Georgia Sae Universiy ScholarWorks @ Georgia Sae Universiy Risk Managemen and Insurance Faculy Publicaions Deparmen of Risk Managemen and Insurance 2015 On he Ineracion beween Transfer Resricions and Crediing Sraegies in Guaraneed Funds Eric R. Ulm Georgia Sae Universiy, eulm@gsu.edu Follow his and addiional works a: hp://scholarworks.gsu.edu/rmi_facpub Par of he Insurance Commons Recommended Ciaion Ulm, Eric R., "On he Ineracion beween Transfer Resricions and Crediing Sraegies in Guaraneed Funds" (2015). Risk Managemen and Insurance Faculy Publicaions. Paper 2. hp://scholarworks.gsu.edu/rmi_facpub/2 This Aricle is brough o you for free and open access by he Deparmen of Risk Managemen and Insurance a ScholarWorks @ Georgia Sae Universiy. I has been acceped for inclusion in Risk Managemen and Insurance Faculy Publicaions by an auhorized adminisraor of ScholarWorks @ Georgia Sae Universiy. For more informaion, please conac scholarworks@gsu.edu.
On he Ineracion beween Transfer Resricions and Crediing Sraegies in Guaraneed Funds Eric R. Ulm 1 Georgia Sae Universiy Absrac Guaraneed funds wih crediing raes for fixed periods deermined by a Pension Provider or Insurance Company are common feaures of accumulaion annuiy conracs. Policyholders can ransfer money back and forh beween hese accouns and Money Marke accouns which give hem feaures similar o demand deposis and ye hey frequenly credi a higher rae han he Money Marke. Transfer resricions are commonly employed o preven arbirage. In his paper, we model he ineracion beween company and policyholder as a muliperiod game in which he company maximizes risk-neural expeced presen value of profis and he policyholder maximizes his expeced discouned uiliy. We find ha he opimal sraegy on he par of he company is o credi a rae higher han money marke rae in he firs period o enice he policyholder o inves in he guaraneed fund. The company hen credis he floor in he remaining periods as he policyholder ransfers ou he maximum amoun. This does beer for he policyholder in low ineres rae environmens and worse in high ineres rae environmens and acs as a ype of ineres rae insurance for he policyholder. Keywords: Annuiy Crediing Sraegies, Opimal Policyholder Behavior 1 Eric Ulm, FSA, is an Assisan Professor in he Deparmen of Risk Managemen and Insurance, J. Mack Robinson College of Business, Georgia Sae Universiy, Alana, GA. Phone: 404-413-7485. E-mail: eulm@ gsu.edu.
1. Inroducion One of he major problems in modern financial planning is accumulaing asses over a working lifeime o provide sufficien income in reiremen. Defined Conribuion Pension Plans have become increasingly common in recen years. Employees deposi money a regular inervals ino a designaed accoun. These conribuions are frequenly mached a some level by he employers. The employee can direc he funds o a number of differen accouns. Subjec o only a few resricions, hey can rebalance heir porfolio whenever hey wan. Mos DC plans have sock funds, bond funds and mixed funds, all of which have he possibiliy of losing money in bad markes. In addiion, many DC plans have a money marke accoun which credis a shor-erm ineres rae and canno 1 lose money. A significan number of plans also conain a Guaraneed Fund which credis a rae guaraneed for a fixed period, ofen monhly or quarerly. These funds are backed by longer erm asses and he rae quoed for he ime period is usually dependen on he book reurn of hese asses less a spread ha covers expenses and insurer profis. Unlike bond funds, which can lose money if ineres raes rise and he bond marke values fall, hese funds are usually redeemable a book value and canno lose money. In addiion, here is a minimum crediing rae for hese conracs. This rae is required by sae non-forfeiure laws bu he insurance company could se a higher rae for markeing reasons. 1 I is heoreically possible for a money marke fund o lose money. This had happened o only hree funds in he 37 years prior o he recen financial crisis. Evens of Sepember 2008 promped he US Treasury o guaranee Money Marke funds.
Prevenion of arbirage beween money marke funds and Guaraneed Funds is a major issue for insurance companies. If here were no resricions on ransfers beween hese accouns, savvy policyholders would ransfer heir money ino he highes earning accoun. The Money Marke accoun would respond quickly o rises in ineres raes, while he Guaraneed Funds would respond wih a lag. Money would be ransferred ou of he Guaraneed Fund when raes are high, exacly he momen when he asse marke value is lower han book value and asses would need o be sold a a loss. In pracice, insurance companies ry o miigae his reacion by imposing ransfer resricions, whereby an individual can ransfer ou only a fixed percenage of his Guaraneed Fund in any given ime period. In his paper, we deermine he opimal crediing sraegy on Guaraneed Funds from he perspecive of maximizing he risk-adjused profi o he pension provider. We hen compare his o crediing sraegies observed in pracice. 2. The Model We will use a game-heoreical model o analyze he inerplay beween he acions of he Pension Provider (hereafer PP) and he Policyholder (hereafer PH). PP s goal is o maximize his presen value of he expeced fuure book profi sream under he Q measure. PH s goal is o maximize he expeced discouned uiliy under he P measure. I could be argued ha in he absence of fricions, PH should insead maximize he expeced presen value under he Q measure as well. There are, however, fricions in his case. The policyholder is unable o sell his pension o a hird pary and is ypically unable
o inexpensively hedge his risk. In hese siuaions, using expeced uiliy under he P measure is arguably correc (see, for insance, Gao and Ulm (2012), Leung and Sircar (2009) or Shreve (2003) page 70). A ime, he universe is in sae (filraion) F. This includes he curren ineres rae environmen, he insurers curren asses and he policyholders curren allocaion. Le s i, represen he curren zero-coupon rae for a duraion of i years. Le, ij A represen he dollar amoun in a zero-coupon asse wih a remaining duraion of i years and r, ij represen he book ineres rae on ha asse. j is an index ha runs over all possible purchase daes for asses wih a remaining duraion of i years. For insance, a curren bond wih a wo year duraion could be a hree-year bond purchased las year, a four-year bond purchased wo years ago, and so on. These bonds would have differen book raes since hey were purchased a differen imes. Le represen he percenage of asses currenly allocaed o he money marke accoun. The game proceeds as follows. A ime : 1. PP picks r c, he rae he will credi for he nex ime period. 2. PH picks his allocaion, 1, which becomes a sae variable for he nex period. 3. PP buys asses, which become sae variables for he nex period. 2.1 Zero-Sum Analysis wih no Transfer Resricions and no Crediing Floor
To moivae he imporance of a risk-averse policyholder who maximizes his expeced uiliy under he P measure, we will here analyze he zero-sum case where he policyholder maximizes his expeced value under he Q measure. Proposiion 2.1: PP s asse purchase sraegy is independen of his crediing sraegy and independen of PH s choices. Proof: PP aemps o maximize BVA BVL BVA BVL E E E. This can be done by choosing Q Q Q 1 1 s 1 1 s 1 1 s asses o maximize he firs sum, and playing he game wih PH in order o minimize he second sum. Therefore, he insurer acs o maximize he asse values and minimize he liabiliy values. The resul of Proposiion 2.1 migh be counerinuiive for acuaries, since Guaraneed Funds frequenly credi a rae ha is ied, a leas loosely, o he reurns on he underlying asse porfolio and Proposiion 2.1 says his is no opimal. Proposiion 2.1 holds even if he insurer is required o back money-marke funds wih shor-erm asses in a separae accoun. If he desires less shor-erm exposure han his, PP can adjus he overall asse porfolio by borrowing shor-erm o buy exra longerm asses in he General Accoun backing he Guaraneed Fund. Proposiion 2.2: PP is indifferen o his asse sraegy.
Proof: This is a basic consequence of he above proposiions and he Modigliani-Miller Theorems (1958, 1961) saing ha companies are indifferen o capial srucure and dividend policy. The resul of Proposiion 2.2 migh be counerinuiive o acuaries who are used o aemping o mach he duraions of asses o he duraions of liabiliies, bu indifference o asse sraegy is common in he financial lieraure as seen in he Modigliani-Miller heorem (1958). This indifference o asse sraegy will no hold in he presence of fricions regarding borrowing coss, differenial ax reamen, or bankrupcy coss. We assume here ha his conrac is a small enough piece of PP s overall porfolio ha he firm can borrow inernally and he conrac has a negligible effec on PP s overall brankrupcy probabiliy. As a consequence of Proposiion 2.2, we will allow he insurer o inves 100% in shor-erm asses for ease of analysis. In his case, he only sae variable needed a a paricular ime is he shor-erm rae and a full yield curve model is unnecessary. Proposiion 2.3: If here are no ransfer resricions, PP will credi a rae rc r,1 and PH will allocae 1 1 or PP will credi rc r,1 and PH will allocae 0 1 1. Proof: Suppose he PP credis rc r,1. If he policyholder invess 1 0 he earns a rae less han r c and can improve by invesing 1 0 in he curren period. This doesn affec his choice se in he nex period, so he policyholder gains by a sraegy change. If
he policyholder invess 1 0, he PP could improve his resul by lowering his crediing rae. Therefore, no Nash equilibrium exiss wih rc r,1. If he PP credis rc r,1 and PH allocaes 0 1 1, he PH earns r,1 and canno improve by a deviaion. If he PP lowers his crediing rae, he PH ransfers o and here is no improvemen. Therefore, his is an equilibrium. 1 0 If he PP credis rc r,1 and PH allocaes 1 1, he PH earns less han r,1 and canno improve by allocaing 1 1. Therefore, his is no an equilibrium. If he PP credis rc r,1 and PH allocaes 1 1, he PH earns r,1 and neiher benefis from a deviaion. Therefore, his is an equilibrium. Proposiion 2.4: A any given ime and sae wih 1, he expeced presen value of fuure book profis under Q is he marke value of he asses less he book value of he asses. Specifically, he expecaion a iniiaion of he conrac is 0. Proof: The proof is by inducion. We assume from proposiion 2.4 ha liabiliies always earn r 1,1 in period 1. The expeced presen value of fuure book profis a ime 1 is equal o he book profis earned in he nex period plus he discouned expeced presen value of book profis a ime. Assume ha in all saes j a ime, he expeced presen value of book profis from ha momen forward is MV (j) BV (j). Assume here are n asses of book value A i and book reurn r i. The value of book profis is he change in
book value of asses, n i 1 A(1 r), less he change in book value of liabiliies, i i n Ai(1 r 1,1) since he sum of he book asse values is he accoun value. Therefore, he i 1 expeced presen value of fuure book profis a ime 1 is: 1 1 A ( r r ) (1 r ) (1 r ) (1 r ) n i i 1,1 EQ[ MV ( j)] EQ[B V ( j)] (1) 1,1 1,1 i 1 1,1 Now, he firs erm is jus he definiion of he marke value of asses MV 1. The book value of asses in he nex period is independen of sae and equal o n Ai(1 ri) so he i 1 expeced presen value of fuure book profis a ime 1 is: n A(1 r) A ( r r ) MV MV A BV MV i i n n i 1 i i 1,1 1 1 i 1 1 (1 r 1,1) i 1 (1 r 1,1) i 1 (2) 2.2 Zero-Sum Analysis wih Transfer Resricions We will now exend he analysis o include he exisence of ransfer resricions. A he end of any period, he money in he money marke accoun is free o be ransferred in whole or in par o he guaraneed fund. On he oher hand, only a percenage x can be ransferred ou of he guaraneed accoun ino he money marke accoun. A percenage 1 x mus remain in he guaraneed accoun for he nex period.
Proposiion 2.5: In he presence of ransfer resricions, he only reasonable allocaions in he period 1 are 1 0 and 1 (1 x) x (or complee indifference o allocaion). The decision of which allocaion o choose is independen of he curren allocaion. Proof: Imagine he PH has hree independen accouns: 1. A guaraneed accoun of (1 x)(1 ) which mus remain in he guaraneed accoun and canno be affeced by he PH s curren choice. 2. A guaraneed accoun of x(1 ) currenly allocaed o he guaraneed accoun bu fully allocaable in he nex period. 3. A money marke accoun of currenly allocaed o he money marke accoun bu fully allocaable in he nex period. This is idenical o he siuaion in he presence of ransfer resricions. Since Funds 2 and 3 are idenical going forward hey should be allocaed idenically in he nex period and should have he same presen value o he PH. Consider Fund 3 firs. In he nex period, some of i will be allocaed o Fund 1, some o Fund 2 and some o Fund 3. The oal value is he weighed average of he amoun allocaed o Funds 1, 2 and 3. Fund 2 and 3 are equally valuable, so if Fund 1 going forward is more valuable han Fund 3, all of Fund 3 should be moved o Fund 1. Oherwise, i should all be reained in Fund 3. The same is rue of Fund 2. Therefore, eiher all of Funds 2 and 3 should be moved o Fund 1 or all should be move o Fund 3. These siuaions correspond o 1 0 and 1 (1 x) x respecively. Indifference is obained if Funds 1 and 3 are equally
valuable going forward. The decision is based enirely on he fuure values of Funds 1 and 3 and is herefore independen of curren allocaion. The argumens in he above proofs are very useful because hey shows ha he PP s sraegy can be analyzed solely by he effec i produces on he acions of a policyholder invesed in Fund 3, i.e. he Money Marke Fund. This will be valuable in he proof of he main resul in his secion. Proposiion 2.6: In he firs period, he policyholder is free o inves a any value of 0 1 1. If here are ransfer resricions, PP will credi a rae r c r where r r1,1 and depends on ime and sae. PH will allocae 1 1 if r c r and 0 1 1 if r c r. Proof: If he PP credis rc r1,1, here is no advanage o PH o invesing 1 1 since he profi in he firs period would be less han (or equal o) he profis a 1 1 and he opions are limied in he nex period. In fac, 1 1 is a sric resul even a rc r1,1 since an allocaion wih 1 1 allows he PP o credi 0 in subsequen periods and he policyholder akes he loss as he slowly ransfers his Guaraneed Funds back o he money marke 2 To compensae for he losses when rapped, he PP will have o credi an amoun greaer han r 1,1 o induce PH o ransfer any funds a all ino he Guaraneed 2 We have no ye shown ha he wo paries do no have superior sraegies o his one, bu he mere exisence of his sraegy is sufficien o prove he Proposiion.
Accoun. There will be a rae r c r in which he profi in he firs period exacly compensaes for he expeced presen value of losses from he rap. If r c r policyholder will inves 1 1 since he firs period gains are insufficien o cover he, he expeced losses in fuure periods. If r c r, PH is indifferen o choice of fund allocaion and can choose any 0 1 1. PP will no credi r c r since i gives away money in he firs period wihou changing PH behavior beyond ha produced by r c r. Proposiion 2.6 implies ha he conrac has a value of 0 a iniiaion, since crediing rc r,1 and allocaing 1 is always a possible equilibrium and has a value of 0. Crediing r c r makes PH indifferen o his oucome, and herefore mus also have a value o PH of 0 and, by he zero-sum propery of he game, o PP as well. Proposiion 2.7: The value of r is independen of he sae variable. Proof: This is a direc consequence of Proposiion 2.5, i.e. ha a sraegy can be evaluaed only by is effec on policyholders invesed solely in Money Marke Funds and ha PP opimal sraegies are independen of allocaion. Proposiion 2.8: If 0, PP should se rc 0
Proof: We will firs prove ha he only raes ha PP should credi are rc 0 and r c r and hen show ha rc 0 gives he more favorable resul o PP. From Proposiions 2.5-2.7, if PP credis 0 rc r, he policyholder will wish o inves Funds 2 and 3 in Money Marke accouns and herefore no value of r c in his range will change behavior, or aler he profi on Funds 2 and 3. On he oher hand, he lower he value of r c, he greaer he gain on Fund 1 o PP. Therefore rc 0 does beer for PP han any oher value of r c r. Similarly, if PP credis r c r, he policyholder will wish o inves in Fund 1 and herefore no value of r c in his range will change behavior. On he oher hand, he lower he value of r c, he lower he loss on Fund 1 o PP. Therefore r c r does beer han any oher value of r c r. We herefore need only evaluae rc 0 or r c r from he perspecive of PP. Now, from Proposiions 2.6 and 2.7, r c r is he rae ha makes a person indifferen beween Money Marke and Guaraneed Accouns and crediing r c r is revenue-neural relaive o crediing rc r,1 in perpeuiy 3. Now, clearly, crediing r 0 followed by crediing rc r,1 in perpeuiy is beer han his, and crediing r 0 followed by opimal crediing in fuure periods is, by definiion, a leas as good as c c 3 Crediing rc r,1 in perpeuiy is no opimal according o his argumen, bu i does no need o be for he argumen o carry hrough. I need only show ha rc r followed by subsequen opimal crediing is equivalen o crediing rc r,1 in perpeuiy and ha crediing rc 0 in he firs period followed by opimal crediing does beer han crediing rc r,1 in perpeuiy.
crediing rc r,1 in fuure periods. In fac, i is sricly beer since we ve shown ha rc r,1 is no opimal in he nex period, only c r r or r 0 could be. c Proposiions 2.6-2.8 are ineresing resuls, as hey allows he PP o credi a rae on he Guaraneed Accoun ha is higher han ha on he Money Marke Accoun, which is seen empirically. On he oher hand, hey imply ha he insurer will credi 0 on Guaraneed Accouns afer he firs year, which disagrees wih real PP pracice. They also imply ha PH will place no funds in he Guaraneed Accoun a iniiaion of he conrac unless r c r exacly. I is possible ha companies can overcome he implicaion ha hey mus credi 0 hrough conracual precommimens. This could explain he prevalence of siuaions where PP credi a spread below heir porfolio raes. I also explains siuaions where policyholders can exi he General Accoun wih an annuiy whose rae is relaed o he curren marke raes. Proposiion 2.9: If PP credis an ineres rae larger han r, and PH can borrow and lend a prevailing raes ouside he pension plan, an arbirage opporuniy exiss for PH. Proof: Neiher he PP or PH sraegy is sae dependen if PP credis r c r followed by r 0 in subsequen periods and PH pus 100% of his money in he Guaraneed Accoun c a ime 0 and moves x percen deerminisically o he money marke every period aferwards. The presen value of his under he Q measure is 1 rc 1since Proposiion 1 r
2.7 implies ha PH is indifferen beween he Guaraneed Accoun and he Money Marke Accoun worh $1 if rc r. To se up he arbirage, PH borrows $1 o inves in he Pension Guaraneed Accoun. He borrows a prevailing raes in such a way as o repay x(1 x) (1 r ) a ineger imes 0. The presen value of his sream is $1 from Proposiion 2.7. He repays hese values by borrowing a money-marke raes, and accumulaes an ousanding deb a reiremen equal o he value of hese cash-flows accumulaed a shor-erm raes. Inside he accoun, PH receives cash flows of x(1 x) (1 r c ) o inves in he Money Marke. These funds accumulae a reiremen o a value 1 rc 1 r imes his accumulaed exernal deb. When he asses and debs are need a reiremen, he amoun is guaraneed o be posiive. This case, where PP credis 0 in subsequen periods is wors case for PH. If PP credis rc 0, he inernal invesed cash flows are even higher and he ne amoun available a reiremen is an even larger posiive number. 3. Analysis Assuming Uiliy Maximizing Policyholders We now consider he non-zero sum case where he PP can hedge and herefore aemps o maximize he expeced value of fuure profis under he Q measure whereas he PH aemps o maximize expeced value of he uiliy of his ending fund under he P measure.
Proposiion 3.1: The resuls of Proposiions 2.3-2.4 hold even when PH aemps o maximize expeced uiliy under he P measure. Proof: The equilibrium argumens for PP deviaions in he proof of Proposiion 2.4 are sill valid. Also, as long as uiliy is increasing in money amoun, he PH deviaion argumens in he proof of Proposiion 2.3 remain valid. Therefore, he assumpion ha liabiliies always earn r 1,1 in period 1 remains valid and he argumen in Proposiion 2.4 carries hrough unchanged. Now, an individual who is even risk-averse in fund oucomes will ofen prefer a crediing sraegy of r in period 1 followed by 0 in subsequen periods o a sraegy where PP credis he money marke rae a all periods. You migh expec from he definiion of r, hese wo sraegies would have equal mean oucomes, bu he oucomes will ypically be lower on average for he credi r sraegy due o he effecs of ime-dependen discoun raes. On he oher hand, he firs sraegy produces beer (worse) ending fund values in low (high) ineres rae scenarios because he cos of crediing 0 is less (more) in hese scenarios. Therefore, he firs sraegy migh easily be preferred by a risk-averse invesor. Also, if he P measure has larger probabiliies for low ineres scenarios relaive o he Q measure which is ypical given he bias oward rising yield curves, an individual who maximizes expeced values under he P measure could easily prefer a crediing sraegy of r in period 1 followed by 0 in subsequen periods o a sraegy where
PP credis he money marke rae a all periods. The P measure overweighs low-ineres rae scenarios where he firs sraegy produces larger values han he second sraegy and underweighs high-ineres rae scenarios where he firs sraegy produces smaller values han he second sraegy. This increases he mean oucome of he firs sraegy, and herefore raises is desirabiliy o a risk-neural invesor who values under he P measure. These resuls sugges ha PPs in perfec compeiion will credi r in period 1 followed by 0 in subsequen periods if fricions are such ha *any* of heir policyholders are influenced by he expecaion of he uiliy of he fund under he real world probabiliy measure. Of course, if he PP credis he full value of r in he firs period, he has a zero profi and he enire surplus goes o he consumer. He could lower his firs-period crediing rae o P r, he rae ha would make a risk averse policyholder who values under he P measure infiniesimally prefer he Guaraneed Accoun. In his case all he surplus is capured by he producer. In realiy, some value r r r would be P c credied depending on he bargaining power of he wo agens. 4. Analysis including he Effec of Minimum Guaranees Now assume here is a minimum credied rae r min which is eiher se by law or conracually guaraneed. The resuls of Proposiion 2.6 follow hrough unchanged. Proposiion 2.9 could be resaed as If 0, PP should se rc rmin, bu he proof is
similar. The argumens used in Secion 3 regarding risk-averse policyholders under he P measure are sill reasonable. I is possible now, however, for r min o exceed r a some imes in some saes of he world. In his case, PH will move all funds o he Guaraneed Accouns. Since PP credis more han r, he expeced profis o PP under he Q measure are negaive. This conrac, herefore, has a negaive expecaion a issue. This would seem o imply ha he PP would no issue such a conrac. On he oher hand, his bargaining power may allow him o lower he firs period crediing rae far enough o creae an expecaion of a posiive profi and allow he conrac o be issued. While his siuaion does exis in pracice, i is also similar o one where wihdrawals are allowed by way of a ransfer payou annuiy wih a fixed erm and rae. The minimum rae in his case is usually ime-dependen and ied o he marke in some fashion. If his rae is conracually ied o a reference rae, his is a way PP can precommi o crediing more han 0 and reduce he value of r necessary o enice policyholders o choose he Guaraneed Accoun. 5. Numerical Examples We now urn our aenion o some numerical calculaions of he ical rae. The behavior of PP and PH is fully deerminisic and no ineres sensiive when r min 0, so r is compleely deermined by oday s yield curve and is no dependen on an ineres
rae model. I does, however, depend on he ransfer resricion x. When r min 0, PH sraegy does depend on he sae of he world and we will need a full ineres rae model. In addiion, P r does depend on he ineres rae model used because i depends on he full disribuion of final oucomes which is model sensiive. P r also depends on he precise form of he PH uiliy funcion and he Radon-Nikodym Derivaive of he Q measure relaive o he P measure. 5.1 Deerminaion of r when r min 0 The case of a level yield curve wih rae r can be sraighforwardly evaluaed and demonsraes he mehod ha will be used for non-level yield curves. The value of any money in he money marke accoun a 0is $1. Puing $1 ino he Guaraneed Accoun produces $(1 r ) in one year. The Guaraneed Accoun hen no longer grows in fuure years. A fracion x is ransferred ou every year and he presen value of hese ransfers mus equal $1 for PH o be indifferen beween he funds. Tha is, 1 x(1 x) 1 (1 r ) (3) (1 r) 1 Which solves nicely for: r r (4) x
When x 1, r r which agrees wih Proposiion 2.4. Now, in cases where he yield curve is no fla, he denominaor in Equaion (3) is easily adjused by replacing (1 r) by (1 s ) where s represens he year spo rae a he iniiaion of he conrac. If he one-year forward raes afer he firs year are level a f 1 and he one-year spo rae is s 0, Equaion (3) becomes: 1 (1 r ) x(1 x) 1 1 1 (1 s 0)(1 f1) (5) which again solves nicely for 1 s r * x (1 x) f xs 0 1 0 1 f1 (6) 1 s 0 This moves linearly from f1 1 f1 when ransfers are compleely disallowed o s 0, in agreemen wih Proposiion 2.4, when here are no ransfer resricions. This general paern of movemen from long-erm o shor erm raes when ransfer resricions are removed is a general feaure of he model for arbirary yield curves. Figure 1 shows he behavior of he ical rae from 1/1990 o 5/2014 when x 25%. We also examined he correlaions beween r (for x =1%, 5%, 10%, 25% and 50%) and reasury raes (a 1 year, 5 year, 10 year and 30 year duraions). We find large correlaions in general. The larges correlaions for x 1%, x 5% and x 10% are wih he 10 year rae, x 25% wih he 5 year rae and x 50% wih he 1 year rae.
All of hese maximum correlaions are above 0.989. These correlaions are shown in Table 1. 5.2 Deerminaion of r when r min 0 We firs examine he case where r r a all imes and in all fuure saes of min world (he saic case). In his siuaion, PP should credi r min in fuure periods and he policyholder should wihdraw he maximum amoun possible. In he case of a level yield curve, Equaion (3) becomes: 1 1 x(1 x) (1 rmin ) 1 (1 r ) (7) (1 r) 1 which solves for: r r r x min rmin (8) which again equals he shor-erm rae when x 1, consisen wih Proposiion 2.4. The equivalen of Equaion (5) now solves for: 1 s r * x (1 x)( f r ) xs 1 0 1 min 0 f1 (9)
Which can be eiher posiive or negaive depending on wheher he long-erm rae is larger or smaller han he minimum crediing rae. If he credied rae a ime 0 is larger han his saic amoun, an arbirage opporuniy analogous o he one in Proposiion 2.10 exiss. In realiy, his saic case ignores a number of imporan opions possessed by PH. For insance, PH can empy his Guaraneed Accoun as described above and sill reain he opion o ransfer back o he Guaraneed Accoun if r min exceeds r a some poin in he fuure, which increases he value of Guaraneed Accoun funds. In addiion, Money Marke funds are worh more han $1 as he PH has he opion o move money o he Guaraneed Accoun if r min ever exceeds r. This implies ha he rue, dynamic r mus equal or exceed he shor-erm rae, oherwise he money marke would be preferable as he opion value on money marke funds exceeds ha on Guaraneed Funds. To see he effec of hese opions on r we calibrae a Black-Derman-Toy (BDT) model wih volailiy 14% o he reasury curves. This volailiy is consisen wih values in Coleman, Fisher and Ibboson (1991), Radhakrishnan (1998) and Damberg and Gullnäs (2012). The resuls are no paricularly sensiive o he choice of volailiy parameer. Figure 2 shows he dynamic and saic values of r for a minimum crediing rae of 3%. The values are nearly indisinguishable excep in hose cases where he dynamic rae is essenially equal o he one-year reasury rae. Alhough no visible in he graph, he dynamic value is abou 3-5 bp below he saic value reflecing he opion value of reenering he Guaraneed Accoun. If PP credis above he saic value, an arbirage
opporuniy exiss. If PP credis above he dynamic value bu below he saic one, he BDT model suggess ha PH should pu funds in he Guaraneed Accoun. This conclusion is dependen on he accuracy of he model, however, and does no necessarily represen an arbirage opporuniy for PH. 5.3 Deerminaion of P r. As in secion 5.2, we will use he calibraed Black-Derman-Toy Model. We assume CRRA policyholders. Tha is, we assume hey have a uiliy funcion 1 w Uw ( ). Figure 3 shows he values of 1 P r for 0 and 3. As expeced, he values for risk-neural individuals ( 0 ) are above he ical raes in figure 1 and he values for reasonably risk-averse individuals ( 3 ) are below boh he risk-neural and Figure 1 values. 6. Crediing in Pracice We now urn our aenion o an empirical analysis of he ypical crediing sraegies of pension providers. Our analysis spans he period from 1990-2011 inclusive. We look a wo quesions. Firs, were here any companies and ime periods where he
arbirage relaionship in Proposiion 2.9 was presen? Second, wha aspecs of a company s asses and he ineres rae environmen predic crediing raes? We deermine a companies credied rae from he informaion provided in publicly available NAIC saemens. We ake ineres credied o be he abular ineres in he Group Annuiies column of he Analysis of Increase in Reserves and Deposi Funds During he Year. Prior o 2000 his was divided ino Reserves and Deposi Funds bu in 2000 and laer he wo amouns were combined. Because of his, i is unclear in some cases wheher a paricular rae is one ha is credied on policyholder conrolled funds. Relevan ransfer resricions are also unavailable. The crediing rae was esimaed by dividing he abular ineres ino he average of he beginning and ending reserves for he year. This will be he dependen variable in he laer regression analysis. Figure 4 shows he median credied rae, as well as he 90 h and 10 h perceniles, for hose companies wih posiive (non-zero) group annuiy reserves. I also shows some of he ical raes from Figures 1-3 as well as a shor erm rae. Companies ypically, bu no always, credi more han money marke raes. For example, many policyholders beween 1993-2000 as well as 2005-2007 would do well o ransfer as much money as allowed ino he money-markes. I also seems likely ha a leas some arbirage possibiliies exised beween 2001-2004 and, more recenly, 2011-2012. We performed a leas-squares regression on he daa o find he deerminans of company crediing sraegies. The dependen variable was he amoun credied and he independen variables were: Asses, NII on line, Proporionally Allocaed Company NII, Shor Term Ineres Crediing, 5 Year Ineres Crediing and 10 Year Ineres
Crediing. The Asses were calculaed as he average of he saring and ending reserves used in he denominaor of he credied rae calculaion. NII on line was obained from he Analysis of Operaions by Line of Business page of he NAIC saemen for he Group Annuiies column. Proporionally Allocaed Company NII calculaed wha he NII would have been on he line of business had hey had he same NII rae as he company as whole. The Shor Term Ineres Crediing, 5 Year Ineres Crediing and 10 Year Ineres Crediing variables were he amouns ha would have been earned by he line if he asses had earned exacly he Shor Term, 5 Year and 10 Year reasury raes respecively. The resuls of he regression are shown in Table 2. The R- squared of he regression is quie high, 0.9813. All coefficiens are quie saisically significan. The resuls are reasonable and can be inerpreed sraighforwardly. The negaive inercep implies ha a ypical company builds in abou $3,500,000 of profi (afer he effecs of he various NII and ineres raes) regardless of size. The average company has abou $1,100,000,000 in group annuiy asses so his abou 0.32% of asses for a ypical company. The coefficien on Asses is 0.0061, implying ha a ypical company credis abou 0.61% independen of exernal raes or is own invesmen performance. The credied rae averaged over all years and companies is abou 6.23% so only abou 1/10 h of he ineres credied is consan independen of company or economic circumsances. The ypical spread profi per company per year (NII on line less Ineres Credied) is abou $33,800,000 per year or abou 3.08% of asses, mosly because ineres raes have been declining hrough mos of he period and Credied Raes have fallen faser han NII raes.
I appears ha exernal raes maer more han inernal invesmen performance, as suggesed by Proposiion 2.1. For example, if he company wide NII rae rises by 1% (in a way which causes he Line NII rae o also rise 1%), he Credied Rae will only rise by 0.12%. On he oher hand, if he exernal yield curve rises by 1% (all hree raes in parallel), he Credied Rae will rise by 0.86%. If all raes, inernal and exernal, rise by 1%, he Credied Rae will rise by 0.986% so almos, bu no quie all, of he exra reurn is passed hrough o policyholders. The paern of coefficiens on he NII variables suggess ha increasing he reurn on eiher line specific or company-wide non-line specific asses resuls in higher crediing raes alhough he effec of line specific asses is larger. The paern of coefficiens on Treasury Rae variables suggess ha he 10-year rae is a srong deerminan of Crediing Raes. In addiion, he slope beween he 5-year and 10-year rae is also quie imporan, suggesing ha expecaions of increasing reurns in he fuure migh produce higher Crediing Raes oday (or ha he Crediing Raes migh depend on an even longer rae, say he 30 year, which is no always available). 7. Conclusion In his paper, we examine he ineracion beween crediing sraegies on guaraneed funds and ransfer resricions. We show ha he opimal sraegy for a pension provider is o credi a ical rae during he firs year and credi he lowes
possible legal or conracually allowed rae hereafer. The policyholder s opimal sraegy is o ener he guaraneed fund a iniiaion of he conrac and hen ransfer he maximum possible amoun ino he money marke unil he guaraneed fund is empied. If he pension provider credis more han he ical rae during he firs year, an arbirage opporuniy exiss for he policyholder. This has likely happened during some years for some companies since 1990. The effec of he arbirage is miigaed somewha since i is no scalable (policyholders have a maximum amoun hey can deposi in axdeferred accouns) and policyholders canno, in pracice, borrow a he money-marke rae. We also examine how Credied Raes are deermined in pracice for U.S. insurance companies. We find ha he effec of exernal reasury raes is far larger han he effec of inernal invesmen reurns, consisen wih heoreical expecaions. 8. Acknowledgmen Financial suppor from he Sociey of Acuaries under he CAE research gran is grealy appreciaed.
References: Coleman, T.S., Fisher, L., and Ibboson, R.G. (1992). "Esimaing he erm srucure of ineres raes from daa ha include he prices of coupon bonds." The Journal of Fixed Income 2(2): 85-116. Damberg, P. and Gullnäs, A. (2012). "Ineres rae derivaives: Pricing of Euro- Bund opions: An empirical sudy of he Black Derman and Toy model (1990)." Working Paper. Gao, J. and Ulm, E. R. (2012). Opimal consumpion and allocaion in variable annuiies wih Guaraneed Minimum Deah Benefis, Insurance: Mahemaics and Economics, 51:586-598. Leung, T. and Sircar, R. (2009). Accouning for Risk Aversion, Vesing, Job Terminaion Risk and Muliple Exercises in Valuaion of Employee Sock Opions, Mahemaical Finance, 19(1):99-128. Modigliani, F. and Miller, M.H. (1958). "The Cos of Capial, Corporaion Finance and he Theory of Invesmen", American Economic Review 48 (3): 261 297. Miller, M. H., and Modigliani, F. (1961). Dividend Policy, Growh, and he Valuaion of Shares. The Journal of Business, 34(4), 411-433.
Radhakrishnan, A.R. (1998). Mispricing of discoun bond opions in he Black Derman Toy model calibraed o erm srucure and cap volailiies: an empirical sudy. Working paper, New York Universiy. Shreve, S., (2003). Sochasic Calculus for Finance I - The Binomial Asse Pricing Model. Springer-Verlag, New York.
Figure 1. r vs. ime for r min 0
Figure 2. Saic and Dynamic Criical Raes for r min 3%
Figure 3. P r vs. Time for Risk-Averse Policyholders.
Figure 4. Acual and Criical Credied Raes.
Table 1. Correlaion Beween Criical Raes and Treasury Raes for Varying Duraions and Transfer Resricions. Transfer Treasury Resricion Duraion 1% 5% 10% 25% 50% 1 0.862 0.892 0.919 0.962 0.989 5 0.971 0.984 0.994 0.999 0.985 10 0.996 0.999 0.998 0.980 0.946 30 0.996 0.987 0.975 0.937 0.885
Table 2. Regression Analysis Coefficiens Sandard Error P-value Inercep -$3,547,190 $446,845 2.38E-15 Asses 0.006 0.001 5.57E-17 NII on Line 0.076 0.003 4.7E-101 NII Proporional 0.049 0.009 1.59E-07 Shor Term 0.288 0.019 1.38E-49 5 Year -1.634 0.073 3.9E-108 10 Year 2.208 0.065 2.5E-234