The Impact of Stochastic Volatility on Pricing, Hedging, and Hedge Efficiency of Variable Annuity Guarantees

Transcription

1 The Impac of Sochasic Volailiy on Pricing, Hedging, and Hedge Efficiency of Variable Annuiy Guaranees Alexander Kling *, Frederik Ruez and Jochen Ruß This Version: Augus 14, 2009 Absrac We analyze differen ypes of guaraneed wihdrawal benefis for life, he laes guaranee feaure wihin Variable Annuiies. Besides an analysis of he impac of differen produc feaures on he cliens payoff profile, we focus on pricing and hedging of he guaranees. In paricular, we invesigae he impac of sochasic (implied) equiy volailiies on pricing and hedging. We consider differen dynamic hedging sraegies for dela and vega risks and compare heir performance. We also examine he effecs if he hedging model (wih deerminisic volailiies) differs from he daa-generaing model (wih sochasic volailiies). This is an indicaion for he risk an insurer akes by assuming consan volailiies in he hedging model whils in he real world, volailiies are sochasic. Keywords Variable Annuiies, Guaraneed Minimum Benefis, Pricing, Hedging, Hedge Performance, Sochasic Volailiy * Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, Ulm, Germany, phone: , fax: , a.kling@ifa-ulm.de Corresponding auhor. Ulm Universiy, Helmholzsraße 22, Ulm, Germany, phone: , fax: , frederik.ruez@uni-ulm.de Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, Ulm, Germany, phone: , fax: , j.russ@ifa-ulm.de

2 1 Inroducion Variable Annuiies are fund-linked annuiies. Such producs were inroduced in he 1970es in he Unied Saes. In he 1990es, insurers sared o include cerain guaranees in such policies, so-called guaraneed minimum deah benefis (GMDB) as well as guaraneed minimum survival benefis ha can be caegorized in hree main groups: guaraneed minimum accumulaion benefis (GMAB), guaraneed minimum income benefis (GMIB) and guaraneed minimum wihdrawal benefis (GMWB). GMAB and GMIB ype guaranees provide he policyholder some guaraneed mauriy value or some guaraneed annuiy benefi, respecively. The hird and currenly mos popular ype of guaraneed minimum living benefis are GMWB. Under cerain condiions, he insured can wihdraw money from heir accoun, even if he value of he accoun is zero. Such wihdrawals are guaraneed as long as boh, he amoun ha is wihdrawn wihin each policy year and he oal amoun ha is wihdrawn over he erm of he policy say wihin cerain limis. Recenly, insurers sared o include addiional feaures in GMWB producs. The mos prominen is called GMWB for life : guaraneed lifelong annual wihdrawals. The oal amoun of such wihdrawals is no limied as long as each annual wihdrawal amoun does no exceed some maximum value and he insured is sill alive. For hese lifelong wihdrawal guaranees, annual wihdrawals of abou 5% of he (single iniial) premium are commonly guaraneed for insured aged 60+. A he same ime, he insured can a any ime access he remaining value of he underlying funds (if posiive) by surrendering he conrac. Also, in case of deah any remaining fund value is paid o he insured s dependans. Usually, he policyholder can choose from a variey of differen muual funds. Therefore, from an insurer s poin of view, hese producs conain an ineresing combinaion of financial risk and longeviy risk ha is difficul o hedge. As a compensaion for he guaranee, he insurer usually charges a guaranee fee ha is deduced from he policy s fund value. Due o he significan financial risk ha is inheren wihin he insurance conracs sold, risk managemen sraegies such as dynamic hedging are commonly applied. During he recen financial crisis, insurers have suffered from inefficien hedge porfolios wihin heir books. 1 Among oher effecs, volailiies have significanly increased leading o a remendous increase in opion values. In paricular for insurers wih no or no sufficien vega hedge (i.e. a hedge agains he risk of changing volailiy), he hedge porfolio did no increase accordingly leading o a loss for exising business (and less aracive condiions, i.e. higher guaranee fees, for new conracs). There already exiss some lieraure on he pricing of differen guaraneed minimum benefis and in paricular GMWB: Valuaion mehods have been proposed by e.g., Milevsky and Posner (2001) for he GMDB-Opion, Milevsky and Salisbury (2006) for he GMWB-Opion, and Holz e al. (2007) for a GMWB for life. Bauer e al. (2008) have presened a general model framework ha allows for he simulaneous and consisen pricing and analysis of differen variable annuiy guaranees. They also give a comprehensive analysis over nonpricing relaed lieraure on variable annuiies. To our knowledge, here exiss lile lieraure on he performance of differen sraegies for hedging he marke risk of variable annuiy 1 Cf. e.g. differen aricles and papers in Life and Pensions : A challenging environmen (June 2008), Variable Annuiies Flawed produc design coss Old Muual 150m (Sepember 2008), Variable annuiies Milliman denies culpabiliy for cliens' hedging losses (Ocober 2008), Variable Annuiies Axa injecs $3bn ino US arm (January 2009).

3 guaranees. Coleman e al. (2005 and 2007) provide such analyses for deah benefi guaranees under differen hedging and daa-generaing models. However, o our knowledge, he performance of differen hedging sraegies for GMWB for Life conracs under sochasic equiy volailiy has no ye been analyzed. The presen paper fills his gap. The remainder of his paper is organized as follows. In Secion 2, we describe differen designs of GMWB for Life conracs ha will be analyzed in he numerical secion and describe he model framework for insurance liabiliies used for our analyses. The liabiliy model we describe is akin o he one presened by Bauer e al. (2008). In Secion 3, we provide he framework for he numerical analyses, saring wih a descripion of he asse models used for pricing and hedging of insurance liabiliies. For he sake of comparison, we use he classic Black-Scholes model (wih deerminisic volailiy) as a reference and he Heson model for he evoluion of an underlying under sochasic volailiy. We also describe he financial insrumens involved in he hedging sraegies described below, and how we deermine heir fair prices and sensiiviies under boh models. The numerical resuls of our conrac analyses are provided in Secion 4, saring wih he deerminaion of he fair guaraneed wihdrawal rae in Secion 4.1 for differen GMWB for Life producs under differen model assumpions, firs under he Black-Scholes model wih deerminisic ineres raes and volailiy, and, secondly, under he Heson model wih sochasic volailiy. We proceed wih an analysis of he disribuion of wihdrawal amouns in Secion 4.2 and rigger imes, i.e. he poin of ime when guaraneed benefis are paid for he firs ime, in Secion 4.3 and finally analyze he so called Greeks in Secion 4.4. In Secion 5, we firs give an overview over differen dynamic and semi-saic hedging sraegies ha can be used o manage he risks emerging from he financial marke and analyze and compare he hedging performance of he sraegies menioned above under boh asse models. We also examine he effecs if he hedging model differs from he daageneraing model. 2 Model framework In Bauer e al. (2008), a general framework for modeling and a valuaion of variable annuiy conracs was inroduced. Wihin his framework, any conrac wih one or several living benefi guaranees and/or a guaraneed minimum deah benefi can be represened. In heir numerical analysis however, only conracs wih a raher shor finie ime horizon were considered. Wihin he same model framework, Holz e al. (2008) describe how GMWB for Life producs can be included in his model. In wha follows, we inroduce his model framework focusing on he peculiariies of he conracs considered wihin our numerical analyses. We refer o Bauer e al. (2008) as well as Holz e al. (2008) for he explanaion of oher living benefi guaranees and more deails on he model. 2.1 High-level descripion of he considered insurance conracs Variable Annuiies are fund linked producs. The single premium P is invesed in one or several muual funds. We call he value of he insured s individual porfolio he accoun value and denoe i by AV. All fees are aken from he accoun value by cancellaion of fund unis. Furhermore, he insured has he possibiliy o surrender he conrac or, of course, o wihdraw a porion of he accoun value. 3

4 Producs wih a GMWB opion give he policyholder he possibiliy of guaraneed wihdrawals. In his paper, we focus on he case where such wihdrawals are guaraneed lifelong (GMWB for life or Guaraneed Lifeime Wihdrawal Benefis, GLWB). The guaraneed wihdrawal amoun is usually a cerain percenage x WL of he single premium P. Any remaining accoun value a he ime of deah is paid o he beneficiary as deah benefi. If, however, he accoun value of he policy drops o zero while he insured is sill alive, he insured can sill coninue o wihdraw he guaraneed amoun annually unil deah. The insurer charges a fee for his guaranee which is usually a pre-specified annual percenage of he accoun value. Ofen, GLWB producs conain cerain feaures ha lead o an increase of he guaraneed wihdrawal amoun if he underlying funds perform well. Usually, on every policy anniversary, he curren accoun value of he clien is compared o a cerain wihdrawal benefi base. Whenever he accoun value exceeds ha wihdrawal benefi base eiher he guaraneed annual wihdrawal amoun is increased (wihdrawal sep-up) or (a par of) he difference is paid ou o he clien (surplus disribuion). In our numerical analyses in Secions 4 and 5, we have a closer look on four differen produc designs ha can be observed in he marke: No Rache: The firs and simples alernaive is one where no raches or surplus exis a all. In his case, he guaraneed annual wihdrawal is consan and does no depend on marke movemens. Lookback Rache: The second alernaive is a rache mechanism where a wihdrawal benefi base a ouse is given by he single premium paid. During he conrac erm, on each policy anniversary dae he wihdrawal benefi base is increased o he accoun value if he accoun value exceeds he previous wihdrawal benefi base. The guaraneed annual wihdrawal is increased accordingly o x WL muliplied by he new wihdrawal benefi base. This effecively means ha he fund performance needs o compensae for policy charges and annual wihdrawals in order o increase he guaraneed annual wihdrawals. Remaining WBB Rache: Wih he hird rache mechanism, he wihdrawal benefi base a ouse is also given by he single premium paid. The wihdrawal benefi base is however reduced by every guaraneed wihdrawal. On each policy anniversary where he curren accoun value exceeds he curren wihdrawal benefi base, he wihdrawal benefi base is increased o he accoun value. The guaraneed annual wihdrawal is increased by x WL muliplied by he difference beween he accoun value and he previous wihdrawal benefi base. This effecively means ha he fund performance needs o compensae for policy charges only bu no for annual wihdrawals in order o increase guaraneed annual wihdrawals. This rache mechanism is herefore c.p. somewha richer han he Lookback Rache. Performance Bonus: For his alernaive he wihdrawal benefi base is defined exacly as in he Remaining WBB rache. However, on each policy anniversary where he curren accoun value is greaer han he curren wihdrawal benefi base, 50% of he difference is paid ou immediaely as a so called performance bonus. The guaraneed annual wihdrawals remain consan over ime. For he calculaion of he wihdrawal benefi base, only guaraneed annual wihdrawals are subraced from he benefi base and no he performance bonus paymens. 4

5 2.2 Model of he liabiliies Throughou he paper, we assume ha adminisraion charges and charges are deduced a he end of each policy year as a percenage φ admin and φ guaranee of he accoun value. Addiionally, we allow for upfron acquisiion charges φ acquisiion ha are charges as a acquisiion AV = P ϕ. percenage of he single premium P. This leads o 0 ( 1 ) guaraneed W We denoe he guaraneed wihdrawal amoun a ime by and he wihdrawal benefi base by WBB. A incepion, for each of he considered producs, he iniial guaraneed guaraneed wihdrawal amoun is given by W0 = xwl WBB0 = xwl P. The amoun acually wihdrawn by he clien is denoed by W. 2 Thus, he sae vecor y = AV, WBB, W, W guaraneed a ime conains all informaion abou he conrac a ha ( poin in ime. ) Since we resric our analyses o single premium conracs, policyholder acions during he life of he conrac are limied o wihdrawals, (parial) surrender and deah. During he year, all processes are subjec o capial marke movemens. For he sake of simpliciy, we allow for wihdrawals a policy anniversaries only. Also, we assume ha deah benefis are paid ou a policy anniversaries if he insured person has died during he previous year. Thus, a each policy anniversary = 1,2,..., T, we have o disinguish beween he value + of a variable in he sae vecor ( ) immediaely before and he value ( ) afer wihdrawals and deah benefi paymens. In wha follows, we describe he developmen beween wo policy anniversaries and he ransiion a policy anniversaries for differen conrac designs. From hese, we are finally able o deermine all benefis for any given policy holder sraegy and any capial marke pah. This allows for an analysis of such conracs in a Mone-Carlo framework Developmen beween wo Policy Anniversaries We assume ha he annual fees φ admin and φ guaranee are deduced from he policyholder s accoun value a he end of each policy year. Thus, he developmen of he accoun value beween wo policy anniversaries is given by he developmen of he underlying fund S afer deducion of he guaranee fee, i.e. S AV AV e ad min guaranee = ϕ ϕ + 1 S. (1) A he end of each year, he differen rache mechanism or he performance bonus are applied guaraneed afer charges are deduced and before any oher acions are aken. Thus W develops as follows: 2 Noe ha he clien can chose o wihdraw less han he guaraneed amoun, hereby increasing he probabiliy of fuure raches. If he clien wans o wihdraw more han he guaraneed amoun, any exceeding wihdrawal would be considered a parial surrender. 5

6 - + No Rache: = = guaraneed - guaraneed + WBB+ 1 WBB P and W+ 1 = W = x WL P. - + WBB = + 1 max { WBB ; AV+ 1} { V + 1} Lookback Rache: and W 1 = x WBB 1 = max W ; x A. guaraneed - guaraneed + + WL + WL Remaining WBB Rache: Since wihdrawals are only possible on policy anniversaries, he wihdrawal benefi base during he year develops like in he Lookback Rache - + WBB = max WBB ; AV and case. Thus, we have + 1 { + 1} guaraneed - guaraneed + W+ 1 = xwl WBB+ 1 = max { W ; xwl A V + 1}. Performance Bonus: For his alernaive a wihdrawal benefi base is defined similarly WBB 1 WBB + = = accoun value and he wihdrawal benefi base is paid ou as a performance bonus. guaraneed Thus, we have W = x P max 0; AV WB. + o he one in he Remaining WBB Rache: + = and guaraneed - guaraneed + W WL P 1 W x. Addiionally, 50% of he difference beween he + 1 WL Transiion a a Policy Anniversary { B } A he policy anniversaries, we have o disinguish he following four cases: a) The insured has died wihin he previous year (-1,] If he insured has died wihin he previous policy year, he accoun value is paid ou as deah benefi. Wih he paymen of he deah benefi, he insurance conrac maures. Thus, AV + = 0, WBB + = 0, W + = 0, and W guaraneed + = 0. b) The insured has survived he previous policy year and does no wihdraw any money from he accoun a ime If no deah benefi is paid ou o he policyholder and no wihdrawals are made from he conrac, i.e W = 0, we ge AV = AV, WBB = guaraneed + = guaraneed - WBB, and W W. In he Performance Bonus produc, he guaraneed annual wihdrawal amoun is rese o is original guaraneed - level since W migh have conained performance bonus paymens. Thus, for his guaraneed alernaive we have W + = x P. WL c) The insured has survived he previous policy year and a he policy anniversary wihdraws an amoun wihin he limis of he wihdrawal guaranee If he insured has survived he pas year, no deah benefis are paid. Any wihdrawal W below guaraneed - he guaraneed annual wihdrawal amoun W reduces he accoun value by he wihdrawn amoun. Of course, we do no allow for negaive policyholder accoun values and + hus ge AV = max 0; AV W. { } For he alernaives No Rache and Lookback Rache, he wihdrawal benefi base and he guaraneed annual wihdrawal amoun remain unchanged, i.e. WBB, and + = WBB 6

7 W W. For he alernaive Remaining WBB Rache, he wihdrawal guaraneed + guaraneed - = + benefi base is reduced by he wihdrawal aken, i.e. { =max 0; } WBB WBB W and he guaraneed + guaraneed - guaraneed annual wihdrawal amoun remains unchanged, i.e. W = W. For he alernaive Performance Bonus, he wihdrawal benefi base is a a maximum reduced by he iniially guaraneed wihdrawal amoun (wihou performance bonus), i.e. + WBB = max 0; WBB min W ; x P and he guaraneed annual wihdrawal amoun is { { WL }} guaraneed + se back o is original level, i.e. = WL P W x. d) The insured has survived he previous policy year and a he policy anniversary wihdraws an amoun exceeding he limis of he wihdrawal guaranee In his case again, no deah benefis are paid. For he sake of breviy, we only give he formulae for he case of full surrender, since parial surrender is no analyzed in wha follows. In case of full surrender, he complee accoun value is wihdrawn, we hen se AV + = 0, + = 0 + guaraneed + WBB, W = AV, and W = 0 and he conrac erminaes. 2.3 Conrac valuaion We denoe by x 0 he insured s age a he sar of he conrac, p x 0 he probabiliy for a x 0 - year old o survive he nex years, q x + he probabiliy for a ( x 0 + ) -year old o die wihin 0 he nex year, and le ω be he limiing age of he moraliy able, i.e. he age beyond which survival is impossible. The probabiliy ha an insured aged x 0 a incepion passes away in he year (,+1] is hus given by px q 0 x0 +. The limiing age ω allows for a finie ime horizon T = ω - x0. In our numerical analyses below, we assume ha moraliy wihin he populaion of insured happens exacly according o hese probabiliies. Assuming independence beween financial markes and moraliy and risk-neuraliy of he insurer wih respec o moraliy risk, we are able o use he produc measure of he riskneural measure of he financial marke and he moraliy measure. In wha follows, we denoe his produc measure by Q. In his seing, conracs can be priced as follows: We already menioned ha for he conracs considered wihin our analysis, policyholder acions during he life of he policyholder are limied o wihdrawals and (parial) surrender. In our numerical analyses in Secions 4 and 5, we do no consider parial surrender. To keep noaion simple, we herefore here only give formulae for he considered cases (cf. Bauer e al. for formulae for he oher cases). We denoe by s he poin of ime a which he policyholder surrenders if he insured is sill alive and le s=t for a policyholder ha does no surrender. For any given value of s, and under he assumpion ha he insured dies in year { 1,2,..., x 0} imes i { 1,2,..., } and all guaranee fees Zi ( s ; ) a imesi { 1,2,..., } capial marke pah. By Φ τ ( s ; ) all fuure guaranee paymens Y ( ; s ) minus guaranee fees Z ( ; ) i ( ; ) ω, all conracual cash flows and hus all guaranee paymens Y s a i are specified for each, we denoe he so called ime τ opion value, i.e. he value of i s in his case: ri ( τ) ri ( τ) Vτ ( ; s) = EQ Yi( ; s) e Fτ EQ Zi( ; s) e Fτ i= τ+ 1 i= τ+ 1 (2) 7

8 Thus, he im e τ value of he opion assuming he moraliy raes defined above (sill for a given ime of surrender) is T Vτ ( s) = p τ q Vτ ( ; s). (3) 1 x0+ x0+ 1 = τ + 1 We finally assume ha policyholders surrender heir conracs wih cerain surrender probabiliies per year and denoe he probabiliy h a a policyholder surrenders a im e s by ps. Then, he ime τ value of he opion is given by T s= 1 s ( ) V = pv s. (4) τ τ 3 Framework for he Numerical Analysis 3.1 Models For our analyses we assume wo primary radable asses: he fund's underlying, whose spo price we will denoe by ), and he money-marke accoun, denoed by B( ). We assume he ineres spread o be zero and he money-marke accoun o evolve a a consan risk-free rae of ineres r: db( = B( = rb( d B(0) exp( r (5) For he dynamics of ), w e will use wo differen models: firs we will assume he equiy volailiy o be deerminisic and consan over ime, and hence use he Black-Scholes model for our simulaions. To allow for a more realisic equiy volailiy model, we will use he Heson model, in which boh, he underlying iself and is volailiy, are modeled by sochasic processes Black-Scholes Model In he Black-Scholes (1973) model, he underlying s spo price ) follows a geomeric Brownian moion whose dynamics under he real-world measure (also called physical measure) P are given by he following sochasic differenial equaion (SDE) d = μ d + σ dw (, 0) 0, (6) BS where µ is he (consan drif of he underlying, σ BS is consan volailiy and W( ) denoes a P-Brownian moion. By Iō's lemma, ) has he soluion 2 = 0)exp ( ) μ σ BS +, (0) 0 2 σ BSW S. (7) Heson Model There are various exensions o he Black-Scholes model ha allow for a more realisic modeling of he underlying's volailiy. We use he Heson (1993) model in our analyses 8

9 where he insananeous (or local) volailiy of he asse is sochasic. Under he Heson model, he marke is assumed o be driven by wo sochasic processes: he underlying s price ), and is insananeous variance V( ), which is assumed o follow a one-facor square-roo process idenical o he one used in he Cox-Ingersoll-Ross (1985) ineres rae model. The dynamics of he wo processes under he real-world measure P are given by he following sysem of sochasic differenial equaions dv ( = κ 2 ( ρdw1 ( + 1 ρ dw ( ) d = μ d + V ( 2 ( θ V ( ) d + σ V ( dw (, V (0) 0,, 0) 0 v 1 (9) (8) where µ again is he drif of he underlying, V( is he local variance a ime, κ is he speed of mean reversion, θ is he long-erm average variance, σ v is he so-called vol of vol, or (more precisely) he volailiy of he variance, ρ denoes he correlaion beween he 2 underlying and he volailiy, and W 1/2 are P-Wiener processes. The condiion 2κθ σ v ensures ha he variance process will remain sricly posiive almos surely (see Cox, Ingersoll, Ross (1985)). There is no analyical soluion for ) available, hus numerical mehods will be used in he simulaion. 3.2 Valuaion Risk-Neural Valuaion In order o deermine he values (i.e. he risk-neural expecaions) of he asses in our model, we firs have o ransform he real-world measure P ino is risk-neural counerpar Q, i.e. ino a measure under which he process of he discouned underlying s spo price is a (local) maringale. While he ransformaion o such a measure is unique under he Black-Scholes model, i is no under he Heson model. If no dividends are paid on he underlying, he dynamics of he underlying s price wih respec o he risk-neural measure under he Black-Scholes model is given by he following equaion (see for insance Bingham and Kiesel (2004)): Q d = r d + σ dw (, 0) 0, (10) BS where r denoes he risk-free rae of reurn and W Q is a Wiener process under he risk-neural measure Q. In he Heson model, as here are wo sources of risk, here are also wo marke-price-of-risk processes, denoed by γ 1 and γ 2 (corresponding o W 1 and W2). Heson (1993) proposed he following resricion on he marke price of volailiy risk process, assuming i o be linear in volailiy, γ ( = λ V ( ). (11) 1 Provided boh measures, P and Q, exis, he Q-dynamics of and V(, again under he 9

10 assumpion ha no dividends are paid, are given by Q 2 Q ( ρdw1 ( + 1 ρ ( ) d = r d + V ( dw2 Q ( θ V ( ) d + V ( dw (, V (0) 0 0) 0 (12) dv ( = κ (13) σ v 1, where W Q 1 ( ) and W Q 2 ( ) are wo independen Q-Wiener processes an d where κθ κ =, θ = (14) ( κ + λσ ) v ( κ + λσ ) v are he risk-neural counerpars o κ and θ (see, for insance, Wong and Heyde (2006)). Wong and Heyde (2006) also show ha he equivalen local maringale measure ha corresponds o he marke price of volailiy risk, λ V (, exiss if he inequaliy κ / σ v λ < is fulfilled. They furher show ha, if an equivalen local maringale measure Q exiss and κ + λσ v σ v ρ, he discouned sock price is a Q-maringale. B( Valuaion of he GMWB for Life Producs For boh equiy models, we use Mone Carlo Simulaions o compue he value of he GMWB opion value V defined in Secion 2.3, i.e. he difference beween expeced fuure guaranee paymens made by he insurer and expeced fuure guaranee fees deduced from he policyholders fund asses. We call he conrac fair, if V 0 = Sandard Opion Valuaion In some of he hedging sraegies considered in Secion 5, European "plain vanilla" opions are used. Under he Black-Scholes model, closed form soluions exis for he price of European call and pu opions. For srike price K and mauriy T, he call opion price a ime is given by he Black (1976) formula Call BS where ( S, P(, T )[ FN( d ) KN ( )] ( 1 d 2 =, (15) d 1 d 2 : = d F : = e P(, T ) : = e ln( F / K) + : = σ 1 σ r( T BS ( r q)( T, 2 ( σ / 2)( T BS T T (16) and N( ) denoes he cumulaive disribuion funcion of he sandard normal disribuion. The price of a European pu opion is given by 10

11 Pu BS ( S, P(, T )[ KN ( d ) FN ( )] ( 2 d1 = (17) For he Heson sochasic volailiy model, Heson (1993) found a semi-analyical soluion for pricing European call and pu opions using Four ier inversion echniques. The formulas have he form Call Pu Heson Heson (, V (, P(, T )[ F P1 K P2 ] (, V (, = P(, T )[ F ( P 1) K ( P 1) ] P 1/ 2 ϕ( u) : = E = (18) 1 1 : = + f 2 π e f1( u) : = Re e f 2 ( u) : = Re Q 0 iu ln iu ln 1/ 2 1 ( u) du K ϕ( u i) iuf K ϕ( u) iu iu ln S ( T ) [ e ] 2 (19) (20) (21) (22) (23) This means φ( ) is he log-characerisic funcion of he underlying s price under he riskneural measure Q. As Kahl and Jäckel (2006) poin ou, he compuaion of he erms P 1/2 includes he evaluaion of he logarihm wih complex argumens which may lead o numerical insabiliies for cerain ses of parameers and/or long-daed opions. Therefore, we use he scheme proposed in heir paper, which should allow for a robus compuaion of he fair values of European call and pu opions for (pracically) arbirary parameers. As in he proposed scheme, we use he adapive Gauss-Lobao quadraure mehod for he numerical inegraion of P 1 and P Compuaion of Sensiiviies (Greeks) Where no analyical soluions for he sensiiviy of he opion s or guaranee s value o changes in model parameers (he so-called Greeks, cf. e.g. Hull (2008)) exis, we use Mone Carlo mehods o compue he respecive sensiiviies numerically. We use finie differences as approximaions of he parial derivaives, where he direcion of he shif is chosen accordingly o he direcion of he risk, i.e. for he dela we shif he sock downwards in order o compue he backward finie difference, and shif he volailiy upwards for he vega, his ime o compue a forward finie difference. 11

12 4 Conrac Analysis 4.1 Deerminaion of he Fair Guaraneed Wihdrawal Rae In his secion, we firs calculae he guaraneed wihdrawal rae x WL ha makes a conrac fair, all oher parameers given. In order o calculae x WL, we perform a roo search wih x WL as argumen and he value of he opion V 0 as funcion value. For all of he analyses we use he fee srucure given in Table 1. Acquisiion fee Managemen fees Guaranee fees Wihdrawal fees 4.00 % of lump sum 1.50 % p.a. of NAV 1.50 % p.a. of NAV 0.00 % of wihdrawal amoun Table 1: Assumed fee srucure for all regarded conracs. We furher assume he policy holder o be a 65 years old male. For pricing purposes, we use bes-esimae moraliy probabiliies given in he DAV 2004R able published by he German Acuarial Sociey (DAV) Resuls for he Black-Scholes model All resuls for he Black-Scholes model have been calculaed assuming a risk-free rae of ineres of r = 4%. Table 3 displays he fair guaraneed wihdrawal raes for differen rache mechanisms, differen volailiies and differen policyholder behavior assumpions: We assume ha as long as heir conracs are sill in force policy holders every year wihdraw exacly he maximum guaraneed annual wihdrawal amoun. Furher, we look a he scenarios no surrender (no surr), surrender according o Table 2 (surr 1) and surrender wih wice he probabiliies given in Table 2 (surr 2). Year Surrender rae 1 6 % 2 5 % 3 4 % 4 3 % 5 2 % 6 1 % Table 2: Assumed deerminisic surrender raes. 12

13 Rache Mechanism Volailiy σ BS =15 % σ BS =20 % σ BS =22 % σ BS =25 % I (No Rache II (Lookback Rache III (Remaining WBB Rache IV (Performance Bonus) No surr 5.26 % 4.80 % 4.43 % 4.37 % Surr % 5.00 % 4.62 % 4.57 % Surr % 5.22 % 4.83 % 4.79 % No surr 4.98 % 4.32 % 4.01 % 4.00 % Surr % 4.50 % 4.18 % 4.19 % Surr % 4.71 % 4.38 % 4.40 % No surr 4.87 % 4.13 % 3.85 % 3.85 % Surr % 4.30 % 4.01 % 4.03 % Surr % 4.50 % 4.20 % 4.24 % No surr 4.70 % 3.85 % 3.61 % 3.62 % Surr % 4.01 % 3.76 % 3.81 % Surr % 4.20 % 3.94 % 4.01 % Table 3: Fair guaraneed wihdrawal raes for differen rache mechanisms, differen policyholder behavior assumpions and under differen volailiies. A comparison of he differen produc designs shows ha, obviously, he highes annual guaranee can be provided if no rache or performance bonus is provided a all. If no surrender is assumed and a volailiy of 20% is assumed, he guaranee is similar o a 5 for life produc (4.98%). Including a Lookback Rache would need a reducion of he iniial annual guaranee by 66 basis poins o 4.32%. If a richer rache mechanism is provided such as he Remaining WBB Rache, he guaranee needs o be reduced o 3.61%. Abou he same annual guaranee (3.62%) can be provided if no rache is provided bu a Performance Bonus is paid ou annually. Throughou our analyses, he Remaining WBB Rache and he Performance Bonus allow for abou he same annual guaranee. However, for lower volailiies, he Remaining WBB Rache seems o be less valuable han he Performance Bonus and herefore allows for higher guaranees while for higher volailiies he Performance Bonus allows for higher guaranees. Thus, he relaive impac of volailiy on he price of a GLWB depends on he chosen produc design and appears o be paricularly high for rache ype producs (II and III). This can also observed comparing he No Rache case wih he Lookback Rache. While when volailiy is increased from 15% o 25% for he No Rache case, he fair guaraneed wihdrawal decreases by jus over half a percenage poin from 5.26% o 4.7%, i decreases by almos a full percenage poin from 4.8% o 3.85% in he Lookback Rache case (if no surrender is assumed). The reason for his is ha for he producs wih rache, high volailiy leads o a possible lock in of high posiive reurns in some years and hus is a raher valuable feaure if volailiies are high. If he insurance company assumes some deerminisic surrender probabiliy when pricing GLWBs, he guaranees increase for all model poins observed. The increase of he annual guaranee is raher similar over all produc ypes and volailiies. The annual guaranee increases by around basis poins if he surrender assumpion from Table 2 is made and increases by anoher 20 basis poins if his surrender assumpion is doubled. 13

14 4.1.2 Resuls for he Heson model We use he calibraion given in Table 4, where he Heson parameers are hose derived by Eraker (2004), and saed in annualized form for insance by Poulsen (2007). Parameer Numerical value r 0.04 θ κ 4.75 σ v 0.55 ρ V(0) θ Table 4: Benchmark parameers for he Heson model. One of he key parameers in he Heson model is he marke price of volailiy risk λ. Since absolue λ-values are hard o be inerpreed, in he following able we show long-run local variance and speed of mean reversion for differen parameer values of λ. Marke price of Speed of mean Long-run local volailiy risk reversion κ variance θ λ = λ = λ = λ = λ = λ = λ = Table 5: Q-parameers for differen choices of he marke price of volailiy risk facor. Higher values of λ correspond o a lower volailiy and a higher mean reversion speed while lower (e.g. negaive) values of λ correspond o high volailiies and lower speed of mean reversion. λ = 2 implies a long-erm volailiy of 19.8% and λ = -2 implies a long-erm volailiy of 25.1%. In he following able, we show he fair annual wihdrawal guaranee under he Heson model for all differen produc designs and values of λ beween -2 and 2. 14

15 Rache Mechanism Marke price of volailiy risk I (No Rache II (Lookback Rache III (Remaining WBB Rache λ = % 4.36 % 4.03 % 4.00 % λ = % 4.27 % 3.95 % 3.93 % λ = % 4.17 % 3.86 % 3.84 % λ = % 4.05 % 3.75 % 3.74 % λ = % 3.90 % 3.62 % 3.62 % IV (Performance Bonus) Table 6: Fair guaraneed wihdrawal raes for differen rache mechanisms and volailiies when no surrender is assumed. Under he Heson model, he fair annual guaraneed wihdrawal appears o be he same as under he Black-Scholes model wih a comparable consan volailiy. E.g. for λ = 0, which corresponds wih a long-erm volailiy of 22%, he fair annual guaraneed wihdrawal rae for a conrac wihou rache is given by 4.87%, exacly he same number as under he Black- Scholes model. In he Lookback Rache case, he Heson model leads o a fair guaraneed wihdrawal rae of 4.17%, he Black-Scholes model of 4.13%. For he oher wo produc designs, again, boh asse models almos exacly lead o he same wihdrawal raes. Thus, for he pricing (as opposed o hedging, see Secion 5) of GLWB benefis, he long-erm volailiy assumpion is much more crucial han he quesion wheher sochasic volailiy should be modeled or no. 4.2 Disribuion of Wihdrawals In his secion, we compare he disribuions of he guaraneed wihdrawal benefis (given he policyholder is sill alive) for each policy year and for all four differen rache mechanisms ha were presened in Secion 2. We use he Black-Scholes model for all simulaions in his chaper and assume a risk-free rae of ineres r = 4%, an underlying s drif μ = 7% and a consan volailiy of σ BS = 22 %. For all four rache ypes, we use he guaraneed wihdrawal raes derived in secion (wihou surrender). In he following figure, for each produc design we show he developmen of arihmeic average (red line), median (yellow poins), 10 h - 90 h percenile (ligh blue area), and 25 h - 75 h percenile (dark blue area) of he guaraneed annual wihdrawal amoun over ime. 15

16 value value year year I (No Rache II (Lookback Rache value value year year III (Remaining WBB Rache IV (Performance Bonus) Figure 1: Developmen of perceniles, median and mean of he guaraneed wihdrawal amoun over policy years 0 o 30 for each rache ype. Obviously, he differen considered produc designs lead o significanly differen risk/reurnprofiles for he policyholder. While he No Rache case provides deerminisic cash flows over ime, he oher produc designs differ quie considerably. Boh rache producs have poenially increasing benefis. For he Lookback Rache, however, he 25 h percenile remains consan a he level of he firs wihdrawal amoun. Thus, he probabiliy ha a rache never happens is higher han 25%. The median increases for he firs 10 years and hen reaches some consan level implying ha wih a probabiliy of more han 50% no wihdrawal incremens will ake place hereafer. Produc III (Remaining WBB Rache provides more poenial for increasing wihdrawals: For his produc, he 25 h percenile increases over he firs few years and he median is increasing for around 20 years. In he 90 h percenile, he guaraneed annual wihdrawal amoun reaches 1,500 afer slighly more han 25 years while he Lookback Rache hardly reaches 1,200. On average, he annual guaraneed wihdrawal amoun more han doubles over ime while he Lookback Rache doesn, of course his is only possible since he guaraneed wihdrawal a =0 is lower. A compleely differen profile is achieved by he fourh produc design, he produc wih Performance Bonus. Here, annual wihdrawal amouns are raher high in he firs years and 16

17 are falling laer. Afer 15 years, wih a 75% probabiliy no more performance bonus is paid, afer 25 years, wih a probabiliy of 90% no more performance bonus is paid. For all hree produc designs wih some kind of bonus, he probabiliy disribuion of he annual wihdrawal amoun is raher skewed: he arihmeic average is significanly above he median. For he produc wih Performance Bonus, he median exceeds he guaranee only in he firs year. Thus, he probabiliy of receiving a performance bonus in laer years is less han 50%. The expeced value, however, is more han wice as high. 4.3 Disribuion of Trigger Times In he following figure, for each of he producs, we show he probabiliy disribuion of rigger imes, i.e. of he poin in ime where he accoun value drops o zero and he guaranee is riggered, (if he insured is sill alive). Any probabiliy mass a =56 (i.e. age 121 which is he limiing age of he moraliy able used), refers o scenarios where he guaranee is no riggered. 18% 18% 16% 16% 14% 14% 12% 12% frequency 10% 8% frequency 10% 8% 6% 6% 4% 4% 2% 2% 0% year 0% year I (No Rache II (Lookback Rache 18% 18% 16% 16% 14% 14% 12% 12% frequency 10% 8% frequency 10% 8% 6% 6% 4% 4% 2% 2% 0% year 0% year III (Remaining WBB Rache IV (Performance Bonus) Figure 2: Disribuion of rigger imes for each of he produc designs. For he No Rache produc, rigger imes vary from 7 o over 55 years. Wih a probabiliy of 17%, here is sill some accoun value available a age 121. For his produc, on he one hand, he insurance company s uncerainy wih respec o if and when guaranee paymens have o be paid is very high; on he oher hand, here is a significan chance ha he guaranee is no 17

18 riggered a all, which reduces longeviy (ail) risk 3 from he insurer s perspecive. For he producs wih rache feaures, very lae or even no riggers appear o be less likely. The more valuable a rache mechanism is for he clien, he earlier he guaranee ends o rigger. While for he Lookback Rache sill 2% of he conracs do no rigger a all, he Remaining WBB Rache almos cerainly riggers wihin he firs 40 years. However, on average he guaranee is riggered raher lae, afer around 20 years. The leas uncerainy in he rigger ime appears o be in he produc wih Performance Bonus. While he probabiliy disribuion looks very similar o ha of he Remaining WBB Rache for he firs 15 years, rigger probabiliies hen increase rapidly and reach a maximum a =25 and 26 years. Laer riggers did no occur a all wihin our simulaion. The reason for his is quie obvious: The Performance Bonus is given by 50% of he difference beween he curren accoun value and he Remaining WBB. However, he Remaining WBB is annually reduced by he iniially guaraneed wihdrawal amoun and herefore reaches 0 afer 26 years (1 / 3.85%). Thus, afer 20 years, almos half of he accoun value is paid ou as bonus every year. This, of course, leads o a remendously decreasing accoun value in laer years. Therefore, here is no much uncerainy wih respec o he rigger ime on he insurance company s side. On he oher hand, he complee longeviy ail risk remains wih he insurer. Whenever he guaranee is riggered, he insurance company needs o pay an annual lifelong annuiy equal o he las guaraneed annual wihdrawal amoun. This is he guaranee ha needs o be hedged by he insurer. Thus, in he following secion, we have a closer look on he Greeks of he guaranees of he differen produc designs. 4.4 Greeks Wihin our Mone Carlo simulaion, for each scenario we can calculae differen sensiiviies of he opion value as defined in Secion 2.3, he so called Greeks. All Greeks are calculaed for a pool of idenical policies wih a oal single premium volume of US$100m under assumpions of fuure moraliy and fuure surrender. All he resuls shown in his secion are calculaed under sandard moraliy and no surrender assumpions. In he following figure, we chose o show differen perceniles as well as median and arihmeic average of he so called dela, i.e. he sensiiviy of he opion value as defined in wih respec o changes in he price of he underlying: 3 This risk is no modelled in our framework. 18

19 value (in millions) value (in millions) year 40 year I (No Rache II (Lookback Rache value (in millions) value (in millions) year 40 year III (Remaining WBB Rache IV (Performance Bonus) Figure 3: Developmen over ime of he perceniles of he dela for a pool of policies muliplied by he curren spo value. Firs of all, i is raher clear ha all producs hroughou do have negaive delas since he value of he guaranee increases wih falling sock markes and vice versa. Once he guaranee is riggered, no more accoun value is available and hus, from his poin on, he dela is zero. Thus, in wha follows, we call dela o be high whenever is absolue amoun is big. A ouse, he produc wihou any rache or bonus does have he bigges dela and hus he highes sensiiviy wih respec o changes in he underlying s price. The reason for his mainly is ha he guaranee is no adjused when fund prices rise. In his case, he value of he guaranee decreases much sronger han wih any produc where eiher rache lead o an increasing guaranee or a performance bonus leads o a reducion of he accoun value. On he oher hand, if fund prices decrease, he firs produc is deeper in he money since i does have he highes guaranee a ouse. Over ime, all perceniles of he dela in he No Rache case are decreasing. For producs II and III, he guaranee can never be far ou of he money due o he rache feaure. Thus dela increases in he firs few years. All perceniles reach a maximum afer en years and end o be decreasing from hen on. For he produc wih Performance Bonus, dela exposure is by far he lowes. This is 19

20 consisen wih our resuls of he previous secion where we concluded ha he uncerainy for he insurance company is he highes in he No Rache case and he lowes in he Performance Bonus case. 5 Analysis of Hedge Efficiency In his Secion, we analyze he performance of differen (dynamic) hedging sraegies, which can be applied by he insurer in order o reduce he financial risk of he guaranees (and hereby he required economic risk capial). We firs describe he analyzed hedging sraegies, before we define he risk measures ha we use o compare he simulaion resuls of he hedging sraegies, which are presened in he las par of his Secion. 5.1 Hedge Porfolio We assume ha he insurer has sold a pool of policies wih GLWB guaranees. We denoe by Ψ( ) he opion value for ha pool, i.e. he sum of he opion values V defined in Secion 2.3 of all policies. We assume ha he insurer canno influence he value of he guaranee Ψ( ) by changing he underlying fund (i.e. changing he fund's exposure o risky asses or forcing he insured o swich o a differen, e.g. less volaile, fund). We furher assume ha he insurer invess he guaranee fees in some hedge porfolio Π Hedge ( ) and performs some hedging sraegy wihin his hedging porfolio. In case a guaranee is riggered, guaraneed paymens are made from ha porfolio. Thus, Hedge Π ( = Ψ(,,...) + Π ( (24) is he insurer s cumulaive profi/loss (in wha follows someimes jus denoed as insurer s profi semming from he guaranee and he corresponding hedging sraegy. The following hedging sraegies aim a reducing he insurer's risk by implemening cerain invesmen sraegies wihin he hedge porfolio Π Hedge ( ). Noe ha he value Ψ( ) of he pool of policies a ime does no only depend on he number and size of conracs and he underlying fund's curren level, bu also on several rerospecive facors, such as he hisorical prices of he fund a previous wihdrawal daes, and on model and parameer assumpions. The insurer s choice of model and parameers can also have a significan impac on he hedging sraegies. Therefore, we will differeniae in he following beween he hedging model ha is chosen and used by he insurer, and he daa-generaing model ha we use o simulae he developmen of he underlying and he marke prices of European call and pu opions. This allows us, e.g., o analyze he impac on he insurer s risk siuaion if he insurer bases pricing and hedging on a simple Black-Scholes model (hedging model) wih deerminisic volailiy whereas in realiy (daa-generaing model) volailiy is sochasic. We assume he value of he guaranee o be marked-o-model, where he same model is used for valuaion as he insurer uses for hedging. All oher asses in he insurer's porfolio are markedo-marke, i.e. heir prices are deermined by he (exernal) daa-generaing model. We assume ha, addiional o he underlying ) and he money-marke accoun B( ), a marke for European plain vanilla opions on he underlying exiss. However, we assume ha only opions wih limied ime o mauriy are liquidly raded. As well as he underlying and he money-marke accoun, we assume he opion prices (i.e. he implied volailiies) o be driven by he daa-generaing model, and presume risk-neuraliy wih respec o volailiy risk, i.e. he marke price of volailiy is se o zero in case he Heson model is used as daa- 20

21 B generaing model. Addiionally, we assume he spread beween bid and ask prices/volailiies o be zero. For all considered hedging sraegies we assume he hedging porfolio o consis of hree asses, whose quaniies are rebalanced a he beginning of each hedging period: a posiion of quaniy Δ ) in he underlying, a posiion of quaniy ΔB( ) in he money-marke accoun and a posiion of ΔX( ) in a 1-year ATMF sraddle (i.e. an opion consising of one call and one pu, boh wih one year mauriy and a he money wih respec o he mauriy s forward, ATMF). We assume he insurer o hold he posiion in he sraddle for one hedging period, hen sell he opions a hen-curren prices, and se up a new posiion in a hen 1-year ATMF sraddle. For each hedging period, he new sraddle is denoed by X( ). We assume ha he porion of he hedge porfolio ha was no invesed in eiher ) or X( ) is invesed in (or borrowed from) he money marke. Thus, he hedge porfolio a ime has he form Π Hedge ( = Δ ( + Δ ( B( + Δ ( X ( S B X, (25) where Δ B Π ( : = Hedge ( Δ S ( Δ B( X ( X (. (26) 5.2 Dynamic Hedging Sraegies For boh considered hedging models, Black-Scholes and Heson, we analyze hree differen ypes of (dynamic) hedging sraegies. No Hedge (NH) The firs sraegy simply invess all guaranee fees in he money-marke accoun. The sraegy is obviously idenical for boh models. Dela Hedge (D) The second ype of hedging sraegy uses a posiion in he underlying in order o immunize he porfolio agains small changes in he underlying s level. In he Black-Scholes framework wihou ransacion coss, such a posiion is sufficien o perform a perfec hedge. In realiy however, ime-discree rading and ransacion coss cause imperfecions. Using he Black-Scholes model as hedging model, in order o immunize he porfolio agains small changes in he underlying's price (i.e. o aain dela-neuraliy), Δ S is chosen as he dela of Ψ( ), i.e. he parial derivaive of Ψ( ) wih respec o he underlying. While dela hedging under he Black-Scholes model (given he ypical assumpions), consiues a heoreically perfec hedge, i does no under he Heson model. This leads o 21

22 (locally) risk minimizing sraegies ha aim o minimize he variance of he insananeous change of he porfolio. Under he Heson model 4, he problem ( Π( ) min, Δ IR, Δ 0 var d = (27) S X has he soluion (see e.g. Ewald, Poulsen and Schenk-Hoppe, 2007) Δ S Ψ ( = Heson (,, V (,...) ρσ v + Ψ Heson (,, V (,...). (28) V ( To keep noaion simple, his (locally) risk minimizing sraegy under he Heson model is also referred o as dela hedge. Dela and Vega (DV) The hird ype of hedging sraegies incorporaes he use of he sraddle X( ), exploiing is sensiiviy o changes in volailiy for he sake of neuralizing he porfolio s exposure o changes in volailiy. Under he Black-Scholes model, volailiy is assumed o be consan; herefore using i o hedge agains a changing volailiy appears raher counerinuiive. Neverheless, following Taleb (1997), we analyze some kind of ad-hoc vega hedge in our simulaions, ha aims a compensaing he deficiencies of he Black-Scholes model: For performing he vega hedge, we do no compue he Black-Scholes vega of he guaranee Ψ( ) and compare i o he corresponding Black-Scholes vega of he opion X( ), bu, insead, we will be using he socalled modified Vega of Ψ( ) for comparison. Since all mauriies canno be expeced o reac he same way o changes in oday s volailiy, he modified Vega applies a differen weighing o he respecive vega of each mauriy. We use he inverse of square roo of ime as simple weighing mehod and use he mauriy of he hedging insrumen X( ), i.e. one year, as benchmark mauriy. The modified vega of Ψ( ) a a policy calculaion dae τ hen has he form ModVega( τ ) = T 1 ν = τ + 1 τ (29) where he ν denoe he respecive Black-Scholes vega of each discouned fuure cash flow of he pool of policies. This deermines he opion posiion (i.e. he quaniy of sraddles) required o achieve vega neuraliy. Under he Heson model, we compare he wo derivaives of Ψ( ) and X( ) wih respec o he curren local variance V( ) and hen analogously deermine he opion posiion required o achieve vega neuraliy. Of course, under boh hedging models, he posiion in he underlying mus be adjused for he dela of he opion posiion Δ X X( ). 4 Noe ha a (ime-coninuously) Dela-hedged porfolio under he Black-Scholes model is already risk-free. Therefore for he Black-Scholes model, he Dela-hedging sraegy coincides wih he locally risk minimizing sraegy. 22

23 The hedge raios for all hree sraegies used in our simulaions are summarized in Table 7 for he Black-Scholes model, and in Table 8 for he Heson model. (NH) ΔS Δ 0 0 X (D-BS) Ψ BS (,, σ BS,...) 0 (DV-BS) Ψ BS (,, σ BS,...) Δ X X BS (,, σ BS,...) ModVega( X BS (,, σ X V ( BS,...) Table 7: Hedge raios for he differen sraegies if he Black-Scholes model is used as hedging model. (NH) (D-H) ΔS Δ 0 0 Heson Heson Ψ (,, V (,...) ρσ v Ψ (,, V (,...) + 0 V ( X (DV-H) Ψ Heson ρσ v Ψ + (,, V (...) Δ Heson X Heson (,, V (...) Δ V ( X X (,, V (...) X Heson (,, V (...) V ( Ψ Heson (,, V (...) V ( X Heson (,, V (...) V ( Table 8: Hedge raios for he differen sraegies if he Heson model is used as hedging model. Addiionally, for all dynamic hedging sraegies (Dela and Dela-Vega), we assume ha he hedger buys 1-year European pu opions a each policy anniversary such ha he possible guaranee paymens for he nex policy anniversary are fully hedged by he pu opions (assuming surrender and moraliy raes are deerminisic and known). This sraegy aims a avoiding having o hedge an opion wih shor ime o mauriy and hence having o deal wih a poenially rapidly alernaing dela (high gamma) if he opion is near he srike. This is possible for all four rache mechanisms, since he guaraneed wihdrawal amoun is known one year in advance. For all considered hedging sraegies we assume ha he hedge porfolio is rebalanced on a monhly basis. 5.3 Simulaion Resuls We use he following hree raios o compare he differen hedging sraegies, all of which will be normalized as a percenage of he sum of he premiums paid o he insurer a =0: [ ] rt Π E P e T, he discouned expecaion of he final value of he insurer s final profi under he real-world measure P, where T= ω-x 0. This is a measure for he insurer s expeced profi and consiues he performance raio in our conex. A value of 1 means ha, in expecaion, for a single premium of 100 paid by he clien, he 23