Money-Back Guarantees in Individual Pension Accounts: Evidence from the German Pension Reform

Transcription

1 Money-Back Guaranees in Individual Pension Accouns: Evidence from he German Pension Reform Raimond Maurer and Chrisian Schlag PRC WP Pension Research Council Working Paper Pension Research Council 3641 Locus Walk, 34 CPC The Wharon School, Universiy of Pennsylvania Philadelphia, PA Tel: 215/ Fax: 215/ hp://prc.wharon.upenn.edu/prc/prc/hml This paper is o be published in The Pension Challenge: Risk Transfers and Reiremen Income Securiy Eds. Olivia S. Michell and Ken Smeers. Philadelphia: Pension Research Council Press, forhcoming. Pension Research Council Working Papers are inended o make research findings available o oher researchers in preliminary form, o encourage discussion and suggesions for revision before final publicaion. Opinions are solely hose of he auhors. 22 Pension Research Council of he Wharon School of he Universiy of Pennsylvania. All Righs Reserved.

2 Money-Back Guaranees in Individual Pension Accouns: Evidence from he German Pension Reform Raimond Maurer and Chrisian Schlag Absrac The German Reiremen Saving Ac insiued a new funded sysem of supplemenary pensions coupled wih a general reducion in he level of sae pay-as-you-go old-age pensions. In order o qualify for ax relief, he providers of supplemenary savings producs mus offer a guaranee of he nominal value a reiremen of conribuions paid ino hese saving accouns. This paper explores how his money-back guaranee works and evaluaes alernaive designs for guaranee srucures, including a life cycle model (dynamic asse allocaion), a plan wih a pre-specified blend of equiy and bond invesmens (saic asse allocaion), and some ype of porfolio insurance. We use a simulaion mehodology o compare hedging effeciveness and hedging coss associaed wih he provision of he moneyback guaranee. In addiion, he guaranee has imporan implicaions for regulaors who mus find an appropriae solvency sysem for such saving schemes. This research is par of he Research Program Insiuional Invesors of he Cener for Financial Sudies, Frankfur/M. This chaper was wrien in par during Dr. Maurer s ime as he Mezler Visiing Professor a he Wharon s School Pension Research Council. The auhors would like o hank Manfred Laux, David McCarhy, Olivia S. Michell, Alex Muermann, Rudolf Siebel, Wolfgang Raab, and Ken Smeers for helpful commens. Opinions and errors are solely hose of he auhors.

3 Money-Back Guaranees in Individual Pension Accouns: Evidence from he German Pension Reform Raimond Maurer and Chrisian Schlag The German Reiremen Saving Ac ( Alersvermögensgesez 1 ) which passed he German legislaive body in May of 21 insiued a new funded sysem of supplemenary pensions coupled wih a general reducion in he level of sae pay-as-you old-age pensions. The goal of his new pension sysem is o cap and o sabilise he conribuions of German employees o he sae pension sysem, which currenly cos 19.1% of salary. For compulsory members of he sae pension sysems no already in reiremen, he maximum firs pillar sae pension level will be gradually cu from 7% o 67% of he las ne salary before reiremen by To compensae for he cu in sae pension payous, individuals will be able o inves volunarily and on a pre-ax basis a par of heir income in individual pension accouns ( Alersvorsorgeverrag, called here IPAs). 3 Addiional incenives o inves ino he IPAs are given by he governmen in he form of a ax relief on pension conribuions, direc subsidies for low income earners, and exra conribuions for children. In order o ge he full benefis, households will have o inves abou one percen of heir income (up o he social securiy ceiling) ino he pension sysem in 22, increasing every wo years by one percen reaching a maximum of four percen in 28. The invesmen income during he accumulaion period is no subjec o income ax, whereas he paymens from he IPA during he disribuion phase will be fully subjec o income ax. In general, individuals are free o make IPA pension invesmens in a wide array of producs offered by privae-secor financial insiuions. This allows paricipans o choose an invesmen porfolio ha is consisen wih heir individual preferences for risk and reurn. In order o qualify for a ax credi, however, he IPA producs have o saisfy a number of crieria. These condiions are codified in a special law concerning he cerificaion of individual pension producs ( Alersvorsorge-Zerifizierungsgesez ) and supervised by a

4 2 special auhoriy ( Zerifizierungsselle ) belonging o he German Federal Financial Supervisory Agency. The inenion of he cerificaion requiremens is wofold: Firs, he governmen wans o ensure ha individuals only use he (ax-suppored) accumulaed savings for a lifelong income sream in he pos-reiremen phase, and no for consumpion during pre-reiremen. Second, privae (and ofen uninformed) invesors paying ino he new individual pension plans should be proeced agains he risk of making oo bad invesmen decisions. In he spiri of he firs inenion, invesmens in he personal pension accouns mus be preserved unil employees reach he age of 6 4, and no disribuions may be made during he accumulaion period. When he age of reiremen is reached, he accumulaed asses be drawn down in he form of a lifelong annuiy or a capial wihdrawal plan which mus (parly) rever ino an annuiy a he age of To provide ransparency, he providers of IPAs mus disclose he naure and level of fees (e.g. o cover disribuion and/or adminisraive coss). If disribuion fees are no charged as a percenage of he periodic conribuion ino he plan, hey mus be spread equally over a period of a leas en years. 6 During he accumulaion phase, he policyholder has he righ o suspend he conrac as well as o erminae he conrac by swiching he cash value of he policy o a new provider. In line wih a cerain minimum level of invesor proecion, only regulaed financial insiuions, like banks, life insurance companies, and muual fund companies, are allowed o offer IPAs. In principle, hese providers are free o design heir IPA. In paricular, he Cerificaion Ac imposes no resricions concerning he asses in which he providers inves he conribuion ha back he pension accouns. 7 In addiion, he supervisory auhoriy does no check wheher he risk and reurn characerisics of an IPA are economically feasible. Ye, he provider of an IPA mus promise he plan paricipan ha he conrac cash value a reiremen is a leas equal o conribuions made o he IPAs, including all exra paymens by

5 3 he governmen. 8 This money-back guaranee, which was he core of an inense and conroversial debae during he social securiy reform in Germany, is he focus of his chaper. Advocaes of he guaranee argue ha i proecs plan paricipans agains a porion of he downside volailiy of capial marke reurns, by providing hem wih a minimum rae of reurn wih respec o heir lifeime conribuions. However, he guaranee shapes he design of saving producs offered by he providers and raises a quesion abou he economic coss of such a promise. 9 Depending on he asses used o back he pension accouns, providers may be exposed o shorfall risk due o adverse movemens in capial markes. If, a reiremen, he value of he pension asses is lower han he sum of he conribuions paid ino he plan, he IPA provider mus fill he gap wih is own equiy capial. The problem faced by money managers is herefore o find a produc design, in conjuncion wih an appropriae invesmen sraegy, ha proecs he credibiliy of he guaranee in scenarios of negaive invesmen reurns (hedging effeciveness), while sill allowing for sufficien upside poenial if capial markes are booming (and hus avoiding excessive hedging coss). In addiion, he guaranee has imporan implicaions for regulaors who mus find an effecive and efficien solvency sysem for such saving schemes, especially for muual funds. The objecive of his chaper is o explore how his money-back guaranee works for producs offered by he German muual fund indusry. We evaluae alernaive designs for guaranee srucures including a life cycle model (dynamic asse allocaion), a plan wih a pre-specified blend of equiy and bond invesmens (saic asse allocaion), and some ype of porfolio insurance. We use simulaion o compare hedging effeciveness and hedging coss associaed wih he provision of he money-back guaranee. Long Term Shorfall-Risk and Reurn of Saving Plans In order o make appropriae invesmen decisions under uncerainy, individuals mus be able o compare he risk and rewards of differen asse classes. Ye policymakers,

6 4 regulaors, and providers are also ineresed in he long run performance of financial asses ha back he new pension producs. The impac of he invesmen horizon on he risk of he various financial asses is sill a subjec of inense and conroversial debae wihin he academic communiy and among invesmen professionals. 1 For example, a popular saemen is ha socks have a lower downside side risk in he long run han in he shor run. A pracical guideline based on his argumen is ha people should inves a higher fracion of heir money in socks he younger hey are, independen of preferences. 11 If he ime horizon is long enough, his approach would imply ha people should inves 1% in socks. To jusify his view, proponens call on he law of large numbers which (seemingly) forces a phenomenon called ime diversificaion. Inuiively, his means ha over a sufficienly long invesmen horizon, losses resuling from he high downside flucuaions will be compensaed by gains resuling from he high upside flucuaions of shor-erm sock reurns. Some invesmen advisors press his argumen by poining o hisorical reurns and demonsraing ha socks have ouperformed bonds for every 1, 15, or 2-year period on record. Neverheless, i is well known in he academic lieraure ha his is a misleading argumen. For example, Samuelson (1963) uses uiliy heory, Levy/Cohen (1998) use sochasic dominance, and Bodie (1995, 21) applies opion pricing heory o demonsrae he logical flaw in his conjecure. In addiion, he use of hisorical reurn series implies ha he 1, 15, or 2-year periods used are srongly overlapping, so he resuling rollover muliperiod reurns have a high degree of correlaion, boh of which resul in a serious esimaion bias. Shorfall Risk Measures This secion provides addiional evidence concerning he impac of he ime horizon on he risk of he major financial asse classes in he German conex, i.e. socks and bonds. To do so, we use alernaive shorfall risk measures. The concep of shorfall risk is

7 5 associaed wih he possibiliy of somehing bad happening, in oher words, falling shor compared o a required arge (benchmark) reurn. 12 Reurns below he arge (losses) are considered o be undesirable or risky, while reurns above he arge (gains) are desirable or non-risky. In his sense, shorfall risk measures are called relaive or pure measures of risk. A popular measure o examine he downside risk of differen invesmen vehicles is he shorfall probabiliy. Formally, le R denoe he cumulaive (muliyear) reurn of an invesmen a a specific poin in ime. Then he shorfall probabiliy is given by SP = Prob(R < z), (1) where z is he arge (benchmark) which ranslaes he oal invesmen reurns ino gains or losses. In he special case of a money back guaranee, he arge is se equal o zero; i.e. he shorfall occurs when he cash value of he policy is lower han he premiums paid ino he saving plan. Despie he populariy of his risk measure in he invesmen indusry, i has a major shorcoming. As Bodie (21, p. 38) poins ou i compleely ignores how large he poenial shorfall migh be. If he same invesmen sraegy can be repeaed many imes, he shorfall probabiliy only answers he quesion how ofen a loss migh occur, bu no how bad such a loss migh be. To provide informaion abou he poenial exen of a loss, we calculae he Mean Excess Loss (MEL), also known as he condiional shorfall expecaion, as an addiional measure o evaluae he long erm shorfall-risk of financial asses. Formally, his risk index is given by MEL = E[ z R R < z], (2) and i indicaes he expeced loss wih respec o he benchmark, under he condiion ha a shorfall occurs. Therefore, given a loss, he MEL answers he quesion how bad on average he loss will be. 13 In his sense, he MEL can be considered a wors case-risk measure, since he measure only considers he consequences of he mean shorfall-level

8 6 assuming ha a shorfall happens. A shorfall risk measure which connecs he probabiliy and he exen of he condiional shorfall in an inuiive way is he shorfall expecaion (SE): SE = E[max( z R,)] = SP * MEL (3) The shorfall expecaion is he sum of losses weighed by heir probabiliies, and hence i is a measure of he uncondiional average loss. As equaion (3) shows, he mean shorfall level is simply he produc of he shorfall probabiliy and he mean level of shorfall given he occurrence of a shorfall. In addiion, he SE is, in a cerain way, relaed o he price of an insurance conrac which would cover he shorfall. For example, he provider may have he possibiliy of ransferring he shorfall risk o he capial marke by using appropriae arbirage-free pu opions. Then he shorfall expecaion beween he cash value of he pension asses and he guaranee paymen wih respec o he risk adjused ( maringale ) probabiliies discouned back a he risk-free ineres rae, resuls in he (modified) Black/Scholes (1973) opion pricing formula. 14 If he provider ransfers he shorfall risk o a reinsurance company, he shorfall expecaion could be seen as an imporan elemen of an appropriae premium. Calibraion Nex we quanify and compare he shorfall risk (in he sense defined above) wih respec o he preservaion of principal of wo saving plans. The firs invess he conribuion ino sock index fund unis, represened by he German sock index (DAX). The oher saving plan is based on bond index fund unis, represened by he German bond index (REXP). The DAX sands for an index porfolio of German blue chips, and he REXP represens porfolio of German governmen bonds. Each of hese indices is adjused for capial gains as well as dividends and coupon paymens (on a pre-ax basis). We assume a series of equal conribuions paid a he beginning of each monh up o he end of he accumulaion period,

9 7 which ranges from 1 o 2 years. To gain informaion abou he relevan risk measures, we employ an ex ane approach by imposing an exogenous srucure on he probabiliy disribuion governing he uncerainy of fuure asse reurns. Wih such a model, i is possible o look ino he fuure and compue he risk measures in which we are ineresed. Due o he complexiy of he underlying paymen srucure of saving plans, here are no analyical closed form expressions for hese risk-measures. 15 Therefore we use Mone-Carlo simulaion o generae a large number of pahs for he evoluion of he saving plans. The relevan saisics for he shorfall risk measures are hen evaluaed on he basis of hese scenarios. The sochasic dynamics of he (uncerain) marke values of invesmen fund unis are posied o follow geomeric Brownian moion, a sandard assumpion in financial economics ha may be raced back o Bachelier (19). This implies ha he log reurns of each ype of index fund are i.i.d. normally disribued. For he esimaion of he process parameers (drif/diffusion), we use he hisorical monhly log-reurns of he DAX and REXP over he period 1: :21. The mean log raes of reurn for socks (bonds) are.7967 percen per monh (.5683 percen p.m.) and he corresponding sandard deviaion 5.58 percen p.m. (1.12 percen p.m.). To ake poenial adminisraion coss ino accoun, we subrac he equivalen of.5 percen p.a. from he monhly average reurn on he invesmens. 16 Compaible wih he curren German muual fund fee srucure, we ake markeing coss ino consideraion by assuming fron end sales charges of five percen for he sock and hree percen for he bond fund unis. Wih respec o hese parameers and consisen wih he model of a geomeric Brownian moion, we generaed 3,, random pahs for he developmen of he pension plan wih an invesmen horizon of 2 years (24 monhs). 17 For each simulaion pah i (i = 1,, 3,,) we compue for each monh ( = 1,, 24) he (uncerain) compounded (muliyear) reurn, i.e.:

10 8 R i, Vi, P =. (4) P Here V i, sands for he cash value of he IPA in monh ( = 1,., T) in simulaion run i (i = 1,, n) and P for he sum of conribuions paid unil monh. According o he money-back guaranee, we se he benchmark reurn equal z =. Wih respec o his arge, he relevan risk parameers are hen deermined on he basis of he specrum of possible fuure developmens. Resuls We sar wih he resuls for he developmen of he expeced (muliyear) reurn and he shorfall probabiliy of he sock and bond index fund over ime. The graphs in Figure 1 indicae ha a German invesor would have he poenial o receive a subsanially higher expeced reurn by invesing in socks insead of bonds. For example, in he case of socks a he end of a 2-year accumulaion period, he invesor can expec a compounded reurn wih respec o his conribuions of 27 percen. For a saving plan based on bond index funds, he expeced reurn is only 19 percen. However, purchasing such an invesmen exposes he plan paricipan o he volailiy and herefore he downside risk of financial markes. Figure 1 here Nex we illusrae he resuls for he developmen of he shorfall probabiliy of a saving plan using sock and bond index funds over ime. Figure 2 shows he well-known effec of ime diversificaion, implying ha he risk of no mainaining nominal capial decreases monoonically wih an increasing invesmen period for bonds and socks. Ye he rae and he exen of he risk reducion differ noably beween he wo invesmen vehicles. For bond index funds, he shorfall probabiliy is 37 percen for a yearly invesmen, and close o zero (i.e. lower han.1 percen) wih an invesmen horizon of seven years onwards. By conras, he shorfall probabiliy of a sock index fund does no converge as rapidly

11 9 owards zero. Thus, even for longer ime horizons, he shorfall probabiliy remains a a subsanial level. For example, he shorfall probabiliy for a 12-monh saving plan is 48.9 percen, and for accumulaion period of 2 years i is sill 2.72 percen. In principle, hese resuls confirm a characerisic which Leibowiz/Krasker (1988) call persisence of risk. Figure 2 here Corresponding resuls for he mean excess loss (MEL) are presened in Figure 3. Saving plans in socks have an MEL ha increases monoonically wih he lengh of he accumulaion period, in conras o bonds. For example, for an accumulaion period of one year (i.e. 12 monhs) he condiional expeced loss is 8.62 percen of he sum of he conribuion paid ino he sock pension plan, while for a holding period of 2 years (i.e. 24 monhs) his risk index increases o percen. For a pension plan using bond index funds, he MEL is 1.63 percen afer one year, while for accumulaion periods of 13 years onwards, none of he 3,, simulaion pahs produces a shorfall. Hence, wih respec o he magniude of a poenial shorfall, he popular argumen ha socks become less risky in he long run is no rue. Figure 3 here This resul is in line wih Samuelson s (1963) finding concerning he fallacy of he law of large numbers. In addiion, hese resuls make clear ha he use of he shorfall probabiliy alone is a misleading risk measure of sock invesmens in he long run. The wors-case aspec of a long erm invesmen in socks is parly hidden by only aking he shorfall probabiliy ino consideraion. Recenly Bodie (22) provided he following very inuiive explanaion for his resul (..) he probabiliy of a bad hing happening is only par of he risk equaion. The oher par is he severiy of ha bad hing, and he furher ou you go, he more severe i could be. Thus, he elucidaion of he wors-case risk embodied in a long-erm invesmen in socks represens an addiional piece of informaion ha migh be essenial for invesors.

12 1 Figure 4 shows how he uncondiional shorfall expecaion develops over ime. For a saving plan in bond index funds, boh he probabiliy of loss and he mean excess loss decrease wih he lengh of he ime horizon. Because he shorfall expecaion measures he ne effec of boh risk componens, i is also decreasing in ime. For a sock-based saving plan, his risk measure is also decreasing, i.e. he decreasing shorfall probabiliy overcompensaes he increasing MEL, o a cerain exen. However, in conras o bonds, we can observe a risk persisence-characerisic in he sock fund: even for very long ime horizons, he shorfall expecaion remains a a subsanial level. Figure 4 here In summary, even for long invesmen horizons, a pure sock invesmen is no free of he downside risk of losing money. Hence i is no possible o perfecly smooh he negaive shor-run flucuaions of sock reurns over long horizons and simulaneously o keep expeced excess reurns wih cerainy. Consequenly, asses wih low volailiy and low expeced reurns, like bonds, are no superfluous in he design of long-erm saving producs. Insurance conracs covering he shorfall of a principal guaranees are no cosless for a pure sock invesmen, even for long invesmen horizons (see Lachance and Michell, his volume). For low volailiy asses like a porfolio of governmen bonds, he probabiliy and he severiy of losing money decreases over ime. Over long invesmen horizons, he price o insure he downside risk of a principal guaranee for a pure bonds invesmen is very low (close o zero). Hence bond pension plans are very effecive vehicles for producing principal guaranees. Of course, his does no mean ha wih a pure bond pension plan, he economic coss of downside proecion is zero. Providers of bond-based IPA s mus give up a subsanial par of he upside reurns ha are possible wih socks. From an ex ane poin of view, a measure of hese economic (hedging) coss - in he sense of a smaller upside poenial can bee seen as he difference beween he expeced reurn of boh invesmen vehicles. 18

13 11 Regulaory Framework of Money-Back Guaranees for IPA: The Case of Muual Funds The money-back guaranee as described in he German Cerificaion Ac can be represened as a fixed liabiliy of he provider, when i issues he IPA. If he cash value of he financial asses backing he liabiliies a he beginning of he reiremen phase is lower han he sum of he conribuions paid ino he policy, he provider mus fill he gap wih equiy capial. From his poin of view, i is clear ha he money-back guaranee should be subjec o solvency regulaion. Saving producs offered by commercial banks (e.g. saving accouns) or insurance companies (e.g. life insurance producs) in Germany are usually designed (a leas in par) wih fixed ineres raes. Neverheless such is no he case for muual funds. The fundamenal idea of a collecive invesmen scheme such as a muual fund is o collec money from many privae invesors via he offering of fund unis, and o inves his money in a well-diversified porfolio of socks, bonds, and/or real esae. The unis of he muual fund are liquid in he sense ha hey are raded on an acive secondary marke (e.g. for so-called exchange-radedfunds) or invesors can ask for redempion of heir holdings a ne-asse value prices, a any poin in ime. The invesmen managemen company usually assumes no obligaion oher han ha of invesing he funds in a reasonable and pruden manner, solely in he ineres of he invesors. I provides no guaranees wih respec o a rae of invesmen reurn. Hence, he invesor bears all capial marke risk and receives he full reward of he financial asse ha backs he muual fund unis. Because he balance shees of muual fund providers are no exposed o financial marke flucuaions, hey are excluded from risk-based solvency capial regulaion requiremens in Germany, in conras o insurance companies and commercial banks. 19 By conras, if he provider of an IPA is an invesmen managemen company which uses is own muual funds, he German Federal Banking Supervisory Auhoriy (BAKred) 2 requires (condiional) solvency capial because of he sauory money back guaranee. This

14 12 solvency requiremen, published in December 21, can be modelled in he following way. Le V denoe he cash value of an IPA a ime, and le P be he sum of he conribuions (including all exra paymens by he governmen) paid ino he policy unil ime. Furhermore, le r f (,T) =: r f, be he yield a ime on a zero coupon bond mauring a ime T (i.e. he planned age of reiremen), aken from he curren erm srucure of German ineres raes. For each IPA, he invesmen managemen company mus build solvency capial equal o eigh percen of he oal conribuions paid ino he plan 21, in each period in which he risk-adjused cash value of he policy is lower han he presen value of he conribuion: V exp(2.33σ ) P (1 + r ) f, T 1. (5) In his formula, 22 σ sands for he monhly volailiy of he muual fund unis backing he pension accoun. The volailiy mus be esimaed from hisorical ime series reurns of he fund uni prices using a window beween wo and five years. If he policy consiss of more han one ype of muual fund (e.g. equiy and bond funds), σ is compued as he weighed sum of he individual fund volailiies according o he curren asse allocaion of he policy. The economic raionale behind his formula is as follows. A every poin in ime, he IPA issuer has he safe invesmen alernaive of invesing some par of he conribuions in zero bonds, so ha a he end of he invesmen period a ime T he proceeds would equal he paricipan s conribuions during he accumulaion phase. The necessary amoun o mee he oal conribuion guaranee of P a ime is P / (1 + r f, ) T-, which is he righ hand side of formula (5). If he provider does no use zero bonds, bu insead employs only socks o back he IPA, nohing happens as long as he cash value of he policy is subsanially higher han he presen value of he conribuions. Subsanially higher means, under he German solvency rule, ha given a curren cash value of V here is a probabiliy of only one percen (noe 2.33 is he 99 percen quanile of he sandard normal disribuion) ha he uncerain

15 13 cash value of he policy one monh laer V +1 is lower han he presen value of he conribuions. This explains he risk adjusmen on he lef hand side of he solvency formula. Hence, wihou capial requiremens, an underfunding of he principal liabiliy during he accumulaion period is possible. The amoun o which such an underfunding is allowed depends on he volailiy of he pension asses and he ime remaining o he end of he accumulaion period. For example (see Table 1), if he monhly reurns of he pension asses have a volailiy of 7.22 percen per monh, which annualized is abou 25 percen per year (a ypical value for German sock funds), he risk-free ineres rae is four percen per annum, and he remaining accumulaion period is 3 years (36 monhs), hen he criical level is only 35.8 percen. This means ha as long as he cash value of he policy exceeds 35.8 percen of he conribuion paid ino he plan, no risk-based-solvency capial is necessary. If he ime o reiremen is only five years (6 monhs), he criical level increases o 97.2 percen. However, he provider has he possibiliy of reducing he volailiy of he IPA and he possible amoun of underfunding by invesing more of he pension asses in low volailiy asses such as bonds. Table 1 here In summary, wih an appropriae asse allocaion and depending on he age of he paricipan, i is possible for he provider of muual fund-based IPA o avoid capial requiremens wihou jeopardizing he credibiliy of he principal guaranee. However, he burden of such a condiional solvency sysem is he implemenaion of an efficien risk monioring sysem for each IPA. Hedging Coss and Hedging Effeciveness of Muual Fund Producs In view of our resuls concerning he long-run risks of pure sock invesmens, and given he regulaory environmen placing a significan capial charge on a fund wih oo much shorfall risk, i is clear ha a sensible sraegy for a muual fund mus conain some elemen

16 14 of risk managemen or hedging. As menioned above, he problem is o provide sufficien credibiliy for promised paymens (hedging effeciveness), while a he same ime reducing he upside poenial of he invesmen as lile as possible, o keep hedging coss low. Noe ha he erm hedging coss refers neiher o he regulaory capial he muual fund company has o provide, nor o poenial expendiures for he purchase of derivaive conracs. The only source of hedging coss for he producs considered below is a reducion in average expeced wealh or, equivalenly, in he oal reurn on he conribuions paid ino he IPA. Because of he subsanial posiive correlaion of he financial asses backing he pension plans, an IPA provider canno manage he risk resuling from guaranees by using radiional insurance pooling echniques. 23 Hence i is necessary o manage underlying risk of he principal guaranee for each IPA individually. Mehodology Focusing on producs currenly offered by German muual fund companies, we compare hem o he simple sraegies of invesing in socks or bonds exclusively. In oal, we analyze five sraegies wih respec o heir long-run risk-reurn profile: a. Pure sock sraegy This sraegy was discussed above wih respec o is long-run risks. Given he parameer values used in our simulaion sudy, his sraegy will likely produce he highes expeced wealh a he end of he invesmen period. On he oher hand, his sraegy can be quie cosly for he muual fund company if i mus pu up subsanial solvency capial o render credible is paymen promises. b. Pure bond sraegy A pure bond sraegy follows he opposie approach. To reduce he risk of falling shor of he promised wealh a he end of he accumulaion period, his sraegy invess only in bonds or broadly diversified governmen bond porfolios. One migh expec ha his

17 15 reduces or even compleely eliminaes he shorfall risk, bu his benefi also comes a he cos of lower expeced reurns. c. Saic porfolio sraegy This sraegy is a mixure of he pure bond and he pure sock sraegy. The porfolio remains unchanged over he whole period, and i conains boh socks and bonds from he sar. Wih reference o he ypical asse allocaion of German reiremen funds (AS- Funds), 24 our simulaions for he 15-year horizon use an equally weighed sock and bond porfolio, whereas for he 3-year invesmen period we use 75 percen socks and 25 percen bonds. d. Life-cycle sraegy Popular advice ofen given o invesors is o aler he porfolio composiion wih age. People are usually advised o hold a larger share of he porfolio in socks when young, and hen o shif ino bonds laer on. The idea behind his sraegy is ha i would be hard o compensae unfavorable movemens on he sock marke occurring lae in he accumulaion period, since lile ime is lef, so ha his ype of risk could be avoided by invesing in bonds. The life-cycle sraegy is an uncondiional hedge in he sense ha more volaile reurn opporuniies are generally considered oo dangerous lae in he invesmen period, irrespecive of he performance of socks before he rebalancing dae. We implemen his sraegy by defining fixed poins in ime a which he porfolio composiion is changed, wih more and more weigh on bonds insead of socks. The exac daes and composiions are as follows: For an invesmen horizon of 15 years, he plan is assumed o sar wih 4 percen of he allocaions going ino equiy and 6 percen ino bonds. Afer five years his allocaion changes, and for he remaining ime 1 percen go ino socks and 9 percen ino bonds. In he case of a 3 year plan, here is an iniial period of en years wih pure sock invesmen, followed by five years wih an allocaion of 7 percen equiy and 3 percen bonds. Afer anoher five years, he

18 16 allocaion of he conribuions is again changed o 4 percen equiy and 6 percen bonds. Over he remaining en years, 9 percen of he conribuions go ino bond funds and he remaining 1 percen ino socks. e. Condiional hedging sraegy This sraegy aims a combining he performance advanage of a pure sock sraegy wih he risk-reducing effec of a pure bond sraegy. As opposed o he life cycle sraegy, however, he decision o shif from one invesmen ino he oher is no driven by an exogenous variable like age, bu raher by he performance of he respecive invesmens. For his reason, such a sraegy represens a condiional hedging approach. Usually one sars ou wih a pure sock invesmen and shifs o bonds as soon as a cerain criical level of wealh is reached. In his case, subsequen conribuions go in bonds unil he safey level is again exceeded, when he sraegy swiches back o a 1 percen sock invesmen. An imporan parameer for his ype of sraegy is he criical level of wealh a which he invesmen rule (for subsequen conribuions) changes. To link his criical value o he inervenion line se by he regulaory auhoriies in Germany (see equaion 5), we se he criical level of wealh (as an example) o 75% above he inervenion value defined according o he solvency formula (5). A possibiliy no discussed up o now is he use of derivaive asses o proec he value of an invesmen plan agains shorfall risk. The appropriae insrumen here would be a pu opion on he value of he plan, wih a srike price equal o he sum of he nominal paymens. However, he applicaion of pu opions in his conex is no wihou problems. Firs of all, due o he very long mauriy of he savings plans, any opion would be very expensive, and he cos would have o be paid up fron, (a he beginning of he accumulaion period) which raises financing quesions. Second, i seems unlikely ha a pu wih such a long ime o mauriy would be offered a all, so ha a roll-over sraegy would become necessary wih all

19 17 he risks involved in erms of prices and liquidiy. Third, for he pu opion o be of real value o he insiuion holding i, he seller would have o demonsrae i could acually cover is liabiliies a he end of he accumulaion period. In pracice, here would always be doubs concerning he acual risk-reducion poenial of such an opion. Finally, here is a significan operaional problem in using pu opions, since all he accouns have o be proeced individually. This means ha for every IPA, he provider would have o hold a pu opion wih he appropriae srike price and ime o mauriy. This seems oo cosly and complicaed for he ypical insiuion, so ha hedging sraegies using physical financial derivaives will no be considered furher in he following analysis. We analyze he five sraegies described above in erms of wealh levels (or oal reurns) and required regulaory capial. Since here are no closed-form expressions for he saisics of ineres, we use Mone Carlo simulaion o generae a large number of pahs for he evoluion of he savings plans. The relevan saisics for oal reurns (relaive o a benchmark) and regulaory capial are hen evaluaed on he basis of hese scenarios. The key ingredien in such a simulaion is a suiable model o describe he dynamics of he relevan funds and he shor rae of ineres. For he funds we use he sandard capial marke model, represening asse price movemens by means of correlaed Wiener processes. The dynamics of he shor rae are given by he one-facor model suggesed by Cox/Ingersoll/Ross (CIR, 1985). While we assume consan correlaions beween he risk facors, he covariances will vary due o he fac ha he condiional sandard deviaion for he shor rae will in general no be equal o he uncondiional value. The ime series used o esimae he process parameers (mean reurns, volailiies, correlaions) are he monhly log reurns of he German sock index DAX represening he sock index fund, he log reurns of he bond performance index REXP as he bond index fund as well as he 1-year ineres rae as a proxy for he shor rae. Parameers were esimaed via a maximum-likelihood approach, he esimaes are presened in Table 2 and 3.

20 18 As discussed above we subraced he equivalen of.5 percen p.a. from he monhly average reurn on he invesmens o ake poenial adminisraion coss ino accoun. Table 2 here Table 3 here The CIR process is very popular in ineres rae modeling. This is mainly due o he fac ha i is able o generae boh mean-reversion in ineres raes as well as non-negaive raes wih probabiliy one. Since he process exhibis mean-reversion, he sign of he drif componen (i.e., he expeced change in he shor rae over he nex ime inerval) depends on wheher he process is currenly above or below is long-run mean. How quickly he process revers back o his long-run mean is deermined by he speed of mean-reversion. In Table 3, κ (kappa) represens his speed of mean reversion, θ (hea) sands for he long-run mean of he ineres rae, while σ (sigma) denoes he volailiy of changes in he shor rae. The marke price of ineres rae risk was se equal o zero for reasons of simpliciy. To ensure sabiliy of he simulaion resuls, we base our analysis on 3,, simulaions for each of he respecive sraegies. We hen compue: saisics relaed o hedging coss: he mean of he oal reurn generaed by he respecive sraegies for he differen poins in ime (monhs) saisics relaed o hedging effeciveness: he shorfall risk and he required regulaory capial for he respecive sraegies. The model assumes equal conribuions ino he plan occurring a he beginning of each monh, and a fron-end load of five percen proporional o he uni price for he sock fund and hree percen for he bond fund. These loads are comparable o he curren German muual fund fee srucure. V i, denoes he uncerain oal wealh of he IPA in monh ( = 1,., T) in simulaion run i (i = 1,, n), and z i, represens he criical level of wealh in monh according o formula (5) deermined by he federal banking supervisory auhoriy (BAKred). 25 If P represens he sum of paymens ino he plan unil ime, he average

21 compounded (muliyear) reurn (EW ) of he IPA a he end of monh is given by: 19 EW 1 V P n i, =. n i= 1 P (6) The probabiliy of a solvency capial charge (CP ) in monh is esimaed by: n 1 CP = max[ zi, Vi, ] I (, z ) ( V, ),, i (7) i n i= 1 where he indicaor variable I (a, b) (X) is equal o one if X (a, b) and zero oherwise. The mean solvency capial charge (MC ) a ime monh afer he beginning of he plan normalized by he sum of he conribuions P paid ino he IPA, is given by MC = 1 n n i= 1 C P i,. (8) According o he regulaory auhoriies, he solvency capial charge C i, depends on how far he muual funds based IPA wealh falls shor of he criical level. The rule says ha he capial charge is a leas 8% when wealh falls below he criical level. If he amoun of he shorfall exceeds 8%, he capial charge is increased accordingly o cover he gap. Hence C i, mus o be calculaed according o he following formula: C i,.8 P = (1 Vi, / z i, ) P < 1 V 1 V i, 1 V i, i, / z / z / z i, i, i,.8 >.8 <. (9) The mean condiional capial charge (MCC ) a monh given ha a capial charge has occured is compued according o: MCC = MC /CP. (1) Resuls Our resuls for he expeced oal reurn of savings plans based on he differen invesmen sraegies are given in Table 4. I is no surprise ha he pure sock sraegy does bes in erms of his measure, since socks have he highes expeced monhly reurn. This

22 2 also causes he differences beween he respecive sraegies o increase wih ime. Neverheless, i is ineresing o noe how close he condiional hedge sraegy comes in erms of expeced oal reurn. Even afer 3 years, he difference o he pure sock sraegy is only abou 3.5 percen of he conribuions paid. This can be aken as a firs indicaion ha his ype of sraegy migh an ineresing compromise beween he reurn poenial of a pure sock sraegy and he risk-avoiding propery of a pure bond approach. Table 4 here Neverheless, expeced wealh is jus one measure o be considered; any sensible comparison of he given producs mus also focus on risk measures. The risk of he differen sraegies is measured by he regulaory capial charge ha he muual fund company adoping hese sraegies would face. Table 5 indicaes ha, for an invesmen horizon of 3 years, no regulaory capial is needed over he firs five years for any of he sraegies. The pure bond sraegy can even be regarded as enirely risk-free wih respec o regulaory capial charges. The life-cycle approach also exhibis very low capial charges on average. This sraegy seems o be an ineresing alernaive o a pure bond invesmen, given is advanage in erms of expeced reurn. Table 5 here As expeced, he pure sock sraegy balances is high reurn poenial wih an expensive regulaory capial level. I requires more han hree imes he regulaory capial han he condiional hedge sraegy, which is in second place wih respec o his crierion. Furhermore, Table 5 provides insigh ino he impac of he invesmen horizon. Long-erm sraegies generally exhibi lower risk han he 15-year plans wih he only excepion being he life-cycle sraegy. The fundamenal reason for longer-erm sraegies requiring less regulaory capial han shorer-erm sraegies is he discouning embedded in he criical solvency raio se by he regulaory auhoriies. This means ha he required minimum wealh level of a plan is he lower he longer he remaining ime o mauriy of he plan. For he life-

23 21 cycle sraegy, however, here are wo effecs o be considered. For he shorer horizon plan, he period of pure sock invesmen is raher shor, so he risk in general decreases. To ake he mos pronounced example for he usual impac of he invesmen horizon, consider he pure sock sraegy. The average capial increases dramaically compared o he 3 year-plan, so ha his approach looks very cosly when i is implemened over his shor invesmen horizon. The probabiliy ha he muual fund company will be required o pu capial aside is given in Table 6, and he resuls are qualiaively similar o Table 5. The pure bond sraegy never forces he provider o pu up capial, whereas he pure sock does wih a significan likelihood. Again, ime is an imporan facor here. For he 3 year horizon, he probabiliy of a capial charge for any sraegy never exceeds 1.4 percen, bu for a 15 year pure sock plan, his probabiliy is almos nine percen a he end of he invesmen period. For he oher sraegies, he raios of 15-year o 3-year probabiliies are also quie high, so ha if a plan acually exhibis he risk of a capial charge, his risk ends o increase for shorer horizons. Table 6 here We also noe ha here is an imporan difference beween shorfall probabiliy and he probabiliy of a regulaory capial charge. As shown above, he shorfall probabiliy of a pure sock invesmen acually falls wih a longer invesmen horizon, whereas he probabiliy of a capial charge goes up. Again his is due o he fac ha he criical level of wealh se by he German regulaory auhoriies conains a discouning componen. Thus his criical level will go up wih decreasing ime o mauriy, hereby causing a higher likelihood for a capial charge. The average condiional regulaory capial charge depiced in Table 7 shows how much capial will be needed given ha he cash value of he IPA falls below he criical BaKred value. Noe ha when he empirical probabiliy of a capial charge is zero, his measure is no defined. The Table shows resuls qualiaively similar o he general long-run

24 22 risk-reurn profile of he various asse classes. The risk of a pure sock sraegy becomes obvious, since if regulaory capial is needed, i will probably be a significan amoun. For example, a he end of 3 years he muual fund company would need on average almos 2 percen of he conribuions as regulaory capial in hose scenarios where wealh falls below he criical BaKred value. The benefis of flexibiliy become obvious when he saic sraegy is compared wih he condiional hedge. The condiional hedge produces, on average, higher wealh over he whole invesmen period and he average condiional regulaory capial is also lower. So if one were o compare he differen producs on he basis of hese wo measures only, he saic sraegy would be dominaed. Noe, however, ha he probabiliy of a capial charge is lower for he saic sraegy. Table 7 here The analyses hus far focus on he saisical oupu, bu i is also imporan o assess he adminisraive coss generaed by each plan. Cos will no be major for he sraegy producs such as he pure bond, he pure sock plans, he saic, and he life-cycle sraegies. However, for he condiional hedge, he need o shif incoming disribuions across asse classes depending on how much wealh has been accumulaed in he plan, migh imply considerable adminisraive effor. I is herefore of ineres o examine he relaive frequency of shifs from one asse class o he oher, when a condiional hedge sraegy is run. Here again, ime is an imporan facor, since for he 15-year plan he muual fund company mus change he asse class in 98 percen of he pahs generaed by he simulaion, whereas his need arises in only 26 percen of he cases for he longer horizon. In any case, coss mus be aken ino accoun when sraegies are compared wih respec o heir pracical applicaion. In summary, i is no possible o idenify he overall dominaing invesmen produc or sraegy. Wih a few excepions, higher poenial in erms of average wealh usually comes a he cos of higher regulaory capial. I is imporan in his conex o look a he average amoun of regulaory capial condiional on he even ha capial acually has o be pu aside.

25 23 Here i becomes obvious ha sraegies wih a fixed sock invesmen can produce significan risks. This risk is miigaed when he condiional hedge sraegy is employed. Neverheless, addiional adminisraive cos mus be considered. Conclusions Due o he severe financing problems of sandard pay-as-you-go pension sysems in many counries, alernaive vehicles for reiremen financing have o be developed. In Germany, such a new sysem was recenly insalled when he German Reiremen Saving Ac was passed by he legislaive body. The governmen offers significan ax relief for invesmen producs meeing cerain requiremens, he mos imporan of hese being a guaranee promising ha he cash value of he IPA a he end of he accumulaion period will be a leas as high as he nominal sum of he conribuions. To lend sufficien credibiliy o he paymen promises made by insiuions providing invesmen producs for hese savings plans, he regulaory auhoriies in Germany have imposed a capial charge in case he value of he savings plan falls below a cerain criical level. A firs sigh, i seems ha in order o implemen such a principal guaranee, complicaed and expensive financial producs like derivaives are needed. However, as we have shown, here are oher ways of achieving a someimes pracically risk-free posiion wihou using opions or similar insrumens. We analyze in deail various sraegies aimed a combining he poenially reurn-increasing properies of equiy invesmen wih he riskreducing characerisics of bond invesmens. These sraegies offer a real-world applicaion of he ools and mehods of capial marke heory. Of course, he rade-off beween reurn and risk is always a he core of he analysis. Ye in he conex of his chaper i is imporan o recognize ha variance is no he mos imporan measure of risk. As opposed o a more radiional approach we consider shorfall and he need of regulaory capial as he wo mos imporan ypes of risk. The sraegies analyzed here range from simple pure bond or equiy invesmens, o

26 24 mixed equiy-bond funds and producs offering a change in porfolio composiion a predefined poins in ime, o highly sophisicaed producs wih condiionally changing invesmen syles. One of he key resuls of our sudy is ha hese dynamic sraegies, swiching from socks ino bonds whenever he value of he savings plan falls below some criical solvency raio se by regulaory auhoriies, perform raher well in erms of expeced oal reurns for long invesmen horizons. They come close o pure sock invesmens wih respec o he average value hey generae for he invesor. However, i is also very imporan for he financial insiuion o keep an eye on he expeced amoun of regulaory capial required by a cerain invesmen sraegy. Due o he condiional change in allocaion when he criical regulaory value is reached, he expeced capial charge is significanly smaller han in he case of a pure equiy invesmen. Besides he basic ype of sraegy, he lengh of he invesmen horizon is an imporan facor for he risks and rewards of alernaive sraegies. In general, he longer he mauriy of he plan, he lower he expeced capial charge, since he criical level se by he auhoriies in Germany conains a discoun facor, and he higher he expeced oal reurn. Neverheless, i is imporan o consider oher risk variables as well in his. We are far from claiming ha one of he sraegies discussed here should be seen as uniformly superior o any oher. Raher we seek o poin ou he benefis and risks offered by he differen ypes of producs, o provide a basis for a horough discussion of he issues involved in produc design and regulaion. This research is par of he Research Program Insiuional Invesors of he Cener for Financial Sudies, Frankfur/M. This chaper was wrien in par during Dr. Maurer s ime as he Mezler Visiing Professor a he Wharon s School Pension Research Council. The auhors would like o hank Manfred Laux, David McCarhy, Olivia S. Michell, Alex Muermann, Rudolf Siebel, Wolfgang Raab, and Ken Smeers for helpful commens. Opinions and errors are solely hose of he auhors.

27 25 Appendix: Derivaion of he Solvency Formula Consider an invesmen plan where paymens ino an IPA are made a equally spaced poins in ime =, 1,...,T (e.g. monhs). Le P denoe he sum of paymens up o ime, T he planned erminal dae of he plan (equal o he beginning of he payou phase), and q(r f,, T-) = (1+r f, ) -T he discoun facor wih risk-free rae r f, and remaining ime o mauriy T-. Wihou loss of generaliy we assume ha he invesor holds exacly one share of he fund a ime. We are ineresed in he solvency raio V / P a ime, which makes sure ha he uncerain marke value of he shares V +1 a ime +1 is less han he sum of paymens P ino he plan discouned up o ime T, i.e. less han P q(r f,, T--1), wih a probabiliy of a mos ε. To be able o quanify his shorfall risk, we have o specify a model for he random evoluion of he value of he invesmen shares. Here we make he sandard assumpion ha he dynamics of his value can be described by a geomeric Brownian moion. This implies ha he relaive change in value (i.e. he log-reurn) ln(v +1 ) ln(v ) is normally disribued wih mean µ und variance σ². Formally we obain he desired solvency raio as he soluion of he following inequaliy: Prob[V +1 < P q(r f,, T--1)] = Prob[ln(V +1 ) < ln(p q(r f,, T--1))] ε (A1) Using he above disribuional assumpion inequaliy (A1) is equivalen o (A2) ln(v ) + µ ln(p ) + ln[q(r f,, T--1)] + N 1-ε σ, where N 1-ε is he (1-ε)-quanile of he cumulaive sandard normal disribuion. Under he addiional (conservaive) assumpion 26 ha he one-period expeced reurn is equal o zero (i.e, µ = ) inequaliy (A2) can be wrien as (A3) V / exp[n 1-ε σ] P q(r f,, T--1). Seing N 1-ε = 2.33, which implies a oleraed shorfall probabiliy of no more han one percen, his represens he equaion for he solvency raio (5) presened in he main ex.