Risk Management of Variable Annuities

Transcription

1 Fakulä für Mahemaik und Wirschafswissenschafen Insiu für Versicherungswissenschafen Risk Managemen of Variable Annuiies Disseraion zur Erlangung des akademischen Grades eines Dokors der Wirschafswissenschafen (Dr. rer. pol.) an der Fakulä für Mahemaik und Wirschafswissenschafen der Universiä Ulm Vorgeleg von: Dipl.-Mah. oec. Frederik Ruez Amierender Dekan: Prof. Dr. Alexander Lindner Vorsizende des Promoionsausschusses: Prof. Dr. Sandra Ludwig Guacher: apl. Prof. Dr. Hans-Joachim Zwiesler apl. Prof. Dr. Jochen Ruß Tag der Promoion: 2. Mai 2017

2 Conens Overview of Research Papers Co-auhorship iii iv Research Conex and Summary of Research Papers 1 1 Field of Research Moivaion and Objecives Summary of Research Papers Research Papers 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency of Wihdrawal Benefi Guaranees in Variable Annuiies 17 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies: The Impac of Regulaor-Imposed Guaranees 50 3 Variable Annuiies wih Guaraneed Lifeime Wihdrawal Benefis: An Analysis of Risk-Based Capial Requiremens 73 Curriculum Viae 105 ii

3 Overview of Research Papers Research papers included in his disseraion 1. Kling, A., Ruez, F. and Ruß, J., The impac of policyholder behavior on pricing, hedging, and hedge efficiency of wihdrawal benefi guaranees in variable annuiies. European Acuarial Journal, 4(2), pp DOI: /s Kling, A., Ruez, F. and Ruß, J., Guaraneed minimum surrender benefis in variable annuiies: he impac of regulaor-imposed guaranees. The final version of his paper will appear in he European Acuarial Journal. DOI: /s Ruez, F., Variable annuiies wih guaraneed lifeime wihdrawal benefis: an analysis of risk-based capial requiremens. Submied o ASTIN Bullein. iii

4 Co-auhorship Dr. Alexander Kling Alexander Kling is parner a he Insiu für Finanz- und Akuarwissenschafen (ifa). He received his docoral degree in 2007 from he Universiy of Ulm. apl. Prof. Dr. Jochen Ruß Jochen Ruß is parner a he Insiu für Finanz- und Akuarwissenschafen (ifa) and adjunc professor a he Universiy of Ulm. He received his docoral degree in 1999 from he Universiy of Ulm and his pos-docoral lecure qualificaion (Habiliaion) in 2009 from he Universiy of Ulm. iv

5 Research Conex and Summary of Research Papers 1 Field of Research Variable annuiies Variable annuiies are uni-linked life insurance conracs where ypically an iniial invesmen amoun is invesed in one or several muual funds. On op of his basic srucure, cerain guaranee riders are offered by he insurer, adding differen ypes of financial proecion o he conrac. Therefore, variable annuiies allow policyholders o benefi from he upside poenial of he underlying fund invesmen and, a he same ime, offer some kind of proecion when he fund loses value. Variable annuiies have experienced a growh in sales in US and Japan since he 1990s and are also becoming increasingly widespread over Europe (cf. EIOPA, 2011). Variable annuiy providers offer a variey of guaranee riders: Besides guaraneed minimum deah benefi riders (GMDB), hree main ypes of guaraneed living benefi riders (GLB) exis: guaraneed minimum accumulaion benefi riders (GMAB), guaraneed minimum income benefi riders (GMIB) and guaraneed minimum wihdrawal benefi riders (GMWB). GMAB and GMIB offer he policyholder some guaraneed mauriy value or some guaraneed annuiy benefi, respecively, while GMWB allow policyholders o (emporarily or lifelong) wihdraw money from heir accoun, even afer he accoun s cash value has dropped o zero. GMWB wih a lifelong guaranee are called "GMWB for life" or guaraneed lifeime wihdrawal benefi riders (GLWB). They offer policyholders a lifelong income and, hus, proecion from ouliving heir savings, wih he invesed amoun sill benefiing from poenial fund growh and remaining under he conrol of he policyholder. In conras o, for insance, radiional life insurance in Germany, GLWB do no require he policyholder o annuiize heir 1

6 Research Conex and Summary of Research Papers savings in order o hedge agains longeviy risk: The iniial invesmen amoun remains in conrol of he policyholder and may be cashed ou (via surrendering of he conrac) a any ime during he conrac s lifeime. Also, beneficiaries will receive a poenial remaining accoun value in case of deah of he policyholder. In oher words, his ype of variable annuiy embeds a varian of ruin-coningen life annuiy (cf. Huang e al., 2014), where he guaranee provider sars o pay a lifelong annuiy as soon as he accoun value (reduced by pre-defined wihdrawals) his zero. Modern GLWB riders ypically also include a form of rache mechanism, hrough which he guaraneed wihdrawal amoun may increase during he lifeime of he conrac if he underlying fund performs well. In conras o more radiional offers, variable annuiy providers usually receive an explici compensaion for he guaranees: Typically, hey receive a guaranee charge ha is periodically deduced from he policyholder s accoun, for insance a cerain percenage of he invesed amoun, annually. Risk profile of variable annuiies From a risk manager s perspecive, he complexiy of he guaranees offered wihin variable annuiies also means ha here are several imporan risks ha need o be managed a he same ime, including financial risk, behavioral risk, biomeric risk, as well as regulaory risk. These risks are accompanied by a variey of addiional risks ha come wih mos insurance conracs (e.g. operaional and repuaional risk) and are amplified by he usually very long erm of ypical (variable) annuiy conracs. Financial risk inheren in variable annuiies wih guaranees comes from he direc exposure o marke movemens via he fund invesmen as well as from he impac ineres raes have on he presen value of fuure benefis. A decrease of ineres raes, for insance, causes he presen value of fuure guaraneed benefis o increase, which could negaively affec he variable annuiy provider s (marke value) balance shee if he provider is no hedged agains such changes. Movemens in he spo prices of he underlying fund s asses, like, for insance, equiy shares, direcly affec he likelihood (and he exen) of he guaranee coming ino effec: An increase in spo prices increases he accoun value and, hus, usually reduces he likelihood of he variable annuiy provider needing o make guaranee paymens. If he guaranee comes ino effec, he exen usually is reduced by he increase in he accoun value. As a consequence, he value of he guaranee (and herefore 2

7 Research Conex and Summary of Research Papers he value of liabiliies on he variable annuiy provider s marke-value balance shee) decreases wih increasing spo prices. Vice versa, decreasing spo prices ypically resul in a higher value of liabiliies on he marke-value balance shee of a variable annuiy provider. In order o lessen he impac marke movemens have on he balance shee of he provider, risk managemen ypically also includes he adminisraion and rebalancing of a so-called hedging porfolio on he asse side. The purpose of such a hedging porfolio is o replicae changes in he value of liabiliies on he liabiliies side of he balance shee wih according increases or decreases in value on he asse side of he balance shee. Ideally, wih such a hedging program in place, he provider s (marke-value) equiy is no affeced by changes in he value of liabiliies. Such hedging programs can be quie effecive in miigaing he financial risks inheren in variable annuiy riders, bu hey usually do no allow for a perfec replicaion of he changes in he value of liabiliies, due o discree rebalancing and oher imperfecions (cf. Ledlie e al., 2008). On op of imperfecions in he hedging program, here are risks ha influence he provider s profi and loss aribuion (P&L) ha are no easily hedgeable, like for insance behavioral risks. Hence, he provider s P&L wih regard o is variable annuiy business remains subjec o flucuaions, even if a hedging program is implemened. Policyholder behavior risk sems from he fac ha variable annuiies usually offer he policyholder many choices, e.g. surrender, parial surrender, he decision wheher or no and when o annuiize (in GMIB producs) or he decision wheher or no and how much o wihdraw each year (in GMWB producs). Several auhors (cf. e.g. Milvesky & Salisbury, 2006, or Bauer e al., 2008) come o he conclusion ha insurers assume wha hey call "subopimal" policyholder behavior when pricing he guaranees. This means ha (a leas some) policyholders are assumed o no behave in a way ha would maximize he value of he insurer s liabiliies arising from he financial guaranees embedded in he producs. From an insurer s risk managemen perspecive, "opimal" policyholder behavior in his sense would consiue a wors-case scenario wih respec o policyholder behavior. Therefore, his kind of behavior can also be described as "loss-maximizing" behavior from he viewpoin of he provider of he guaranee (cf. Azimzadeh e al., 2014). Such behavior is o be expeced from insiuional invesors who buy policies in a secondary marke and, subsequenly, opimize he opions embedded in he policies. Bauer e al., 2008, sae in paricular ha he value of cerain guaranees under opimal policyholder behavior significanly exceeds ypical prices charged in many 3

8 Research Conex and Summary of Research Papers insurance markes, whereas he value of he same guaranees assuming subopimal behavior (using e.g. ypical surrender probabiliies and independence beween surrender behavior and financial markes) are in line wih observed prices. This appears o bear significan risks for he insurers. There are several examples where insurance companies had o updae heir policyholder behavior assumpions leading o significan increases in liabiliies, for insance ING, Manulife Financial and Sun Life Financial (cf. Knoller e al., 2016). Oher insurers even compleely sopped heir variable annuiy business in cerain markes, cf. for insance The Harford, On op of marke and behavioral risks, variable annuiy providers also face regulaory challenges: Firs, providers have o comply o capial requiremens, which can rigger he need for capial injecions, lessen he reurn on equiy of he company and, as a resul, can make i harder o run a profiable variable annuiy business. Capial requiremens, in paricular risk-based capial requiremens, can also change over ime and may be marke-dependen, such ha difficul marke condiions may be accompanied by addiional sress o he provider from a simulaneous increase in capial requiremens. Second, for reasons of consumer proecion, regulaors may impose changes o he way some benefis of variable annuiies are calculaed, for insance surrender benefis. Such mandaory changes ha are sipulaed by a regulaor can also apply o conracs ha are already in force, i.e., effecively, he produc design of he variable annuiy is changed afer incepion of he conrac and, herefore, was likely neiher considered in he pricing nor he (iniial) hedging of he conrac. In conclusion, he risk managemen of variable annuiies wih guaranees covers a variey of aspecs and risks, which are hard o conrol and which are no easily hedged agains. Therefore, risk managemen of a variable annuiy produc already sars in he produc developmen process, where he design of he produc s feaures has o be carefully analyzed and weigh agains differen objecives, boh, from he provider s perspecive as well as from he perspecive of a poenial fuure cusomer. To accommodae for he complexiy of he produc and is inheren risks, usually a quaniaive analysis is necessary o assess he consequences of cerain design choices. In his process, fuure risk managemen and he risk-miigaing effec of fuure hedging should be considered. While for he economic risk only he rue effeciveness of he hedging program seems relevan, for he calculaion of (fuure) capial requiremens (and herefore, fuure capial coss) i is also relevan o which exen his risk-miigaing effec is allowed o be considered in he calculaion of fuure capial requiremens. Also, possible fuure mandaory changes made 4

9 Research Conex and Summary of Research Papers o he produc design of already exising conracs imposed by he regulaor can have a subsanial impac on he profiabiliy and should be assessed when designing a new produc. Behavioral risk is a main risk ha risk managers of variable annuiies have o consider when assessing he risk profile and when seing up a hedging porfolio. This risk can be inensified by he presence and acions of insiuional invesors ha ry o profi from (from heir perspecive) underpriced conracs, which hey subsequenly opimize from he provider s perspecive his represens loss-maximizing behavior, which should be assessed and considered in he risk managemen of variable annuiies. 2 Moivaion and Objecives The pricing and valuaion of variable annuiy conracs wih guaranees have been sudied in grea deail, wih Milvesky & Salisbury, 2006, being he firs o analyze he valuaion of guaraneed minimum wihdrawal benefis, and wih Bauer e al., 2008, as well as Bacinello e al., 2011, providing general frameworks for he valuaion of variable annuiies wih all ypes of guaranees. Regarding he risk managemen of variable annuiies, Cahcar e al., 2015 provide schemes o efficienly calculae he "Greeks" of a variable annuiy liabiliy via Mone Carlo simulaion, while Coleman e al., 2006, and Coleman e al., 2007, provide schemes o hedge variable annuiies under differen assumpions regarding he capial marke. Forsyh & Vezal, 2014, presen an opimal sochasic conrol framework, in which hey analyze he sensiiviy of he cos of hedging a variable annuiy wih GLWB o various economic and conracual assumpions. The hedging coss for variable annuiies wih combined guaraneed lifelong wihdrawal and deah benefis (GLWDB) is analyzed in Azimzadeh e al., 2014, in which he auhors also argue ha, when analyzing dynamic policyholder behavior from an insurer s perspecive, i is beer o use he erm "loss-maximizing sraegy" insead of "opimal sraegy". In Kling e al., 2011, he auhors analyze he pricing and risk profile (from a provider s perspecive) of GLWB riders wih differen produc designs. In paricular, hey analyze how resuls change if equiy volailiy is modeled sochasic insead of deerminisic. In heir analysis, capial marke models are used for several purposes: Firs, hey are used in he pricing of he conrac, i.e. he calculaion of a "fair" guaranee charge as compensaion for he guaranee of a GLWB rider. Second, afer incepion of a variable annuiy conrac, marke models are used as a means o 5

10 Research Conex and Summary of Research Papers calculae he consiuion of he hedging porfolio, i.e. o calculae he weighs of he (financial) insrumens used for hedging. Depending on he considered hedging sraegy, he insrumens hey modeled in heir analysis include a money marke accoun, a posiion in he underlying equiy fund and a posiion in a pu opion (wih he fund as underlying). Third, hey use marke models o calculae risk measures of a sylized pool of policies of variable annuiy conracs wih GLWB riders. In heir simulaion sudy, he considered hedging programs are projeced over he lifeime of he variable annuiy conracs and risk measures of he provider s resuling P&L are compued. For all hree applicaions, Kling e al., 2011, analyze how he modeling of equiy volailiy influences he resuls and how hese resuls are affeced by he differences in he considered produc designs. They find ha boh, he probabiliy ha guaraneed paymens have o be paid and heir amoun vary significanly for he differen considered produc designs. The developmen over ime of dela, rho and vega i.e. he sensiiviy of he value of liabiliies wih respec o changes in he underlying s price, he ineres rae level and he level of equiy volailiy, respecively was found o be also significanly differen beween he produc designs, resuling in boh, he consiuion of a hedging porfolio (following a cerain hedging sraegy) and he provider s risk afer hedging o differ significanly for he differen produc designs. Thus, risk managemen already sars during he developmen process of a new variable annuiy produc. The auhors also find ha he fair prices of he considered guaranees hardly change when sochasic volailiy is inroduced, while he provider s risk changes dramaically. They analyze differen hedging sraegies (no hedging, dela hedging, and dela and vega hedging) o deal wih his risk and analyze he disribuion of he provider s P&L and cerain risk measures hereof. They find ha he provider s risk can be reduced significanly by implemening suiable hedging sraegies. Risks caused by policyholder behavior, however, is no par of heir analysis. The impac of policyholder behavior on he pricing of guaranees embedded in insurance conracs has been analyzed by several auhors, e.g. by Grosen & Jorgensen, 2000, Seffensen, 2002, Bacinello, 2003, Bacinello, 2005, Bacinello e al., 2010, Bacinello e al., 2011, and Gao & Ulm, 2012, and wih focus on he opimal sopping ime wihin he conex of GMWB guaranees for example by Chen e al., 2008, and Yang & Dai, Bernard e al., 2014, analyze so-called "opimal" policyholder behavior for variable annuiies wih GMAB riders. De Giovanni, 2010, uses a "Raional Expecaion" model describing he policyholder s behavior in surrendering he con- 6

11 Research Conex and Summary of Research Papers rac, which also allows for irraional policyholder behavior (as opposed o "opimal" behavior). Knoller e al., 2016, analyze individual policy daa from a Japanese variable annuiy produc and find evidence ha confirms heir "moneyness hypohesis": In heir saisical analysis, he fund performance and hence he value of he financial opions and guaranees has he larges explanaory power for he surrender rae. However, o our knowledge, here exiss no simulaneous analysis of he impac of policyholder behavior on he pricing, hedging and hedge efficiency of GLWB riders wih paricular emphasis on differen produc designs. Therefore, in our firs research paper, we exend he analysis conduced in Kling e al., 2011, wih regard o he modeling of policyholder behavior and wih a special focus on he risks ha arise from behavior ha differs from anicipaed behavior. In our second research paper, we perform a similar analysis for variable annuiies wih GMAB riders and analyze he risks ha arise if a regulaor imposes cerain guaraneed surrender benefis for variable annuiy conracs ha are already in force. We also analyze he impac such a mandaory change would have on he pricing and he risk profile of new variable annuiy conracs. The impac he presence of insiuional invesors has on he resuls is also par of he analysis in our second research paper. Finally, in our hird research paper, we have a closer look on he risk profile of a pool of variable annuiies wih GLWB riders and analyze he corresponding capial requiremens under a risk-based regulaory regime like Solvency II in he European Union. In paricular, we analyze he risk of simulaneous changes in he value of liabiliies (which are likely o be hedged) and changes in capial requiremens (which are likely no hedged agains) und differen assumpions regarding he exen o which he riskmiigaing effec of he hedging program can be considered in he calculaion of he capial requiremens. In summary, he following research quesions are considered in his hesis: (1) From a variable annuiy provider s perspecive, how does policyholder behavior impac he risk profile (before and afer hedging) of a pool of variable annuiies wih GLWB riders under differen assumpions regarding he produc design of he GLWB riders? (2) Wha is he impac of regulaor-imposed guaraneed minimum surrender benefis on he risk profile of exising conracs wih GMAB riders and how are new conracs affeced? (3) How does he risk profile of variable annuiies wih GMAB and he impac of mandaory guaraneed minimum surrender benefis change if insiuional 7

12 Research Conex and Summary of Research Papers invesors buy conracs from policyholders who are willing o surrender heir conrac? (4) How do risk-based capial requiremens for a pool of variable annuiies wih GLWB riders change depending on he marke environmen and he level of recogniion of he acual hedging program? 3 Summary of Research Papers Research Paper 1: The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency of Wihdrawal Benefi Guaranees in Variable Annuiies The pricing of guaranees in variable annuiies is usually performed under cerain assumpions for fuure surrender raes. Such assumpions can be, for insance, deerminisic surrender or (ypically) pah-dependen surrender (where assumed surrender raes depend on marke parameers and/or he value of he guaranee). However, he pricing is usually no performed under he assumpion of "opimal" surrender. This reduces he price of such guaranees since in simplified erms fuure profis he insurer expecs from sub-opimal policyholder behavior are given o he clien by means of a reduced price for he guaranee. The possibiliy o allow for sub-opimal policyholder behavior in pricing and hedging of such producs is a reason why hese (ofen primarily financial) guaranees can be offered by insurers a compeiive prices when compared o similar producs offered by banks. This opens opporuniies for insiuional invesors o purchase such policies in a secondary marke a a price ha exceeds he surrender benefi from policyholders who are willing o surrender heir conrac. In his siuaion, selling he conrac o he insiuional invesor insead of surrendering i is beneficial for he policyholder. Afer acquiring he conrac, he insiuional invesor hen maximizes (opimizes) he value of he conrac, which ypically resuls in loss-maximizing behavior from he insurer s perspecive. In Kling e al., 2014, we exend he model used in Kling e al., 2011, and incorporae differen models of dynamic, so-called pah-dependen policyholder behavior. The analyzed models include purely deerminisic surrender raes, pah-dependen 8

13 Research Conex and Summary of Research Papers surrender raes ha are influenced by a cerain observable quaniy (for insance he "moneyness" of he guaranee), as well as a model of "opimal" policyholder behavior, i.e. behavior ha maximizes he value of liabiliies of he guaranee. We find ha he popular mehod of using he "moneyness", while being a huge improvemen (from a risk manager s perspecive) over purely deerminisic behavior, is no enough o capure he full risk opimal behavior poses o he insurer. We also found ha he produc design is a powerful ool o minimize behavior risk: wih an appropriae design of he rache mechanism of he GLWB rider, he guaranee is never fully "ou of he money" (i.e. wih lile value) and herefore, he value of liabiliies, from he provider s perspecive, is less sensiive o surrender raes. In hese cases, for pricing purposes, opimal behavior was very close o he assumpion of he policyholders no surrendering a all, as he surrender benefi and he value of coninuing he conrac (he so-called coninuaion value) are more closely ogeher. However, in he analysis of he risk profile, a proper modeling of loss-maximizing behavior while cerainly a wors-case scenario seems indispensable for a full risk analysis. This paper is join work wih Alexander Kling and Jochen Ruß and has been published in he European Acuarial Journal. I answers he firs research quesion lised above. Research Paper 2: Guaraneed Minimum Surrender Benefis in Variable Annuiies: The Impac of Regulaor-Imposed Guaranees Surrender risk is no only influenced by dynamic policyholder behavior, bu also by he way surrender benefis hemselves are calculaed. The produc design of variable annuiies usually sipulaes ha he surrender value of such producs coincides wih he policyholder s accoun value (minus surrender charges, if applicable). The "fair value" of he guaraneed benefis or he marke value of cerain hedge asses is ypically no par of he individual policyholder s accoun value and hus, wih he usual produc design, no par of he surrender value. This in consequence means ha he surrender value in general is differen from a "fair" marke value of he conrac. In paricular, he surrender benefi will no be reduced if ineres raes rise, alhough boh, he asses backing he conrac and he "fair value" of he conrac, 9

14 Research Conex and Summary of Research Papers would drop. The resuling risk has been discussed e.g. in Feodoria & Försemann, Addiionally, regulaor-imposed minimum surrender benefis, like hey are discussed and imposed in Germany, pose a relevan risk o he providers of variable annuiies. This is especially he case if he guaraneed minimum surrender benefis (GMSB) are imposed afer incepion of he conracs and, hus, were no incorporaed in he pricing and (iniial) hedging process of he produc. We analyze he accompanying risk in Kling e al., 2016, where we analyze differen discussed and proposed models for deermining a "ime value" of he guaranee as a minimum for he surrender benefi a policyholder would receive in case of surrendering heir conrac. For his purpose, we analyze variable annuiy conracs wih a GMAB rider under differen assumpions regarding he GMSB as well as differen policyholder behavior models. A model for deermining a GMSB is especially harmful o he provider if i is sysemaically uilized by an insiuional invesor like hedge funds in a poenial secondary marke (cf. e.g. Cenral Bank of Ireland, 2010). We inroduce a model where policyholders who are willing o surrender heir conrac sell hem insead o an insiuional invesor, if hey will receive a selling price ha is higher han he curren surrender benefi. Of course, he insiuional invesor is only able o offer a higher price in an economically sound way, if he coninuaion value of he conrac exceeds he surrender benefi. We find ha, while he impac of GMSBs on marke risk is relaively low, he impac on he fair guaranee charge, he value of liabiliies and he risk resuling from changes in policyholder behavior is subsanial. If he GMSB is already considered when pricing he conrac, he resuling advanage for policyholders who surrender he conrac comes a he price of increased guaranee charges for all policyholders, adversely affecing especially hose who keep he conrac unil mauriy. As a consequence, he same proecion level wih regard o old-age provision becomes more expensive when GMSBs are in place. If a GMSB is inroduced afer incepion of he conrac, e.g. because of a regulaory change, he insurer will suffer an immediae loss on is marke-value balance shee. While he value of he conrac increases wih he value added by he GMSB, he sensiiviy wih regard o surrender raes decreases, as, from a valuaion perspecive, i becomes less imporan wheher policyholders decide o surrender or no. As a consequence, he poenial for mispricing of he conracs wih respec o incorrec surrender assumpions is reduced. 10

15 Research Conex and Summary of Research Papers Our analyses wih regard o he impac of a secondary marke show ha, in a marke wihou GMSBs, he presence of an insiuional invesor creaes a loss for he insurer and also increases marke risk. A he same ime, he impac of inroducing GMSBs is reduced and he specific design of he GMSB is less relevan. On he oher hand, if GMSBs are already in place, he poenial for a successful secondary marke is reduced, since he difference beween he surrender benefi of a conrac and is coninuaion value is ypically lower and, hus, insiuional invesors less likely are able o offer prices ha exceed he surrender benefi. This paper is join work wih Alexander Kling and Jochen Ruß and answers he second and hird research quesion lised above. Research Paper 3: Variable Annuiies wih Guaraneed Lifeime Wihdrawal Benefis: An Analysis of Risk-Based Capial Requiremens Under risk-based regulaory regimes like Solvency II in he EU and he Swiss Solvency Tes in Swizerland, he risk profile of a variable annuiy direcly affecs he amoun of capial ha providers are required o hold. Therefore, providers of variable annuiies no only face he challenge o hedge agains changes in he value of embedded guaranees (i.e. he value of liabiliies), bu are also exposed o poenial addiional capial needs due o changes in heir capial requiremens. Boh, he (marke) value of liabiliies as well as corresponding risk-based capial requiremens, are dependen on marke parameers and, hus, subjec o changes. Therefore, under risk-based regulaory regimes, no only is he valuaion and hedging relevan when designing and profi-esing new variable annuiy producs, bu also (fuure) capial requiremens. No only does he produc design direcly influence he risk profile and, hereby, he capial requiremens under risk-based regulaory regimes, bu i also influences he risk of changing capial requiremens in he fuure. A change in he value of he guaranee rider from he provider s perspecive, for insance, is likely o be accompanied by a change in he risk-based capial requiremens. Furhermore, while he hedging program migh prove reliable in realiy, i is no clear o which exen i can be considered when calculaing he capial requiremens. If cerain regulaory requiremens are no fulfilled i is likely ha he risk-miigaing effec of fuure hedging is only allowed o be parially considered in 11

16 Research Conex and Summary of Research Papers he calculaion. While he change in he value of liabiliies is likely (a leas parially) hedged, he change in capial requiremens ypically is no. In Ruez, 2016, we analyze he risk profile and corresponding risk-based requiremens of a pool of variable annuiy policies wih Guaraneed Lifeime Wihdrawal Benefi (GLWB) riders wih regard o he pool s key financial risk drivers: equiy reurns, implied equiy volailiy and ineres raes. In a simulaion sudy, we analyze he effeciveness of differen sylized hedging programs over a one-year ime horizon and compue indicaors for risk-based capial requiremens. The approach we use is comparable o an inernal model ype approach under Solvency II. We also analyze he impac changing marke environmens have on risk profile, hedge effeciveness and capial requiremens, similar o a forward-looking analysis in he conex of he mandaory Own Risk and Solvency Assessmen (ORSA) under Solvency II. We find ha, in addiion o he sress from poenially unhedged increases in he value of liabiliies, changes in he marke environmen can have a subsanial impac on capial requiremens. As a resul, GLWB providers face he risk of increases in heir risk-based capial requiremens and, hus, he need for capial injecions even wihou pricing errors or malfuncioning of he hedging program. We also find ha, while he impac of he level of ineres raes on he effeciveness of he modeled hedging program is raher low, a higher volailiy level has a disinc adverse effec on he hedge effeciveness, leading o a furher increase of risk-based capial requiremens. However, here are also cases where an increase in he value of liabiliies was accompanied by a decrease of capial requiremens, reducing he overall impac on he insurer. This is he case for some risk measures if no allowance of he hedging program is made in he calculaion of capial requiremens and equiy volailiy is increased. As he sensiiviy of capial requiremens o marke parameers is no easily assessable, horough numerical analyses appear necessary for a proper assessmen of his risk. In such analyses, also he effec of a poenially reduced hedge performance in adverse marke environmens and a reduced level of recogniion of he hedging program s risk-miigaing effec should be considered, as his may lead o subsanial addiional increases of capial requiremens during he lifeime of he variable annuiy conrac. In summary, his research paper answers he fourh research quesion lised above. 12

17 Research Conex and Summary of Research Papers References Azimzadeh, P., Forsyh, P.A. & Vezal, K.R., Hedging coss for variable annuiies under regime-swiching. In Hidden Markov Models in Finance Volume II, R. Mamon and R. Ellio, eds. Springer, New York. pp Bacinello, A.R., Fair valuaion of a guaraneed life insurance paricipaing conrac embedding a surrender opion. Journal of Risk and Insurance, 70, pp Bacinello, A.R., Endogenous Model of Surrender Condiions in Equiy-Linked Life Insurance. Insurance: Mahemaics and Economics, 37(2), pp Bacinello, A.R., Biffis, E. & Millossovich, P., Regression-Based Algorihms for Life Insurance Conracs wih Surrender Guaranees. Quaniaive Finance, 10, pp Bacinello, A.R., Millossovich, P., Olivieri, A. & Piacco, E., Variable annuiies: A unifying valuaion approach. Insurance: Mahemaics and Economics, 49(3), pp Bauer, D., Kling, A. & Ruß, J., A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies. ASTIN Bullein, 38(2), pp Bernard, C., MacKay, A. & Muehlbeyer, M., Opimal surrender policy for variable annuiy guaranees. Insurance: Mahemaics and Economics, 55(1), pp Cahcar, M.J., Lok, H.Y., McNeil, A.J. & Morrison, S., Calculaing Variable Annuiy Liabiliy Greeks Using Mone Carlo Simulaion. ASTIN Bullein, 45(2), pp Cenral Bank of Ireland, Requiremens on Reserving and Risk Governance for Variable Annuiies. [Online] Available a: hp:// ie/regulaion/indusry-secors/insurance-companies/documens/ requiremens%20on%20reserving%20and%20risk%20governance%20for% 20variable%20annuiies%20-%20december% pdf [Accessed 31 July 2016]. Chen, Z., Vezal, K. & Forsyh, P.A., The effec of modelling parameers on he value of GMWB guaranees. Insurance: Mahemaics and Economics, 43(1), pp

18 Research Conex and Summary of Research Papers Coleman, T.F., Kim, Y., Li, Y. & Paron, M., Hedging Guaranees in Variable Annuiies under Boh Equiy and Ineres Rae Risks. Insurance: Mahemaics and Economics, 38, pp Coleman, T.F., Kim, Y., Li, Y. & Paron, M., Robusly Hedging Variable Annuiies wih Guaranees under Jump and Volailiy Risks. Journal of Risk and Insurance, 74(2), pp De Giovanni, D., Lapse Rae Modeling: A Raional Expecaion Approach. Scandinavian Acuarial Journal, 1, pp EIOPA, Repor on Variable Annuiies. [Online] Available a: hps://eiopa. europa.eu/publicaions/repors/repor-on-variable-annuiies.pdf [Accessed 31 July 2016]. Feodoria, M. & Försemann, T., Lehal lapses how a posiive ineres rae shock migh sress German life insurers. [Online] Available a: hps:// Discussion_Paper_1/2015/2015_06_22_dkp_12.hml [Accessed 31 July 2016]. Forsyh, P. & Vezal, K., An opimal sochasic conrol framework for deermining he cos of hedging of variable annuiies. Journal of Economic Dynamics and Conrol, 44, pp Gao, J. & Ulm, E.R., Opimal Consumpion and Allocaion in Variable Annuiies wih Guaraneed Minimum Deah Benefis. Insurance: Mahemaics and Economics, 51(3), pp Grosen, A. & Jorgensen, P., Fair valuaion of life insurance liabiliies: The impac of ineres rae guaranees, surrender opions, and bonus policies. Insurance: Mahemaics and Economics, 26, pp Huang, H., Milevsky, M.A. & Salisbury, T.S., Valuaion and Hedging of he Ruin-Coningen Life Annuiy (RCLA). Journal of Risk and Insurance, 81(2), pp Kling, A., Ruez, F. & Ruß, J., The impac of sochasic volailiy on pricing, hedging, and hedge efficiency of wihdrawal benefi guaranees in variable annuiies. ASTIN Bullein, 41(2), pp

19 Research Conex and Summary of Research Papers Kling, A., Ruez, F. & Ruß, J., The impac of policyholder behavior on pricing, hedging, and hedge efficiency of wihdrawal benefi guaranees in variable annuiies. European Acuarial Journal, 4(2), pp Kling, A., Ruez, F. & Ruß, J., Guaraneed Minimum Surrender Benefis in Variable Annuiies: The Impac of Regulaor-Imposed Guaranees. Working paper, Ulm Universiy. Knoller, C., Krau, G. & Schoenmaekers, P., On he Propensiy o Surrender a Variable Annuiy Conrac: An Empirical Analysis of Dynamic Policyholder Behavior. Journal of Risk and Insurance, 83(4), pp Ledlie, M.C. e al., Variable Annuiies. Briish Acuarial Journal, 14, pp Milvesky, M.A. & Salisbury, T.S., Financial valuaion of guaraneed minimum wihdrawal benefis. Insurance: Mahemaics and Economics, 38, pp Ruez, F., Variable Annuiies wih Guaraneed Lifeime Wihdrawal Benefis: An Analysis of Risk-Based Capial Requiremens. Working paper, Ulm Universiy. Seffensen, M., Inervenion opions in life insurance. Insurance: Mahemaics and Economics, 31(1), pp The Harford, s quarer 2009 sakeholder message of The Harford. [Online] Available a: hp:// Message.pdf [Accessed 18 December 2016]. Yang, S.S. & Dai, T.-S., A flexible ree for evaluaing guaraneed minimum wihdrawal benefis under deferred life annuiy conracs wih various provisions. Insurance: Mahemaics and Economics, 52(2), pp

20 Research Papers 16

21 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency of Wihdrawal Benefi Guaranees in Variable Annuiies The final publicaion is available a link.springer.com hps://link.springer.com/aricle/ /s Kling, A., Ruez, F. and Ruß, J., The impac of policyholder behavior on pricing, hedging, and hedge efficiency of wihdrawal benefi guaranees in variable annuiies. European Acuarial Journal, 4(2), pp DOI: /s

22 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency of Wihdrawal Benefi Guaranees in Variable Annuiies Alexander Kling Insiu für Finanz- und Akuarwissenschafen, Lise-Meiner-Sr. 14, Ulm, Germany, phone: +49 (731) , Frederik Ruez Corresponding auhor. Ulm Universiy, Lise-Meiner-Sr. 14, Ulm, Germany, phone: +49 (731) , Jochen Ruß Insiu für Finanz- und Akuarwissenschafen, Lise-Meiner-Sr. 14, Ulm, Germany, phone: +49 (731) , Absrac We analyze he impac of policyholder behavior on pricing, hedging and hedge efficiency of variable annuiies wih guaraneed lifeime wihdrawal benefis. We consider differen produc designs, marke models and approaches for modeling policyholder behavior in our analyses, covering deerminisic behavior, behavior depending on he moneyness of he guaranee, and opimal (value maximizing) behavior. Firs, we assess he risk of mispricing he guaranee due o inaccurae assumpions regarding fuure policyholder behavior. Comparing producs wih differen rache mechanisms, we find ha his poenial for mispricing is he smalles for he produc design wih he mos valuable rache mechanism. We furher quanify he impac of differen behavior models on he efficiency of he insurer s hedging sraegy and he risk ha resuls if he insurer's assumpion for policyholder behavior deviaes from acual behavior. Our analyses indicae significan differences beween he considered producs in erms of hedgeabiliy and he sensiiviy of he guaranee s value owards policyholder behavior and owards changes in he underlying asse s volailiy. Also, we show ha a simple pahdependen behavior model may no be suiable o fully assess he risk arising from policyholder behavior. Keywords Variable Annuiies, Guaraneed Lifeime Wihdrawal Benefis, Policyholder Behavior, Pricing, Hedging, Hedge Performance, Model Risk 18

23 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency 1 Inroducion Variable annuiies are fund-linked annuiies where he policyholder ypically pays a single premium ino he policy and he money is hen invesed in one or several muual funds. Variable annuiies usually offer a wide range of invesmen opions for he policyholder o choose from. On op of his basic srucure, cerain guaranee riders are offered by he insurer, adding differen ypes of financial proecion o he conrac. There are several ypes of guaranee riders ha come wih variable annuiies, including guaraneed minimum deah benefis (GMDB) as well as guaraneed minimum living benefis, which can be caegorized ino hree main subcaegories: guaraneed minimum accumulaion benefis (GMAB), guaraneed minimum income benefis (GMIB) and guaraneed minimum wihdrawal benefis (GMWB). A GMAB guaranee provides he policyholder wih some guaraneed value a one or several fuure poins in ime, while he GMIB guaranee provides a guaraneed annuiy benefi, saring afer a cerain defermen period. Wih he GMWB rider, if cerain condiions are me, he policyholders may coninue o wihdraw money from heir accoun, even afer he value of he accoun has dropped o 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. Insurers also sared o include addiional feaures in GMWB producs. The mos prominen is called GMWB for Life (also known as guaraneed lifeime wihdrawal benefis, GLWB). Wih his ype of guaranee, he oal amoun of wihdrawals is unlimied. However, he annual amoun ha may be wihdrawn while he insured is sill alive may no exceed some maximum value; oherwise he guaranee will be affeced. The wihdrawals made by he policyholder are deduced from heir accoun value as long as his value is posiive. Aferwards, he insurer has o provide he guaraneed wihdrawals for he res of he insured s life. In reurn for his guaranee, he insurer receives guaranee charges, which are deduced from he policyholder s accoun value (as long as his value is posiive). These charges are ypically calculaed as a fixed annual percenage of he so-called wihdrawal benefi base (explained below) or of he accoun value. In a few producs, annual guaranee charges are calculaed as a fixed percenage of he single premium. In conras o a convenional annuiy, where he asses covering he liabiliies are owned by he pool of insured, in a GLWB policy, he fund unis of he conrac are owned by he individual policyholder and remain accessible o he policyholder even in he payou phase. The policyholder may access he remaining fund asses a any ime by (parially) surrendering he conrac. In case of deah of he insured, any remaining fund value (or a guaraneed minimum deah benefi if such a rider is included and he corresponding value exceeds he fund value) is paid ou o he beneficiary. From an insurer s poin of view, such producs conain an ineresing and challenging combinaion of several risks, resuling from policyholder behavior (wih regard o surrender and wihdrawal), financial markes, and longeviy, alongside a variey of addiional risks ha come wih mos insurance conracs (e.g. operaional and repuaional risk). This combinaion of risks makes hese guaranees challenging o hedge and has been in he focus of boh, academics and praciioners. Policyholder behavior risk sems from he fac ha variable annuiies usually offer he policyholder many choices, e.g. surrender, parial surrender, he decision wheher or no and when o annuiize (in GMIB producs) or he decision wheher or no and how much o wihdraw each year (in GMWB producs). Several auhors (cf. e.g. Milevsky and Salisbury, 2006, or Bauer e al., 2008) come o he conclusion ha insurers assume wha hey call "subopimal" 19

24 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency policyholder behavior when pricing he guaranees. This means ha (a leas some) policyholders are assumed o no behave in a way ha would maximize he value of he insurer s liabiliies arising from he financial guaranees embedded in he producs. From an insurer s risk managemen perspecive, opimal policyholder behavior in his sense would consiue a wors-case scenario wih respec o policyholder behavior. Bauer e al. (2008) sae in paricular ha he value of cerain guaranees under opimal policyholder behavior significanly exceeds ypical prices charged in many insurance markes, whereas he value of he same guaranees assuming subopimal behavior (using e.g. ypical surrender probabiliies and independence beween surrender behavior and financial markes) are in line wih observed prices. This appears o bear significan risks for he insurers. There are several examples where insurance companies had o updae heir policyholder behavior assumpions leading o significan increases in liabiliies, see e.g. ING (2011), Manulife Financial (2011), and Sun Life Financial (2011). Oher insurers even compleely sopped heir variable annuiy business in cerain markes, cf. for insance The Harford (2009). The effec of policyholder behavior no only on pricing bu also and much more imporanly on hedging and hedge efficiency of variable annuiy guaranees should herefore be of ineres o academics, produc providers and regulaors. The impac of policyholder behavior on he pricing of guaranees embedded in insurance conracs has been analyzed by several auhors, e.g. by Grosen and Jørgensen (2000), Seffensen (2002), Bacinello e al. (2003, 2005, 2011) and Gao and Ulm (2012) and wih focus on he opimal sopping ime wihin he conex of GMWB guaranees for example by Chen e al. (2008) and Yang and Dai (2013). Bernard e al. (2014) analyze opimal policyholder behavior for variable annuiies wih a GMAB. De Giovanni (2010) uses a Raional Expecaion model describing he policyholder s behavior in surrendering he conrac, which also allows for irraional policyholder behavior. Knoller e al. (2013) analyze individual policy daa from a Japanese variable annuiy produc and find evidence ha confirms heir moneyness hypohesis : In heir saisical analysis he fund performance and hence he value of he financial opions and guaranees has he larges explanaory power for he surrender rae. They find ha surrender raes increase wih decreasing value of he guaranee and ha policyholders apparen raionaliy increases wih increasing conrac volume. To our knowledge, here exiss no simulaneous analysis of he impac of policyholder behavior on he pricing, hedging and hedge efficiency of GLWB riders wih paricular emphasis on differen produc designs. The presen paper fills his gap: We exend he model presened in Kling e al. (2011) o incorporae non-deerminisic policyholder behavior and for differen produc designs analyze he impac policyholder behavior has on pricing, hedging and hedge efficiency, and how resuls change if he capial marke model incorporaes sochasic insead of deerminisic equiy volailiy. The remainder of his paper is organized as follows. In Secion 2, we describe our model framework ha consiss of hree pars: he financial model, where for he sake of comparison we use boh, he classic Black-Scholes model (wih deerminisic equiy volailiy) and he Heson model for he evoluion of an underlying wih sochasic equiy volailiy; he liabiliy model ha describes he differen considered variable-annuiy conracs wih differen GLWB opions; and he valuaion framework including he policyholder-behavior model, which allows for differen policyholder sraegies wih regard o surrendering he conrac. We paricularly consider opimal policyholder behavior, as well as several subopimal sraegies, where, in boh cases, opimal as explained above denoes he behavior ha maximizes he value of he insurer s liabiliies. In Secion 3, we presen he resuls of our analyses regarding he pricing of 20

25 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency he guaranee. In paricular, we analyze he differences in he opion value for differen produc designs and how he opion value depends on assumed policyholder behavior. This is a firs indicaion for an insurer s poenial loss arising from an inaccurae assessmen of policyholder behavior. Secion 4 deals wih hedging sraegies and hedge efficiency. Here, we paricularly analyze how he insurer s expeced profi and risk change if acual policyholder behavior deviaes from he behavior assumed wihin he hedging sraegy. Finally, Secion 5 concludes. 2 Model Framework In Bauer e al. (2008), a general framework for modeling and 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. Holz e al. (2012) describe how GLWB producs can be included in his model. In wha follows, we apply he general framework of Bauer e al. (2008). However, in our concree specificaion, addiionally o he simple Black-Scholes model used in Bauer e al. (2008), we also consider a model which allows for sochasic equiy volailiy (Secion 2.1). In Secion 2.2, we inroduce and define he specific produc designs considered wihin our numerical analyses. Differen models for policyholder behavior are inroduced in Secion 2.3, where also our valuaion approach is summarized. 2.1 Financial Marke The valuaion framework in his secion follows in some pars he one used in Bacinello e al. (2010) and in ohers Bauer e al. (2008). We ake as given a filered probabiliy space,,f,p F is a in which P is he real-world (or physical) probabiliy measure and F 0 filraion wih F, and 0 0 F. We assume ha rading akes place coninuously over ime and wihou any ransacion coss or spreads. Furhermore, we assume ha he price processes of he raded asses in he marke are adaped and of bounded variaion. For our analyses we assume wo primary radable asses: he underlying fund (or baske of funds), whose spo price a ime will be denoed by S, and he money-marke accoun, whose value a ime will be denoed by B. We assume he money-marke accoun o evolve a a consan risk-free rae of ineres r: db rb d B B 0 exp( r) (1) For he dynamics of S, we use wo differen models. Firs, we 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 also use he Heson model, in which boh, he underlying and is (insananeous) variance, are sochasic processes. These wo models will be explained in he following wo subsecions Black-Scholes Model In he Black-Scholes (1973) model, he underlying s spo price S follows a geomeric Brownian moion whose dynamics under he real-world measure P are given by ds S d S dw, S 0 0, (2) BS 21

26 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency where µ is he (consan) drif of he underlying, σbs is consan volailiy and W denoes a P- Brownian moion. By Iō's lemma, S has he soluion S S 2 exp BS BSW, S. (3) 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 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 S, 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 ineres rae model (Cox e al., 1985). The dynamics of he wo processes under he real-world measure P are given by he following sysem of sochasic differenial equaions: 1 ds S d V S dw, S 0 (4) dv 2 V d V dw, V 0, v 0 0 (5) 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 variance, σv is he so-called volailiy of volailiy, and W are correlaed P-Brownian moion processes (wih correlaion parameer ρ). The condiion 1/ v ensures ha he variance process will remain sricly posiive almos surely (see Cox e al., 1985) Equivalen Maringale Measure Assuming he absence of arbirage opporuniies in he financial marke, here exiss a probabiliy measure Q ha is equivalen o P and under which he gain from holding a raded asse is a Q-maringale afer discouning wih he chosen numéraire process, in our case he money-marke accoun. Q is called equivalen maringale measure. While under he usual assumpions he ransformaion o such a measure is unique under he Black-Scholes model (cf. e.g. Bingham and Kiesel, 2004), i is no under he Heson model. Wihin he Heson model, since here are wo sources of risk, here are also wo marke-price-of-risk processes, denoed by and (corresponding o W and W ). Heson (1993) proposed he following resricion on he marke price of volailiy risk process, assuming i o be linear in volailiy, 1 V. (6) Provided boh measures, P and Q, exis, he Q-dynamics of S and V, again under he assumpion ha no dividends are paid, are hen given by 22

27 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency ds rs d dv V S dw, Q,1 S 0 Q,2 V d V dw, V 0 v 0 0 (7) (8) Q,1 Q,2 where W and W are wo correlaed Q-Brownian moion processes (wih correlaion parameer ρ) and where v, (9) v are he risk-neural counerpars o κ and θ (cf., for insance, Wong and Heyde, 2006). 2.2 Model of he Liabiliies Wih variable annuiies, he single premium P is invesed in one or several muual funds. We call he value of he policyholder s individual porfolio he accoun value and denoe is value a ime by AV. All charges are aken from he accoun value by cancellaion of fund unis. Furhermore, he policyholder has he possibiliy o surrender he conrac or o wihdraw a porion of he accoun value. Producs wih a GMWB opion give he policyholder he possibiliy o perform 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 iniially guaraneed wihdrawal amoun is usually a cerain percenage xwl of he single premium P. In mos producs, xwl depends on he age when wihdrawals sar. 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 policyholder can sill coninue o wihdraw he guaraneed amoun unil deah of he insured. The insurer charges a fee for his guaranee, which is usually a pre-specified annual percenage of he wihdrawal benefi base, he accoun value or he single premium. In wha follows, we will assume ha he guaranee charge is a percenage of he accoun value and ha wihdrawals may only occur on he policy s anniversary daes. Ofen, GLWB producs conain cerain feaures ha lead o an increase of he guaraneed wihdrawal amoun if he underlying funds perform well. Typically, on every policy anniversary, he curren accoun value is compared o a cerain reference value, which we refer o as wihdrawal benefi base. Whenever he accoun value exceeds he wihdrawal benefi base, he guaraneed annual wihdrawal amoun is increased (sep-up or rache). In our numerical analyses in Secions 3 and 4, we consider hree differen produc designs ha can be observed in he marke: No Rache (Produc I): The firs and simples alernaive is one where no raches exis a all. In his case, he guaraneed annual wihdrawal amoun is consan and does no depend on marke movemens. Lookback Rache (Produc II): The second alernaive is a rache mechanism where he 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 23

28 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency o he accoun value, if he accoun value exceeds he previous wihdrawal benefi base. The guaraneed annual wihdrawal amoun is increased accordingly o xwl muliplied by he new wihdrawal benefi base. This effecively means ha he fund performance needs o compensae for charges and annual wihdrawals in order o increase fuure guaraneed wihdrawals. Increases in he guaraneed wihdrawal amoun are permanen, i.e. over ime, he guaraneed wihdrawal amoun may only increase, never decrease. Remaining WBB Rache (Produc III): The basic idea of he hird produc is o provide a rache mechanism where, in order o increase guaraneed annual wihdrawals, he fund performance needs o compensae only for charges, bu no for annual wihdrawals. In his produc, he wihdrawal benefi base a ouse is also given by he single premium paid. However, a each wihdrawal dae, he wihdrawal benefi base is reduced by he wihdrawn amoun (if his amoun does no exceed he guaraneed wihdrawal amoun). If on a policy anniversary he curren accoun value exceeds his reduced wihdrawal benefi base by a cerain amoun, he guaraneed annual wihdrawal is increased by xwl Afer such an increase, he wihdrawal benefi base is rese o he accoun value. This rache mechanism is herefore c.p. somewha richer han he Lookback Rache. As a consequence, he iniially guaraneed wihdrawal amoun should c.p. be lower han wih a produc offering a Lookback Rache. As wih he Lookback Rache design, increases in he guaraneed amoun are permanen. Throughou he paper, we assume ha adminisraion charges and guaranee charges are deduced a he end of each policy year as a percenage φ adm and φ guar of he accoun value. Addiionally, we allow for upfron acquisiion charges φ acq as a percenage of he single acq AV P 1 premium P. This leads o 0. guar We denoe he guaraneed wihdrawal amoun a ime by W and he corresponding wihdrawal benefi base by WBB. A incepion, for each of he considered producs, he iniial wihdrawal benefi base is se o P and hence he guaraneed wihdrawal amoun for he iniial guar wihdrawal is given by W0 xwlwbb 0 xwl P. The amoun acually wihdrawn by he clien is denoed by W. 1 Since we resric our analyses o single premium conracs, policyholder acions during he life of he conrac are limied o wihdrawals and (parial) surrender. During he year, all processes are subjec o capial marke movemens. As menioned above, 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 ( immediaely before and he value ) ( afer wihdrawals, (parial) surrender, and deah ) 1 Noe ha he clien can choose 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. 24

29 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency benefi paymens. For he laer, we assume ha no addiional guaraneed minimum deah benefi rider is included in he policy, i.e. in case of deah he remaining fund value is paid ou. In wha follows, in he spiri of Bauer e al. (2008), we describe he developmen beween wo policy anniversaries and he ransiion a policy anniversaries for he considered conrac designs. From hese, we are finally able o deermine all benefis for any given policyholder 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 adm and φ guar 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 AV S 1 adm guar 1 AV exp. (10) S A he end of each year, he differen rache mechanisms are applied afer deducion of charges and before any oher acions are aken. Thus W develops as follows: guar guar guar No Rache: WBB 1 WBB P and W W xwl P 1. Lookback Rache: WBB 1 max WBB, AV 1 guar guar and x WBB max W x AV W. 1 WL 1, WL 1 Remaining WBB Rache: Since wihdrawals are only possible on policy anniversaries, he wihdrawal benefi base during he year develops like in he Lookback Rache case. WBB max WBB AV and Thus, we have 1, 1 guar guar W W x AV WBB,0. 1 WL max Transiion a a Policy Anniversary 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 guar, WBB 0, W 0, and W 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, guar WBB WBB, and guar W W. 25

30 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency 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 up o guar 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 WBB, and W guar W guar. For he alernaive Remaining WBB Rache, he wihdrawal benefi base is reduced by he wihdrawal aken, i.e. WBB 0; WBB W guar wihdrawal amoun remains unchanged, i.e. guar W W. max and he guaraneed annual 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 formulas for he case of full surrender, since parial surrender is no analyzed in wha follows. 2 In case of full surrender, he complee accoun value is wihdrawn. We hen se AV 0, WBB 0, guar W AV, and W 0 and he conrac erminaes. However, he policyholder does no surr receive he full asse value as surrender benefi, since surrender fees are deduced from he cash amoun exceeding he guaraneed wihdrawal amoun. 2.3 Valuaion Le Q be an equivalen maringale measure of he financial marke (cf. secion 2.1.3). Assuming independence beween financial markes and moraliy as well as risk-neuraliy of he insurer wih respec o moraliy and behavioral risk, we are able o use he produc measure of Q and he moraliy measure. In wha follows, we denoe his measure by Qˆ. As menioned earlier, for he conracs considered wihin our analyses, policyholder acions are limied o wihdrawals and (parial) surrender. In our numerical analyses in Secions 3 and 4, we only consider wo possible policyholder acions: wihdrawal of he guaraneed wihdrawal guar amoun, i.e. W W, or full surrender, i.e. W AV. This also means ha we assume ha wihdrawals begin a he earlies anniversary possible and, hence, ha here is no iniial waiing period before he firs wihdrawal. To keep noaion simple, we only give formulas for he considered cases (cf. Bauer e al. (2008) for formulas for he oher cases). We denoe by x 0 he insured s age a he sar of he conrac, a x0 -year old o survive he nex years, qx p x 0 he probabiliy under Qˆ for ( x 0 -year old 0 he probabiliy under Qˆ for a ) 2 For deails on parial surrender, we refer he reader o Bauer e al. (2008). 26

31 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency o die wihin he nex year, and le ω be he limiing age of he moraliy able, i.e. he age beyond which survival is deemed impossible. The probabiliy under Qˆ ha an insured aged x0 a incepion passes away in he year (,+1] is hus given by p q. The limiing age ω allows for a finie ime horizon T x 1. 0 x 0 x0 For pricing purposes, we consider a pool of policyholders who hold idenical conracs and in which each insured has he same age, same gender and same moraliy probabiliy. We assume he number of policyholders o be large enough such ha he assumpion ha deahs occur exacly according o he probabiliies qx 0 is jusified. The policyholders in he pool may however differ in heir (surrender) behavior. We model he (surrender) behavior of he policyholders in he pool as a Fˆ -adaped family of random variables, where 0 T 1,..., T 1, 1,...,, represens he fracion of he remaining policyholders a ime who surrender heir conrac a ime. Afer he guaranee has been riggered, i.e. W guar AV for some =1,,T, here is no raional reason for a policyholder o surrender heir conrac, hence we se 0, G, where G represens a Fˆ -sopping ime indicaing he policy anniversary a which he guaranee of he GLWB rider riggers, i.e. he smalles =1,,T for which W guar AV holds. If he guaranee does no rigger during he conrac s lifeime, we se G T. For a given behavior assumpion 1,..., T, all conracual cash flows of he pool of policies are specified for any given capial marke scenario. Thus boh, he guaranee paymens (i.e. paymens made by he insurer afer he accoun value has dropped o zero) a imes i 1,2,...,T, denoed by G P ( ), and he guaranee fee paymens G F ( ) made by he i i policyholder (including surrender fee paymens), again a imes i 1,2,...,T, are known. For any given, he ime- value V G ( ) of he GLWB rider is hen given by he expeced presen value of all fuure guaranee paymens P i ( ) G (), i 1,2,...,T, V F i G P F G ( ) G ( ) Fˆ G, i 1,2,...,T, minus fuure guaranee fees T r( i) ( ) EQˆ e i i. (11) i1 In he following numerical secion, his value is calculaed using (nesed) Mone-Carlo simulaions. Wihin our numerical analyses, we consider five differen assumpions regarding policyholder behavior: 27

32 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency a) No surrender Under his assumpion, policyholders never surrender heir conrac, i.e , 0 1,..., T, 1,..., T. b) Deerminisic surrender Surrender under his assumpion occurs according o pre-specified deerminisic (imedependen) percenages s, 0 s 1, 1,..., T, as long as he guaranee has no been 1,..., T riggered. In formulas, d d s, d : 0, 1,..., T, 1 else G c) Longsaff-Schwarz approximaion o opimal surrender In order o compue he fair value of an American opion using Mone-Carlo echniques, Longsaff and Schwarz (2001) inroduced a mehod in which opimal behavior is approximaed via leas-squares regression of he condiional expecaion of he opion's payoff, given some pah- and ime-dependen variables. Essenially, when applied o pricing of he GLWB rider, heir algorihm works as follows: 1. Define a se of base funcions ha ake some sae variables of he conrac and he scenario as argumen and reurn a real number. 2. Creae a se of N scenarios under Qˆ. 3. Saring a T-1, a each policy anniversary, compue he presen value of he cash flow beween and T for each scenario in which he guaranee has no been riggered a ime. Fi he linear leas-squares regression wih hese presen values as dependen variables and he base funcions wih he corresponding sae variables of he conrac and he scenario as inpu variables. 4. Evaluae he resuling approximaion of he GLWB rider's coninuaion value for each scenario and decide wheher he policyholder should surrender or no. If hey surrender, he cash flow following is se o zero and he cash flow a o minus he surrender fee paid. 5. Repea seps 3-5 for -1 unil =0 is reached. As base funcions we use weighed Hermie polynomials up o a degree of hree for each sae variable and cross producs hereof, again up o a degree of hree, as well as a consan. Before simulaion and/or pricing, we firs execue he Longsaff-Schwarz algorihm wih a separae se of scenarios in order o avoid an upward bias. The surrender behavior of he policyholder can be considered opimal in he sense ha i G maximizes he opion value V of he GLWB rider if he policyholder decides o surrender he conrac whenever he benefi from disconinuing he conrac (i.e. he negaive of he coninuaion value) exceeds he surrender fees. 28

33 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency G Wih Vˆ denoing he approximaed coninuaion value of he GLWB rider a ime, he policyholder behavior is modeled as follows: LS LS 1, : 0, LS 1,..., T Vˆ G, surr AV, 1 else G In wha follows, we refer o surrender behavior according o his algorihm as being opimal, alhough we have o keep in mind ha i is only an approximaion for he value maximizing * G sraegy defined as arg max V0 ( ), where denoes he se of all admissible sraegies. d) Funcion of moneyness Wihin his approach, we model he fracion of he policyholders who surrender heir conrac as a funcion of ime and he in-he-moneyness of he guaranee (as, for example, described in American Academy of Acuaries, 2005). We define he moneyness of he guaranee a ime as he raio of he surrender value (accoun value less surrender fees) and he srike price of he guaranee, for which we use he ne presen value of an immediae annuiy paying he curren guaraneed wihdrawal amoun annually unil he insured s deah. Because his annuiy s ne presen value is a lower limi for he sum of asse value and opion value of he GLWB rider, will be upward biased and no reside around 1 ("a-he-money") as desired. To correc for his, we use 0, he moneyness a incepion of he conrac, as benchmark and use he relaive deviaion of hereof as measure. The basis for he surrender funcion is a se of given pre-specified deerminisic percenages s, 0 s 1, 1,..., T (as in he deerminisic surrender scenario). However, we now 1,..., T model he fracion of he surrendering policyholders a ime as s muliplied by a facor ha depends on he moneyness-variable. In deail, we model he behavior according o he following formulas: ITM, ITM 1,..., T s 1( / 0), 1 ITM G :, 0, else 1/ 3, x , 0.95 x ( x) : 3, 1.05 x , x 1.15 e) Funcion of opion value Here, we use a similar approach as for he funcion of moneyness, excep ha we now use he sum of he rider s (approximaed) coninuaion value and he surrender charge as decision 29

34 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency variable. Wihin he Longsaff-Schwarz algorihm, i is opimal for he policyholder o disconinue he conrac whenever his value becomes negaive. Using again pre-specified probabiliies s, 0 s 1, 1,..., T, his modeling approach is defined as follows: OV OV 1,..., T 1,..., T ˆ G 0 surr OV s 2( V ( ) AV ), : 0, 1/ 3, x , 0.01 x ( x) : 3, 0.01 x , x else Noe ha he las wo models for policyholder behavior, d) and e), allow for he following inerpreaion which appears o be he moivaion for he use of such models in pracice: If a cerain percenage of he policyholders follow a more or less opimal sraegy (in he sense ha G hey inuiively or wih he help of professional advisors aim a maximizing he value V of he embedded guaranee) and he res of he policyholders are assumed o follow a subopimal sraegy wih deerminisic surrender raes, hen a pool would show paerns similar o models d) and e). 3 Conrac Analysis 3.1 Assumpions For all of he analyses we use he fee srucure given in Table 1. G Acquisiion charges Managemen charges Guaranee charges 4.00 % of single premium 1.50 % p.a. of AV 1.50 % p.a. of AV Table 1: Fee srucure for he considered conracs. We furher assume he policyholder o be a 65 year old male. For pricing purposes, we use besesimae annuian moraliy probabiliies given in he DAV 2004R able published by he German Acuarial Sociey (DAV). As described in Secion 2.3, we use differen assumpions for he policyholder behavior. In he case where surrender is assumed o be deerminisic, we use he surrender paern given in Table 2. As observed for many producs in many markes, we assume higher surrender raes in earlier years and some base surrender in laer years. 30

35 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency Year Surrender rae p S 1 6 % 2 5 % 3 4 % 4 3 % 5 2 % 6 1 % Table 2: Assumed deerminisic surrender raes. Besides deerminisic surrender (in wha follows denoed by DS), we also analyze he oher ypes of policyholder behavior inroduced in Secion 2.3, i.e. Longsaff-Schwarz-opimal surrender behavior (opimal), surrender behavior depending on he opion value (OV), and surrender behavior depending on he in-he-moneyness of he opion (ITM). We also consider he case wihou any surrender (NS). 3.2 Deerminaion of he Fair Guaraneed Wihdrawal Rae For he pricing of he conrac, i.e. for he deerminaion of he guaraneed wihdrawal rae xwl G ha makes he conrac fair a incepion in he sense ha V 0 holds, we perform a roo G search wih xwl as argumen and he value of he opion V 0 as funcion value, cf. e.g. Bauer e G al. (2008) or Kling e al. (2011). In his process, V 0 is compued via Mone-Carlo simulaion, where 100,000 pahs are used per valuaion Resuls for he Black-Scholes model In Table 3, we show he fair guaraneed wihdrawal raes xwl for differen rache mechanisms, volailiies, raes of ineres, surrender fees and policyholder-behavior assumpions. Noe ha we here analyze he impac of he policyholder behavior assumpions used for pricing he conrac. Effecs resuling from a poenial deviaion beween acual policyholder behavior and behavior assumed in pricing and hedging will be analyzed in Secions 3.3 and

36 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency σbs, r σbs = 15%, r = 4% σbs = 20%, r = 4% σbs = 22%, r = 4% σbs = 25%, r = 4% σbs = 22%, r = 2% σbs = 22%, r = 3% σbs = 22%, r = 5% Produc I (No Rache) surr = 1% surr = 3% Rache mechanism Produc II (Lookback) surr = 1% surr = 3% Produc III (Remaining WBB) surr = 1% surr = 3% Behavior Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Table 3: Fair guaraneed wihdrawal raes xwl in percen under he Black-Scholes model for differen rache mechanisms, policyholder behavior assumpions, volailiies, raes of ineres and surrender fees. Obviously, he produc design wihou any rache allows for he highes wihdrawal raes hroughou, while he remaining WBB rache (which consiues he riches ype of rache) allows for he lowes. I is also obvious ha fair wihdrawal raes are decreasing wih increasing volailiy and/or decreasing ineres raes, since he corresponding guaranees increase in value 32

37 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency wih increasing volailiy or decreasing ineres raes. Our main focus, however, is on he analysis of differen assumpions abou policyholder behavior: G As defined in Secion 2.3, opimal surrender behavior maximizes he value V of he conrac. Thus, he fair wihdrawal raes are he lowes in his case. For all considered parameer combinaions, he assumpion of deerminisic surrender raes leads o he highes fair wihdrawal rae. The difference beween he fair wihdrawal raes in hese wo cases can exceed a full percenage poin. For a volailiy of 25%, for example, and in he produc wihou rache, he fair wihdrawal rae assuming deerminisic surrender amouns o 4.90% in he case of a surrender fee of 1% (and 4.95% for a surrender fee of 3%) while for opimal policyholder behavior, he fair wihdrawal rae is only 3.74% (4.04%). Hence, an insurer assuming deerminisic behavior would be willing o provide policyholders lifelong guaraneed wihdrawal amouns ha are (c.p.) more han 30% higher han he raes offered by a more conservaive insurer assuming opimal policyholder behavior. For he produc design wih no rache, his difference in wihdrawal raes is increasing wih increasing volailiy and wih increasing ineres raes. Thus, he poenial for mispricing resuling from oo aggressive assumpions for policyholder behavior is also increasing. For he produc designs wih rache, he difference of he fair wihdrawal rae assuming deerminisic policyholder behavior and opimal policyholder behavior, respecively, is significanly smaller and much less sensiive o changes in volailiy or ineres raes. For he lookback rache, he difference is abou 30 o 40 basis poins, in he case of he remaining WBB rache, he difference amouns o 20 o 25 basis poins. Thus, he poenial for mispricing by assuming incorrec policyholder behavior is he smalles for he produc design wih he mos valuable rache mechanism. One reason for his can be seen by comparing he fair wihdrawal rae assuming no surrender and opimal surrender. If he rache mechanism is quie valuable (i.e. remaining WBB rache), here is very lile or even no difference in he corresponding fair wihdrawal raes. Thus, no surrender seems o be very close o an opimal policyholder behavior. Hence, by assuming some deerminisic (and fairly low) surrender rae, he assumpion basically is ha almos all policyholders behave opimally (by no surrendering) and only very few behave subopimally. For a produc design wihou any rache on he oher hand, surrender can become opimal if funds perform well. In all hese scenarios, deerminisic surrender raes imply he assumpion of a high porion of cusomers behaving subopimally by no surrendering and only a low porion of cusomers displaying opimal behavior. The wo pah-dependen assumpions abou policyholder behavior (OV and ITM) show a raher similar paern. For he produc design wihou rache, boh show a significan poenial for mispricing. Even if volailiy is only 15%, he difference in he fair wihdrawal raes beween pah-dependen assumpions and opimal behavior is around 40 basis poins (roughly 30 basis poins for a surrender fee of 3%). Again, wih increasing volailiy or ineres raes, his difference also increases. However, for his produc design, he considered pah-dependen assumpions lead o lower guaraneed wihdrawal raes han assuming no surrender. Thus, in his case he poenial for mispricing is lower if pah-dependen assumpions are made. This changes if raches are included ino he produc. Then again, he differences in wihdrawal raes decrease. A he same ime, he fair wihdrawal raes assuming no surrender are lower han he fair wihdrawal raes assuming pah-dependen surrender. Thus, wihin his modeling framework, even hough an insurer assumes surrender behavior ha is somehow linked o he 33

38 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency opion value or he in-he-moneyness of he opion, he poenial for mispricing is higher han when assuming no surrender. Fair wihdrawal raes obviously increase wih increasing surrender fees. I is, however, worh noing ha for produc I he difference in fair wihdrawal raes beween a surrender fee of 3% and 1% increases wih increasing opimaliy of he policyholders behavior. If a srong rache mechanism is included (e.g. produc III), however, here is almos no difference if policyholders behavior opimally. This again is due o he fac ha for his produc design, no surrendering is close o opimal, even for he lower surrender fee Resuls for he Heson model For he Heson model, we use he model parameers given in Table 4, ha were derived by Eraker (2004), and saed in annualized form for insance by Ewald e al. (2007). Parameer Numerical value r 0.04 θ (0.22) 2 κ 4.75 σv 0.55 ρ V(0) θ Table 4: Parameers for he Heson model. One of he key parameers in he Heson model is he marke price of volailiy risk. Since absolue -values are hard o inerpre, in he following able we show he values of he longerm variance and he speed of mean reversion for differen values of Marke price of volailiy risk Speed of mean reversion κ * Long-erm variance θ * λ = (0.190) 2 λ = (0.198) 2 λ = (0.208) 2 λ = (0.220) 2 λ = (0.234) 2 λ = (0.251) 2 λ = (0.272) 2 Table 5: Q-parameers for differen values of he marke price of volailiy risk. Higher values of correspond o a lower volailiy and a higher mean-reversion speed, while lower (and negaive) values of correspond o high volailiies and a lower speed of mean reversion. E.g., = 2 implies a long-erm volailiy of 19.8% and = -2 implies a long-erm volailiy of 25.1%. In he following able, we show he fair annual guaraneed wihdrawal raes under he Heson model for all differen produc designs using he same assumpions regarding policyholder behavior and ineres raes as for he Black-Scholes model, and values of beween -2 and 2. 34

39 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency λ, r λ = 2, r = 4% λ = 1, r = 4% λ = 0, r = 4% λ = -1, r = 4% λ = -2, r = 4% λ = 0, r = 2% λ = 0, r = 3% λ = 0, r = 5% Produc I (No Rache) surr = 1% surr = 3% Rache mechanism Produc II (Lookback) surr = 1% surr = 3% Produc III (Remaining WBB) surr = 1% surr = 3% Behavior Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Table 6: Fair guaraneed wihdrawal raes xwl in percen under he Heson model for differen rache mechanisms, policyholder behavior assumpions, marke price of volailiy risk parameers λ, raes of ineres and surrender fees. 35

40 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency The resuls under he Heson model are very similar o hose observed wihin he Black-Scholes model: Fair wihdrawal raes are higher for he produc design wihou rache and for lower long-erm volailiy assumpions. The poenial for mispricing is also higher for he produc design wihou any rache. Comparing he resuls of he Heson model wih he corresponding resuls using he Black- Scholes model shows ha he assumpion of sochasic equiy volailiy seems o have only lile influence on pricing resuls for GLWB riders (which is consisen o findings in Kling e al., 2011). 3.3 Quanifying he Risk resuling from Behavioral Assumpions In a nex sep, we analyze he loss poenial an insurer faces if pricing assumpions for policyholder behavior deviae from acual policyholder behavior Resuls for he Black-Scholes model Table 7 shows he GLWB rider s value a incepion from he insurance company s perspecive as a percenage of he single premium if he acual fuure policyholder behavior as well as equiy volailiy or ineres raes differ from he pricing assumpions. Negaive values herefore represen he equivalen of an immediae loss for he insurance company if he insurer charges a cerain price for he guaranee ha was calculaed using assumpions ha differ from acual behavior and/or marke parameers in a negaive way. The resuls in his able are given for a surrender fee of 3%. We assume ha he producs are priced assuming an equiy volailiy in he Black-Scholes model of 22% alongside an ineres rae of 4%, and ha he acual parameers are eiher 22% or 25% for he volailiy and eiher 4% or 3% for he rae of ineres. 36

41 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency Behavior Pricing: σbs = 22%, r = 4% Acual: σbs = 22%, r = 4% Pricing: σbs = 22%, r = 4% Acual: σbs = 25%, r = 4% Pricing: σbs = 22%, r = 4% Acual: σbs = 22%, r = 3% Pricing Acual I II III I II III I II III Opimal OV Opimal ITM NS DS Opimal OV OV ITM NS DS Opimal OV ITM ITM NS DS Opimal OV NS ITM NS DS Opimal OV DS ITM NS DS Table 7: GLWB rider value a incepion as percenage of he single premium if acual policyholder behavior and/or parameers in he Black-Scholes model differ from pricing assumpions. Pricing assumpion DS We firs look a he case where deerminisic surrender probabiliies are assumed in he pricing of he conrac. Clearly, he poenial loss is he highes if policyholders behave opimally. In paricular, if assumpions abou equiy volailiy and ineres raes are correc, he insurance company s loss is 4.7% of he single premium paid if no rache is included, 2.5% of he single premium paid in he case of he lookback rache and 2.4% of he single premium paid for he remaining WBB rache. In line wih he resuls from Secion 3.2, he loss poenial if only he assumpion abou policyholder behavior is incorrec is significanly lower if raches are included ino he produc and is he lowes for he produc design wih he mos valuable rache mechanism. If policyholders in realiy do no behave opimally bu eiher surrender according o one of he pah-dependen rules (OV or ITM) or do no surrender a all (NS), hen he poenial loss roughly lies beween 1% and 3% of he single premium paid if he assumpions regarding he marke parameers are correc. Again, he riskies produc design is he one wihou rache. However, he produc designs wih raches are more sensiive o changes in volailiy: 37

42 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency If (addiionally o policyholder behavior assumpions) volailiy assumpions are also wrong (i.e. σbs = 25%), he insurance company s loss increases by a leas 2% for he producs wih rache, while for he produc wihou rache, he increase is beween 1.1% and 1.6% of he single premium paid. If (addiionally o policyholder behavior assumpions) ineres rae assumpions are also wrong (i.e. r = 3%), he insurance company s loss increases by a leas 3.5% of he single premium paid. This increase is raher similar across he hree produc designs. Losses can now go up o almos 10% of he premium paid if deerminisic policyholder behavior is assumed and acual behavior is opimal. If policyholders in realiy do no surrender a all (NS), independen of volailiy and ineres raes, he loss poenial is higher han wih any pah-dependen behavior (OV and ITM) for he produc designs wih raches and lower for he produc design wihou rache. A he same ime, no surrendering seems o be very close o he opimal sraegy for produc III, which has a rich rache mechanism. Also, he ineres rae sensiiviy is he highes if policyholders do no surrender. Pricing assumpion NS We now look a he case where no surrender is assumed in pricing. In his case (if assumed and acual marke parameers coincide), for he producs wih rache, he insurer would realize a gain if any of he oher non-opimal policyholder behavior paerns occurs, i.e. OV, ITM or DS. Thus, he insurance company can reduce he risk resuling from policyholder behavior by including a srong rache mechanism ino he produc and a he same ime assuming no surrender in pricing he conrac. A rich rache mechanism can preven high values of he opion o surrender under almos all circumsances. This can be a very effecive means o manage policyholder behavior risk. The effec of wrong volailiy assumpions on he insurance company s loss is similar o he one observed when deerminisic surrender is assumed: The loss increases by a leas 2% for he producs wih rache, while for he produc wihou rache he increase is beween 1.1% and 1.5% of he single premium paid. Similar increases can also be observed for all oher pricing assumpions. The absolue increase caused by wrong ineres rae assumpions, however, is less pronounced han if deerminisic surrender is assumed in pricing. The increase is sill similar for he differen produc designs. Pricing assumpions OV and ITM A common pah-dependen assumpion abou policyholder behavior suggess ha surrender raes are influenced by he in-he-moneyness (as e.g. described in American Academy of Acuaries, 2005) or (more direcly) he value of he guaranee. Alhough more conservaive han assuming purely deerminisic behavior, hese assumpions can sill be quie dangerous: If, for insance, a remaining WBB rache is in place and policyholders do no surrender a all, he poenial loss amouns o almos 1% of he single premium paid even if assumed marke parameers are correc. A slighly smaller loss occurs in case of a lookback rache, whereas if no rache is in place, here even is a profi. However, if policyholders behave opimally, he poenial loss, again, is he highes for he produc design wihou rache and amouns o 2.2% of he single premium paid. 38

43 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency If addiionally acual marke parameers deviae from assumed marke parameers, poenial losses increase up o over 5% of he single premium paid. The srucure of he increase is similar o he effecs observed previously, only he differences beween he differen produc designs wih respec o heir ineres rae sensiiviy are now slighly higher. This also holds for he following opimal behavior assumpion. Pricing assumpion Opimal The mos conservaive assumpion abou he policyholders behavior is of course o assume ha hey follow an opimal surrender sraegy. As a resul, losses due o mispricing only occur if addiionally o policyholder behavior also marke parameers are differen han assumed. In his case, he profi from he poenially over-conservaive behavior assumpion is reduced by hese losses. Of course, he losses are he highes if acual behavior is eiher opimal or, in he case of he remaining WBB rache, where no surrendering yields very similar resuls o opimal behavior, if policyholders do no surrender a all. Summarizing, we find ha he produc design wihou rache shows he highes sensiiviy o deviaions from assumed policyholder behavior. On he oher hand, i is he design wih he leas sensiiviy o deviaions from assumed volailiy, while he negaive effec of an overesimaed level of ineres raes is roughly he same for all hree produc designs. 39

44 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency Resuls for he Heson model Table 8 shows similar resuls o hose presened in he previous secion bu now using he Heson model. For all resuls in his able, he surrender fee was se o 3% and, for pricing, he marke-price-of-risk facor was se o λ=0 and a rae of ineres of 4% was assumed. Behavior Pricing: λ=0, r = 4% Acual: λ=0, r = 4% Pricing: λ=0, r = 4% Acual: λ=-2, r = 4% Pricing: λ=0, r = 4% Acual: λ=0, r = 3% Pricing Acual I II III I II III I II III Opimal OV Opimal ITM NS DS Opimal OV OV ITM NS DS Opimal OV ITM ITM NS DS Opimal OV NS ITM NS DS Opimal OV DS ITM NS DS Table 8: GLWB rider value a incepion as percenage of he single premium if acual policyholder behavior and/or parameers in he Heson model differ from pricing assumpions. Again, he resuls observed under he Heson model are very similar o hose observed under he Black-Scholes model. Thus, he poenial for mispricing arising from wrong assumpions abou policyholder behavior or a wrong level of volailiy or ineres raes seems o be much higher han he poenial loss arising from ignoring he sochasiciy of equiy volailiy. However, by solely calculaing he rider value of he guaranee we implicily assume perfec hedge effeciveness which is no given in realiy. 40

45 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency Kling e al. (2011) have shown ha he impac of sochasic volailiy on hedge efficiency of such producs is ypically much higher han on pricing. Therefore, we now analyze how his resul relaes o differen assumpions abou policyholder behavior. 4 Analysis of Hedge Efficiency In his secion, we analyze he performance of a hedging program an insurer migh apply in order o reduce he financial risk and hus also he required economic capial resuling from selling GLWB guaranees. We analyze his performance under differen assumpions regarding policyholder behavior and paricularly analyze he case where he policyholder behavior assumed by he insurer for pricing and hedging differs from he acual behavior of he policyholders. In wha follows, we firs describe he analyzed dela-hedging sraegy; we hen define he risk measures ha we use o compare he (simulaed) hedge performance, and finally, we presen he simulaion resuls in he las par of his secion. The mehodology we use is similar o he one used by Kling e al. (2011). 4.1 Hedge Porfolio We assume ha an insurer has sold a pool of policies wih GLWB guaranees. We denoe by Ψ he cumulaive opion value for ha pool of guaranees, i.e. he sum of he opion values V of each policy as defined in Secion 2.3. We assume ha he insurer canno influence he value of Ψ by changing he underlying fund (e.g. changing he fund's exposure o risky asses or forcing he policyholder o swich o a differen, e.g. less volaile, fund). We furher assume ha he H insurer invess he guaranee fees as well as surrender fees in a hedge porfolio and applies some hedging sraegy wihin his porfolio. In case he guaranee of a policy is riggered, he guaraneed paymens due are deduced from his porfolio. Thus, : (12) H is he insurer s cumulaive profi/loss (in wha follows someimes jus denoed as he insurer s profi) a ime semming from he guaranee and he corresponding hedging sraegy. We assume he value of he guaranee o be marked-o-model, where he same model he insurer uses for hedging is used for he valuaion of Ψ. For he simulaions in he following secion, we assume ha he insurer uses he Black-Scholes model for hedging purposes and applies a simple dela-hedging sraegy wihin he hedge H porfolio : In order o immunize he porfolio agains small changes in he underlying's spo price S (i.e. o aain dela-neuraliy), he quaniy of exposure o he underlying wihin he insurer s hedge porfolio is deermined as he dela of Ψ, i.e. he parial derivaive of Ψ wih respec o S. We assume ha he hedge porfolio is rebalanced on a monhly basis, using cenral finie differences calculaed via Mone-Carlo simulaion as approximaion for he parial derivaive of Ψ wih respec o S. G 41

46 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency 4.2 Risk Measures We use he following hree measures o compare he differen hedging sraegies. All measures will be normalized as a percenage of he premium volume a =0: E rt T, he expecaion of he discouned final value of he insurer s profi under he real-world measure P. This is a measure for he insurer s expeced profi and consiues he performance measure in our conex. A value of 1 means ha, in expecaion, for a single premium of 100 paid by he clien, he insurance company s discouned profi from selling and hedging he guaranee is 1. P e CTE1 ( ) EP VaR ( ) r variable, where min e 0,1,..., T, he condiional ail expecaion of he random is defined as he minimum of he discouned values of he insurer s profi/loss a all policy calculaion daes and VaR () denoes he Value a Risk of he variable a he level. This is a measure for he insurer s risk resuling from a cerain hedging sraegy: i can be inerpreed as he addiional amoun of money ha would be necessary a ouse such ha he insurer s porfolio would never become negaive over he life of he conrac, even if he marke develops according o he average of he (e.g. 10%) wors scenarios in he sochasic model. Thus a value of 1 means ha, in expecaion over he wors scenarios, for a single premium of 100 paid by he clien, he insurance company would need o hold 1 addiional uni of capial upfron. rt rt rt rt CTE ( e ) E e e VaR ( e ) 1 T P T T T, he condiional ail expecaion of he discouned profi/loss final value. This is also a risk measure which, however, focuses on he value of he profi/loss a ime T, i.e. afer all liabiliies have been me, and does no accoun for negaive porfolio values over ime. Thus, a value of 1 means ha, in expecaion over he wors scenarios, for a premium of 100 paid by he clien, he insurance company s expeced loss is 1. By definiion, of course, rt CTE1 ( ) CTE1 ( e T ). 4.3 Simulaion Resuls In he numerical analyses below, we se =10% for boh risk measures and assume a pool of idenical policies wih parameers as given in Secion 3. We assume ha moraliy wihin he populaion of insured occurs according o he bes-esimae probabiliies given in he DAV 2004R able. As our analysis focuses on model risk raher han parameer risk, we use he parameers for he capial marke models presened in Secion 3 for boh, he hedging and he daa-generaing model. The resuls are calculaed using 10,000 Mone-Carlo pahs for he simulaion, 1,000 pahs for each of he valuaions used in he calculaion of he cenral finie differences and 10,000 pahs for each of he valuaions of Ψ used o calculae Resuls for he Black-Scholes model Table 9 gives he resuls for differen combinaions of behavioral assumpions made by he insurer and acual behavior wihin he pool of policies. The Black-Scholes model wih σbs=22% 42

47 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency and r=4% is hereby used as he hedging model of he insurer and as a model of he real-world progression of he capial marke (wih μ=7%). Pricing / Hedging Opimal OV ITM NS DS Behavior Produc I Produc II Produc III Acual E P [e rt Π T ] CTE[χ] CTE[e rt Π T ] E P [e rt Π T ] CTE[χ] CTE[e rt Π T ] E P [e rt Π T ] CTE[χ] CTE[e rt Π T ] Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Table 9: Hedge efficiency resuls using he Black-Scholes model as daa-generaing model. Pricing assumpion DS We firs look a he case where he insurer assumes deerminisic surrender in pricing and hedging he conrac. If, in realiy, he pool of policyholders behaves exacly according o he same paern, he insurer s expeced profi is close o zero for all differen produc designs. (Noe ha we assume ha he insurer priced he conracs wihou incorporaing any profi margin.) On average over he 10% wors scenarios, he presen value of he insurer s final loss averages o 1.0% of he single premium for produc design I. The corresponding values are 2.2% and 2.0% for produc designs II and III, respecively. The CTE of he presen value of he maximum loss over all policy calculaion daes is slighly higher. Similar resuls are observed for oher assumpions abou he policyholder behavior as long as assumed policyholder behavior and realized policyholder behavior coincide. If he insurer assumes deerminisic surrender bu policyholders acually behave according o he considered funcion of he in-he-moneyness (ITM) or he considered funcion of he opion 43

48 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency value (OV), he insurer s expeced loss significanly increases. The expeced loss for he produc design wihou rache is roughly 3% of he single premium paid and hence more han wice as high han for he produc designs wih rache. In he case of opimal surrender, he expeced loss furher increases o 5.5% of he single premium paid in he case wihou rache and abou half ha value for he producs wih rache. Wih he expeced loss, also he risk increases. Assuming deerminisic surrender for produc I resuls in a CTE of final losses beween 7.6% and 13.3% of he single premium paid if acual policyholder behavior is pah-dependen or even opimal. This risk is reduced by roughly 50% by including raches ino he produc design. If policyholders do no surrender a all (NS), he risk is almos he same for all produc designs. Consisen wih he pricing resuls above, for produc design III, he resuls for no surrender and opimal policyholder behavior are almos idenical since opimal surrender for his produc is close o no surrender. For produc design I, he risk if policyholders do no surrender is lower han for any pah-dependen behavior. Pricing assumpion NS If no surrender is assumed in pricing and hedging, he resuls are raher diverse. Acual deerminisic policyholder behavior leads o a posiive expeced profi for all produc designs and raher limied risk for produc designs II and III. However, risk measures for produc design I are almos 4%. If policyholders acually behave according o he considered funcion of he in-he-moneyness (ITM) or he considered funcion of he opion value (OV), he insurer s expeced profi is slighly posiive for produc designs II and III and around -2% for produc design I. Ineresingly, while for he produc designs wih rache, he risk also is raher limied, he risk measures for produc design I exceed 10%. Thus, if no rache is included in he produc design, he assumpion of no surrender is raher risky. If policyholder behavior is opimal, he risk for his produc even increases o 15%. Pricing assumpions OV and ITM If policyholder behavior is assumed o occur according o he considered funcion of he in-hemoneyness (ITM) or he considered funcion of he opion value (OV) and raches are included ino he produc design (producs II and III), he expeced loss for he insurer (even in he case of opimal policyholder behavior) is below 1.5% of he single premium paid. Furhermore, he considered risk measures remain below 5%. Again, for produc design III, acual policyholder behavior wihou surrender urns ou o be almos as risky as opimal policyholder behavior. For produc design I, however, no surrender leads o expeced profis of 2.9% of he single premium paid and risk measures below 2.3% while opimal behavior leads o a risk of 7.5% and a negaive expeced profi. Deerminisic behavior under boh assumpions and for all produc designs leads o expeced profis and raher low risk. Pricing assumpion Opimal No very surprisingly, he mos conservaive assumpion of opimal policyholder behavior always leads o he highes expeced profi. If acual behavior is deerminisic or no surrender, for produc design I he expeced profi reaches 7.8% and 8.7%, respecively. Also, risk is raher limied and for all produc designs below 3.4%. However, i is worh noing ha he risk if policyholders do no surrender for produc design III (3.4%) is slighly higher han in he case of opimal surrender. We aribue his o he fac ha in he case of opimal surrender, all policyholders surrender a he same ime. Thus, hedging is needed for a poenially shorer period of ime, resuling in a reduced hedging error. Also, for produc III, assuming opimal surrender resuls in a slighly higher risk han if no surrender is assumed. We aribue his o a 44

49 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency more sable hedging in he case of no surrender (as here are no decisions wheher all of he policyholders eiher say or leave) and some imperfecions in he Longsaff-Schwarz algorihm we used Resuls for he Heson model Table 10 shows he same resuls as in Table 9, bu now using he Heson model as daageneraing model insead of he Black-Scholes model, wih parameers as saed in Table 4 and wih μ=7%. Pricing / Hedging Opimal OV ITM NS DS Behavior Produc I Produc II Produc III Acual E P [e rt Π T ] CTE[χ] CTE[e rt Π T ] E P [e rt Π T ] CTE[χ] CTE[e rt Π T ] E P [e rt Π T ] CTE[χ] CTE[e rt Π T ] Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Opimal OV ITM NS DS Table 10: Hedge efficiency resuls using he Heson model as daa-generaing model. Changing he daa-generaing model from Black-Scholes o Heson does no have any subsanial impac on he expeced profi, independen of he produc design and he assumed policyholder behavior. Also, he srucure of he resuls, i.e. he relaion beween he resuls for he differen producs and he differen clien behavior paerns is very similar. However, he absolue values of he risk measures change. While produc design I appears o be less risky in case of pah-dependen behavior under he Heson model if deerminisic or no surrender is assumed, he risk for produc designs II and III increases. The resuls show ha produc designs II and III display a higher sensiiviy o volailiy han he design wihou rache (I). This is in line wih he resuls of Secion

50 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency We can conclude ha assumpions abou policyholder behavior can bear significan risk for he insurer, especially if such assumpions are oo aggressive, i.e. if policyholders behavior is closer o opimal behavior han assumed. However, his risk can be significanly reduced by means of produc design and making appropriae behavioral assumpions. The laer, however, also increases he price of he produc and may resul in a lower compeiiveness of he produc. While he produc designs wih rache feaures (II and III) appear o be less sensiive o policyholder behavior, our resuls indicae ha hey may be harder o hedge and are more sensiive o changes in volailiy and/or model risk, respecively. 5 Conclusions In he presen paper, we have analyzed he impac of policyholder behavior on pricing, hedging and hedge efficiency of differen GLWB guaranees in variable annuiies. We have considered several ypes of policyholder behavior ranging from deerminisic surrender over pahdependen surrender o opimal sraegies. We have found ha he price of he guaranee srongly depends on he assumed policyholder behavior and here is a significan poenial for mispricing if acual policyholder behavior deviaes from assumed behavior. Comparing producs wih differen rache mechanisms, we find ha his poenial for mispricing is he smalles for he produc design wih he mos valuable rache mechanism. Analyses of an insurer s hedging sraegy showed ha boh, he insurer s expeced profi and he insurer s risk (quanified by CTE measures), depend heavily on he deviaion beween assumed and acual policyholder behavior as well as he chosen produc design. We find ha he produc design wihou rache shows he highes sensiiviy o changes in policyholder behavior. On he oher hand, i is he design wih he leas sensiiviy o changes in volailiy and he poenially easies one o hedge. We also find ha he impac of sochasic volailiy on hedging (and he insurer s risk) is much higher han on pricing (and he insurer s expeced profi). In fuure research, i would be ineresing o combine he analyses of model risk performed in Kling e al. (2011) wih he analyses of policyholder behavior risk and quanify how he insurer s risk depends on a simulaneous deviaion from realiy of assumpions regarding policyholder behavior and he capial marke model. I migh also be worhwhile o analyze differen ypes of variable annuiy guaranees and see wheher differen ypes of guaranees (e.g. GMAB or GMIB) display higher or lower behavioral risk han he GLWB designs considered in his paper. Our analyses so far have been performed on he level of an individual policy. Since hedging errors are no necessarily addiive over a pool of policies, i would be worhwhile o analyze how he resuls wih respec o risk managemen and hedge efficiency change for a heerogeneous pool of policies. References [1] American Academy of Acuaries (2005), Recommended Approach for Seing Regulaory Risk-Based Capial Requiremens for Variable Annuiies and Similar Producs, June 2005, Boson. [2] Bacinello, A.R. (2003) Fair valuaion of a guaraneed life insurance paricipaing conrac embedding a surrender opion. Journal of Risk and Insurance, 70,

51 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency [3] Bacinello, A.R. (2005) Endogenous Model of Surrender Condiions in Equiy-Linked Life Insurance. Insurance: Mahemaics and Economics, 37 (2), [4] Bacinello, A.R., Biffis, E. and Millossovich, P. (2010) Regression-Based Algorihms for Life Insurance Conracs wih Surrender Guaranees. Quaniaive Finance, 2010, 10, [5] Bacinello, A.R., Millossovich, P., Olivieri, A., and Piacco, E. (2011) Variable Annuiies: A Unifying Valuaion Approach. Insurance: Mahemaics and Economics, 49 (3), [6] Bauer, D., Kling, A., and Ruß, J. (2008) A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies. ASTIN Bullein, 38 (2), November 2008, [7] Bernard, C., MacKay, A., Muehlbeyer, M. (2014) Opimal surrender policy for variable annuiy guaranees. Insurance: Mahemaics and Economics, 55 (1), [8] Bingham, N.H. and Kiesel, R. (2004), Risk-Neural Valuaion: Pricing and Hedging of Financial Derivaives, Springer, Berlin. [9] Black, F., and Scholes, M. (1973) The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy, 81, [10] Chen, Z., Vezal, K. and Forsyh, P.A. (2008) The effec of modelling parameers on he value of GMWB guaranees. Insurance: Mahemaics and Economics, 43 (1), [11] Coleman, T.F., Kim, Y., Li, Y., and Paron, M. (2006) Hedging Guaranees in Variable Annuiies under Boh Equiy and Ineres Rae Risks. Insurance: Mahemaics and Economics, 38, [12] Coleman, T.F., Kim, Y., Li, Y., and Paron, M. (2007) Robusly Hedging Variable Annuiies wih Guaranees under Jump and Volailiy Risks. The Journal of Risk and Insurance, 74 (2), [13] Cox, J.C., Ingersoll, J.E., and Ross, S.A. (1985) A Theory of he Term Srucure of Ineres Raes. Economerica, 53, [14] DAV (2004) Herleiung der DAV-Serbeafel 2004 R für Renenversicherungen. Deusche Akuarvereinigung, available online a: hp://akuar.de/cusom/download/dav/veroeffenlichungen/2004-uag- Rennerserblichkei-DAV-2004R.pdf. [15] De Giovanni, D. (2010) Lapse Rae Modeling: A Raional Expecaion Approach. Scandinavian Acuarial Journal, 1, [16] Eraker, B. (2004) Do sock prices and volailiy jump? Reconciling evidence from spo and opion prices. Journal of Finance, 59, [17] Ewald, C.-O., Poulsen, R., and Schenk-Hoppe, K.R. (2009) Risk Minimizaion in Sochasic Volailiy Models: Model Risk and Empirical Performance. Quaniaive Finance, 9 (6), Sepember 2009,

52 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency [18] Gao, J. and Ulm, E.R. (2012) Opimal Consumpion and Allocaion in Variable Annuiies wih Guaraneed Minimum Deah Benefis. Insurance: Mahemaics and Economics, 51(3), [19] Glasserman, P. (2003), Mone Carlo Mehods in Financial Engineering. Springer, Berlin. [20] Grosen, A., and Jorgensen, P. (2000) Fair valuaion of life insurance liabiliies: The impac of ineres rae guaranees, surrender opions, and bonus policies. Insurance: Mahemaics and Economics, 26, [21] Heson, S. (1993) A Closed Form Soluion for Opions wih Sochasic Volailiy wih Applicaions o Bond and Currency Opions. Review of Financial Sudies, 6 (2), [22] Holz, D., Kling, A., and Ruß, J. (2012) GMWB For Life An Analysis of Lifelong Wihdrawal Guaranees. Zeischrif für die gesame Versicherungswissenschaf, 101 (3), [23] Hull, J.C. (2008), Opions, Fuures and Oher Derivaives. 7 h ediion. Prenice Hall, New Jersey. [24] ING (2011), Press release, December 7 h 2011, available online a: hp:// [25] Kahl, C., and Jäckel, P. (2006) No-so-complex Logarihms in he Heson Model. Wilmo Magazine, Sepember 2005, [26] Kling, A., Ruez, F., and Ruß, J. (2011) The Impac of Sochasic Volailiy on Pricing, Hedging, and Hedge Efficiency of Wihdrawal Benefi Guaranees in Variable Annuiies. Asin Bullein, 41(2), [27] Knoller, C., Krau, G., and Schoenmaekers, P. (2013) On he Propensiy o Surrender a Variable Annuiy Conrac: An Empirical Analysis of Dynamic Policyholder Behavior. Working Paper, Munich Risk and Insurance Cener. [28] Longsaff, F.A. and Schwarz, E.S. (2001) Valuing American Opions by Simulaion: A Simple Leas-Squares Approach. The Review of Financial Sudies, 14 (1), [29] Manulife Financial (2011), Press release, November 3 rd 2011, available online a: hp:// [30] Mikhailov, S., and Nögel, U. (2003) Heson s Sochasic Volailiy Model. Implemenaion, Calibraion and Some Exensions. Wilmo Magazine, July 2003, [31] Milevsky, M., and Posner, S.E. (2001) The Tianic Opion: Valuaion of he Guaraneed Minimum Deah Benefi in Variable Annuiies and Muual Funds. The Journal of Risk and Insurance, 68 (1), [32] Milevsky, M., and Salisbury, T.S. (2006) Financial Valuaion of Guaraneed Minimum Wihdrawal Benefis. Insurance: Mahemaics and Economics, 38,

53 1 The Impac of Policyholder Behavior on Pricing, Hedging, and Hedge Efficiency [33] Seffensen, M., (2002) Inervenion opions in life insurance. Insurance: Mahemaics and Economics, 31(1), [34] Sun Life Financial (2011), Press release, November 2 nd 2011, available online a: hp:// un_life_financial_repors_hird_quarer_2011_resuls_nov2.pdf. [35] Taleb, N. (1997) Dynamic Hedging: Managing Vanilla and Exoic Opions. Wiley Finance, New York. [36] The Harford (2009) 1 s quarer 2009 sakeholder message of The Harford, available online a: hp:// [37] Wong, B., and Heyde, C.C. (2006) On Changes of Measure in Sochasic Volailiy Models. Journal of Applied Mahemaics and Sochasic Analysis, Volume 2006, [38] Yang, S.S., Dai, T.-S. (2013) A flexible ree for evaluaing guaraneed minimum wihdrawal benefis under deferred life annuiy conracs wih various provisions. Insurance: Mahemaics and Economics, 52(2),

54 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies: The Impac of Regulaor-Imposed Guaranees The final publicaion is available a link.springer.com hps://link.springer.com/aricle/ /s Kling, A., Ruez, F. and Ruß, J., Guaraneed minimum surrender benefis in variable annuiies: he impac of regulaor-imposed guaranees. To appear in European Acuarial Journal. DOI: /s

55 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies Guaraneed Minimum Surrender Benefis in Variable Annuiies: The Impac of Regulaor-Imposed Guaranees Alexander Kling Insiu für Finanz- und Akuarwissenschafen (ifa) Lise-Meiner-Sraße Ulm, Germany Frederik Ruez (corresponding auhor) Insiu für Finanz- und Akuarwissenschafen (ifa) & Insiue of Insurance Science, Universiy of Ulm Lise-Meiner-Sraße Ulm, Germany phone: fax: Jochen Ruß Insiu für Finanz- und Akuarwissenschafen (ifa) & Insiue of Insurance Science, Universiy of Ulm Lise-Meiner-Sraße Ulm, Germany Absrac We analyze he impac of regulaor-imposed minimum surrender benefis on variable annuiies wih a Guaraneed Minimum Accumulaion Benefi (GMAB) rider. Based on recen discussions in he German marke, we consider differen models how hese Guaraneed Minimum Surrender Benefis (GMSB) are deermined: A minimum surrender benefi given by he presen value of he GMAB calculaed using marke ineres raes, he presen value of he GMAB calculaed using some echnical rae of ineres, and he marke-consisen value of he GMAB. We look a he case where he GMSB is inroduced before he conrac is sold and considered in he pricing of he GMAB rider. We also consider he case if he GMSB is imposed afer he conrac has been sold and analyze he impac on he echnical provisions and capial requiremens of already exising conracs. Finally, we analyze how our resuls change in he presence of a secondary marke. Our resuls show ha (if considered in he pricing of he conrac) a GMSB can significanly affec he fair guaranee charge of variable annuiies. We also find a significan impac on he echnical provisions and capial requiremens of already exising conracs. Finally, our resuls indicae ha a secondary marke adversely affecs he insurer s profiabiliy bu reduces he impac of he considered GMSBs on he insurers. Keywords Variable Annuiies, Guaraneed Minimum Accumulaion Benefi, Guaraneed Minimum Surrender Benefi, Pricing, Capial Requiremens, Secondary Marke 51

56 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies 1 Inroducion Variable annuiies are uni-linked life insurance conracs ha ofen come wih invesmen guaranees. Therefore, hey allow policyholders o benefi from he upside poenial of he underlying fund and, a he same ime, offer proecion when he fund loses value (cf. EIOPA, 2011). Such producs offer a variey of guaranees. Besides guaraneed minimum deah benefis (GMDB), hree main ypes of guaraneed living benefis (GLB) exis: guaraneed minimum accumulaion benefis (GMAB), guaraneed minimum income benefis (GMIB) and guaraneed minimum wihdrawal benefis (GMWB). GMAB and GMIB offer he policyholder some guaraneed mauriy value or some guaraneed annuiy benefi, respecively, while GMWB allow policyholders o (emporarily or lifelong) wihdraw money from heir accoun, even afer is cash value has dropped o zero. Variable annuiies have experienced a growh in sales in US and Japan since he 1990s and are also becoming increasingly widespread over Europe (cf. EIOPA, 2011). The produc design of variable annuiies usually sipulaes ha he surrender value of such producs coincides wih he policyholder s accoun value (minus surrender charges, if applicable). The fair value of he guaraneed benefis or he marke value of cerain hedge asses is ypically no par of he individual policyholder s accoun value and hus, wih he usual produc design, no par of he surrender value. The pricing of he guaranees in variable annuiies is usually performed under cerain assumpions for fuure surrender raes. Such assumpions can be, for insance, deerminisic surrender or (ypically) pahdependen surrender (where assumed surrender raes depend on marke parameers and/or he value of he guaranee). However, he pricing is usually no performed under he assumpion of opimal surrender (in he sense of loss-maximizing behavior from he insurer s perspecive, cf. Azimzadeh e al., 2014). This reduces he price of such guaranees since in simplified erms fuure profis he insurer expecs from sub-opimal policyholder behavior are given o he clien by means of a reduced price for he guaranee. The possibiliy o allow for sub-opimal policyholder behavior in pricing and hedging of such producs is a reason why hese (ofen primarily financial) guaranees can be offered by insurers a compeiive prices when compared o similar producs offered by banks. This opens opporuniies for insiuional invesors o purchase such policies in a secondary marke a a price ha exceeds he surrender benefi from policyholders who are willing o surrender heir conrac. In his siuaion, selling he conrac o he insiuional invesor insead of surrendering i is beneficial for he policyholder. Afer acquiring he conrac, he insiuional invesor hen maximizes (opimizes) he value of he conrac, which ypically resuls in loss-maximizing behavior from he insurer s perspecive. Of course, his creaes risks for he insurer, mos noably he risk ha policyholders behave differenly han assumed. In Kling e al., 2014, he auhors have analyzed he resuling risk in deail. In his paper, we use heir model o addiionally analyze a new, regulaor-imposed risk ha migh arise in cerain insurance markes: The risk arising from Guaraneed Minimum Surrender Benefis (GMSB). To analyze his risk, we use differen exemplary approaches a regulaor migh choose for including GMSBs in variable annuiies. The models resul from discussions in Germany where differen approaches have been discussed among consumer proecion organizaions, insurance companies and he local Acuarial Associaion. Even hough he specific approaches are based on his discussion in Germany, he basic inuiion behind each model could be inroduced in any marke. Since is revision in 2008, 169 of he German Insurance Conrac Law (Versicherungsverragsgesez) requires guaraneed minimum surrender values for all insurance conracs where boh, in case of deah or survival, an insurance benefi is paid. For radiional life insurance conracs wih an ineres rae guaranee, he law even prescribes how his guaraneed minimum surrender value has o be calculaed: 52

57 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies i is given by he prospecive policy reserve which is he presen value of fuure benefis. As a discoun rae, he echnical rae used for he calculaion of premiums and benefis of he conrac has o be applied. Therefore, he surrender value does no allow for any kind of adjusmens o changing marke condiions. This in consequence means ha he surrender value in general is differen from a fair marke value of he conrac. In paricular, he surrender value will no be reduced if ineres raes rise, alhough boh, he asses backing he conrac and he fair value of he conrac, would drop. The resuling risk has been discussed e.g. in Feodoria & Försemann, For uni-linked conracs wihou guaranees, he law requires he surrender value o coincide wih he ne asse value of he fund. For uni-linked conracs wih guaranees, however, he law is no very clear, since i uses he erm ime value of he conrac, which is no properly defined. In paricular, he quesion if and how his law has o be applied o guaraneed minimum benefis in variable annuiies is conroversially discussed. We consider four differen poenial inerpreaions of he law: Firs, here is a group of legal expers saing ha he corresponding secion of he insurance conrac law defining guaraneed minimum surrender values is no applicable a all for ypical USsyle variable annuiies. Under his inerpreaion, he surrender value is given by he policyholder s fund value and neiher fuure guaraneed benefis nor guaranee charges are aken ino accoun. Second, since he law uses he word ime value, some marke paricipans demand ha a marke-consisen value of he conrac has o be paid ou as surrender benefi. Third, he German Acuarial Associaion (Deusche Akuarvereinigung e.v.) has issued a paper inroducing an easy-o-implemen mehod ha could serve as an approximaion for his marke-consisen value if he value of he guaranee has o be considered in he surrender benefi (see Deusche Akuarvereinigung e.v., 2011). Noe ha his paper does no give an opinion on he quesion wheher he value of he guaranee has o be considered or no. Finally, based on he inerpreaion of minimum reserves required in he German Insurance Supervisory Law (Versicherungsaufsichsgesez) given in Herde, 1996, for cerain oher uni-linked insurance producs wih a mauriy guaranee, a minimum reserve for he guaraneed mauriy value (which is given by he guaraneed mauriy value discouned wih some echnical ineres rae) migh also have o be paid ou as a minimum surrender benefi. This echnical ineres rae is se when he conrac is concluded and will no change wih changing marke ineres raes a similar approach as described above for radiional life insurance conracs. We herefore consider his minimum surrender benefi as a furher GMSB-model in our analysis. We analyze he effec of he considered GMSB-models on pricing, profiabiliy, and marke as well as behavioral risk. We paricularly consider he effec on an insurer s profiabiliy if a GMSB is imposed afer a produc has been sold. Furhermore, when assessing policyholder behavior and lapse risk, insurers are required o consider aciviy by insiuional invesors like hedge funds in a poenial secondary marke (cf. e.g. Cenral Bank of Ireland, 2010). Therefore, we also invesigae he impac of he differen ypes of GMSB on a secondary marke for variable annuiies. To our knowledge, such analyses, in paricular wih respec o variable annuiies wih regular premium paymen, have no ye been performed. The paper is organized as follows. In Secion 2, we presen he model framework ha we use o conduc our analyses, including he modeling of he pool of policies, he assumed hedging sraegy of he insurer, and, of course, he considered models of he GMSB. In Secion 3, we presen our numerical resuls regarding he impac of he considered GMSBs on differen key figures from he insurer s perspecive, such as marke risk, sensiiviy o changes in surrender raes and he guaranee value of he conrac. In Secion 4 we inroduce insiuional invesors ino our model framework. We firs presen an exension of he model given in Secion 2 and, subsequenly, presen numerical resuls for he exended model. Finally, Secion 5 concludes. 53

58 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies 2 Model 2.1 Produc design of he considered variable annuiy We consider a variable annuiy conrac ha offers he policyholder a Guaraneed Minimum Accumulaion Benefi (GMAB, see, e.g., Bauer e al., 2008), where he policyholder is eniled o a minimum accoun value B A,g a mauriy T of he conrac. We assume all ransacions and evens wihin he conrac o happen a one of he conrac anniversary daes, represened by he se T = { 0, 1,, N }, where 0 = 0 and N = T. A hese daes, poenial premium paymens are made by he policyholder or benefis are paid ou by he insurer in case he policyholder decided o surrender he conrac, he insured person has died or he conrac has maured. A a conrac anniversary prior o T, he premium P is paid by he policyholder, provided he conrac is sill acive (i.e. he insured person is sill alive and he conrac has no been surrendered) and he policyholder has no decided o surrender he conrac a ime. For a single premium conrac, we le P 0 > 0, P = 0 > 0. The minimum accumulaion benefi B A,g guaraneed a mauriy T is defined as a percenage γ of he sum of premiums paid by he policyholder, i.e. N 1 B A,g γ P i and he accumulaion benefi B A is he larger of he accoun value F T and he guaraneed minimum accumulaion benefi: i=0 B A max(f T, B A,g ). In reurn for his guaranee, he insurer receives an ongoing guaranee charge as a percenage η g of he policyholder s accoun value. The ongoing adminisraion charges, also deduced from he accoun value, are denoed by he percenage η a. Addiionally, acquisiion and adminisraion charges are deduced from each premium paymen, denoed by he percenage η a,u. The accoun value direcly afer incepion is herefore given by F 0 P 0 (1 η a,u ). A any conrac anniversary i T { 0 }, he accoun value is calculaed as F i (F i 1 + P i 1 (1 η a,u )) e (ηa +η g ) ( i i 1 ), S i 1 where S denoes he price of one share of he variable annuiy s underlying fund a ime. In case of deah of he insured, he policyholder receives he sipulaed deah benefi a he subsequen conrac anniversary and he conrac expires. In wha follows, B D denoes he deah benefi paid a T and τ D denoes he firs conrac anniversary afer he insured s deah. If τ D > T hen he insured is sill alive a he conrac s mauriy. Wih he considered produc design, B D F, T. S i 54

59 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies We assume ha he policyholder has he righ o (fully) surrender he conrac a any ime during he conrac s lifeime. If he policyholder decides o surrender he conrac, he sipulaed surrender benefi B S is paid ou by he insurer a he subsequen dae T and he conrac expires. We assume ha he policyholder always wais unil a conrac anniversary before deciding wheher o surrender or coninue he conrac. This implies ha he policyholder knows he exac amoun of he poenial surrender benefi before deciding on wheher o surrender or no. The dae a which he policyholder surrenders he conrac is denoed by τ S wih 1 τ S T. If he policyholder does no surrender he conrac, we se τ S = T. We also le B T S B A. Noe ha he conrac expires a he ime τ min(τ D, τ S ) wih 1 τ T and he cash flow o he policyholder (or he beneficiaries) is nonzero only a ime τ and equals eiher B τ S or B τ D Guaraneed minimum surrender benefis We consider four differen ypes of guaraneed minimum surrender benefi (GMSB). As explained in Secion 1, hese four approaches are based on ideas discussed in Germany. Since hese ideas cover a wide range of poenial ideas, he resuls may be of relevance for any marke where he inroducion of GMSBs is discussed. Even if in some marke a differen GMSB-model is being considered, qualiaively, he effecs will likely be similar o our resuls. The GMSB a ime (before deducion of surrender charges) is denoed by B S,g,j, where he superscrip j {1,2,3,4} indicaes he ype of he considered GMSB. In all four cases, he surrender benefi is calculaed as follows: B S (1 η S ) max(f, B S,g,j ), T {N }, j {1,2,3,4}, where η S represens a ime-consan surrender charge. In he firs considered case, denoed as no GMSB, here is no guaraneed surrender benefi, i.e. B S,g,1 0, T { N }. The second case is denoed as marke-rae GMSB. This model is similar o he approximaion for he fair value given by he German Acuarial Associaion. Here, he policyholder receives a leas he discouned guaraneed accumulaion benefi B A,g ha would resul if all following premium paymens P were zero, i.e. only premium paymens made prior o are considered: where 1 i < denoes he indicaor funcion. N 1 B A,g γ P i 1 i <, i=0 In order o calculae he guaraneed minimum surrender benefi, his hypoheical guaraneed minimum accumulaion benefi B A,g is discouned wih he hen-curren marke rae and muliplied by he probabiliy of he insured o survive unil mauriy of he conrac. Le Z (T ) denoe he price of a riskless zero-coupon bond a ime wih mauriy a ime T and le q(s, ) represen he expeced percenage of he insured who are alive a ime s and die wihin he ime inerval ]s, ]. The guaraneed minimum surrender benefi is hen defined as B S,g,2 B A,g Z (T ) (1 q(, T)), T { N }. 55

60 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies The hird version of he GMSB is denoed as echnical-rae GMSB. This is he model ha is based on he prospecive minimum reserve using a echnical ineres rae. Here, again he presen value of he hypoheical guaraneed minimum accumulaion benefi B A,g is used; however, i is now discouned wih a echnical, ime-consan rae ξ and again weighed wih he survival probabiliy. The guaraneed surrender benefi is hen defined as B S,g,3 B A,g e (T ) ξ (1 q(, T)), T { N }. In he fourh design, denoed as MCV GMSB, he GMSB is he marke-consisen value of he GMAB from he insurer s perspecive, i.e. he marke-consisen value of he guaraneed minimum accumulaion benefi less he marke-consisen value of he fuure guaranee charges o be received by he insurer (no considering he opion o surrender a a fuure dae). As wih he previous wo ypes of GMSB, he valuaion implies ha here are no fuure premium paymens. In our analyses, his marke-consisen value of he GMAB is approximaed by he value of a European pu opion on he accoun value wih srike price B A,g minus he presen value of fuure guaranee charges. Boh, he guaranee a mauriy as well as he fuure guaranee charges, are weighed wih he corresponding survival probabiliies of he insured. Le F,s denoe he accoun value a ime s assuming ha since < s here were no more premium paymens, i.e. F,s F S s S e (ηa +η g )(s ). A a conrac anniversary T, for he purpose of his GMSB, he value of he guaranee is calculaed as V g E Q [ C max (B A,g F,T ] (1 q(, T)) C T = F O P S (T, B A,g F S, η g + η a ) (1 q(, T)), where C denoes he value of he cash accoun a ime and O P (s, K, φ) denoes he price of a European pu opion on he underlying S wih ime o mauriy s, srike price K and a drain φ due o charges (ha are assumed o have he same effec on he opion price as a dividend yield). A any ime T, for he purpose of his GMSB, he value of he fuure guaranee charges deduced from he policyholder s accoun value, denoed by V c, is defined as 1 N 1 ηg V c ( η g + η a (E Q [ C F,i ] E C Q [ C F,i+1 ]) (1 q(, i C i )) 1 i >) i+1 i=1 N 1 = ( ηg η g + η a (F e (ηa +η g )( i ) F e (ηa +η g )( i+1 ) ) (1 q(, i )) 1 i >) i=1 η g 1 The reasoning behind he erm is he following (cf. DAV 2011): E η g +η a Q [ F s η g C d ] = i 1 C s η g F (1 η g +η a e (ηa +η g )( i+1 i ) ). 56 i

61 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies N 1 = F ( ηg η g + η a (1 e (ηa +η g )( i+1 i ) ) e (ηa +η g )( i ) (1 q(, i )) 1 i >). i=1 The marke-consisen value of he GMAB, from he policyholder s viewpoin, is hen V g V c, which is added o he accoun value F in order o deermine he fourh ype of considered GMSB: B S,g,4 F + V g V c, T { N }. This GMSB is a represenaion of an acual (marke-consisen) ime value of he GMAB. However, as he surrender benefi is he maximum of he fund value and GMSB less surrender charges, he GMSB only increases he surrender benefi if he ime value is posiive, bu does no lead o a reduced surrender benefi if he ime value is negaive. Noe also, ha he acual liabiliy of he insurer includes fuure premium paymens, while his GMSB only considers he sum of premiums paid so far. 2.2 Pool of policies For our analyses on a porfolio level, we assume a pool of policies wih idenical conrac parameers wih regard o incepion and mauriy dae, guaranee level, charges, ec. We also assume he pool of insured o be homogeneous and large enough o jusify he applicaion of he law of large numbers such ha moraliy henceforh is only expressed as a percenage of he pool of insured. We denoe he number of conracs in he considered pool of policies a ime by π. The oal number of conracs ha expire a ime i due o deah of he insured person is given by π D i q( i 1, i ) π i. The policyholders are assumed o surrender according o deerminisic base probabiliies, which are increased by a facor of 2 if he conrac is ou-of-he-money, i.e. if he conrac s discouned guaranee (assuming no fuure premium paymens) is lower han he curren surrender benefi. This represens policyholders who are no able or willing o fully opimize heir conrac, bu will have an increasing endency o surrender heir conrac if he guaranee appears less valuable. 2 The fracion of policyholders who surrender heir conrac a he end of he ime inerval ] i 1, i ] is given by s i { 2 s i, if B S i > B A,g Z i (T i ), s i, else where s i represen he deerminisic base surrender probabiliies. The oal number of policyholders who surrender heir conrac a ime i < T, is hen given by π S i = s i (π i 1 π D i ). In line wih he approach in Secion 2.1, we model he mauriy of he conrac as all remaining policyholders leaving he pool via surrender, i.e. we se π S T = π N 1 π D N. The number of conracs immediaely afer i is given by π i = π i 1 π D i π S i. 2 See Secion 4 for an exension of his modeling approach, where opimal (loss-maximizing) behavior by insiuional invesors is explicily modeled. 57

62 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies 2.3 Hedging We assume he insurer o have a hedging program in place ha aims a miigaing he effecs he key financial risk drivers have on he insurer s P&L. Guaranee charges from he pool of policies are used o finance he hedge porfolio. In reurn, guaranee paymens are aken from he hedge porfolio s funds. A incepion, here is no cash flow o or from he hedge porfolio, i.e. Φ π 0 = 0 π The cash flow Φ i a subsequen daes i T, i > 0 is given by π = π i F i η g + η a (1 e (ηa +η g )( i i 1 ) ) + π S i (B S i F i ) Φ i η g If he surrender benefi is less han he accoun value (due o surrender charges), his is also used for financing he hedge porfolio. Le Φ π denoe he cash flow a an arbirary ime beween incepion and mauriy, i.e. 0 T, Φ π = { Φ π, if T. 0, else We denoe by V π he marke-consisen value a ime of he cash flow {Φ u π, u T, u > }, i.e. he value for which he pool s guaranee-relaed cash flows a all fuure daes can be raded. In order o replicae he changes in he value of V π due o movemens in he underlying fund and changing ineres rae environmen, we assume a hedging sraegy using hree hedging insrumens and he hedge porfolio o be rebalanced on a regular basis. The considered hedging insrumens are: cash (overnigh lending/borrowing), he underlying fund (long/shor exposures) and a zero-coupon paying bond wih he same mauriy as he variable annuiy conrac. The value of he hedge porfolio a ime is denoed by Ψ. We assume he hedge porfolio o sar wih a value of zero, i.e. Ψ 0 = Φ π 0 = 0. We assume a simple dynamic hedging sraegy ha aims a offseing changes in he value of he pool s liabiliy resuling from changes in he underlying s price ( Dela ) and shifs in he ineres rae srucure ( Rho ). For an arbirary ime, le λ S denoe he number of shares of he underlying fund in he hedge porfolio, λ Z denoe he corresponding number of zero-coupon bonds wih mauriy in T and λ C denoe he sum invesed in a cash accoun. For a rebalancing dae, le s denoe he ime when he las rebalancing occurred. The value Ψ hen calculaes as Ψ = Ψ s + Φ π + λ s C C C s + λ s S S S s + λ s Z Z (T ) Z s (T s). A a rebalancing dae, he weighs of he hedge posiions in he underlying and he bond, λ S and λ Z, are calculaed as follows: π V λ S = V π, λ Z r S =. Z (T ) r 58

63 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies In he simulaion, λ S and λ Z are calculaed numerically by compuing he finie difference wih respec o changes in he marke-consisen value V π. The posiion in cash, λ C, is a funcion of he oher wo posiions: λ C = Ψ (λ S S + λ Z Z (T )). We assume ha he insurer neiher injecs nor exracs any money from he hedge porfolio, such ha he final value of he hedge porfolio, Ψ T gives an indicaion of he insurer s profi or loss wih regard o pricing and hedging he conracs guaranees. We assume ha acquisiion and adminisraion charges and expenses are no par of his consideraion. In he following analyses, we will analyze he probabiliy disribuion of Ψ T and use corresponding (risk) measures as indicaors for he marke risk of he insurer. 2.4 Financial marke model For he valuaion as well as he simulaion, we need o projec he price dynamics of he following asses: he price of one share of he variable annuiy s underlying fund, S ; he price of a risk-free (wih regard o defaul) zero-coupon bond wih ime o mauriy of τ, Z (τ); he price of he cash accoun, C ; and he prices of simple pu opions on he underlying fund, O P (τ, K, φ), where K denoes he srike level of he respecive opion and φ he drain due o charges (or he dividend yield, respecively). We use a similar approach and similar model as in Ruez, However, we use an exension of he Black-Scholes model ( Black & Scholes, 1973) wih sochasic ineres raes via he Cox-Ingersoll-Ross model ( CIR, Cox e al., 1985). Therefore, he dynamics of he marke s sae variables under he riskneural measure Q are given by dr = κ r Q (θ r Q r )d + σ r Q r dw Q,r, ds = r S d + σ S Q S dw Q,S dc = r C d where W Q,r and W Q,S are wo independen Wiener processes under he risk-neural measure Q. The fair value of a zero-coupon bond can be compued by closed-form formulas given in Cox e al., The value of he pu opion is approximaed via he Black-Scholes formulas wih he assumpion of deerminisic ineres raes. For he real-world simulaion used o projec he hedging program of he insurer, we use he same sysem of sochasic differenial equaions for he dynamics under he real-world measure P, dr = κ r P (θ r P r )d + σ r P r dw P,r, ds = (r + μ)s d + σ S P S dw P,S dc = r C d where W P,r and W P,S are wo independen Wiener processes under P. 3 Numerical resuls In his chaper, we analyze he impac of he considered GMSB on he produc s pricing, expeced profi and is risk profile wih respec o marke and lapse risk. The parameers for he ineres-rae model used for valuaion are aken from Bacinello e al., We use he same se of parameers for he real-world 59

64 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies projecion his implies a marke ha is risk-neural wih regard o ineres-rae risk. For he equiy process, we se he volailiy o 10% for boh, valuaion and real-world projecion, and we use 3% for he parameer μ in he real-world projecion (as in Kling e al., 2014). Table 1 summarizes he parameers used for he marke model in he base case (sensiiviy analyses follow). Parameer Value r 0, θ P Q r, θ r 0.03 κ r P, κ r Q 0.60 σ r P, σ r Q 0.03 σ P Q S, σ S 0.10 μ 0.03 Table 1: Marke parameers used in he base case. The parameers for he variable annuiy conrac are summarized in Table 2. Parameer Value T 10 γ 100.0% η a,u 5.0% η a 1.0% η S 1.0% Table 2: Conrac parameers used in he base case. The rebalancing of he hedge porfolio is assumed o happen on a monhly basis. We assume he insured person o be 60 years old and male. We use he bes-esimae moraliy probabiliies for annuians given in he DAV 2004R moraliy able published by he German Acuarial Associaion (DAV). We assume a base surrender rae of 10% in he firs year ha is subsequenly reduced by 1% per year unil i reaches 2%, i.e. s i = max(2%, 10% (i 1) 1%), i = 1,2, For he analyses wih regard o surrender risk, we also use scenarios wih increased and decreased lapse, where he base surrender raes are muliplied by 1.5 and 0.5, respecively, as well as a scenario where no surrender occurs a all. For he echnical-rae GMSB, we use ξ = 1.25%. We use 25,000 Mone Carlo pahs for he valuaions and 10,000 pahs for he real-world simulaion. Wihin he real-world simulaion, we use 1,000 pahs o compue he finie differences used in he modeling of he hedging program. 3.1 Impac of he GMSB on conrac pricing We sar our analyses wih a comparison of he fair guaranee charges for he four considered GMSBs (cf. Secion 2.1.1). For he purpose of his analysis, he fair guaranee charge is he guaranee charge for which he marke-consisen value V 0 π of he variable annuiy s guaranee-relaed cash flow is zero, i.e. a incepion of he conrac, he value of he guaranee charges coincides wih he value of he poenial guaranee paymens. In order o illusrae he sensiiviy wih regard o lapse, we use he differen lapse assumpions defined above. 60

65 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies Noe ha in his secion, he fair guaranee charge η g is calculaed under he assumpion ha he ype of GMSB is already known a he ime he insurer prices he variable annuiy, i.e. he produc is being offered afer he GMSB has been required by he regulaor. Figure 1 shows he fair guaranee charges for a single premium produc (lef) and a regular premium produc (righ). Figure 1 Fair guaranee charges for he single-premium produc (lef) and he regular-premium produc (righ). Even hough he fair guaranee charge significanly differs beween he single premium and he regular premium conrac, he paern wih respec o changes in surrender raes and he differen ypes of GMSB is similar. As expeced, if no GMSB is in place, he fair guaranee charge is he lowes for all considered surrender assumpions. Under he base-lapse assumpion i amouns o 0.48% for a single premium conrac and 1.04% for regular premium paymens. Surrender is on average profiable for he insurer and, hus, he fair guaranee charge decreases if he likelihood of surrender increases. The addiion of a GMSB causes surrender o be less profiable for he insurer. Consequenly, he fair charge increases by roughly 10 bp if a marke-rae GMSB is enforced and by 20 bp if a echnical-rae GMSB is enforced. In case of he MCV GMSB, he fair guaranee charge increases by more han 30 bp. Thus, if he regulaor imposes a GMSB in order o rea surrendering cusomers beer, he producs will become more expensive for all cusomers. While a change in surrender assumpions has a considerable impac on he fair guaranee charge if no GMSB is in place, his sensiiviy is reduced if a GMSB is in place. As a consequence, he poenial for mispricing (by using incorrec surrender assumpions) is he highes if no GMSB is in place. Wihou GMSB, he fair guaranee charge changes by 15 bp for he single premium case and 20 bp for regular premiums if surrender raes are increased or decreased. In he case of he MCV GMSB, he fair guaranee charge is he highes ou of all considered GMSBs and almos independen of surrender assumpions. In urn, he poenial for mispricing wih regard o he assumed fuure surrender raes is fairly low. For he single premium produc, he fair guaranee charge wih surrender is even higher han wihou surrender. This means ha, on average, surrender wih his GMSB means a loss for he insurer, despie he earned surrender charges. 3.2 Impac of he GMSB on he guaranee value of exising conracs In a nex sep, we analyze how he value of he variable annuiy conrac changes if a cerain GMSB is enforced by he regulaor (immediaely) afer he conrac has been sold. For his and all following analyses, we assume a fixed guaranee charge of η g = 1.0% p.a. for he single premium conrac and 61

66 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies η g = 1.5% p.a. for he regular premium conrac has been used by he insurer when he conrac was sold. We hen analyze he change in he value V 0 π caused by he inroducion of a GMSB. This can be inerpreed as an immediae loss (in case of an increase of V 0 π ) or an immediae profi (in case of a decrease of V 0 π ) for he insurer caused by he regulaory change. Figure 2 shows he value of he guaranee for he single-premium produc (lef) and for he regularpremium produc (righ) assuming a fixed guaranee charge. Here and in all following figures, values are given as a percenage of he single premium or he sum of premiums, respecively. Figure 2 Value of he guaranee for he single-premium produc (lef) and he regular-premium produc (righ). We firs have a look a he resuls for he base lapse raes. The iniial value of he guaranee if no GMSB is in place is roughly -2% of he premium for he single-premium conrac and roughly -0.65% of he sum of premiums for he regular-premium conrac. The inroducion of a marke-rae GMSB immediaely afer he sar of he conrac causes an increase of he value of he guaranee and hence a loss for he insurance company of roughly 0.5% for he single-premium conrac and almos 0.2% for he regular-premium conrac. The loss caused by he inroducion of a echnical-rae GMSB is wice as high while ha caused by he inroducion of a MCV GMSB is roughly hree imes as high. Thus, requiring GMSBs for business in force can have a significan impac on he insurer s profiabiliy and can in paricular immediaely wipe ou any safey loadings or profi margins. The loss caused by he inroducion of a GMSB obviously depends on he surrender assumpions used when he conrac was priced. The higher he assumed surrender raes, he higher he impac of a GMSB on he value of he guaranee, which is consisen wih he resuls in he previous secion. We finally analyze he sensiiviy of he value of he guaranee wih respec o changes in surrender raes, i.e. we consider he immediae loss / profi caused by higher / lower surrender raes. As wih he fair guaranee charges (cf. Figure 1), he sensiiviy o surrender differs significanly beween he four GMSB models and also beween single and regular premium paymen. Wihou GMSB, surrender on average is profiable for he insurer. For regular premiums surrender is always profiable alhough any GSMB reduces he profiabiliy. For he single premium produc, his changes: if he mos valuable GMSB, he MCV GMSB, is considered, addiional surrender causes a loss for he insurer, i.e. he addiional value of his GMSB ouweighs he surrender charges. The difference beween single premium and regular premium paymens mosly resuls from he calculaion of he GMSB in he regular premiums case. For he purpose of calculaing he GMSBs, he conrac is assumed o receive no more premiums afer he surrender dae. Simply pu, if a fla guaranee 62

67 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies charge is used, hen he guaranee is overpriced for early premiums while i is underpriced for laer conribuions. If fuure premiums are no considered and he guaranee charge remains he same, he conrac ends o be overpriced and, hus, ofen becomes less valuable o he policyholder. This is refleced in he value of he MCV GMSB in he regular-premiums case and makes surrender less valuable for he policyholder han in he single-premium case. 3.3 Impac of he GMSB on capial requiremens In marke-based solvency regimes, solvency capial requiremens are ofen calculaed as an immediae loss resuling from some sress scenario. The above sensiiviies can herefore also be inerpreed as an indicaion for solvency capial requiremen (SCR) for lapse risk, where only he direcion of change (increase or decrease of lapse raes) ha causes he highes loss for he insurer has o be considered. Addiionally, we will now give an indicaion for he SCR for marke risk via he following approach: In conras o e.g. he Solvency II sandard formula, where only he risk resuling from an immediae (or one year) shock is being considered, we consider a full lifeime projecion of he pool of policies in order o assess he marke risk resuling from he oal remaining lifeime of he conracs, cf. Secion 164 in EIOPA, 2011 in connecion wih Aricle 122 of he Direcive 2009/138/EC. This includes he risk from accumulaed hedging errors, paricularly hedging errors ha occur close o mauriy of he conracs. We accoun for he longer ime horizon by seing he VaR level a 95% and consider he respecive percenile of he real-world disribuion of he (discouned) profi/loss Ψ T C T for he insurer afer hedging is applied (cf. Secion 2.3). We inerpre he difference of his 95 h percenile and he presen value of he guaranee V 0 π as (an indicaor for) he SCR for marke risk, cf. also Cenral Bank of Ireland, The resuling number shows how much solvency capial needs o be added on he value of he guaranee in order o mee all liabiliies unil he mauriy of he conrac wih a probabiliy of 95%. In Figure 3, we show he surrender sensiiviies for lapse risk as well as he considered capial requiremen for marke risk for he single-premium produc (lef) and for he regular-premium produc (righ). Figure 3 Surrender sensiiviies for lapse risk and capial requiremen for marke risk for he single-premium produc (lef) and he regular-premium produc (righ). For all considered GSMB models, he SCR for marke risk remains below 1% of he sum of premiums. While for he regular-premium produc marke risk is more or less he same for all ypes of GMSB, i increases wih he inroducion of GMSBs for he single-premium produc. This is in line wih he findings regarding he impac on he guaranee value. For regular-premium producs he SCR for marke 63

68 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies risk is roughly 0.5% of he sum of premiums for all GMSBs. For he single premium produc, i flucuaes beween 0.6% and 0.8% of he single premium. The considered SCR for surrender risk overall is smaller, bu depends more heavily on he GMSB. Again, we can observe ha, depending on he GMSB ype, he risk of he insurer can lie in eiher increased or decreased surrender. I is worh noing ha he risk arising from he inroducion of a cerain GMSB afer he produc has been sold (see Figure 2) is significanly higher han he considered SCR for he risk arising from changes in surrender behavior once a GMSB is in force. 3.4 Sensiiviy o ineres raes In a nex sep, we will perform capial marke sensiiviies. We sar wih an analysis of he impac of lower ineres raes on he value of he guaranee and capial requiremens. For his, we se r 0, θ r P and θ r Q o 1.5% (as opposed o 3.0% in he base case). As wih he inroducion of he GMSBs, we assume ha he change happens afer he variable annuiy has been sold, i.e. he pricing is no adjused o he new ineres rae level. This represens a scenario in which, afer he variable annuiy has been sold wih he guaranee charge used in he previous secions, he embedded guaranee (and also he modeled hedging porfolio) becomes raher valuable. In such a scenario, wihou a GMSB, i is highly profiable for he insurer if he policyholder decides o surrender, since in his case he value of he guaranee remains wih he insurer. The addiion of a GMSB reduces his effec and is hus poenially especially harmful for he insurer in such a scenario. Figure 4 shows he value of he guaranee for he single-premium produc (lef) and for he regularpremium produc (righ) for low ineres raes. In Figure 5, we show surrender sensiiviies for lapse risk as well as he considered capial requiremen for marke risk for he single-premium produc (lef) and for he regular premium produc (righ) for low ineres raes. Figure 4 Value of he guaranee for low ineres raes, single-premium produc (lef) and regular-premium produc (righ). 64

69 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies Figure 5 Surrender sensiiviies for lapse risk and capial requiremen for marke risk for low ineres raes, single-premium produc (lef) and regular-premium produc (righ). Obviously, a change of ineres raes has a remendous effec on he resuls. We can see from Figure 4 ha he value of he guaranee has increased and is now posiive, independen of he assumed surrender behavior and he GMSB model. This means he presen value of fuure expeced guaranee paymens exceeds he presen value of fuure expeced guaranee charges. As such, surrender is highly profiable for he insurer. Overall, wihou a GMSB, he sensiiviy wih respec o surrender has srongly increased in comparison o he base case. While marke risk has increased only slighly in comparison wih he base case and sill does no exceed 1% of he sum of premiums, surrender risk can be as high as 1.7% if no GMSB is in place. As expeced, wih he higher value of he guaranee, he immediae loss resuling from inroducing a GMSB also increases. In he case of he single-premium produc, he immediae loss from inroducing he MCV GMSB is roughly 4% of he premium. Also he sensiiviy for lapse risk is significanly reduced, i.e. from he insurer s perspecive i is now more or less irrelevan if he policyholders do surrender or no. This does no hold for regular premiums, again due o he assumpion of a conrac wih no more fuure premium paymens. Wih he oher, less valuable, GMSB models, surrender is sill profiable for he insurer. Tha means he corresponding guaraneed surrender benefis are below he marke value of he conrac. 3.5 Sensiiviy o equiy volailiy Finally, we perform a similar sensiiviy analysis wih respec o he equiy volailiy. We assume he equiy volailiy (i.e. σ S P and σ S Q ) o be 12.5% (as opposed o 10.0% in he base case). As wih he scenario of lower ineres raes, we assume he change o happen immediaely afer he variable annuiy has been sold wih he guaranee charge used in he previous secions. Figure 6 shows he value of he guaranee for he single-premium produc (lef) and for he regularpremium produc (righ) for he increased equiy volailiy. In Figure 7, we show surrender sensiiviies for lapse risk as well as he considered capial requiremen for marke risk for he single-premium produc (lef) and for he regular-premium produc (righ). 65

70 2 Guaraneed Minimum Surrender Benefis in Variable Annuiies Figure 6 Value of he guaranee for increased equiy volailiy, single-premium produc (lef) and regular-premium produc (righ). Figure 7 Surrender sensiiviies for lapse risk and capial requiremen for marke risk for increased equiy volailiy, singlepremium produc (lef) and regular-premium produc (righ). The value of he guaranee as well as boh considered risk indicaors increase wih higher equiy volailiy. The considered SCR for lapse risk increases more srongly han he SCR for marke risk and, for some GMSBs, boh reach similar levels. The immediae loss caused by he inroducion of a GMSB increases only slighly bu, a he same ime, sensiiviy wih respec o surrender increases. Changing volailiy, herefore, no only has an impac on he insurer s marke risk, bu can have an even higher impac on he insurer s lapse risk. 4 Impac of a secondary marke In his secion, we analyze he impac of a secondary marke wih raional, i.e. value maximizing, invesors. We assume ha hose policyholders who are willing o surrender heir conrac have he possibiliy o alernaively sell heir policy o some insiuional invesor in he secondary marke. Afer purchasing he conrac, he invesor hen acs raionally. We assume ha he presence of a secondary marke has no been considered in he pricing of he conrac and herefore use he same conrac and guaranee charges as in Chaper Model descripion We use he model from Secion 2 unless saed oherwise in his secion. 66