Fee Structure and Surrender Incentives in Variable Annuities

Transcription

1 Fee Srucure and Surrender Incenives in Variable Annuiies by Anne MacKay A hesis presened o he Universiy of Waerloo in fulfillmen of he hesis requiremen for he degree of Docor of Philosophy in Acuarial Science Waerloo, Onario, Canada, 2014 c Anne MacKay 2014

2 Auhor s Declaraion I hereby declare ha I am he sole auhor of his hesis. This is a rue copy of he hesis, including any required final revisions, as acceped by my examiners. I undersand ha my hesis may be made elecronically available o he public. ii

3 Absrac Variable annuiies (VAs) are invesmen producs similar o muual funds, bu hey also proec policyholders agains poor marke performance and oher risks. They have become very popular in he pas weny years, and he guaranees hey offer have grown increasingly complex. Variable annuiies, also called segregaed funds in Canada, can represen a challenge for insurers in erms of pricing, hedging and risk managemen. Simple financial guaranees expose he insurer o a variey of risks, ranging from poor marke performance o changes in moraliy raes and unexpeced lapses. Mos guaranees included in VA conracs are financed by a fixed fee, paid regularly as a fixed percenage of he value of he VA accoun. This fee srucure is no ideal from a risk managemen perspecive since he resuling amoun paid ou of he fund increases as mos guaranees lose heir value. In fac, when he accoun value increases, mos financial guaranees fall ou of he money, while he fixed percenage fee rae causes he fee amoun o grow. The fixed fee rae can also become an incenive o surrender he variable annuiy conrac, since he policyholder pays more when he value of he guaranee is low. This incenive deserves our aenion because unexpeced surrenders have been shown o be an imporan componen of he risk faced by insurers ha sell variable annuiies (see Kling, Ruez, and Russ (2014)). For his reason, i is imporan ha he surrender behaviour be aken ino accoun when developing a risk managemen sraegy for variable annuiy conracs. However, his behaviour can be hard o model. In his hesis, we analyse he surrender incenive caused by he fixed percenage fee rae and explore differen fee srucures ha reduce he incenive o opimally surrender variable annuiy conracs. We inroduce a sae-dependen fee, paid only when he VA accoun value is below a cerain hreshold. Inegral represenaions are presened for he price of differen guaranees under he sae-dependen fee srucure, and parial differenial equaions are solved numerically o analyse he resuling impac on he surrender incenive. From a heoreical poin of view, we sudy cerain condiions ha eliminae he incenive o surrender he VA conrac opimally. We show ha he fee srucure can be modified o design conracs whose opimal hedging sraegy is simpler and robus o differen surrender behaviours. The las par of his hesis analyzes a differen problem. Group self-annuiizaion schemes are similar o life annuiies, bu par, or all, of he invesmen and longeviy risk is borne by he annuian hrough periodic adjusmens o annuiy paymens. While hey may decrease he price of he annuiy, hese adjusmens increase he volailiy of he iii

4 paymen paerns, making he produc risky for he annuian. In he las chaper of his hesis, we analyse opimal invesmen sraegies in he presence of group self-annuiizaion schemes. We show ha he opimal sraegies obained by maximizing he uiliy of he reiree s consumpion may no be opimal when hey are analysed using differen merics. iv

5 Acknowledgemens Firs, I would like o hank my advisors, Professors Carole Bernard and Mary Hardy, for he research ideas ha form he backbone of his hesis. I am graeful for he guidance, he advice and he encouragemens hey have provided me over he pas hree years. I wan o hank Carole for always being available, wherever she was in he world. Her enhusiasm and her energy were a grea moivaion, and wihou her I would no have been able o complee my degree so quickly. Mary s dedicaion o acuarial science is a consan source of inspiraion. I am very hankful for all he help, advice and commens she has provided me over he pas hree years. These have been insrumenal in he expansion of my research horizons and professional growh. I am also graeful o Professor Phelim Boyle for giving me he opporuniy o collaborae on a projec ha became a chaper of his hesis. I am very lucky o have had he chance o learn from him, hrough lecures, discussions, and while wriing his hesis. Professor David Saunders has also been very helpful hroughou his process. I especially appreciae his suggesions, which grealy improved my work. I wan o hank him for his ime over he pas year. I am also indebed o Professors Jusin Wan and Pierre Devolder, who graciously acceped o be par of my examining commiee. I wan o hank hem for heir ime and heir helpful commens. Professor Parice Gaillardez, my masers advisor, has had an imporan influence on my pah as a graduae suden. I wan o hank him for his precious advice and guidance. I also wan o acknowledge all he professors who have augh me since I sared graduae school, as hey all have, in a way, conribued o his hesis. In paricular, I had very helpful discussions wih Professors Adam Kolkiewicz and Bin Li, o whom I am very hankful. I also appreciae all he suppor and advice provided by Professor Jose Garrido. All he commens and suggesions I have received when I presened my research a differen conferences and seminars have also grealy improved he qualiy of his hesis. I am graeful for he commens provided by Dr. Jochen Russ and Dr. Daniel Bauer. I am hankful o have had he chance o know he PhD sudens a Waerloo, who are supporive and inspiring. I also wan o hank he adminisraive saff in saisics and acuarial science, especially Mary Lou Dufon, wihou whom he deparmen would no be he same. Finally, I wan o hank my friends, who always believed in me. In paricular, I am graeful o Mélina and Maciek for he advice and encouragemen. v

6 Thank you o my parens for giving me all he opporuniies I have had, and for eaching me o always aim high. Thanks o he whole family for always being here, if no in person, a leas on he phone. Merci d êre là pour moi. And hank you Chris for your unwavering suppor and for all he ediing. vi

7 Table of Conens Lis of Tables Lis of Figures xi xiii 1 Noaion and seing Marke model Variable annuiy conracs Mauriy benefi Opimal surrender Fair fee Oher assumpions Opimal surrender under consan fee srucure Inroducion Seing Fair Fee for he European Benefi Surrender Opion Derivaion of he opimal surrender boundary Alernaive derivaion of he opimal surrender boundary Pah-dependen payoff Numerical Examples vii

8 2.5.1 Opimal Boundary for he VA sudied in Secion Opimal Boundary for he VA sudied in Secion Concluding Remarks Appendices 39 2.A Opimal Surrender Region for GMAB B Las sep of he proof of Proposiion C Opimal Surrender Region wih Asian Benefis Sae-dependen fees for variable annuiy guaranees Inroducion Pricing wih sae-dependen fee raes Noaion Pricing he VA including guaranees wih sae-dependen fee raes Examples Sae-dependen fee raes for a VA wih GMMB Sae-dependen fee raes for a VA wih GMDB Numerical Resuls Analysis of he Surrender Incenive Consan Fee Sae-Dependen Fee Model Risk Concluding Remarks Appendices 71 3.A Proof of Proposiion B Deails for he GMMB price viii

9 4 Opimal surrender under he sae-dependen fee srucure Inroducion Pricing he GMMB Marke and Noaion Pricing VAs in he presence of a sae-dependen fee and surrender charges Numerical Examples Solving he PDE numerically Numerical Resuls Theoreical analysis of he surrender incenive Surrender incenive for large accoun values when β < Minimal surrender charge o eliminae he surrender incenive Dynamic hedging Calculaion of he ne hedged loss a mauriy Calculaion of Modeling policyholder behaviour Resuls Concluding Remarks Appendices A Proof of Equaion (4.1) B Proof of Equaion (4.3) Opimal surrender under deerminisic fee srucure Inroducion Assumpions and Model Variable Annuiy Benefis ix

10 5.3 Valuaion of he surrender opion Theoreical Resul on Opimal Surrender Behaviour Valuaion of he surrender opion using PDEs Numerical Example Numerical Resuls Concluding Remarks Group Self-Annuiizaion Schemes: How opimal are he opimal sraegies? Inroducion and Moivaion Variable Paymen Life Annuiies Adjusmen facor wihou moraliy Adjusmen facor wih moraliy The Opimizaion Problem The model, assumpions and noaion Solving he Opimizaion Problem Resuls of he Opimizaion Problem Assumpions and Parameers Numerical Resuls of Uiliy Maximizaion Exploring differen assumpions Opimal Sraegies in an Open Group of Reirees Conclusion Appendices A Solving he opimizaion problem using dynamic programming A.1 Simplifying he opimizaion problem by normalizing B Parameer ses and associaed numerical resuls References 155 x

11 Lis of Tables 3.1 Fair fee raes for he GMMB and GMDB wih respec o mauriy when β = G Fair fee raes for he 10-year GMMB wih respec o volailiy when β = G Regime-swiching log-normal parameers used for Mone Carlo simulaions Fair fee raes in he Regime Swiching model and in he Black-Scholes model, fees paid monhly and coninuously Fair fee for differen VA conracs wih T = 10, r = 0.03, and σ = Saisics of he insurer s ne hedging loss Value of he surrender opion for a 10-year variable annuiy conrac for differen fee srucures Value of he surrender opion for 5-year and 15-year variable annuiy conracs for differen fee srucures Parameers used o obain numerical resuls for he opimizaion problem Opimal invesmen a reiremen for differen ineres margins λ as % of wealh a reiremen Opimal invesmen for differen ineres margins λ, when γ = Opimal invesmen a reiremen, wih associaed iniial consumpion in % of fund, for differen ineres margins λ when α V = Composiion of reiree group a = 0 when ω V = Saisics of he disribuion of he annual paymen as a percenage of he iniial consumpion for differen invesmen sraegies xi

12 6.7 Definiion of parameer ses Saisics of he disribuion of he annual paymen as a percenage of he iniial paymen for differen ses of parameers Probabiliies of hiing he povery level before cerain ages for differen ses of parameers xii

13 Lis of Figures 2.1 Opimal surrender boundary in he consan fee case: sensiiviy analysis Opimal surrender surface for a pah-dependen guaranee Opimal surrender boundaries for a pah-dependen guaranee No arbirage price of a 10-year GMMB as a funcion of c Sensiiviy of fair fee raes for GMMB wih respec o volailiy and conrac erm Sae-dependen fee raes for GMMB as a funcion of he fee barrier loading Difference beween he value of he GMMB a mauriy T and he expeced value of he discouned fuure fees (consan fee) Difference beween he value of he GMMB a mauriy T and he expeced value of he discouned fuure fees (sae-dependen fee, β = G) Difference beween he value of he GMMB a mauriy T and he expeced value of he discouned fuure fees (sae-dependen fee, β = 1.4G) Surrender charge funcions κ Opimal surrender region, κ = Surrender charge and opimal surrender region, λ = 0.5, κ = 1 e κ(10 ) Surrender charge and opimal surrender region, κ = 0.05(1 /10) Opimal surrender boundary for differen surrender behaviours, κ = Surrender charges and opimal surrender boundary for differen values κ Opimal surrender boundary for differen values β xiii

14 4.8 Minimal surrender charge o eliminae surrender incenive and values a which i was calculaed Opimal surrender boundary, T = 10, consan and deerminisic fees Opimal surrender boundary, T = 15, consan and deerminisic fees simulaed pahs of he VLA Disribuion of he annual paymen during reiremen, λ = Disribuion of he annual paymen during reiremen, λ = Disribuion of he annual paymen during reiremen, ω F = Disribuion of he annual liquid wealh during reiremen, ω F = 0, i = Disribuion of he annual paymen during reiremen, ω V = Disribuion of he annual paymen during reiremen, ω V = 0, reiree group open xiv

15 Inroducion Overview of he hesis This hesis is divided ino wo pars. The firs par conains Chapers 1 o 5, and is concerned wih he impac of differen fee srucures on he incenive o surrender variable annuiy conracs. The second par is Chaper 6. In his chaper, we analyse he place of group self-annuiizaion schemes in he porfolio of a new reiree who seeks o maximize he uiliy of his consumpion, and sudy he resuling paymen paerns. Fee srucure and he surrender incenive in variable annuiies Inroducion and moivaion Over he pas 15 years, equiy-linked insurance producs have grown in populariy. By offering paricipaion in marke performance while proecing he iniial invesmen, hey are very aracive o many ypes of invesors. While hey used o be considered almos riskless, equiy-linked producs evenually proved o carry heir share of risk, especially during he pas financial crisis. This coincided wih a rapid growh in he lieraure on he subjec. Equiy-linked insurance producs are comprised of differen ypes of conracs ha differ in heir feaures, bu all offer financial guaranees ha may expose heir issuer o differen ypes of risk. In his hesis, we focus on variable annuiies (VAs), which are also referred o as segregaed funds in Canada. They are similar o muual funds, bu have a fixed erm and guaraneed minimum paymens a he ime of deah of he policyholder or a mauriy. These guaranees, along wih he ax advanages hey bring, have made variable annuiies 1

16 very popular. However, hey do presen some challenges, in paricular in erms of design, pricing, valuaion and risk managemen; of course, each of hese is inricaely relaed o he ohers (Hardy (2003), Boyle and Hardy (2003), Palmer (2006), Coleman, Kim, Li, and Paron (2007)). Nowadays, he guaranees ha can be added o variable annuiy conracs are numerous. The range of guaraneed minimum benefis is ofen referred o as GMxBs (Bauer, Kling, and Russ (2008)), o cover GMDBs (deah benefis), GMMBs (mauriy), GMIBs (income) and so on. The more complex guaranees evolved o disinguish he producs from heir compeiors and o reain he policyholders. Addiional guaraneed wihdrawal riders can also be added o a ypical variable annuiy conrac. In his hesis, we focus on GMMBs and apply some resuls o GMDBs. Noe ha GMMBs can also be referred o as guaraneed minimum accumulaion benefis (GMABs), and he wo erms will be used inerchangeably hroughou his hesis. Our resuls could evenually be exended o oher ypes of guaranees. In many cases, he financial guaranees embedded in VAs are analogous o financial opions wrien on sock or indices. Techniques developed o price financial opions have ofen been used o analyse he value of he guaranees embedded in variable annuiies and oher ypes of equiy-linked insurance producs. The firs o do so were Boyle and Schwarz (1977), while Boyle and Hardy (1997) and Barbarin and Devolder (2005) compare and combine acuarial and financial pricing mehods. There are however some differences beween he way financial opions and guaranees embedded in VAs are financed. In paricular, financial opions sold on he marke are paid for upfron, whereas VA guaranees are usually funded via a fee paid ou of he VA accoun, which is also he asse underlying he financial guaranee. The fee is ypically se as a fixed percenage of he accoun value and is paid regularly hroughou he conrac. I is similar o he managemen fee paid ou of a muual fund o cover invesmen and oher expenses. The fee rae charged on VA accouns is usually higher han in he muual fund case, because i also covers he differen financial guaranees. The srucure of he managemen fee in VAs creaes a misalignmen beween he income and he cos of he opion. When he fund value is high, large fees are received, bu he opion value is low because i has a small probabiliy of being riggered a mauriy. The opposie happens when he fund value is low. This discrepancy represens an incenive for he policyholder o surrender he conrac when he guaranee is well ou-of-he-money (see Bauer, Kling, and Russ (2008) and Milevsky and Salisbury (2001) for example). In fac, if he fund value is high enough ha he guaranee has a very low probabiliy of being in-he-money a mauriy, hen here is lile poin in coninuing o pay for ha guaranee. In ha case, he policyholder should lapse and buy a new policy wih he fund value. This 2

17 new policy would be a-he-money for a similar cos. Alhough he mauriy would be exended, i may sill be an opimal sraegy for he policyholder (see Moenig and Bauer (2012)). To reduce he surrender incenive, mos VA conracs include surrender charges during a leas he early par of he conrac duraion. The surrender charge reduces he payoff received on surrender, so he policyholder does no receive he full value accumulaed in he accoun. This surrender charge is also in place o recover he high expenses relaed o he sale of he VA conrac. While his fee does give he policyholder an incenive o remain in he conrac, here are many siuaions where i is opimal o surrender, even afer aking he surrender charge ino accoun. Lieraure abou he surrender opion Over he pas 20 years, numerous papers have been concerned wih pricing differen equiylinked insurance conracs. In paricular, many auhors have analysed he impac of marke assumpions on he price of equiy-linked producs. For example, Lin and Tan (2003) and Gaillardez (2008) use sochasic ineres rae models o price equiy-indexed annuiies, while Kling, Ruez, and Ruß (2011) sudy he impac of sochasic volailiy on VAs. In his hesis, since we wan o focus on he effec of he fee srucure on he surrender incenive, we mosly consider a Black-Scholes marke model. The surrender problem has been reaed in differen ways in he lieraure. Noneheless, all agree ha unexpeced lapses represen a significan risk for he insurer (see Kling, Ruez, and Russ (2014)). This is why policyholder behaviour needs o be accouned for when VA conracs are priced. Differen assumpions can be used o model lapse behaviour, ranging from a simple deerminisic lapse raes o more sophisicaed models, like De Giovanni (2010) s raional expecaion and Li and Szimayer (2014) s limied raionaliy. Under mos of hese assumpions, he policyholder is no able o assess he exac risk-neural value of he conrac. In addiion, exogenous facors can affec her decision. Anoher approach o modeling policyholder behaviour is o assume ha he policyholder is perfecly raional and surrenders he conrac as soon as i is opimal o do so from a financial perspecive. Under his assumpion, he surrender opion can be viewed as an American opion ha can be exercised a any ime before mauriy (see Grosen and Jørgensen (2000)). The value obained for he VA conrac using his assumpion represens an upper bound for is price, as i considers he wors-case scenario for he insurer. Even if he value hus obained is no used as he final price, i sheds ligh on he inrinsic value of surrender opion and on he risk i bears. Furhermore, while here are many oher 3

18 facors why policyholders lapse, Knoller, Krau, and Schoenmaekers (2011) show ha he moneyness of he embedded guaranee plays a role in surrender decisions. This is no dissimilar o surrendering opimally when he guaranee is ou-of-he-money. They also find ha financial lieracy increases sensiiviy owards he moneyness. Pricing a VA conrac assuming opimal surrender sraegy can be jusified if he insurer wans o cover he wors-case scenario. However, opimal surrenders are more complex o hedge and o manage. For his reason insurers can be emped o ignore lapse risk or o make simplifying assumpions ha do no reflec acual lapse behaviour. These flawed assumpions can significanly reduce he efficiency of a hedging sraegy. For example, Kling, Ruez, and Russ (2014) show ha hedging effeciveness can be hreaened when lapse behaviour assumpions fail o predic acual surrenders. Thus, early surrenders are an imporan componen of he risk faced by issuers of VA conracs. Alhough mos variable annuiy conracs charge a consan fee as a percenage of he accoun value o cover embedded guaranees, many auhors assume ha hese benefis are covered by he iniial premium (see, for example, Grosen and Jørgensen (2002), Bacinello (2003a), Bacinello (2003b), Siu (2005), Bacinello, Biffis, and Millossovich (2009), Bacinello, Biffis, and Millossovich (2010), Bernard and Lemieux (2008)). However, as discussed earlier, he managemen fee has an impac on he surrender incenive and should be considered when he policyholder is assumed o lapse opimally. This is menioned by Bauer, Kling, and Russ (2008) and Milevsky and Salisbury (2001). In paricular, Milevsky and Salisbury (2001) argue ha surrender charges are necessary o complee he marke; hey allow he insurer o fairly price he VA conrac and hedge i appropriaely. Under he raionaliy assumpion, he surrender opion embedded in a VA conrac can be analysed wih ools developed for American opions. A vas lieraure has been developed on his opic, so we will no ry o cover i all here. American opions can be priced in many differen ways, each of which has is advanages and disadvanages. In paricular, Kim and Yu (1996) use no-arbirage argumens o derive an inegral form for he value of he early exercise premium. In his hesis, we apply his echnique o VA conracs o isolae he value of he surrender opion and o undersand he differen facors affecing is value. This value can also be obained hrough he mehod developed in Kim (1990). American opions can also be priced using parial differenial equaions (PDEs), such as in Carr, Jarrow, and Myneni (1992). PDEs are also used in he conex of equiy-linked insurance producs (for example in Dai, Kuen Kwok, and Zong (2008), Chen, Vezal, and Forsyh (2008) and Belanger, Forsyh, and Labahn (2009)). In his hesis, we use hem o assess he surrender incenive when he inegral represenaion canno be obained. 4

19 Exploring new fee srucures Chaper 1 of his hesis presens he noaion and conceps used hroughou he firs five chapers of he hesis. I reviews cerain noions of risk-neural pricing, fair pricing and opimal surrender. I also describes he VA conracs considered in his hesis. In Chaper 2, we propose a echnique o isolae he value of he surrender opion in VA conracs wih differen ypes of accumulaion benefis. Relying on he no-arbirage argumens presened by Kim and Yu (1996), we develop an inegral represenaion for he value added by he possibiliy o surrender a VA conrac early. From his represenaion, i becomes obvious ha opimal surrender incenives depend on he value of he fee paid when he accoun value is high above he guaraneed level. In oher words, decreasing he fee paid when he mauriy guaranee is ou-of-he-money would reduce he surrender incenive. Wih his resul in mind, we inroduce a new sae-dependen fee srucure in Chaper 3. Under his new seing, he fee is sill paid as a consan percenage of he VA accoun, bu only when he value of his accoun is below a cerain hreshold. Chaper 3 explores he fair fee for he accumulaion benefi wih his new fee srucure. Using he appropriae change of measure and he necessary rivariae densiy derived in Karazas and Shreve (1984), we obain an inegral represenaion for he value of he mauriy benefi. This allows us o perform differen sensiiviies on he price of he conrac, and o do a preliminary analysis of he surrender incenive. Chaper 4 sudies he effec of he sae-dependen fee on he surrender incenive. Since he problem now includes opimal surrenders, i is no longer possible o obain an analyic formula for he value of he conrac. Insead, he price is obained numerically by solving a parial differenial equaion using finie difference schemes. This also allows us o visualise he opimal surrender region for a VA wih a simple GMAB a mauriy. In paricular, we show ha he sae-dependen fee combined wih early surrender charges is effecive in reducing he opimal surrender incenive. In his chaper, we also explain how o design a markeable conrac for which he opimal behaviour is o keep i unil mauriy. In oher words, we eliminae he surrender incenive compleely, hus grealy reducing he complexiy of he sraegy required o hedge he opimal lapse behaviour. By analysing he hedging errors resuling from he applicaion of such a sraegy, we demonsrae ha i is effecive a miigaing lapse risk. In Chaper 5, we modify he fee srucure and sudy he case where he fee is paid as a fixed amoun (insead of a fixed percenage of he accoun). This fee srucure can be seen as a funcion of he accoun value. In paricular, when he mauriy guaranee is 5

20 ou-of-he-money, he fee rae paid by he policyholder is smaller han when he guaranee is in-he-money. We show ha his fixed fee affecs he shape of he opimal surrender region, also reducing he surrender incenive. Opimal invesmen sraegies a reiremen in he presence of group self-annuiizaion schemes While he firs five chapers are concerned wih a produc ha is ypically used for prereiremen savings, Chaper 6 explores pos-reiremen invesmen. Group-self annuiizaion schemes can be compared o life annuiies wih variable paymens, which depend on he invesmen and moraliy experience of he group. They are aracive o pension plan sponsors because hey ransfer invesmen and longeviy risk o he reirees. For he same reason, hey can resul in very volaile paymen paerns, which is paricularly risky if hey consiue he main source of income for reirees. In Chaper 6, we assume ha a reiree seeks o maximize he expeced uiliy of his consumpion by invesing in one or more of he following: A risk-free bank accoun A balanced fund A fixed life annuiy A self-annuiizaion scheme. Our resuls show ha fixed life annuiies sill have a place in a reiree s porfolio, even when heir price includes a margin for invesmen and longeviy risk. Using a differen meric, we also show ha uiliy maximizaion does no necessarily yield he mos appropriae invesmen sraegies for reirees. 6

21 Chaper 1 Noaion and seing In his chaper, we inroduce he marke model used in he nex four chapers. We also review some noions of financial and acuarial mahemaics and define he main conceps discussed hroughou his hesis. 1.1 Marke model We consider a variable annuiy conrac wih mauriy T and assume ha is accoun racks he value of an index {S } 0 T wih real-world (P-measure) dynamics ds = S ( µd + σdw P ), where W P is a P-Brownian moion. We work on a filered probabiliy space (Ω, F T, F, P), where F = σ({ws P } 0 s ) is he filraion induced by he Brownian moion, and F = {F } 0 T. For 0 T, we le F be he value of he VA accoun a ime and denoe by C he oal fees paid beween imes 0 and. We assume ha he fee paid a ime can be funcion of ime and of he accoun value, so is dynamics are given by dc = F c(, F )d, where C 0 = 0 and c(, F ) represens he fee rae. Since he managemen fee is paid ou of 7

22 he VA accoun, he process {F } 0 T follows df = F S ds dc = F ( (µ c(, F ))d + σdw P The differen fee srucures used in his hesis are as follows: In Chaper 2, we assume ha he fee is paid coninuously ou of he fund a a rae c, so c(, F ) = c. In Chaper 3 and 4, we consider a fee ha is paid only when he accoun value is below a level β. In his case, c(, F ) = c1 {F<β}, where 1 A is he indicaor funcion of he se A. In Chaper 5, we explore a fee se as he sum of a deerminisic amoun p a ime and a fixed percenage c of he VA accoun F. Thus, c(, F ) = c + p F. Throughou his hesis, we consider ha he assumpions of he Black-Scholes model (see Black and Scholes (1973)) hold. In paricular, as demonsraed by Harrison and Pliska (1981), his means ha here exiss an unique equivalen risk-neural measure Q under which discouned price processes are maringales. Under his measure, he dynamics of he VA accoun are given by ( ) df = F (r c(, F ))d + σdw Q, where W Q is a Q-Brownian moion. In he subsequen chapers, o simplify he noaion, we will drop he superscrip indicaing under which measure he Brownian moion is defined whenever he conex is clear. ). 1.2 Variable annuiy conracs Mauriy benefi In his hesis, we focus on a policy wih a simple guaranee effecive a mauriy T. The payoff of he conrac is he maximum beween a pre-deermined amoun G and φ(t, F ) a funcion which may depend on he enire pah of he fund value. In Chaper 2, we discuss 8

23 payoffs ha can be pah-dependen. In paricular, F can be he average of he fund value over ime. In he subsequen chapers, we resric ourselves o he case φ(t, F ) = F T. In all cases, he payoff of he VA a mauriy is hus max(g, φ(t, F )). We ypically express he pre-deermined amoun G as he iniial premium P = F 0 rolled-up o T a a conservaive rae g < r, so ha G = F 0 e gt. We denoe he value of he mauriy benefi a ime by U(, F ) and define i as he risk-neural expecaion of he payoff discouned from T o : U(, F ) = E Q [ e r(t ) max(g, φ(t, F )) ] Opimal surrender In Chaper 3, we focus on pricing he mauriy benefi only. However, in he oher chapers, we also consider he value added by he possibiliy of surrendering he conrac before is mauriy. This requires us o define he concep of opimal surrender and opimal surrender region, which is done here. These conceps, which will be reviewed in he nex few chapers, are analogous o he ones used in he lieraure on American opions (Wu and Fu (2003), for example). We denoe he price of he VA conrac by V (, F ). If his conrac is surrendered a ime [0, T ), he policyholder receives (1 κ )φ(, F ), where κ [0, 1) is he surrender charge a and φ(, ) is a funcion of he fund value or of is pah o ime. Furher assumpions will be made on he form of he surrender charge in he nex chapers. We also le τ be a sopping ime wih respec o F and denoe by T he se of all sopping imes τ greaer han and bounded by T. Denoe he coninuaion value of he VA conrac a ime by V (, F ) and define i by V (, x) = sup τ T E [ e r(τ ) ψ(τ, F ) F = x ], where ψ(τ, F ) denoes he benefi received if he conrac is surrendered or expires a τ. Tha is, { (1 κ )x, if 0 < T ψ(, x) = max(x, G), if = T. 9

24 Heurisically, he coninuaion value is he discouned maximum value ha he policyholder can expec o receive if she holds he conrac a leas one insan more. For each ime (0, T ), we can define he opimal surrender region R as R = {F : V (, F ) ψ(, F )}. (1.1) We assume ha he conrac is surrendered as soon as he accoun value eners he opimal surrender region. Finally, he price of he conrac is equal o he coninuaion value in he opimal surrender region. On he boundary of his region, he policyholder is indifferen beween he coninuaion value V (, F ) and he surrender value ψ(, F ). Ouside of his region, i is simply equal o he benefi received on surrender. Thus, for [0, T ), we have { (1 κ )φ(τ, F ), if F R, V (, F ) = V (, F ), oherwise Fair fee In Chapers 2 o 5, we are concerned wih pricing VA conracs under differen assumpions and fee srucures. Here, we will consider ha he fair fee rae is he smalles rae for which he iniial premium is equal o he risk-neural expecaion of he VA payoff. Denoing he fair fee rae by c, i is he smalles rae ha saisfies P = F 0 = V (c ) (0, F 0 ), (1.2) where he superscrip c represens he dependence of he value of he policy on he fee rae. We will usually drop his superscrip, unless he fee rae used is no clear from he conex. Noe ha Equaion (1.2) also sheds ligh on anoher assumpion used hroughou his hesis he iniial premium P is equal o he iniial fund value F 0. In oher words, here are no upfron fees paid by he policyholder when buying he conrac; he enire premium is deposied in he accoun. Unless oherwise saed, we consider ha he policyholder does no make furher deposis in he VA accoun. 1.3 Oher assumpions Throughou his hesis, we mosly consider VA conracs assuming ha he policyholder is sill alive a mauriy. However, mos policies offer addiional guaranees if she passes away 10

25 before mauriy of he conrac, and he mauriy benefi is only paid if he policyholder is alive. For his reason, insurers need o accoun for moraliy risk when pricing VAs. This is paricularly rue when insurers offer income guaranees, which can be valid as long as he policyholder is alive. In ha case, longeviy risk becomes an imporan par of he risk faced by he insurer, and modelling moraliy improvemens accuraely is crucial. However, since our goal is o concenrae on he surrender incenive for producs wih fixed mauriy, and because we wan o isolae his incenive, we believe ha our simplifying assumpion is jusified. Since we are using he Black-Scholes model, we assume ha he risk-free rae is deerminisic and consan. Long-erm financial guaranees like he ones embedded in VA conracs are sensiive o changes in he ineres rae, so hey should be priced and furher sudied using a model ha allows for sochasic ineres raes. Fuure exensions of our work should include analysis of our conclusions under sochasic ineres rae models. 11

26 Chaper 2 Opimal surrender under consan fee srucure 2.1 Inroducion This chaper is based on a paper ha was wrien in collaboraion wih Dr. Carole Bernard and Max Muehlbeyer (from Ulm Universiy), and ha was published in Insurance: Mahemaics and Economics (see Bernard, MacKay, and Muehlbeyer (2014)). In his chaper, we invesigae he opimal surrender sraegy for a variable annuiy conrac wih a minimum accumulaion benefi, when he fee is paid as a consan percenage of he fund. We firs consider a simple poin-o-poin guaranee and derive an inegral represenaion for he coninuaion value of he conrac, which can be solved o compue he opimal surrender boundary. To do so, we use no-arbirage argumens presened, among ohers, by Kim and Yu (1996) and Carr, Jarrow, and Myneni (1992). This echnique, originally designed for vanilla call opions, can be exended o more complex pah-dependen payoffs linked, for example, o he average fund value. Our objecive is o illusrae a general echnique o compue he opimal surrender sraegy for a possibly pah-dependen conrac. This echnique may help o undersand he effec of complex pah-dependen benefis on surrender incenives and could be useful o reduce he surrender opion value by modifying he ype of benefis offered and assess he riskiness of pah-dependen benefis. The assessmen of he value of he surrender opion is also crucial o developing an upper bound for he price of a VA conrac. The chaper is organized as follows. In Secion 2.2 we sae he seing. The opimal surrender policy is derived in Secion 2.3. Secion 2.4 exends his mehod o pah-dependen 12

27 payoffs. In Secion 2.5 we apply hese resuls o numerical examples and analyse he sensiiviy of he boundary wih respec o a range of parameers. Secion 2.6 concludes. 2.2 Seing Consider a variable annuiy conrac wih a guaraneed minimum accumulaion benefi G a mauriy T. This accumulaion benefi is compued as G = F 0 e gt where g represens he guaraneed roll-up rae. Le F denoe he underlying accumulaed fund value of he variable annuiy a ime. We assume ha he insurance company charges a consan fee c for he guaranee, which is coninuously wihdrawn from he accumulaed fund value F. Furhermore, we assume ha he policyholder pays a single premium o iniiae he conrac. The insurer hen invess his premium in he fund or index ha was chosen by he policyholder. We denoe his underlying fund or index by S and assume ha i follows a geomeric Brownian moion. Therefore, is dynamics under he risk-neural measure Q are given by ds = rs d + σs dw, (2.1) where r is he risk-free ineres rae, σ > 0 he consan volailiy and W he Brownian moion. We denoe by F he naural filraion associaed wih his Brownian moion. In his case, he sock price a ime u > given he sock price a ime has a lognormal disribuion and is explicily given by σ2 (r S u = S e 2 )(u )+σ(wu W) In his chaper, we are only concerned wih pricing he surrender opion and as such, we can rea he whole problem under he risk-neural measure. This choice is also moivaed by he use of no-arbirage argumens in he derivaion of he expression for he surrender opion. I is based on he assumpion ha policyholders opimize over all possible surrender sraegies and will choose o surrender opimally from a financial perspecive. The following resuls (2.2) and (2.3) will be useful o derive he resuls of his chaper. Since he insurance company coninuously wihdraws he fee from he fund value a a rae c, we have he following relaionship beween S u and F u a any ime u [0, T ] F u = e cu σ2 (r c S u = F e 2 )(u )+σ(wu W). (2.2) 13

28 Therefore, he condiional disribuion of F u F for u > is a lognormal disribuion wih log-scale parameer ln(f ) + (r c σ2 )(u ) and shape parameer 2 σ2 (u ). Hence, he risk-neural ransiion densiy funcion of F u a ime u > given F is given by f Fu (x F ) = 1 [ln( x ) (r c σ2 F 2 )(u )]2 2πσ2 (u )x e 2σ 2 (u ), x > 0. (2.3) Noe ha in his chaper we resrain ourselves o he case when he underlying follows a geomeric Brownian moion, which presens a simple closed expression for is ransiion densiy. However, he mehod we presen here can easily be exended o more general marke models. We discuss his poin briefly in he concluding remarks Fair Fee for he European Benefi We assume iniially ha he VA canno be surrendered early. Le c be he fee charged by he insurer beween 0 and T. Noe ha he fund value a ime T depends on his fee. We denoe by FT c he value a T of he fund given ha he fee charged during [0, T ] is equal o c and by φ(t, F c ) he payoff a mauriy T which may depend on he pah of he fund denoed by F c. If he fee c is fair (for he European benefi), we denoe i by c and i saisfies F 0 = E[e rt max(t, φ(f c ), G)], (2.4) where F 0 is he lump sum paid iniially by he policyholder ne of iniial expenses and managemen fees. For φ(f ) = F T, and for oher usual payoff funcions φ(f c ), his fee c exiss and is unique. To compue his fair fee, i is always possible o use Mone Carlo echniques. However when he disribuion of φ(t, F c ) is known, an analyical formula may be derived, which subsequenly can be solved for c. For example when {X } [0,T ] is a Markov process wih X T X LN (M, V ) (a lognormal disribuion wih log-scale parameer M and shape parameer V ), hen E[max(X T, G)] can be compued as ( E[max(X T, G) F ] = e M+ V ln(g) + M + V ) ( ln(g) M ) 2 N + GN V V (2.5) We omi he proof as i is a raher sandard compuaion. The expression (2.5) can be used o compue he value of he mauriy benefi of he VA in a Black Scholes seing when φ(t, F c ) = FT c, which is he simples benefi: a GMAB on he erminal fund value payable a ime T (Secion 2.3). We can hen solve for he fair fee in (2.4). I will also be applied when φ(t, F c ) is he geomeric average of he fund value in Secion

29 2.2.2 Surrender Opion We now assume ha he policyholder is allowed o surrender he policy a any ime [0, T ) for a surrender benefi equal o (1 κ )φ(, F c ) where κ is a penaly percenage charged for surrendering a ime. This is consisen wih he modeling of surrender charges in Milevsky and Salisbury (2001). A sandard penaly is ypically decreasing over ime. Examples of penaly funcions are given in Palmer (2006). In he absence of a surrender penaly (, κ = 0), we will see in he numerical analysis in Secion 2.5 ha he opimal surrender boundary is decreasing as a funcion of c. This resul is inuiive: if he fee c charged on he fund is high, he policyholder has a larger incenive o surrender he conrac when he guaranee is ou of he money, because she is paying more for i. 1 This observaion means ha i may be difficul o pay for he surrender benefi by wihdrawing a higher fixed percenage of he fund. Indeed if, for example, i is opimal o surrender when F > 125 when c = 1%, hen by charging c = 2% i migh be opimal o surrender when F > 100. Increasing he fee c o ake ino accoun he surrender benefi increases he value of he surrender opion. Alernaives include he possibiliy o charge for his benefi iniially as a lump paymen or o design a sufficienly high surrender penaly o decrease he incenive o surrender. This poin is already presen in he analysis of Milevsky and Salisbury (2001). I is clear ha when κ is sufficienly high hen i is never opimal o surrender a ime. For simpliciy, hroughou he chaper, we assume ha κ is exponenially decreasing and equal o 1 exp( κ(t )) so ha he surrender benefi is equal o e κ(t ) φ(, F c ), (2.6) for κ < c. For example when he surrender benefi a ime is e κ(t ) F c, hen he inequaliy κ < c ensures ha i can be opimal o surrender he VA for a sufficienly high value of he fund F c. The coninuaion value of he conrac a ime is indeed always sricly greaer han F c e c(t ) because he policyholder will receive max(ft c, G) a ime T and hus a leas he fund FT c. A ime, he value of receiving F T c a ime T is given by E[FT ce r(t ) F ] = e c(t ) F c. By assuming ha κ < c, we ensure ha for any fixed ime 1 In oher words a a given ime, he higher c, he larger he fuure fees o pay before he mauriy, whereas he final benefi is decreasing in c, so he gap beween he fuure benefi associaed wih he guaranee opion and he fuure expeced fees remaining o be paid increases and hus he incenive o surrender increases as well. 15

30 [0, T ), here exiss a fund value high enough ha he surrender benefi is worh more han he mauriy benefi so ha surrendering he policy migh become opimal. 2.3 Derivaion of he opimal surrender boundary This secion presens he echnique used o derive he opimal surrender boundary. As menioned earlier i is someimes opimal for he policyholder o surrender he conrac before he mauriy T because he fee c is charged as a percenage of he fund value. Thus, assuming he fund value is sufficienly high, he fee paid for he guaranee would exceed he acual value of he guaranee. This mismach leads o an opimal early surrender of he variable annuiy. Consider he variable annuiy conrac from Secion 2.2 wih a payoff of max(f T, G) a mauriy T. Here we assume ha c is given and hus omi he superscrip c in he noaion for he value of he fund a ime. If he conrac is surrendered early, a ime < T, he policyholder receives he accumulaed fund value F reduced by he surrender penaly, so ha he surrender benefi is given by e κ(t ) F (paricular case of (2.6)). Le B denoe he value of he opimal surrender boundary a ime, i.e. if he fund value crosses his value from below, i is opimal for he policyholder o lapse he conrac and receive he amoun B. 2 In order o derive he value of he surrender opion and he opimal surrender boundary we use he same echnique as Kim and Yu (1996) and Carr, Jarrow, and Myneni (1992). We firs seek o calculae he coninuaion value of he VA conrac, which is defined in Chaper 1. I represens he value of he policy given ha i is kep a leas one insan longer, and i is he value used o define he boundaries of he opimal surrender region. Ouside of he opimal surrender region, he coninuaion value of he VA conrac is equal o is price. Throughou his chaper, we use price and coninuaion value inerchangeably. To calculae he coninuaion value a ime, denoed by V (, F ), we decompose i ino a European par and a surrender opion. To undersand he inuiion behind his approach, consider he following rading sraegy which convers he full conrac value ino he corresponding value of he mauriy benefi and he surrender opion. We know ha he price of he VA a ime < T along he surrender boundary is equal o e κ(t ) F, he value of he surrender benefi a. This comes from he definiion of he opimal 2 Here we assume ha he opimal surrender region is of he form {F > B }, in oher words he opimal surrender behaviour is based on a hreshold sraegy where opimal surrender is driven by he value of he underlying fund crossing a barrier. This assumpion is discussed and jusified in Appendix 2.A. 16

31 surrender boundary given in Chaper 1. Moreover, B 0 > F 0 because oherwise i would no be opimal for he policyholder o buy he VA a ime 0 for a price F 0. We neglec all ransacion coss. Assume ha he policyholder has bough he VA a ime = 0. Now whenever he fund value crosses he opimal surrender boundary from below, she exercises he opion and surrenders he conrac. And whenever he fund value crosses he boundary from above, she buys back he VA conrac (given ha he boundary is exacly equal o he value of he VA by definiion). Any profis resuling from his rading sraegy consiue he value added by he possibiliy of lapsing he conrac before mauriy he surrender opion. So assume ha a ime he fund value F crosses he opimal surrender boundary from below. The policyholder surrenders he conrac and receives e κ(t ) F = e κ(t ) e c S which she insananeously invess in he sock S. However, since S is no subjec o he guaranee fee c, S ouperforms F. Therefore, in he case ha he fund value crosses he surrender boundary from above, say a ime u >, he value of he conrac on he boundary is e κ(t u) F u, he policyholder only needs o pay e κ(t u) F u o re-ener, ha is e κ(t u) e cu S u = e κt e (c κ)u S u < e κ(t ) e c S u (because c κ > 0). The profi from his sraegy is he value of he surrender opion. A formal derivaion is given in he proof of Proposiion below. Proposiion The benefi associaed wih he exercise of he surrender opion beween [, + d] is equal o h() = e κ(t ) (c κ)f d + g(d), where g(d) is o(d) as d 0. 3 Proof. Assume he VA conrac is surrendered a ime. Then he policyholder receives an amoun of e κ(t ) F = e κ(t ) e c S, which is invesed in he index S. In order o buy i back a ime + d >, she only needs e κ(t (+d)) F +d = e κt e (c κ)(+d) S +d. Therefore, consider he following decomposiion of he amoun received a ime : e κ(t ) e c S = e κt e (c κ)(+d) S + e κt S (e (c κ) e (c κ)(+d) ) = e κ(t (+d)) e c(+d) S + e κ(t ) e c S (1 e (c κ)d ) (2.7) The firs addend is he amoun invesed in he asse S ha is needed o re-ener he conrac a ime + d (in oher words, i is he no-arbirage price of e κ(t (+d)) e c(+d) S +d paid a ime + d). The second addend is he amoun ha needs o be siphoned off and is invesed in he risk-free asse. This decomposiion is going o be he key sep in generalizing his proof o more general benefis (see Secion 2.4 for an example of pah-dependen benefi). 3 A funcion f(x) is o(g(x)) when x 0 if lim x 0 f(x) g(x) = 0. 17

32 Now we can look a wha happens o his porfolio afer we perform he ime sep from o + d. We use he firs order approximaion o approximae e (c κ)d and e rd. Then he righ hand side of (2.7) becomes e κt e (c κ)(+d) S +d + e κt e (c κ) S e rd (1 e (c κ)d ) = e κt e (c κ)(+d) S +d + e κt e (c κ) S (1 + rd)(c κ)d + o(d) = e κt e (c κ)(+d) S +d + e κt e (c κ) S (c κ)d + o(d) = e κ(t (+d)) F +d + e κ(t ) (c κ)f d + o(d) The firs par of he expression is he cos of buying back he variable annuiy. Then he policyholder is lef wih he benefi of surrender of h() := e κ(t ) (c κ)f + g(d), where g(d) is o(d). Using Proposiion and he rading sraegy explained above we are now able o derive a pricing formula for he variable annuiy conrac wih a surrender benefi similarly o Kim and Yu (1996). Theorem 1. Le V (, F ) denoe he coninuaion value a ime of he variable annuiy wih guaranee G a mauriy and a surrender benefi equal o he accumulaed fund value wih some penaly κ > 0, e κ(t ) F. Then V (, F ) can be decomposed ino a corresponding European par U(, F ) and a surrender opion e(, F ) where V (, F ) = U(, F ) + e(, F ), (2.8) { U(, F ) = e c(t ) F N ( d 1 (F, G, T, ) ) + e r(t ) GN ( d 2 (F, G, T, ) ), e(, F ) = e κt (c κ)f e c T e (c κ)u N ( d 1 (F, B u, u, ) ) du, and N (x) is he sandard normal disribuion funcion wih d 1 and d 2 defined as d 1 (x, y, T, ) := ln( x y )+(r c+ σ 2 σ T 2 )(T ) d 2 (x, y, T, ) := σ T d 1 (x, y, T, )., (2.9) (2.10) Proof. Firs we prove he formula for he European par U(, F ) of he VA. Since F T F LN (ln(f ) + (r c σ2 )(T ), 2 σ2 (T )), we can use (2.5) o calculae he 18