 billion in March 211 in trms of th VA assts undr managmnt 2. In th rcnt yars a similar trnd was obsrvd on th Europan markt. At th sam tim thr appard svral paprs in th acadmic litratur ddicatd to th analysis of ths products. Starting from th pionring work by Brnnan and Schwartz 1976] and thn by Boyl and Schwartz 1977] on quitylinkd lif insuranc contracts, svral xtnsions wr proposd for pricing of variabl annuitis and othr fund- or quity-linkd insuranc products. Exampls of studis in th acadmic litratur daling with pricing issus of spcific GMxBs products includ: Milvsky and Posnr 21], Ulm 28] for GMDBs; Milvsky and Salisbury 22] for GMDB products allowing policy surrndrs by th insurd prson; Marshall t al. 21] for GMIB undr stochastic intrst rats; Milvsky and Salisbury 26], Chn t al. 28], Dai t al. 28] for GMWB options undr th assumption of optimal policyholdr bhavior. 1 $159 billion of variabl annuity sals compard to $81 billion of fixd annuity sals in th U.S. in 211 according to th Lif Insuranc Markting and Rsarch Association LIMRA. 2 Sourc: Milliman, Lif Insuranc Association of Japan. 2

3 2 GUARANTEED MINIMUM ACCUMULATION BENEFIT Anothr milston was st by Baur t al. 28] who providd a gnral framwork for simultanous and consistnt pricing of a larg varity of VA guarants. For simplification purposs th authors stayd in th Black-Schols world, dscribing th volution of th financial markt with a gomtric Brownian motion. Furthr contributions includ th work of Bacinllo t al. 21] and Bacinllo t al. 211], whr a unifying framwork undr mor gnral assumptions for numrical valuation of VA is suggstd. Most of ths acadmic paprs can b dividd into two catgoris, thos intrstd in finding som analytical approximation and thus simplifying th modls usd or th products thmslvs, and th othr part, which ar focusd on th numrical solutions within a mor comprhnsiv and ralistic pricing framwork. This work aims to bridg this gap and provids a hybrid modl Hull-Whit-Black- Schols with tim-dpndnt volatility and stochastic mortality within a gnral stup for pricing of guarants includd in VAs. W focus on th Guarantd Minimum Accumulation Bnfit and xmplarily driv closd-form pricing formulas for this guarant in th prsntd hybrid modl. Spcial attntion will b ddicatd to th modl calibration which is oftn nglctd in th acadmic litratur on VA pricing. Last but not last, w analyz th impact of diffrnt risk factors on th VA prics. Th rmaindr of th papr is structurd as follows. In Sction 2, th gnral structur of a GMAB is xplaind and th rquirmnts on th pricing modl ar discussd. Sction 3 dscribs th valuation framwork, including financial and insuranc markt modls. In Sction 4, xplicit xprssions for GMABs with diffrnt ridrs ar drivd in th prsntd framwork. Thraftr, Sction 5 discusss th data usd in this study and th corrsponding calibration approach. Numrical rsults and snsitivity analysis ar prsntd in Sction 6. Finally, w conclud th papr with a short summary in Sction 7. 2 Guarantd Minimum Accumulation Bnfit Guarantd Minimum Accumulation Bnfit GMAB is a typ of variabl annuity that givs th policyholdr at th rtirmnt dat T th maximum from th account valu AT and th guarantd amount GT 3. Th valu of th GMAB at 3 Most of th products combin this guarant with som paymnt at th tim of dath of th policyholdr. Th corrsponding xtnsion of our modl is in th focus of currnt and futur rsarch. 3

4 3 VALUATION MODEL maturity T, conditiond on th policyholdr survival until tim T, can b writtn as: V T = 1 {τ>t } maxat, GT, whr τ can b intrprtd as th tim of th policyholdr s dath. Th following options ar common on th markt for th guarantd amount GT : Rturn of prmium GT = P Roll-up guarant GT = P δt with som prdfind roll-up rat δ Ratcht guarant GT = max At i <t 1 <...<t n=t Maximum of ratcht and roll-up GT = max max At i, P δt <t 1 <...<t n=t Hr P dnots th initial prmium of th contract. It should b mntiond that th first cas is a spcial form of th scond on for δ =. Thus, in Sction 4, w will concntrat on th roll-up, ratcht and th combination of thm. As it can b sn from th xprssions abov th pric of GMAB is snsitiv to th mortality intnsity and quity rturns. Morovr, as most of th VAs offrd on th markt ar vry long-datd products, and thus vry snsitiv to th changs in intrst rats, w nd to introduc stochastic intrst rats in our modling framwork. In th nxt sction, a hybrid modl for ths thr typs of risk is prsntd.. 3 Valuation modl Th modling framwork usd in this papr is similar to th stup proposd in Marshall t al. 21] for GMIBs. In this papr, w xtnd thir modl by introducing tim-dpndnt quity volatility as wll as stochastic mortality, indpndnt from th financial markt. Morovr, in th nxt sction w driv closd-form xprssions for som of th guarants that ar oftn pricd via Mont-Carlo simulation. 3.1 Financial markt modl Lt Ω, F, F, IP b a filtrd probability spac satisfying th usual conditions of compltnss and right-continuity of th filtration F = F t t with F = {, Ω}. 4

5 3 VALUATION MODEL 3.1 Financial markt modl W assum th xistnc of an adaptd short-rat procss r, such that th monymarkt account Bt, qual to t Bt = xp rsds, rprsnts th amount of mony at tim t rsulting from isk-fr invstmnt of on unit at tim. In th absnc of arbitrag, an quivalnt martingal or risk-nutral masur Q as opposd to th ral-world masur P xists. Undr this masur th gain from holding a scurity S aftr discounting with th monymarkt account is a Q -martingal s.g. Bilcki and Rutkowski 24]: St = IE Q t rsds ST F t ]. W now formulat our modl to dscrib th volution of th financial markt undr th risk-nutral masur. W call this hybrid modl th Hull-Whit- Black-Schols tim-dpndnt volatility HWBS tdv modl, which combins th famous Black-Schols modl for quity prics Black and Schols 1973], xtndd by tim-dpndnt volatility, with th 1-factor Hull-Whit intrst-rat modl as in Hull and Whit 1993]. Hnc, w analyz th following dynamics: whr rt = φt + xt, r = r, dxt = xtdt + σ r dw Q r t, x =, dst = rtstdt + σ S tstdw Q S t, S = S, dw Q S tdw Q r t = ρ Sr dt, 1 φt = f, t + σ2 r t 2 2a 2 r is a dtrministic function usd to fit th initial trm-structur of intrst rats P,t with f, t = dnoting th markt instantanous forward rat at tim t with maturity t s Brigo t al. 26] for mor dtails., σ r ar two positiv constant paramtrs, man-rvrsion lvl and volatility, of th intrst-rat modl. Tim-dpndnt quity volatility is dnotd by σ S t and instantanous corrlation btwn quitis and intrst rats by ρ Sr. Dnoting th log-pric of a scurity S as Y = lns, rsults in th affin procss for Y : dy t = rt 1 2 σ2 St dt + σ S tdw Q S t. 2 5

6 3.2 Insuranc markt modl 3 VALUATION MODEL 3.2 Insuranc markt modl W ar working on th sam filtrd probability spac Ω, F, IF, IP and focus on a x-yar-old prson at tim t =. W dnot th maximum attainabl ag as T and modl th random liftim of an insurd prson as andom variabl τ x, which corrsponds to th first jump of a counting procss N x+t t with andom intnsity λ x+t t. Formally w hav τ x := min T, inf{t, T ] : N x+t t > }. W introduc two subfiltrations of IF : G = G t t by th corrsponding σ -algbras: and H = H t t, gnratd G t = σλ x+s s : s t, H t = σ1 {τx s} : s t. In othr words G rprsnts th volution of th random intnsity associatd with th counting procss N x+t t. It, howvr, dos not includ th information whthr th policyholdr has did by thn, i.. th first jump of th procss N x+t t. This information is givn by filtration H. To improv analytical tractability, w furthr assum that th counting procss N x+t t is a doubl stochastic or Cox procss drivn by th filtration G, with G prdictabl intnsity λ x+t t. This mans that givn any particular trajctory of λ x+t t, th counting procss N x+t t bcoms a conditional Poisson procss with th hazard procss Γ x+t t = t λ x+ssds as dscribd in Bilcki and Rutkowski 24]. Consquntly, IP N x+t T N x+t t = k F t G T = Γ x+t T Γ x+t t k = k! λ x+s sds t k! k Γ x+t T Γ x+t t T λ x+s sds t. 3 Hnc, givn k = and N x+t t =, IPN x+t T = F t G T is th probability that a prson who is aliv a tim t survivs at last up to tim T givn th information on th mortality dvlopmnt up to tim T, whr w do not know if h or sh is still aliv at tim T. Dfinition Survival probability. p x+t t, T F t dnots th probability masurd at tim t that a prson agd x+t at tim t survivs at last up to tim T: p x+t t, T F t := IPτ x > T F t. 4 6

7 3 VALUATION MODEL 3.2 Insuranc markt modl p x+t t, T F t is calld th survival probability. Lmma 3.1. For th survival probability masurd at tim t of a prson who is aliv at tim t it holds that ] T p x+t t, T F t = IE λ x+s sds t Ft. Proof. W us quations 3 and 8 as wll as th law of itratd xpctations and th fact that N x+t t = to prov th lmma: p x+t t, T F t = IPτ x > T F t = IE 1 {τx>t } F t ] = IE IE 1 {τx>t } G T F t ] Ft ] = IE IPτx > T G T F t F t ] = IE IPN x+t T = G T F t F t ] = IE IPN x+t T N x+t t = G T F t F t ] ] T = IE λ x+s sds t Ft. To dscrib th volution of th mortality intnsity λ w propos a two stp approach basd on th work of Dahl 24] and Dahl t al. 26]. First, th initial mortality intnsity λ x+t is paramtrizd via th standard Gomprtz modl s Bowrs t al. 1997] for som gnral dtails or Milvsky and Posnr 21] for its application to th pricing of variabl annuitis: λ x+t = 1 b x+t m b, 5 whr λ x+t dnots th initial mortality intnsity of a prson agd x + t at tim and m, b ar som constant paramtrs that can b stimatd from th corrsponding mortality tabl. Dspit th broad us of th Gomprtz modl in th actuarial litratur, this modl dos not account for potntial improvmnts and fluctuations in mortality rats. That is why in th scond stp th so-calld mortality improvmnt ratio ξ x+t t is introducd as th ratio of th mortality intnsity of a prson agd x + t at tim t and th mortality intnsity of a prson at th ag of x + t at tim : ξ x+t t = λ x+tt λ x+t. 7

8 3.2 Insuranc markt modl 3 VALUATION MODEL Figur 1 shows th historical dvlopmnt of ξ x+t t for mals and fmals from 1968 to W obsrv som fluctuations around th dclining tim-dpndnt Figur 1: Mortality improvmnt ratio drift that rflcts dcrasing mortality for th sam ag x + t ovr tim. W dscrib th dynamics of this mortality improvmnt ratio via xtndd Vasick modl: dξ x+t t = k ξ γ ξ t ξ x+t t dt + σ ξ dw ξ t, 6 whr k ξ is th man rvrting spd, γ ξt tim-dpndnt drift and σ ξ is th volatility of th mortality improvmnt ratio. Thus, to calculat th mortality intnsity at tim t of a x + t -yar old policyholdr, i.. λ x+t t, w just nd to multiply th initial mortality intnsity λ x+t with th corrsponding improvmnt ratio at tim t, ξ x+t t, i.. λ x+t t = λ x+t ξ x+t t. Th corrsponding implications on th mortality intnsity ar summarizd in th following thorm. Thorm 3.2 Mortality intnsity. Th dynamics of th mortality intnsity λ x+t t is dscribd by th following stochastic diffrntial quation: dλ x+t t = c 1 c 2t c 3 λ x+t t dt + c 4 c 5t dw ξ t 7 with c 1 := k ξ b x m b, c 2 := 1 b γ ξ, c 3 := k ξ 1 b, c 4 := σ ξ b x m b, and c 5 := 1 b. 4 Basd on th JP Morgan LifMtrics Historic Indx Data. 8

9 3 VALUATION MODEL 3.2 Insuranc markt modl Th mortality intnsity at tim T, λ x+t T, is normally distributd with man µ λt and varianc σ 2 λt givn by th following quations: µ λt = IEλ x+t T F ] = λ x c 3T + c 1 c 2 + c 3 c 2 T c 3T, σ 2 λt = V arλ x+t T F ] = c 2 4 2c 5 + 2c 3 2c 5 T 2c 3T. W rfr to Appndix 8.1 for th proof of th thorm. Th main drawback of th prsntd modl for stochastic intnsity is that it can bcom ngativ with positiv probability. This probability can b calculatd analytically as λx+t T µ λt IPλ x+t T < = IP < µ λt = Φ µ λt, σ λt σ λt σ λt whr Φ is th cumulativ distribution function of th standard normal distribution. Howvr, it should b mntiond that in practical applications this probability is usually ngligibl 5. Du to analytical tractability of affin modls s.g. Duffi t al. 2], th following closd-form formulas for th survival probabilitis can b drivd: Thorm 3.3 Affin mortality structur. In th affin dynamics dscribd in 7 th survival probabilitis can b xprssd as ] T p x+t t, T F t = IE λ x+s sds t Ft = C λt,t D λ t,t λ x+t t, 8 with C λ t, T and D λ t, T givn by D λ t, T = c 3T t c 3 C λ t, T = c 1 c 2 t c c 2T 1 c 2 t c 3 T t c 2T c 2 c 3 c 3 c 2 + c 3 + c 2 4 2c 5 t 2c 5T c c 5 t c 3 T t 2c 5T 4c 2 3c 5 c 2 3 2c 5 + c 3 + c 2 4 2c 5 t 2c 3 T t 2c 5T. 9 4c 2 3c 3 + c 5 W rfr to Appndix 8.2 for th dtaild proof of th thorm. 5 For th paramtrs usd in our xampl s Sctions 5 and 6 this probability is lss than

10 4 PRICING OF GMAB 4 Pricing of GMAB W considr a GMAB contract issud at tim for th singl prmium P. W assum T to b th nd of th accumulation priod and th account valu At at tim t to b 1% invstd in th undrlying quity fund St : dat = At dst, A = P. St As it has bn xplaind in Sction 2, at th nd of th accumulation priod th policyholdr rcivs th maximum of th account valu AT and th guarantd amount GT. In our drivation procdur w will xtnsivly us masur chang tchniqus, spcially th cas whn quity is usd as a numrair. Thus, w nd to rwrit our financial modl undr th quity masur and driv all rquird momnts undr this masur. 4.1 Chang of numrair W dnot by Q S th T -pricing masur rlvant to ST which is also known as th quity masur. It is dfind by th following Radon-Nikodym drivativ s.g. Gman t al. 1996]: dq S dq ST B = SBT = 1 2 σ2 S udu+ σ SudW Q S u. Undr this nw masur two Brownian motions ar givn by dwr QS = dwr Q ρ Sr σ S tdt dw QS S = dw Q S σ Stdt. Substituting ths xprssions in 1 w rciv th dynamics of x and Y undr th masur Q S : dxt = xt + σ r σ S tρ Sr dt + σ r dwr QS t dy t = rt σ2 St dt + σ S tdw QS S t. Intgrating both ths quations from to T and using th fact that x = w gt, conditiond on F : xt = ρ Sr σ r art u σ S udu + σ r 1 art u dw QS r u 1

11 4 PRICING OF GMAB 4.1 Chang of numrair and Y T = σ r + + ρ Srσ r T u dwr QS u φudu whr φu is givn by quation 2. σ 2 Sudu + σ S udw QS S u T u σ S udu, 11 Thorm 4.1 Distribution undr Q S. In th Black-Schols-Hull-Whit framwork with tim-dpndnt volatility undr quity masur Q S it holds that both rt and Y T ar conditional on F normally distributd with th corrsponding momnts µ rt, σrt 2 and µ Y T, σy 2 T and corrlatd with th corrlation cofficint ρ Y T rt : µ rt = IErT F ] = φt + ρ Sr σ r art u σ S udu σ 2 rt = V arrt F ] = σ2 r 2 2T µ Y T = IEY T F ] = ρ Srσ r + σ2 r 2a 2 r σ 2 Sudu T u σ S udu ln P M, T T 2 T T σy 2 T = V ary T F ] = σ2 r T 2 T + 1 2T 2 + ρ Y T rt = a 2 r σ 2 Sudu + 2 ρsrσ r CovY T rt σ Y T σ rt CovY T, rt = σ r ρ Sr σ S u art u du + σ2 r 1 T 1 2 2T 11 T u σ S udu.

12 4.1 Chang of numrair 4 PRICING OF GMAB Proof. Th fact that rt and Y T ar normally distributd follows dirctly from 1 and 11. Taking th xpctation in 1 and rcalling that rt = xt +φt w immdiatly rciv xprssion for µ rt. Using th Ito isomtry s,.g., Zagst 22], p.24 w can also calculat th varianc: V arrt F ] = σrv 2 ar = σ 2 r art u dw QS r 2arT u du = σ2 r 2 2T. For th xpctd valu of Y T w nd to valuat th intgral ovr th function φu. Using formula 2 and th fact that w rciv: φudu = ln P, T + P, T = f,sds, = ln P, T + σ2 r 2a 2 r = ln P, T + σ2 r 2a 2 r σ 2 r 2a 2 r ] u 2 du 1 2 u + 2aru du T 2 T T Substituting this xprssion in 11 and taking corrsponding xpctations lads to th final xprssion for th first momnt, µ Y T. For th scond momnt w can writ V ary T F ] = σ2 r V ar a 2 r + V ar T u dwr QS u σ S udw QS S u ] + 2 σr Cov T u dwr QS u ]. ]. σ S udw QS S u 12

13 4 PRICING OF GMAB 4.1 Chang of numrair Using Ito isomtry, w gt V ary T F ] = σ2 r + a 2 r = σ2 r a 2 r + T u 2 du σ 2 Sudu + 2ρ Sr σ r σ S u art u du T 1 T T σ 2 Sudu + 2 ρsrσ r T u σ S udu. Furthrmor, w obtain for th covarianc btwn intrst rats and quity rturns CovY T, rt = IE Q σ r + IE Q σ 2 r art u dwr QS u art u dwr QS u = σ r ρ Sr art u σ S udu + σ2 r T u 2arT u du = σ r ρ Sr σ S u art u du + σ2 r 1 T 1 2 2T σ S udw QS S u ] T u dwr QS u. ] Substitution of th constant σ S in th formulas from th Thorm 4.1 and straightforward intgration lad to th following rsult. Corollary 1. For th HWBS modl with constant quity volatility σ S u = σ S holds for th momnts of quity rturns and th covarianc Cov Y T, rt µ Y T = 1 2 σ2 ST + ρ Srσ r σ S T 1ar T ln P M, T + σ2 r σ 2 Y T = σ 2 ST + σ2 r a 2 r + 2 ρsrσ r σ S 2a 2 r T 2 T + 1 2T, 2 T 2 T + 1 T 1 art 2 2T it 13

14 4.2 Explicit xprssion for th GMAB 4 PRICING OF GMAB and CovY T, rt = σ rρ Sr σ S T + σ2 r 1 T 1 2 2T 4.2 Explicit xprssion for th GMAB Th tim t = fair valu of th GMAB can b found as an xpctd valu of th discountd futur payoff, takn undr th risk-nutral masur Q, i.. V = IE Q ] T rsds 1 {τ>t } maxat, GT. 12 Th indicator function 1 {τ>t } in th xprssion abov assurs that th policyholdr is aliv at th rtirmnt tim T. Morovr, w will assum, as it has bn mntiond bfor, that A = P. In th following w considr diffrnt options for th guarantd amount GT as givn in Sction Roll-up guarant For th roll-up, th guarantd amount GT at maturity T, as mntiond in th Sction 2, is givn by GT = P δt, whr δ is th guarantd roll-up rat. For δ = th guarant corrsponds to th rturn of prmium cas, i.. GT = P. Th fair valu of th roll-up guarant in th closd form is givn by th following thorm: Thorm 4.2. µy T δt V = Ap x, T Φ σ Y T ] δt σ 2 + δt Y T µ Y T Φ µ Y T σ2 Y T, σ Y T ar th corrsponding momnts of Y T undr th quity ma- whr µ Y T, σ Y T sur Q S. 14

15 4 PRICING OF GMAB 4.2 Explicit xprssion for th GMAB Proof. Using th law of itratd xpctations and xploiting th doubl stochastic proprty, w obtain V = IE Q max Y T, δt A rsds 1 {τ>t } ] = IE Q IE Q max Y T, δt A rsds 1 {τ>t } G T ]] = IE Q max Y T, δt A rsds IE Q 1{τ>T } G T ] ] = IE Q max Y T, δt A rs+λ x+ssds Using th indpndnc btwn insuranc and financial markts and th rsults from th prvious chaptr, w gt V = IE Q max Y T, δt A ] T rsds IE Q ] T λ x+ssds = IE Q max Y T, δt A rsds ] p x, T. Th last xprssion can b rwrittn as V = p x, T IE Q A Y T 1 {Y T δt } + A δt 1 {Y T δt } rsds ]. Changing to th Q S masur, w obtain ] V = Ap x, T IE Q S 1{Y T δt } + δt Y T 1 {Y T δt } = Ap x, T ] Q S Y T δt + δt IE Q S Y T 1 Y T δt. Straightforward calculations using th Thorm 4.1 giv µy T δt V = Ap x, T Φ σ Y T ]. ] δt σ 2 + δt Y T µ Y T Φ µ Y T σ2 Y T. σ Y T Th corrsponding momnts undr th quity masur Q S ar givn in th Thorm 4.1. Thus, w drivd closd-form xprssion for th pric of a GMAB with rollup guarant in th Hull-Whit-Black-Schols modl with tim-dpndnt volatility. 15

16 4.2 Explicit xprssion for th GMAB 4 PRICING OF GMAB Ratcht guarant Th guarantd amount GT for th ratcht product is dfind s Sction 2 as GT = max At i. <t 1 <...<t n=t Explicit closd-form xprssion for th GMAB with this typ of guarant is givn by th following thorm: Thorm 4.3. V = Ap x, T Φ n 1 ; µ k Y, Σ k Y whr + Ap x, T n 1 k=1 σ Φ n 1 ; µ k Y Σ k Y k, Σ k Y µ 2 n,k Y + n,k Y 2 i,k Y := {Y t k Y t i } i {1,...,n}\{k}, t n := T k Y := { i,k Y } i {1,...,n}\{k}, µ k Y := {µ i,k Y } i {1,...,n}\{k}, Σ k Y := {Σ i,k Y, j,k Y } i,j {1,...,n}\{k}, k is a unit vctor with a k -th lmnt qual to 1 and Φ n 1 u, µ, Σ - is a distribution function of a multivariat normal distribution with man vctor µ and covarianc matrix Σ undr th quity masur Q S., Proof. Similar to th prvious cas w sparat insuranc and financial parts and rwrit th xprssion undr th xpctation ] V = IE Q 1 τ>t ] IE Q rsds max AT, max At i t i = p x, T n k=1 n = p x, T I tk. k=1 IE Q ] T rsds At k 1 Atk At i,i {1,...,n}\{k} 16

17 4 PRICING OF GMAB 4.2 Explicit xprssion for th GMAB Changing to th quity masur Q S, w obtain I tk = IE Q rsds At n At ] k At n 1 At k At i,i {1,...,n}\{k} ] Atk = AIE Q S 1 A ti A tn 1,i {1,...,n}\{k} A tk ] Yt k = AIE Q S 1 Ytn Y ti Y 1,i {1,...,n}\{k} t k = AIE Q S Y tk Y tn ] 1 Yti Ytk = AIE Q S n,k Y 1 i,ky,i {1,...,n}\{k}] with ik Y = Y t k Y t i, i, k {1,..., n}. Th lmnts of th vctor k Y = { i,k Y } i {1,...,n}\{k} ar th diffrncs btwn two normally distributd rturns. Thus, th ntir vctor itslf is also normally distributd, with th corrsponding man µ k Y and covarianc matrix Σ k Y, which can b calculatd analytically s Appndix 8.3. This lads us to th following xprssion for k = n I T = A Φ n 1 ; µ ny, Σ ny, whr Φ n 1 is a cumulativ distribution function of th multivariat n 1 - dimnsional normal distribution. For k < n, it is straightforward to show that: σ I tk = AΦ n 1 ; µ k Y Σ k Y k, Σ k Y µ 2 n,k Y + n,k Y Maximum of ratcht and roll-up Th so-calld ratchup, which w us to dnot th maximum of ratcht and roll-up, provids th following guarantd amount at maturity: GT = max max At i, P δt <t 1 <...<t n=t It turns out that vn for th combination of both guarants th closd-form xprssion can b drivd.. 17

18 4.2 Explicit xprssion for th GMAB 4 PRICING OF GMAB Thorm 4.4. V = Ap x, T Φ n ; µ k Y, Σ k Y + Ap x, T n 1 Φ n ; µ ky Σ ky k, Σ ky µ n,k Y + k=1 + Φ n, µ δ Y Σ δ Y δ, Σ δ Y µ n,δ Y + σ n,δ Y 2 2, 2 σ n,k Y 2 whr k Y := { i,k Y } i {1,...,n,δ}\{k}. with i,δ Y = δt Y t i. µ k Y, Σ k Y Q S. ar th momnts undr th quity masur Proof. Th insuranc part of th contract can b again sparatd from th financial part. Thus w concntrat on th following xprssion ] V = p x, T IE Q rsds max AT, max P δt, max At i. t i Rarranging th trms undr th xpctation lads to ] V = p x, T IE Q rsds max AT, P δt, max At i t i = p x, T IE Q rsds AT 1 AT At1,...,AT At n 1,AT P δt ] n 1 + p x, T IE Q ] T rsds At k 1 Atk At 1,...,At k AT,At k P δt k=1 + p x, T IE Q ] T rsds P δt 1 P δt At 1,...,P δt AT n 1 = p x, T I T + I tk + I δ. k=1 W introduc th nw variabl i,δ Y = δt Y t i, which is normally distributd with varianc σ 2 i,δ Y = σ2 Y t i and a shiftd man, µ i,δ Y = δt µ Y ti. Hnc, w xtnd th vctor k Y by this additional lmnt and dnot it by k Y, i.. k Y := { i,k Y } i {1,...,n,δ}\{k}. 18

19 4 PRICING OF GMAB 4.2 Explicit xprssion for th GMAB W know th man of this vctor and all lmnts of th covarianc matrix xcpt for th Cov i,k Y, δ,k Y ]. This, howvr, can also b found analytically: Cov i,k Y, δ,k Y ] = CovY t k Y t i, δt Y t k ] = V ary t k ] + CovY t i, Y t k ]. 13 Th first summand can b rcivd from th Thorm 4.1 and for th covarianc w procd as follows: V ar i,k Y ] = V ary t k Y t i ] Rarranging th trms, w hav = V ary t k ] + V ary t i ] 2 CovY t k, Y t i ]. CovY t k, Y t i ] = 1/2 V ary t k ] + V ary t i ] V ar i,k Y ]. Finally, substituting this in 13, w obtain Cov i,k Y, δ,k Y ] = 1/2 V ary t i ] V ary t k ] V ar i,k Y ], whr V ar i,k Y ] is givn in Appndix 8.3. For I T and I tk w can rciv vry similar xprssions to th prvious cas whr w us k Y instad of k Y. For I δ w can writ I δ = IE Q ] T rsds P δt 1 P δt At 1,...,P δt AT = IE Q ] T rsds P δt 1 δt Y t1,...,δt Y T. Multiplying th xprssion undr th xpctation by Y T Y T quity masur Q S w hav I δ = IE Q ] T rsds P Y T δt Y T 1 δt Y t1,...,δt Y T δt Y T 1 δt Y t1,...,δt Y T ] = AIE Q S = AIE Q S ] n,δ Y 1 1,δY,..., n,δy. and changing to th Th last xprssion can now b valuatd in closd-form similar to th k-th xpctation in th ratcht guarant. This rsults in 2 σ n,δ I δ = AΦ n, µ δ Y Σ δ Y δ, Σ µ Y n,δ Y + 2. δ Y 19

20 5 CALIBRATION 5 Calibration In this sction w dscrib our approach for calibrating th insuranc and financial parts of our modl as wll as th data usd in th calibration procdur. In th first cas w us historical mortality data and currnt lif tabl; for th HWBS tdv modl today s financial markt data is usd. In th following w dscrib our paramtr stimation procdur in mor dtail. 5.1 Mortality modl W propos a two stp approach for th stimation of th unknown paramtrs. In th first stp, currnt lif tabl 6 is usd and th Gomprtz modl is calibratd. Basd on Baur 28] w calculat th mortality intnsity from a lif tabl and stimat th unknown paramtrs via th last squar mthod. For this w rwrit th Gomprtz quation 5 as λ x+t = d 1+d 2 x+t, 14 whr d 2 = 1/b and d 1 = m/b logb. Taking th logarithm of both sids in quation 14 w hav lnλ x+t = d 1 + d 2 x + t. Now, d 1 and d 2 can b asily obtaind as rgrssion cofficints whr our lnλ x+t srv as th rspons variabl and ag x + t as th rgrssor. Consquntly, ths valus can b usd to stimat paramtrs b and m from th initial formulation of th Gomprtz modl. In th nxt stp w nd to stimat th thr paramtrs of th xtndd Vasick procss. For this purpos w calculat th historical mortality improvmnt ratio ξ x+t t basd on historical intnsitis λ x+t t xtractd from th historical lif tabls 7. Thraftr th Maximum Liklihood approach is usd to stimat th unknown paramtrs. For this w nd th conditional probability dnsity of th obsrvation of ξ x+t+1 t + 1 givn th prvious obsrvation ξ x+t t. It can b shown that, ξ x+t t givn any prvious valu ξ x+s s, s < t is conditionally normally distributd with th corrsponding µ ξt ξs and σ ξt ξs s Appndix 6 Sourc: Fdral Statistical Offic of Grmany Statistischs Bundsamt Dutschland. 7 Sourc: JP Morgan LifMtrics Historic Indx Data. 2

21 5 CALIBRATION 5.2 Financial modl 8.4. Choosing for our cas s := t and t := t+1 with bing tim stp btwn thm w can writ th momnts as µ ξt+1 ξt = ξ x+t t k + k k γ γ t+ γ k σ 2 ξt+1 ξt = σ 2 2k 2k =: ˆσ 2 and consquntly th conditional dnsity function as fξ x+t+1 t + 1 ξ x+t t; k, γ, σ = 1 2πˆσ xp ξt + 1 ξt k k γ t+ γ k k γ Th log-liklihood function of a st of obsrvations ξ,..., ξ n can b drivd basd on th conditional dnsity function n Lk, γ, σ = lnfξ i ξ i 1 ; k, γ, σ i=1 = n ln2π n ln ˆσ 2 1 n 2ˆσ 2 i=1 ξ i ξ i 1 k 2ˆσ 2 k k γ γti 1 γ k 2. This function can now b maximizd with rspct to th unknown paramtr st k, γ, σ. An ovrviw of th stimatd paramtrs for th Grman population is givn in Tabl 1 8. b m k γ σ fmal mal Tabl 1: Paramtr stimation for th mortality modl Financial modl For th financial markt modl w first calibrat th intrst rat modl basd on th initial swap-basd trm structur. W us Euribor cash rats for maturitis lss 8 Basd on th yarly data for mortality improvmnt ratio from 1968 to 28 as was shown on Figur 1. 21

22 5.2 Financial modl 5 CALIBRATION than 6 months, 6 12 and FRAs, as wll as th 6-month Euribor swaps for maturitis from 2 to 3 yars 9 as a st of bnchmark instrumnts to construct th yild curv. Th rsulting trm structur on April 3 th, 212 is prsntd on th Figur 2. Figur 2: Intrst rat trm structur In th nxt stp w us markt implid volatilitis for th Europan at-th-mony swaptions with maturitis and tnors up to 3 yars s Figur 3 to stimat th rmaining paramtrs of th Hull-Whit modl, σ r. For this w compar th swaption prics P B S σb, drivd from th quotd Black s implid volatilitis σ B 1, with th thortical prics in th Hull-Whit modl P HW S, σ r 11. Th rquird paramtrs, σ r can b thrfor found by minimizing th sum of squard diffrncs btwn thos prics. Th rsulting paramtrs ar givn in th Tabl 2. σ r Tabl 2: Paramtr stimation for th Hull-Whit modl W can thn stimat φt such that it fits th initial trm structur basd on th quation 2. 9 All financial data hr and blow ar takn from Bloombrg. 1 S p. 72 in Brigo t al. 26] for th Black s formula for th Europan swaptions. 11 S p. 77 in Brigo t al. 26] for th swaption pricing formula in th Hull-Whit modl. 22

23 5 CALIBRATION 5.2 Financial modl Figur 3: Swaption implid volatilitis To prov th fitting quality w show th valus of th markt implid volatilitis and modl implid volatilitis for th givn swaption st in Figur 4. a Markt b Modl Figur 4: Swaptions implid volatilitis In th nxt stp w dtrmin th function of th instantanous quity volatility for our modl, givn fixd intrst rat paramtrs. Comparing th varianc of quity rturns σy 2 tdv T in our HWBS s Thorm 4.1 modl with th varianc 23

24 5.2 Financial modl 5 CALIBRATION in th standard Black-Schols formula σ 2 BS, T i T i, w hav σ 2 BS, T i T i = σ2 r a 2 r T i 2 T i 1 + 2T i 2 + i σ 2 Sudu + 2 ρ Srσ r i T i u σ S udu, whr T i ar availabl option xpiry dats, i = 1,..., n. In cas of dtrministic intrst rats, i.. σ r = this xprssion simplifis to σbs 2, T i T i = i σ 2 S udu. Assuming a picwis constant form for th instantanous volatility σ S, its valus can b asily xtractd by rcursion. For th gnral cas of stochastic intrst rats, w also assum a picwis constant structur of th volatility, and rciv σ 2 Y T i = i i σst 2 σ r k T k T k 1 + 2ρ Sr σ S T k T k T k 1 k=1 2ρ Sr σ r i 1 k=1 σ S T k 1 T i T k art i T k 1 k=1 + σ S T i 1 T i T i 1 + σ2 r a 2 r T i 2 T i 1 + 2T i 2 W can solv th quation abov for σ S T i and obtain th following rcursiv formula 12 : ρ Sr σ r σ S T i = T i T i 1 at i T i 1 + R, T i T i 1 whr R = ρ Sr σ r T i T i 1 at i T i 1 2 T i T i 1 12 W nglct th ngativ root in th formula, as it might lad to ngativ volatilitis. 24

25 5 CALIBRATION 5.2 Financial modl 1 T i T i 1 i 1 σst 2 k T k T k 1 k=1 σ r i 1 2ρ Sr σ S T k 1 art i T k art i T k 1 σ a BS, 2 T i T i r k=1 1 σ r i 1 2ρ Sr σ S T k T k T k 1 T i T i 1 k=1 + σ2 r T a 2 i 2 T i 1 + 2T i r 2 Thus, w hav dfind th trm structur of th instantanous volatility, which is consistnt with markt implid volatilitis. As th proxy for th implid volatility, th VSTOXX trm structur availabl in Bloombrg for maturitis up to 2 yars s Figur 5 as of April 3 th, 212. Figur 5: Implid volatility trm structur Th rmaining part for a markt consistnt pricing is th instantanous corrlation cofficint ρ Sr, which should b in lin with markt implid corrlation. As thr ar no liquid markt instrumnts from which th implid corrlation btwn quitis and intrst rats can b xtractd, w approximat ρ Sr with th historical corrlation btwn th EuroStoxx5 log-rturns and th absolut diffrncs in short-trm intrst rats ovr th last tn yars on a monthly basis. For th short-trm intrst rats 3-month zro-rats ar usd as an approximation. 25

26 6 VALUATION RESULTS: EXAMPLE 6 Valuation rsults: xampl In th following w will considr a singl prmium Guarantd Minimum Accumulation Bnfit, with maturity of 2 yars for a mal policyholdr at th ag of 45. Thr diffrnt ridrs roll-up, ratcht and th combination of both mbddd in th VA product ar analyzd. Figurs 6 and 7 show th sampl paths of two of th considrd guarants: Figur 6: Ratcht stp = 4 yars Figur 7: Roll-up rat = 2% W will now analyz th bhavior of th prics of th guarants with rspct to changing paramtrs of th guarant itslf, changing paramtrization of th financial markt, and changing paramtrization of th insuranc markt Snsitivitis to th product paramtrization W start with th product paramtrs. For th roll-up guarant, whr thr diffrnt valus for th roll-up rat ar considrd, w obsrv s Tabl 3 incrasing prics of th guarants with incrasing guarantd rat. This is absolutly xpctabl, as th highr th minimum payoff i.. guarantd paymnt policyholdr will gt at maturity th highr th pric of th product. For th ratcht Tabl 4 w obsrv dcrasing prics with incrasing tim stps btwn diffrnt obsrvations. This happns du to th fact, that th largr th ratcht stp th lss frquntly th account valu can b monitord. Hnc, som of th high paks can b xcludd in th obsrvations and, thus, ignord in th calculation. 13 For th as of xposition, in th following w us σ S t = σ S. 26

27 6 VALUATION RESULTS: EXAMPLE 6.1 Product paramtrization Last but not last for th maximum of roll-up and ratcht guarant, which is qual or highr than ach of th singl ons w can obsrv similar bhavior. Prics of guarant incras ovr columns and dcras ovr rows in Tabl 5. Roll-up guarant Roll-up rat GMAB % % % Tabl 3: Roll-up prics Ratcht guarant Ratcht stp GMAB 2 yars yars yars Tabl 4: Ratcht prics Maximum of roll-up and ratcht % 1.5% 3% 2 yars yars yars Tabl 5: Ratchup prics In th following w will analyz financial and insuranc risks for roll-ups and ratchts. All corrsponding statistics rlvant for th maximum of both guarants ar givn in Appndix

28 6.2 Financial markt 6 VALUATION RESULTS: EXAMPLE 6.2 Snsitivitis to th financial markt paramtrs Analyzing financial risk w start with th snsitivitis to implid quity volatility, that ar givn in Tabl 6. Snsitivitis ar calculatd basd on th cntral finit diffrncs for a paralll shift of an absolut chang of.1% in ach dirction. Roll-up rat % 1.5% 3% Snsitivity.75%.98% 1.19% Tabl 6: Volatility snsitivitis for roll-ups Th positiv snsitivity can b obsrvd, as th incras in uncrtainty i.. volatility lads to an incrasing option valu. Morovr, this snsitivity is highr for th products with highr minimum guarantd rat. At th sam tim, for th ratchts, w obsrv dcrasing positiv snsitivity with incrasing ratcht stp s Tabl 7. This snsitivity is gnrally highr than of roll-ups, spcially for th small ratcht stp 2 yars. Ratcht stp 2 yars 4 yars 8 yars Snsitivity 2.24% 1.81% 1.24% Tabl 7: Volatility snsitivitis for ratchts This can b asily xplaind by th fact that for this typ of products w considr not only th final valu of th quity, but svral discrt obsrvations, starting from th scond yar ratcht stp of 2 yars. Thus, as th ffct of changing volatility is mor rlvant for options with shortr maturity, it rsults in th ovrall highr snsitivity for th first ratcht compard to th scond and th third, as wll as to all roll-ups consquntly. On should kp in mind that th snsitivitis rprsnt just th impact of changing risk factors on th guarant prics and not th risk itslf. That is why on should account for th siz of potntial changs in th risk factors thmslvs and us som prdfind scnarios to gt a bttr undrstanding of th undrlying risk. Thus, in th scond stp of our analysis of quity volatility risk w provid two strss scnarios basd on th rcommndations of Solvncy II 14. Ths ar on up 14 S QIS5 calibration papr CEIOPS 21] for mor information on this and subsqunt strss tsts. 28

29 6 VALUATION RESULTS: EXAMPLE 6.2 Financial markt strss with lativ incras in volatility of 5 % from th currnt valu and on down strss with a dcras of 15 %. Th rsults ar givn in Tabl 8 and 9 for roll-ups and ratchts corrspondingly. Similar pattrn can b obsrvd as bfor, i.. incrasing guarant prics in th high volatility rgim and dcrasing prics in th low volatility scnario. Roll-up rat % 1.5% 3% Up strss Currnt valu Down strss Tabl 8: Volatility strss tsts for roll-ups. Ratcht stp 2 yars 4 yars 8 yars Up strss Currnt valu Down strss Tabl 9: Volatility strss tsts for ratchts. For th intrst rat risk w provid snsitivitis in Tabls 1 and 11. Cntral finit diffrncs with a paralll shift of.1% ach way, applid to th ntir curv, ar usd to stimat ths snsitivitis. Roll-up rat % 1.5% 3% Snsitivity -4.19% -6.73% -1.44% Tabl 1: Intrst rat snsitivitis for roll-ups Ratcht stp 2 yars 4 yars 8 yars Snsitivity -6.76% -6.16% -4.74% Tabl 11: Intrst rat snsitivitis for ratchts Much highr ngativ snsitivitis compard to th cas of implid volatility can b obsrvd. This coms from th fact that variabl annuity products ar vry longdatd options for which th drift i.. intrst rat undr th risk-nutral masur 29

30 6.3 Insuranc markt 6 VALUATION RESULTS: EXAMPLE is th most influntial risk factor. In trms of th absolut valu of th snsitivitis, it again incrass with incrasing roll-up rat and dcrass with incrasing ratcht stp. Howvr, in cas of th intrst rat, th snsitivitis ar rathr at th sam lvl for both products. Analog to th quity volatility risk w now us scnarios proposd in th Solvncy II calibration papr CEIOPS 21] to analyz th pric bhavior for an up and down strss tst for intrst rats s Tabls 12 and 13. W rfr to Appndix 8.6 for mor information on th applid strss scnarios. As xpctd, w obsrv lowr prics for th highr intrst rats i.. highr discount factors and th othr way around for th down shift in th curv. Roll-up rat % 1.5% 3% Up strss Currnt valu Down strss Tabl 12: Intrst rat strss tsts for roll-ups Ratcht stp 2 yars 4 yars 8 yars Up strss Currnt valu Down strss Tabl 13: Intrst rat strss tsts for ratchts 6.3 Snsitivitis to th insuranc markt paramtrs Last but not last, w do th sam xrcis for longvity risk. W first calculat th snsitivitis s Tabls 14 and 15, which turnd out to b vry small in comparison to othr risk factors. In this cas w us a on-dirctional diffrnc quotint basd on th 1% rlativ dcras in mortality rats for all ags as an approximation for th snsitivity. Ths low snsitivitis can b attributd to th fact that for this typ of products only on survival probability.g. of a policyholdr agd 45 to surviv for th nxt 2 yars in th considrd xampl is rlvant for th pricing. Longvity risk 3

31 7 CONCLUSION Roll-up rat % 1.5% 3% Snsitivity.11%.12%.14% Tabl 14: Mortality snsitivity for roll-ups Ratcht stp 2 yars 4 yars 8 yars Snsitivity.14%.13%.8% Tabl 15: Mortality snsitivity for ratchts bcoms howvr much mor rlvant for th products with longr maturitis, with mbddd annuity payout.g. GMIBs or for th oldr policyholdrs. Rsults of th strss tsts coming from th Solvncy rquirmnts i.. 25% rduction applid to th ntir mortality tabl ar prsntd in th Tabls 16 and 17, from which th sam conclusion as bfor can b withdrawn. Roll-up rat % 1.5% 3% Initial Rducd Tabl 16: Mortality strss tsts for roll-ups Ratcht stp 2 yars 4 yars 8 yars Initial Rducd Tabl 17: Mortality strss tsts for ratchts 7 Conclusion In this papr w prsntd a hybrid modl for actuarial and financial risks and driv closd-form formulas for Guarantd Minimum Accumulation Bnfits with diffrnt ridrs. This work can b sn as a gnralization to som widsprad modls in th litratur,.g. inclusion of stochastic intrst rats and stochastic mortality compard to Baur t al. 28] and Milvsky and Salisbury 26], 31

32 8 APPENDIX tim-dpndnt volatility as wll as xplicit incorporation of mortality modling in th framwork of Marshall t al. 21]. Furthrmor, as opposd to svral paprs on th valuation of quity-linkd products, whr numrical pricing of th guarants is suggstd, analytical closd-form xprssions ar providd in th prsntd framwork. Alongsid with rathr simpl roll-up guarants, w xamin ratchts and th maximum of both guarants. For all typ of guarant ridrs w conduct a snsitivity analysis to th product spcification as wll as to th financial and insuranc markt paramtrs. Morovr, strss tsts for th corrsponding markts according to Solvncy II ar applid. Furthr rsarch will b ddicatd to an xtnsion to a 2-factor modl for intrst rats, incorporation of policyholdr bhavior risk as wll as th analysis of furthr guarants. 8 Appndix 8.1 Proof of Thorm 3.2 Straightforward application of Ito s lmma to th function λ ξt, with λ givn by quation 5 and dynamics for ξt by quation 6 15, lads us to th dynamics of th intnsity procss λt s quation 7. Furthrmor, using Ito s lmma for th function c 3t λt and solving for λt givs λt = λ c 3T + c 1 c 2u c 3T u du + c 4 c 3T u c5u dw u. Taking th corrsponding xpctations, lads to µ λt = IEλT F ] = λ c 3T + c 1 c 2 + c 3 c 3T c 2+c 3 u T = λ c 3T + c 1 c 2 + c 3 c 2 T c 3T. 15 For convninc w will omit th subindx in th proof. 32

33 8 APPENDIX 8.2 Proof of Thorm 3.3 Applying Ito isomtry w obtain for th varianc σ 2 λt = V ar λt F ] = c 2 4 = c 2 4 2c 3T u2c 5+2c 3 du 2c 3T u 2c 5u du = c 2 4 2c 3T 1 2c 5 + 2c 3 T 2c 5 +2c 3 1 = c 2 4 2c 5 + 2c 3 2c 5 T 2c 3T. 8.2 Proof of Thorm 3.3 Following Duffi t al. 2] it can b asily shown that functions C λ t, T and D λ t, T from th Thorm 3.3 satisfy two ordinary diffrntial quations: C λ t, T t D λ t, T t = c 1 c 2t D λ t, T c2 4 2c 5t D 2 λt, T, 15 = c 3 D λ t, T 1, 16 with boundary conditions C λ T, T = and D λ T, T =. Morovr, it turns out that ths quations can b solvd analytically. As can b asily chckd, th solution to quation 16 is givn by s.g. Zagst 22] D λ t, T = c 3T t c 3. Using th solution for D λ of th trms C λ t, T t in quation 15 w rciv aftr som rarrangmnts = c 1 c2t c 1 c 2t c 3 T t + c2 4 2c 5t c 3 c 3 2c 2 3 Intgrating both parts lads to c2 4 2c 5t c 3 T t + c2 4 2c 5t 2c 3 T t. c 2 3 2c 2 3 C λ t, T = M 3 + c 1 c2t c 1 c 2 c 3 c 3 c 2 + c 3 c 2t c 3 T t + c2 4 2c 5t 4c 2 3 c 5 c 2 4 c 2 3 2c 5 + c 3 2c 5t c 3 T t c c 2 3 c 5 + c 3 2c 5t 2c 3 T t. 33

34 8.3 Equity rturn btwn two obsrvations 8 APPENDIX whr M 3 is just som constant that can b found from th boundary condition C λ T, T = : M 3 = c 1 c2t c 1 + c 2 c 3 c 3 c 2 + c 3 c 2T c2 4 2c 5T 4c 2 3 c 5 + c 2 4 c 2 3 2c 5 + c 3 2c 5T c 2 4 4c 2 3 c 5 + c 3 2c 5T. Substituting this in th abov formula for C λ t, T rsults in th final xprssion for C λ t, T as givn in Equity rturn btwn two obsrvations Basd on th quation 11 and assuming i < k w can writ for i,k Y i,k Y = Y t k Y t i = σ r tk + ρ Srσ r ti = σ r + σ r + σ 2 Sudu + tk φudu 1 2 ρ ti Srσ r ti tk t i tk t i + ρ Srσ r tk tk tk t k u dwr QS u + φudu σ S udw QS S u t k u σ S udu σ r ti σ 2 Sudu ti t i u σ S udu t i u art k u dwr QS u tk t k u dwr QS u + σ S udw QS S u + ρ Srσ r tk t i ti t k u σ S udu. ti σ S udw QS S u t i φudu t i u dwr QS u tk t i σ 2 Sudu t i u art k u σ S udu 34

35 8 APPENDIX 8.3 Equity rturn btwn two obsrvations Taking th corrsponding xpctation w hav for µ i,k Y µ i,k Y = 1 2 tk + ρ Srσ r + ρ Srσ r t i σ 2 Sudu + ti tk t i tk t i φudu t i u art k u σ S udu t k u σ S udu. It should b mntiond that for th cas i > k, w us th fact For t k t i tk φudu w hav t i φudu = tk t i f, udu + µ i,k Y = µ k,i Y. tk t i σ 2 r 2a 2 r = ln P, t k + ln P, t i + σ2 r u 2 du 2a 2 r tk = ln P, t i P, t k + σ2 r t 2a 2 k t i 2 t i art k r + 1 2t i 2art k. 2 t i 1 2 u + 2aru du For th varianc w rciv V ar i,k Y ] = σ2 r a 2 r ti + σ2 r a 2 r tk t i u 2 art i+t k 2u + 2art k u du t k t i 2 t k t i + 1 t i = σ2 r a 2 r tk σ2 r a 2 r σ 2 Sudu + 2 ρsrσ r tk t k t i 2 t k t i + 1 t i σ 2 Sudu + 2 ρsrσ r t i tk t i 2t k t i 2 t k u σ S udu 2t k t i 2 t k u σ S udu 1 2 2t i 1 t k t i art k+t i t k t i 2art k 35.

36 8.3 Equity rturn btwn two obsrvations 8 APPENDIX For th cas i < k varianc rmains th sam. Morovr, w will nd th covarianc btwn two priods i,k Y and j,k Y for th following cass i < j < k, i < k < j and k < i < j : i < j < k Th varianc of i,j Y can b rprsntd as V ar i,j Y ] = V ar i,k Y j,k Y ] = V ar i,k Y ] + V ar j,k Y ] 2 Cov i,k Y, j,k Y ]. Solving th xprssion abov for th covarianc, lads to Cov i,k Y, j,k Y ] = 1/2 V ar i,k Y ] + V ar j,k Y ] V ar i,j Y ]. i < k < j W first writ th xprssion for th varianc of i,j Y V ar i,j Y ] = V ar i,k Y + k,j Y ] = V ar i,k Y ] + V ar k,j Y ] + 2 Cov i,k Y, k,j Y ]. From this w hav for Cov i,k Y, k,j Y ] Cov i,k Y, k,j Y ] = 1/2 V ar i,j Y ] V ar i,k Y ] V ar k,j Y ]. Using th fact that Cova, b = Cova, b and varianc dosn t chang th sign, w rciv Cov i,k Y, j,k Y ] = 1/2 V ar i,k Y ] + V ar j,k Y ] V ar i,j Y ]. k < i < j Similar to th prvious cas w rprsnt th varianc of i,j Y, which is th sam as th varianc j,i Y, as V ar i,j Y ] = V ar k,j Y k,i Y ] = V ar k,j Y ] + V ar k,i Y ] 2 Cov k,j Y, k,i Y ]. Rarranging th trms and using th sam proprtis of varianc and covarianc as bfor, w hav Cov i,k Y, j,k Y ] = 1/2 V ar j,k Y ] + V ar i,k Y ] V ar i,j Y ]. 36

37 8 APPENDIX 8.4 Momnts of th mortality improvmnt ratio 8.4 Momnts of th mortality improvmnt ratio Applying Ito s lmma to quation 6 and solving for ξt givn ξs 16 lads to ξt = ξs kt s + k k γ γt k γt s + σ kt t s ku dw u. Th xprssion abov can b usd to calculat th momnts of th mortality improvmnt ratio µ ξt ξs = IEξt ξs] = ξs k + k k γ γt k γ σ 2 ξt ξs = V arξt ξs] = 2kt σ 2 t whr w usd = t s. s 2ku du = σ2 2k 2k, 8.5 Strss tsts for th maximum of roll-up and ratcht Similar to th singl guarants w provid th rsults of th strss tsts for quity volatility, intrst rat and longvity risks according to th Solvncy II rquirmnts for th maximum of roll-up and ratcht guarant. Equity volatility Rat Stp % 1.5% 3% 2 yars yars yars Tabl 18: Up strss Rat Stp % 1.5% 3% 2 yars yars yars Tabl 19: Down strss Intrst rats Rat Stp % 1.5% 3% 2 yars yars yars Tabl 2: Up strss Rat Stp % 1.5% 3% 2 yars yars yars Tabl 21: Down strss 16 W again omit th subindx in th proof. 37

38 8.6 Intrst-rat strss scnarios REFERENCES Longvity Rat Stp % 1.5% 3% 2 yars yars yars Tabl 22: Initial Rat Stp % 1.5% 3% 2 yars yars yars Tabl 23: Rducd Sam rsults as for th singl guarants roll-up and ratcht ar obtaind: highr prics for high quity volatility, low intrst rats and rducd mortality. 8.6 Intrst-rat strss scnarios Basd on th Solvncy II calibration papr CEIOPS 21] w obtain following strss-tst scnarios for th trm structur of intrst rats usd in our xampl, s Figur 8. Figur 8: Strss tsts for intrst rats Rfrncs Bacinllo t al. 21] Bacinllo, A., Biffis, E. and Millossovich, P.: Rgrssion-basd algorithms for lif insuranc contracts with surrndr guarants, Quantitativ Financ, 1,

39 REFERENCES REFERENCES Bacinllo t al. 211] Bacinllo, A., Biffis, E. and Millossovich, P.: Variabl annuitis: A unifying valuation approach, Insuranc: Mathmatics and Economics, 493, Baur t al. 28] Baur, D., Kling, A. and Russ, J.: A univrsal pricing framwork for guarantd minimum bnfits in variabl annuitis, ASTIN Bulltin, 38, Baur 28] Baur, D.: Stochastic mortality modling and scuritization of mortality risk, IFA-Schriftnrih, Ulm. Bilcki and Rutkowski 24] Bilcki, T. and Rutkowski, M.: Crdit risk: modling, valuation and hdging, Springr Financ. Springr, Brlin, 2nd dition. Black and Schols 1973] Black, F. and Schols, M.S.: Th pricing of options and corporat liabilitis, Journal of Political Economy, 812, Bowrs t al. 1997] Bowrs, N.L., Grbr, H.U., Hickman, J.C., Jons, D.A. and Nsbitt, C.J.: Actuarial mathmatics, Socity of Actuaris, Schaumburg Illinois. Boyl and Schwartz 1977] Boyl, P. and Schwartz, E.: Equilibrium prics of quity linkd insuranc policis with an asst valu guarant, Journal of Risk and Insuranc, 44, Brnnan and Schwartz 1976] Brnnan, M.J. and Schwartz, E.S.: Th pricing of quity-linkd lif insuranc policis with an asst valu guarant, Journal of Financial Economics, 3, Brigo t al. 26] Brigo, D. and Mrcurio, F.: Intrst rat modls - thory and practic, Springr-Vrlag, Brlin. Brunnr t al. 29] Brunnr, B. and Krayzlr, M.: Variabl annuity dilmma, risklab Working Papr. CEIOPS 21] Committ of th Europan Insuranc and Occupational Pnsion Suprvisors: QIS5 Calibration Papr, CEIOPS-SEC-4-1. Chn t al. 28] Chn, Z., Vtzal, K. and Forsyth, P.: Th ffct of modlling paramtrs on th valu of GMWB guarants, Insuranc: Mathmatics and Economics, 431,

40 REFERENCES REFERENCES Dahl 24] Dahl, M.: Stochastic mortality in lif insuranc: markt rsrvs and mortality-linkd insuranc contracts, Insuranc: Mathmatics and Economics, 351, Dahl t al. 26] Dahl, M. and Mollr, T.: Valuation and hdging of lif insuranc liabilitis with systmatic mortality risk, Insuranc: Mathmatics and Economics, 392, Dai t al. 28] Dai, M., Kwok, Y. and Zong, J.: Guarantd minimum withdrawal bnfit in variabl annuitis, Mathmatical Financ, 184, Duffi t al. 2] Duffi, D., Pan, J. and Singlton, K.: Transform analysis and asst pricing for affin jump-diffusions, Economtrica, 686, Gman t al. 1996] Gman, H., Karoui, E. and Rocht, J.C.: Changs of numrair, changs of probability masurs and pricing of options, Journal of Applid Probability, 32, Hull and Whit 1993] Hull, J. and Whit, A.: On factor intrst rat modls and th valuation of intrst rat scuritis, Journal of Financial and Quantitativ Analysis, 282, Marshall t al. 21] Marshall, G., Hardy, M. and Saundrs, D.: Valuation of a guarantd minimum incom bnfit, North Amrican Actuarial Journal, 141, Milvsky and Posnr 21] Milvsky, M. and Posnr, S.: Th titanic option: valuation of guarantd minimum dath bnfits in variabl annuitis and mutual funds, Journal of Risk and Insuranc, 681, Milvsky and Salisbury 22] Milvsky, M. and Salisbury, T.: Th ral option to laps and th valuation of dath-protctd invstmnts, IFID Working Papr. Milvsky and Salisbury 26] Milvsky, M. and Salisbury, T.: Financial valuation of guarantd minimum withdrawal bnfits, Insuranc: Mathmatics and Economics, 381, Ulm 28] Ulm, E.: Analytic solution for rturn of prmium and rollup guarantd minimum dath bnfit options undr som simpl mortality laws, ASTIN Bulltin, 382, Zagst 22] Zagst, R.: Intrst Rat Managmnt, Springr Vrlag, Brlin. 4