Asset and Liability Management, Caisse. a manager of public debt

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1 Asse and Liabiliy Managemen by CADES, a manager of public deb Name Deparmen & affiliaion Mailing Address address(es) Phone number , Fax number Eric Ralaimiadana Asse and Liabiliy Managemen, Caisse d Amorissemen de la Dee Sociale CADES 4 bis, boulevard Didero PARIS (France) eric.ralaimiadana@cades.fr, eric.ralaimiadana@ensae.org Absrac The mehod chosen by CADES o seer he process of paying down he social securiy deb i has assumed is relaed o our paricular asse and liabiliy managemen policy. The economy is ruled by hree facors, he dynamics of which govern he principal classes of negoiable deb insrumen and our only asse, which is he CRDS ax revenue, generaed via a levy on nearly all forms and sources of income in France. Risk is defined as he probabiliy ha we will no achieve an accepable performance level in erms of deb repaymen capaciy, while our aversion o risk is refleced in he convexiy of he relaionship beween performance and he redempion horizon. We implemen he dynamics of our balance shee componens and exhibi he enire se of opimal porfolios under a pre-defined rule of re-balancing. The opimal porfolios will be a sub-se of he fronier of efficien porfolios, condiionally a he hreshold of he chosen risk. Keywords: refinancing, amorizing capaciy, redempion horizon, opimal porfolio, efficiency fronier, risk hreshold. 1/22

2 1. Inroducion The role of CADES (Caisse d Amorissemen de la Dee Sociale) is o reimburse he accumulaed deficis of he French social securiy or healh insurance sysem. To his end, a single and exclusive resource has been allocaed o CADES by law he CRDS (Conribuion au Remboursemen de la Dee Sociale). To respond o he following quesion How can his deb be opimally repaid? we use our asse and liabiliy managemen model. 1 The managemen of deb presens some sriking analogies wih he managemen of asses. For example, a company ha insures a given flee of risks receives premiums and consiues a porfolio of invesed asses. I builds his porfolio o maximize reurn so ha i can mee is policyholder liabiliies, cover is own operaing coss and generae a profi margin. Similarly, CADES receives he proceeds of a ax whose axable base and rae are defined by law. While he axable base flucuaes in erms of he exac naure of is componens, he ax rae has no changed since he ax was firs levied. Accordingly, we manage wha migh be considered a defined conribuion plan, wih he noion of conribuion corresponding o CRDS inflows. Conversely, our liabiliies are made up of he programmed ouflows by which we amorize he deb. While he modeling of our asse and liabiliy managemen is largely inspired by he heoreical ools used in asse managemen, aricles on he managemen of deb srico sensu are rare, since his kind of analysis is primarily conduced by organizaions ha are responsible for sensiive deb, i.e., wihin a governmen s public finance adminisraion, a public service agency or a very large corporaion. The research work ha our modeling srongly resembles is ha of Brennan and Xia (2002)[1]. The auhors defined he opimal invesmen sraegy wihin a universe ha did no conain any insrumens generaing an inflaion-indexed reurn, made up of a savings accoun, a risky asse, and nominal fixed coupon bonds. They demonsraed ha, for an agen wih a finie invesmen horizon T, in he presence of an unanicipaed inflaion componen no hedged by a marke insrumen, he opimal porfolio is he sum of wo porfolios: one providing he reurn mos srongly correlaed o ha of an indexed bond wih a mauriy of T; he oher being he minimum variance porfolio as inended by Markowiz (1959)[2], combining he risky asse and he savings accoun. Their findings revealed a srong sensiiviy o agen risk aversion: he higher i is, (i) he higher he allocaion o a porfolio replicaing he indexed securiy, and (ii) he more he mauriy of he nominal bond diminishes. In heir sudy, hey cie work done by Campbell and Viceira (1999)[3], whose hinking our own closely mirrors. The laer used a numerical mehod o resolve he opimizaion of he sraegy of an invesor wih no horizon limiaion, operaing in he same invesmen universe as Brennan and Xia, using a so-called myopic sraegy, i.e., wih consan proporions. By exhibiing an opimal soluion in an analyic form, Brennan and Xia underscore he loss of value generaed by he myopic sraegy, as well as he sensiiviy of he resuls o wo 1 We would like o hank Jean-François Boulier for having encouraged us o publish his aricle. We would also like o hank he anonymous arbiraor(s) of Banque e Marchés for heir work and heir exremely useful commens, as well as Parice Rac Madoux and Chrisophe Frankel for heir insighful remarks. 2/22

3 characerisics of he model: he invesmen horizon and he mean reurn parameers of variae diffusion processes. A recen aricle in Banque e Marchés (2004)[4] assesses mehods for managing pension funds. We deeced several poins of conac wih our own approach on he level of modeling he processes followed by he variaes under sudy. In paricular, ha of Cairns (1998)[5] inroduces a sochasic reiremen flow ino he modeling of a defined benefis fund. A reiremen enilemen or pension is a fixed percenage of an individual s wage, which he auhor models using a diffusion process ha is no correlaed wih marke noise. In addiion, he analyses developed in aricles by Svensson and Werner (1993)[6], as well as by Koo (1998)[7], resonae direcly wih our own reflecion. Their auhors examine he opimaliy of he porfolio and consumpion in he case of an agen wih a sochasic wage. They inroduce a source of non-duplicable risk via a negoiable insrumen, hereby placing he problem wihin a framework of marke incompleeness. The res of he aricle is srucured as follows. We briefly review he regulaory framework ha governs he funcioning of CADES. Then, we describe our represenaion of he balance shee in simple componens, resuling in an economy regulaed by hree variaes, he nominal shor-erm rae, he rae of inflaion, and he rae of volume growh in he CRDS. Having described he diffusion equaions followed by heir processes, we explain our opimizaion problem and is resoluion. We hen describe he decision suppor ools we have rolled ou based on he resuls of he modeling. The las par of our paper is devoed o a criical review of he model and he changes envisioned, ending wih a few conclusion on deb managemen informed by asse and liabiliy managemen. 2. Review Three exs mark he hisory of CADES: - The seminal ex is he French ordinance daed January 24, 1996, which defines he mission of CADES (i.e., exinguish he French social securiy deb) and ses is life span (i.e., unil January 31, 2009). Deb ousanding oals 21 billion euros, plus annual paymens o he French governmen of 1.9 billion euros, over a period of welve years. - The Social Securiy Financing Ac of December 19, 1997 for year 1998 exends he remi of CADES for an addiional five years, and ransfers an addiional 13.3 billion euros of deb. - Finally, he Healh Insurance Ac of Augus 13, 2004 ransfers 35 billion euros worh of defici accumulaed hrough 2004 o CADES, o which is added esimaed deb of up o bu no more han 15 billion euros. The Ac also srikes all reference o a defined dae on which CADES ceases o exis and holds ha all new deficis from he social securiy sysem mus be financed by a new resource. In wha follows, we will illusrae he way CADES has ackled is asse and liabiliy managemen issues in a changing environmen due o regulaory amendmens. 3/22

4 3. Mehodology 3.1. Balance-shee modeling The CADES balance shee can be broken down ino four major iems. The real esae holdings (asses) ha were inheried when CADES was formed having been disposed of in full, is asses oday consis exclusively of a receivable on he naionwide ax levied on nearly all sources of income (he CRDS), which is allocaed solely o CADES. Is liabiliy is financing deb, and i has no shareholders equiy Asse The axable base for he CRDS is earned income from work (67%), replacemen income (21%), income earned from asses and invesmens (10%), gaming proceeds and he proceeds from he sale of precious meals (2%). To he exen ha his conribuion is levied globally on all forms of income, a very sraighforward way of modeling our revenue is o use gross available income, he naional accouning aggregae, as a proxy for he CRDS axable base. Anoher opion would be o model separaely ransfer and wage income, asse and invesmen income, and assimilae growh in he remainder o a random walk. We oped for he mos sraighforward soluion, noing ha he axable base has undergone numerous changes as well as various specific exempions, and here is no reason o believe his will end: any advanage o be gained hrough more refined modeling by income caegory should be pu ino proper perspecive considering he flucuaions due o changes in scope. The nex quesion, hen, is o model he rae a which hese income inflows grow, using a consan ax rae (0.5%). If we focus on he hree mos significan sources of revenue in he axable base, we see ha wages and old age income have experienced quasi-consan volume growh over he period. Over he same period, invesmen income has undergone volume growh ha can be assimilaed o a rend growh plus a whie noise. A fairly simple modeling of our asses is based on diffusion equaions for wo processes, he real rae of growh in our ax revenue and he rae of inflaion. They follow Ornsein-Uhlenbeck processes. The value growh of our asse is calculaed hrough he composiion produc of he laer. A ime, we noe A he value of he asse, growh rae and i he rae of inflaion. k is value growh rae, g is real The dynamic of A is described by he following diffusion equaion da And is rae of value growh is modeled by e kd e A k d g i d The diffusion equaions followed by hese rae processes will be developed furher on. 4/22

5 Ne deb dynamic Ne deb varies in he following manner: a he end of each year, we noe he balance beween CRDS revenue received and ouflows, eiher ineres paid on ousanding deb, or addiional new deb. If his balance is posiive, we reduce he deb hrough buy-backs. If i is negaive, hen we mus increase he level of our borrowings. By financing balance, we mean he change in ne deb afer employmen of his balance, and we noe i S. In wha follows, i will also be referred o as amorizaion capaciy. To elucidae he ne deb dynamic, we adop he following noaions: le, - ^ L he value of he deb payable year, - L 1 he value in curren euros in, of he inheried deb for year 1 before paymens falling due, - L deb a end of year before reallocaion * - L he value in curren euros in, of ne ousanding The ne deb dynamic is wrien in simple fashion L L 1 S Ne deb observed for year, before payables arriving a mauriy, is he sum of deb payable in and deb a end of year before reallocaion, which is wrien as: ^ * 1 L L L We can show ha ne deb a end of year afer reallocaion is wrien ne L L * S (1) ne where S designaes he balance ne of financing, i.e. he balance S ha is modified by he effecs of ime and marke flucuaions on he inheried deb from year 1, as well as payables arriving a mauriy. Proof Indeed, he ne balance reads ne S S ^ L L 1 L 1 (2) The financing balance is derived by calculaing D, he amoun available a ime, expression in which V designaes an evenual new inflow of deb and c designaes operaing coss D A 4 exp kudu 1 V c If he value of deb was reduced owing o marke flucuaion, hen he financing balance is increased, and vice versa. Accordingly, we add o D, he opposie of his change in value, i.e., o derive S L 1 L 1 5/22

6 The ne balance is hen wrien as S D L 1 L 1 S ne so ha ne deb dynamic can be wrien as L (3) ^ D L L 1 D L 1 L 1 * L L L D which, in accordance wih (4), gives back equaion (1) The ne balance ne S ^ is given by (4). Depending on is sign, i represens eiher a financing capaciy or a borrowing need. I consiues he oal of buy-backs or aps (a he iniial proporions) of exising bonds a heir price, measured in. (4) Liabiliy Our liabiliies are almos exclusively limied o our deb porfolio. CADES does no have shareholders equiy. Deb is classified according o hree ypes: fixed-rae insrumens, bonds pegged o inflaion (in France, inflaion excluding obacco), and floaing rae insrumens, including medium erm noes. The facors which rule liabiliies are nominal ineres raes and he rae of inflaion. We specify a Vasicek[8] model for he yield curve. I offers he dual advanage of inegraing a mechanism of reurn o he mean and of allowing us o rebuild he enire curve from he shor-erm ineres rae alone. I reflecs a characerisic ha has been demonsraed by economeric sudies, i.e., he fac ha changes in he shor-erm ineres rae alone explain abou 80% of all yield curve movemens. The variaes ha rule liabiliies are finally, he shor-erm ineres rae and he rae of inflaion. 3.2.Processes assumed by relevan variaes The nominal shor-erm ineres rae process is described by he following SDE (Sochasic Differenial Equaion) dr a b r d dw r The formulaion of he zero-coupon rae for he [,T] period is 1 at R r 1 e T r 2 at 1 e r R T, r R 2 4a a 2 6/22

7 The inflaion rae process is described by an SDE ha is idenical o ha of he shorerm rae di cd i d i dwi While he growh rae of he CRDS in volume erms is described by he following diffusion equaion dg m g d g dwg The hree sources of risk, he Brownian moions W r, W i, W g, are linked by heir insananeous cross-correlaions r, i, g, i, g, r, respecively. One has o ransform his vecor of Brownian moions ino a new vecor of uncorrelaed Brownian moions. This is obained by ransforming he covariance marix of he iniial vecor ino a riangular marix. In a 2-dimension case, applying he copula heory o he 2-dimension vecor of he firs Brownian moions Wr andw i for insance, would yield he same ransformaion. The dynamics of boh he shor-erm ineres rae and he inflaion rae processes can be re-wrien as dr ab r d r dwr where W r, di c 2 d i d dw dz, 1 r i Z i are uncorrelaed Brownian moions. The dynamic of he CRDS growh rae in volume erms is re-wrien using he same echnique, bu will no be shown here because of he lengh of he SDE. i r r, i i i 4. The Opimizaion Problem 4.1. Formalizaion The financing balances ha are accumulaed year afer year will deermine our capaciy o amorize he deb. By ieraing he dynamic equaion of he ne deb, we see ha he change in he laer beween years 0 and is equivalen o he accumulaed financing balance for each period. Opimizing he amorizaion of he deb can be wrien, as a firs approach, as he maximizaion of he expeced aggregae of annual financing balances. The opimizaion program is wrien as follows max l0 The soluion (or soluions) of he opimizaion program consiss (or consis) of porfolio weighings. The resul depends in paricular on he re-balancing rule ha is adoped. We have oped for he rule of reallocaing each porfolio by mainaining he iniial proporions, which is anamoun o seeking arge porfolio srucures ha are mainained consan hroughou he erm of he mandae. We will noe he vecor of he deb S l k,m 7/22

8 weighings for each class, flagged by he index k, and for each mauriy, flagged by he index m. Indeed, we are rying o deermine one (or more) srucure(s) which, over he long erm, enables us o achieve our objecive i.e., ha of paying down he deb a he lowes possible cos o he axpayer. In he conex of a deb managemen sraegy o be carried over a long period, and which we wan o be as ransparen as possible, choosing he consan proporion rule allows us o presen (for example, o he minisries wih supervisory power over CADES) opimal deb srucures ha in essence do no flucuae along wih fuure evens. Since he deb is ranscribed in a porfolio of zero-coupon bonds, all refinancing or ne buy-back ransacions are carried ou a going marke raes, which, like S and S, are measurable in. We will noe E k, m he final value of a given ouflow paid on he k h class of deb wih mauriy m, and B, he price a ime of a zero-coupon bond of mauriy m. m When he ne financing balance is allocaed, he curren value of he deb becomes ne L L * S L K M ne B, mek, m k, ms k1 m1 where, for a given erm X, X+ designaes he posiive par of X. Indeed, if he porion of he balance allocaed o he amorizaion of ousanding deb E k, m exceeds he marke value of he laer calculaed year, his ousanding will be redeemed in full, and he remainder will be added o he remaining available balance. The deb amorizaion mechanism enails ha, he year in which he ne deb crosses he null value, he financing balance is posiive and exceeds he curren value of he deb observed and recorded a he end of he preceding year. This allows us o represen our opimizaion program under a dual form Dual form of he opimizaion program We will briefly leave behind he paricular case of CADES and ake a look a he sylized case of an indebed corporaion, ha is ordered by is shareholder o pay off is borrowings, by allocaing all of is operaing revenues o repaymen. This corporaion s guaranor of las resor is he Sae. Le s suppose ha he corporaion has made a commimen o he financial communiy o a probable dae of full repaymen H, and ha is las borrowing falls due on his dae. There are wo possible oucomes on dae H : - eiher he corporaion has correcly esimaed he full deb reimbursemen dae and will have i repaid in full or even earlier, which ranslaes as 8/22

9 S H L H 1 - or he corporaion has underesimaed he full deb reimbursemen dae, in which case S H LH 1 In he even of he second oucome, he corporaion runs he risk of seeing is credi raing downgraded, and of finishing repaymen on a dae ha is laer han he esimaed dae of full repaymen. Le s look more closely a oucome number wo: underesimaing he amorizaion period means ha he corporaion s financing requiremen is LH 1 S H, which in urn requires eiher re-borrowing on less appealing credi and liquidiy erms or urning o he guaranor of las resor o absorb he financing requiremen. The envisioned consequences of his oucome do no exis for CADES. Indeed, he agency enjoys he implici backing of he Sae, and is revenues are levied on and aken from he naional income. However, risk analysis via he reimbursemen horizon is valid, and allows us o express he probabiliy ha he reimbursemen arge will no be me, as a funcion of he risk quanile H ( ), in he following manner S H L H 1 0 Accordingly, we can wrie our opimizaion problem as he minimizaion of a risk, in he form min L H 1 S 0 H We do no know he analyic form of he probabiliy densiy of he variae S L - filraion engendered by brownian moions X 1, condiionally a filraion W r, W i, W g. The presence of ime incremen 1 wihin he expression of X shows he «pah-dependency» of X, and ha of is probabiliy densiy. The expression of he laer migh no be rivial. Neverheless, we can simulae drawings ino he condiional probabiliy disribuion of S. This is wha is done in he course of our resoluion process Esimaor of he expeced amorizaion capaciy The condiional probabiliy densiy of X hus depends on he level ha has been reached by his variae on dae. Saring from an iniial level of deb L 0 a he beginning of year 0, X depends on he level reached by he aggregae balances beween years 0 and, noed S c 0,, i.e., of he even S c 0, S l0 l x 9/22

10 Our risk of failure, defined in he preceding paragraph as he risk of no achieving reimbursemen by horizon H( ), grows wih he decrease in x. The lower he aggregae balances, he more difficul i will be o reimburse before due dae. In like manner, we can use he same reasoning, considering he average annual balance (or he average annual amorizaion capaciy) insead of he accumulaed or aggregae balances for he period beween years 0 and, ha we noe _ S and calculae as follows _ 1 S S As a maer of fac, he _ S saisic is a monoonous funcion, sricly increasing, of he accumulaed financing balances over he period exending from years 0 and, scaled by a facor equal o 1. For a given iniial deb, he lower he average annual amorizaion capaciy, he farher he reimbursemen horizon will be. _ S is our esimaor of he expeced annual amorizaion capaciy. c 0, In he example ha follows, of an iniial deb of 100 billion euros reimbursed over 15 years, we illusrae he relaionship beween he level reached by _ S, as observed a he end of he period, and he reimbursemen horizon. S0 Reimbursemen and annual amorizing capaciy an example 100 S1 90 S2 Ne Deb 7,0 Ne Deb (bn EUR) Expeced annual amorizaion capaciy (S0+...+S9)/10 6,5 6,0 5,5 Annual Amorizing Capaciy (bn EUR) 10 0 (S0+S1)/2 S ,0 Horizon (yrs) 4.4. Risk aversion A he same ime, he saisic _ S is a decreasing funcion of horizon, wih a convex shape. The convexiy of his funcion evidences CADES s aversion o risk. Indeed, we can wrie ha, below some hreshold of he annual amorizaion capaciy achieved over he period, 10/22

11 he lower he average annual amorizaion capaciy he greaer he increase in he amorizing period, for a deerioraion in he capaciy of he same magniude. Accordingly, here is a region of risky values ha we wish o avoid, boh for he accumulaed balances beween 0 and and, likewise, for he average annual amorizaion capaciy. The occurrence of such values, a he risk level (%) a ime, enails he exploding rend of he reimbursemen horizon The risk as seen from he perspecive of he invesor/axpayer To shed addiional ligh on he risks we run, we look a he problem from he perspecive of an invesor who holds CADES deb and is a axpayer a he same ime. Le s posi ourselves in he year H, our saed probable reimbursemen horizon. Le s furher suppose ha he invesor/axpayer in quesion was holding a CADES bond reimbursed a horizon H, ha CADES was obliged o issue a new bond o repay he bondholders, and ha he invesor reinvesed in he new issue. The wealh of his invesor, including axes levied, will be reduced. Indeed, assuming ha he decision o pospone he dae of final reimbursemen does no have any impac on he credi spread, he invesor will earn he same ineres when he reinvesmen is made, which is he yield on a CADES securiy. Bu he will be liable for an addiional ax levy over a longer period due o he gap beween he horizon esimaed on dae 0 and ha which will in fac come o pass. The analysis may be made from anoher angle: he Social Securiy Financing Ac ha was passed in Augus 2004 requires ha all deficis be offse by addiional ax revenue so as o mainain unchanged he dae on which reimbursemen is compleed. Insead of bearing an exension of he conribuion, he invesor would be axed up fron for he resources needed o mee he financing needs of CADES on dae H. His profi/loss profile is ha of a pu selling posiion, he loss growing wih he magniude of he error of esimaion commied wih respec o he reimbursemen horizon. This analysis sheds ligh on he imporance of he level of probabiliy. The more averse we are o he risk of making a misake on he reimbursemen horizon, and he consecuive one of having o resor o levying addiional ax, he more we will require ha be small Opimaliy crierion The selecion of an opimaliy crierion is one of he pillars supporing any opimizaion program. Our variae under sudy, he expeced annual financing balance, consiues a gross performance measure. I is inadequae in ha i evacuaes from he crierion he impac of he risk, whose imporance we have measured, and does no ake ino consideraion CADES aversion o risk. Accordingly, we need o use a risk-adjused performance measuremen as our crierion. 11/22

12 A a ime before he regulaory amendmens aforemenioned, when he reimbursemen horizon was fixed, we were focused on he ne financial posiion measured a his horizon. We used o compue he disribuion of his variable, such as simulaed a his fixed horizon. The risk-adjused performance measuremen we used was a Sharpe raio, calculaed over his disribuion. The inermediae values reached by he variae under sudy was no he maer. Wihin he new conex ha has emerged since he iniial end dae assigned o CADES was wihdrawn i.e., he iniial end dae se forh in he decree of December 19, 1997 for he social securiy financing ac for 1998 we sop he dynamic of he annual financing balance process in he period where he ne financial posiion becomes posiive. For a given deb srucure, we calculae he expeced annual balance for each drawing, hereby compuing is simulaed disribuion. Nex, we need o adjus his gross performance for risk, so ha we can ge an idea of he qualiy of he disribuion. Indeed, an effecive sraegy mus boh opimize he expeced annual balance and reduce he previously defined risk, o which we are averse. We use he following crierion S k S This crierion rewards sraegies ha opimize he expeced annual balance and penalizes hose ha allow a large dispersion. According o he srong law of large numbers, if we posulae ha he disribuion of he expeced annual balance is Gaussian, 5% of he disribuion sands ouside he radius inerval 1.96 S cenered in S. We use his value for k, which is ypically rounded off o Solving mehod Using arge porfolios, we bring up he deb porfolio srucures ha urn ou o be opimal under he vas majoriy of scenarios, assuming ha a given porfolio is reallocaed sysemaically according o he weighs of is iniial srucure wih each refinancing or buyback ransacion. By doing his, we follow he mehod ha was popularized in paricular by Black and Perold (1992) [9], called CPPI (Consan Proporion Porfolio Insurance). We use he Mone Carlo mehod o simulae scenarios and numerically consruc he disribuion of he resuls for each deb srucure under he aforemenioned mehod of reallocaion. For his, we discreize he asse and liabiliy dynamics following he Euler diagram, re-balancing each porfolio wih is saring proporions. Our opimizaion process enails, firs of all, choosing he accepable level of risk based on he simulaed resuls profiles for he various eligible porfolios, including he curren deb porfolio. The variae presened above, i.e. he annual amorizaion capaciy, is a good es saisic. I allows us o build he following decision rule: - if he annual amorizaion capaciy is greaer han he fracile T, we accep he sraegy, 12/22

13 - if he annual amorizaion capaciy is less han he fracile T, we rejec he sraegy. The probabiliy level (%) represens he risk of wrongly deciding ha he sraegy under consideraion is admissible. Accordingly, we will decide ha a deb sraegy wih an expeced amorizaion capaciy ha is sufficien o achieve full amorizaion of he deb before horizon H, wih he risk of making an error ; his risk is known as he «firs-order» risk. If such an even comes o realizaion, he consequence of acually underesimaing he amorizaion duraion is an addiional ax levied on he axpayer. The convexiy of he risk profile accenuaes he graviy of his consequence, because he lower he level of he capaciy, he greaer he magniude of he amorizaion period increase, for he same decrease in amorizaion capaciy. Conversely, if he amorizaion period is overesimaed, he consequence is less serious for he axpayer, since he ax levied on he laer will be less han iniially planned. Below, we represen he profiles (or disribuions) of he resuls for wo porfolios. Each profile can be divided ino sraa based on he cumulaive frequency. The sraa are hus delimied by he levels of cumulaive probabiliy: 0%; 5%; 10%, ec. This cumulaive probabiliy is precisely he noion of risk ha was defined above. Curren deb compared o some "opimal" porfolio Risk profiles (*) 6,0 5,5 (*)risk defined as he probabiliy of failing o reach a hreshold in erms of amorizaion capaciy 95% 95% Some opimal porfolio CADES Deb Annual amorizing capaciy (bn EUR) 5,0 4,5 4,0 75% 75% 50% 50% 30% 30% 10% 10% 3,5 5% 5% 3, Redempion Horizon (yrs) The graph shows, for example, ha he porfolio whose profile is represened by he blue curve has a 5% probabiliy ha is expeced annual balance does no exceed 3.5 billion euros, and he same probabiliy of being unable o reimburse wihin 21 years. 13/22

14 5. Decision suppor ools 5.1. Efficiency froniers We hen compare all he resul disribuions a he risk level. By synhesizing hem in a plan represenaion whose axes are a performance measuremen and he fracile of he disribuion of he amorizaion capaciy a (%) risk level, we derive an efficiency fronier. This fracile will be noed as CAaR( %). We have he abiliy o simulae muliple combinaions of feasible srucures, depending on wheher we are sudying he effec of moving of he inflaion indexed weighing, or he floaing rae weighing, or ha of a mauriy exension on borrowings. We hereby derive several porfolio froniers, a selecion of which is shown hereunder. Efficiency froniers in he Expeced Amorizaion Capaciy / CAaR 5% plan 100% Floaing Rae Froniier wih 0% Inflaion Indexed 3,0 3,1 3,2 Increasing Risk Fronier wih 0% Floaing Rae 100% Inflaion Indexed 70% Fixed Cpon 30% Floaing 30% Fixed Cpon 70% Inflaion Indexed 90% Fixed Cpon 10% Floaing Increasing Performance 90% Fixed Cpon 10% Inflaion Indexed 4,0 4,3 4,4 4,5 4,6 4,7 4,8 Expeced Amorizaion Capaciy (bn EUR) 3,3 3,4 3,5 3,6 3,7 3,8 3,9 Amorizaion Capaciy a 5% risk 5.2. Performance and risk measuremens The performance and risk measuremens ha we have oped o use are boh cusomary and criical for deermining an opimum. Looking a he graph in he preceding paragraph, he porfolio achieving he opimum would radiionally be ha locaed in he lowes par of he graph, and furhes o he righ. However, his is no necessarily he porfolio ha maximizes he opimaliy crierion we presened in he previous secion. Firs, he fracile CAaR( %) belongs o he simulaed disribuion of he annual amorizaion capaciy. This may presen disorions compared wih a Gaussian disribuion. Furhermore, he proposed crierion brings ino play he sandard deviaion, square roo of he second momen of he disribuion. We know ha i does no accoun for such disorions. A he same ime, he opimaliy crierion presened offers he advanage of consiuing a scale of measuremen ha will allow us o rank he various porfolios. 14/22

15 5.3. Analysis of risk Amorizaion profiles and dominance We perform risk analysis a he porfolio level. Indeed, each porfolio can be characerized by amorizaion profiles ha are comparable for an equal level of risk. An amorizaion profile is simply he amorizaion pah of he ousanding deb for a given porfolio srucure, corresponding o he quanile of annual amorizaion capaciy of risk (%). Below, we represen he amorizaion profiles of he curren deb porfolio, a differen levels of risk. Deb Amorizaion according o risk(*) 20 (*) risk = probabiliy of failing o reach a hreshold in erms of reimbursemen 0 Ne Deb (bn EUR) % Risk 50% Risk 5% Risk >= 30 Redempion horizon (yrs) A profile will be said o be beer han anoher for he same level of risk, if i crosses he X-axis corresponding o he null value of ne deb sooner. The convenional condiions of sochasic dominance of he firs and second order will be ranslaed as follows: - a porfolio dominaes anoher, in he sense of «firs order dominance», if is profiles are beer a every level of risk, - a porfolio dominaes anoher, in he sense of «second order dominance», if is profiles are beer a risk levels ha are lower han or equal o 50 % Risk region We can hen answer he following quesion: wha are he risky scenarios for a porfolio? Indeed, once we know he (%) risk region of he expeced annual amorizaion capaciy, we are able o infer, for each of he facors ha rule our economy, he regions ha represen he same cumulaive risk percenage (%). Accordingly, we can idenify he risky combinaions of facors for each porfolio and represen, for example, he median pah in he a-risk zone. In he graph ha follows, we show 15/22

16 he median pahs, wihin he region bearing 5% of cumulaive risk, of he shor-erm ineres rae, he inflaion rae and he rae of CRDS volume growh, for CADES porfolio a some simulaion dae. The risky configuraions of facors differ from one porfolio o anoher. Going from one deb srucure o anoher means moving in he described universe of risks. Our simulaions show ha scenarios feauring a sharp rise in nominal shor-erm raes are no ino he region of 5% risk level for one of he recommended opimal porfolios. These have a very low level of exposure o shor-erm ineres raes, wih a near o null weighing in he floaing raes deb class Performance analysis Saic analysis The opimaliy crierion consiues a scale of values wih which we can rank he porfolios in decreasing order. Having sudied a muliude of combinaions of possible srucures, we have consruced hree froniers: - one wih increasing weighings of floaing rae deb a he expense of fixed coupon deb, wih zero exposure o inflaion, - one wih increasing weighings of inflaion-indexed deb a he expense of fixed coupon deb, wih no floaing rae allocaion - one composed of wo halves : (i) one half made of porfolios aiming a arge spliing of deb beween fixed coupon and inflaion-indexed classes, and (ii) he oher half being a varian allocaing 10 % o he floaing rae class par a he expense of he fixed coupon class, boh halves bearing porfolios differing in erms of he disribuion of mauriies. 16/22

17 These froniers allow o sequenially evaluae a arge spliing beween fixed and indexed deb. This spliing needs o be reassessed on a regular basis, and a he very leas we need o es is sabiliy over a sufficienly long period of esimaion. Currenly, his evaluaion is done monhly. Saring from his las fronier, we build a small number of sub-ses of porfolios combining he mauriy breakdowns ha appear o be he bes, and use as a gearing he exposure of he deb o inflaion around he arge. We hen evaluae hem a weekly inervals. Wihin hese sub-froniers, we ry o idenify a cluser formed of he bes porfolios, which we es for sabiliy. In he same process, we assess he curren deb porfolio, and can measure he performance and risk profile gains ha would be procured if i were ransformed ino one of he porfolios from he cluser. A he ime of simulaion, our analyses show ha he curren deb srucure is dominaed by hese porfolios Dynamic analysis When one compares he curren deb from one dae of calculaion o anoher, i is useful o separae hree effecs. The deb porfolio is composed of securiies ha are subjec o marke variaions. The parameers of he model, calibraed upon marke prices and yields, change along wih hem. Accordingly, he simulaions are subjec o his effec from one evaluaion ime o he oher. Nex, i is necessary o separae he effec of a change in he nominal ousanding, from ha of achieved ransacions, i.e. a change of srucure, which mus be measured on a consan ousanding deb basis. This means we need o assess (i) he curren deb porfolio wih he new ousanding under he srucure ha prevailed before he ransacions were compleed i.e., as if we had mainained he deb srucure ha prevailed on he dae of he previous assessmen, and (ii) his same curren deb, bu afer compleion of he ransacions. The gap in performance measuremens beween wo assessmen daes, for he former deb on a consan srucure basis, wih he same ousanding, represens he effecs of ime, marke flucuaions and, where applicable, new daa (inflaion index, CRDS revenue inflow observed). For he same evaluaion dae, he gap beween he measuremens made while keeping consan he former srucure, wih differen nominal ousanding, represens he effec of he change in he nominal ousanding. Finally, for he same evaluaion dae, he gap in measuremens made on he deb wih he new nominal ousanding, beween he old and he new srucure, represens he effec of ransacions. Accordingly, we have simulaed on December 29, 2005 wha CADES deb porfolio would have become if, during he assumpion of 50 billion euros of addiional deb, he 17/22

18 porfolio had been buil along he same srucure as ha of March 7, We compared his evaluaion wih ha of CADES curren deb on he daes of Sepember 30, 2004 (he addiional deb assumpion being provisioned) and December 29, The resuls are presened in condensed form in he following able, via he risk-adjused performance described in sub-secion 4.6. on on wih srucure as of on wih curren srucure 80 bn EUR Table 1- RISK-ADJUSTED PERFORMANCE OF CURRENT DEBT On December 29, 2005, our porfolio afer he assumpion of an addiional 50 billion euros of healh insurance deb package showed a performance surplus of 0.53, according o our risk-adjused performance crierion. 5.5.Measuremen of financial performance: Inernal Rae of Reurn We also presen a new measuremen, raher an acuarial han a saisical one, which allows us o ranslae he performance of a deb porfolio in oher erms, based on he following analysis. For a given porfolio, a pah is a sequence of flows made up of annual balances, calculaed on a curren euro basis, which allowed o achieve (or no achieve) he reimbursemen of an iniial deb. For each pah, one can deermine he discoun rae for which he sum of he discouned annual balances is equal o he presen value of his deb. When we have compleed a se of simulaions, each porfolio can be characerized by a disribuion of inernal raes of reurn or IRR. This acuarial measuremen bears a plain financial meaning, since i simulaneously akes ino accoun reimbursemen capaciy and reimbursemen speed. The greaer he annual balances, he earlier he deb is amorized, and he beer he IRR of he porfolio. This measuremen allows us o draw a parallel beween he agency and an agen who assesses an invesmen projec. The agen knows he oal presen cos of he projec and esimaes he expeced fuure ne revenue flows. This allows our agen o assess he rae of reurn of he projec. In similar fashion, CADES runs a deb projec. I knows he presen value of ha deb and esimaes is fuure amorizaion flows, in order o assess he IRR Insrumens of analysis Our porfolio racking is wofold : 18/22

19 - we assess he curren deb porfolio by aking differen snapshos of i over ime. As we have seen in sub-secion 5.4, his allows us o compare is presen evaluaion wih wha i would have been if no ransacions had been compleed since he las snapsho, - we assess all he arge porfolios, and he disance ha separaes our curren deb from he bes porfolios. This allows us o esimae, over ime, how much of he performance is due o he increase (or decrease) of he proximiy vis-à-vis a arge srucure. In erms of deb managemen, we face wo complemenary direcions of acion : - from a risk perspecive, saring from he exising porfolio and if we consider i wise o immunize ourselves from he risk zone as analyzed by he model, we can consider hedging ransacions ha will proec us agains he risky values of marke facors, - from an opimizaion perspecive, we will seek o ge closer o he cluser of arge porfolios. The direcion in which we decide o move will depend on he prioriy we have se in erms of he risk/reurn radeoff and, of course, on he marke environmen. 6. Asse and liabiliy managemen, and he model 6.1. Modeling assumpions and parameer values The fixed coupon deb class should benefi from posiive or cyclical inflaion, i.e., posiively correlaed o growh as well as o nominal raes. The resuls of he model will depend, a a firs order, on he g reel spread, wih reel designaing he deflaed nominal rae. More generally, if he assumpions of he model yield he generaion of deflaed raes ruling a given deb class, such ha he spread described hereabove is posiive for he greaes par of he borrowing phase, his deb class will be one of he bes candidaes, paricularly in erms of he expeced reimbursemen capaciy. The long-erm rends, he evenual backward forces and, above all, he volailiies of facors, are herefore crucial for performance and risk measuremen, as well as for he resuls of he opimizaion. A a second order, he resuls will depend on correlaions beween risk facors Reallocaion rules and opimaliy of resuling porfolios Wheher we use a crierion like he uiliy of erminal wealh or one like he uiliy of iner-emporal wealh, he resoluion of he opimizaion canno be differen: oherwise, he agency would have wo soluions o achieve a single objecive, one being necessarily less 19/22

20 opimal han he oher. The same by-absurd reasoning can be used if we compare our mehod of opimizaion, which may be described as empirical, and he analyic resoluion mehod of he program formalized in secion 4, wih he help of opimal conrol ools. The refinancing/re-invesmen sraegy adoped (of he CPPI ype) leads us o inegrae ino he opimizaion program an a priori consrain. Such a consrain may be accepable if i corresponds o a rule governing he operaing process of he agency, or if i models is raionaliy in he fuure. For example, in he case of life insurance companies, rules for harvesing capial gains are inegraed ino he reinvesmen sraegy. In our case, his sraegy apparenly leads o a behavior ha is no always opimal Model risk: some responses Unavoidably, he resuls are he produc of he model s fundamenal assumpions, and of he iniial condiions. To proec ourselves from model risk, we have recourse o he following acions: - performing simulaions of he model on conrasing marke configuraions o he exen permied by he daa, - running regular simulaions, - implemening sensiiviy ess o shocks on one or he oher of our fundamenal assumpions, in order o conrol he response of he model, - developing alernaive models 6.4. Conemplaed changes We have buil oher models, in paricular a Vecorial Auoregressive Model wihin a purely economeric framework. We have also developed a quaniaive model for he indexed deb, hanks o he work done by I. Toder (2004)[12], based primarily on he aricle wrien by Jarrow and Yildirim (2002)[10]. This model, wrien under he risk neural probabiliy measure, is sill o be inegraed wihin he modeling of our economy, which is under he real probabiliy measure. Finally, we sill need o solve dynamically he program defined in secion 4 in order o find an opimal porfolio using opimal sochasic conrol echniques Reflecions on deb managemen As he schemaic represenaion of our balance shee demonsraes, we are mainly «asse sensiive». I is possible for us o model our asses in a relaively sraighforward way, over a sufficienly long horizon. This exercise is far more difficul for a number of oher insiuions and companies, eiher wih respec o modeling a porion of heir balance shee, or 20/22

21 because of heir weaker abiliy o make long projecions due o a fairly shor business cycle (2-5 years). We can make full use of our asse and liabiliy managemen capabiliy o monior deb allocaion over a long ime frame. If we compare ourselves o insiuional invesors such as life insurance companies ha sell annuiies, our balance shee is in some respecs a mirror image of heirs. Indeed, hey inves in asses offering a reurn ha is a leas adequae o cover heir policyholder reiremen annuiy liabiliies, which are srongly indexed o wages. Whereas we issue inflaion-linked bonds, among oher insrumens, and our asse grows roughly in line wih wages. Our respecive allocaion processes seek o opimize a similar objecive, of an opposie sign. 7. Conclusion The nominal rae equal o our deb cos is relaed o is susainable naure. The deb is susainable as long as he oal amoun is a mos equal o he sum of he discouned value of expeced revenues. If he nominal rae exceeds an equilibrium value, he deb is no longer susainable. Accordingly, gaining conrol over he expeced fuure coss and he variabiliy of he amorizaion is a he hear of deb managemen. Presenly, i is up o his agency o esimae he dae on which he deb will be fully reimbursed. This is one of he resuls ha we assess wih measured cauiousness. As we saw in sub-secion 4.7, he risk of underesimaion in economerics can be seen as a «firs-order» risk, and is far more serious han he risk of overesimaion. If his risk were o occur, i would lead o an addiional cos o which CADES is, by definiion, averse. 21/22

22 REFERENCES [1] M.J.Brennan, Y.Xia (Juin 2002), «Dynamic asse allocaion under inflaion», Journal of Finance [2] Harry Markowiz (1959), «Porfolio selecion : efficien diversificaion of invesmen», New York, Wiley [3] John Y.Campbell, Luis M.Viceira (1999), «Who should buy long erm bonds», Working Paper, Harvard Universiy [4] M.Grasselli (Oc-Nov 2004), «Recen mehods of pension funds managemen», Banque e Marchés [5] A.J.G.Cairns (1998), «Some noes on he dynamics and opimal conrol of sochasic pension fund models in coninuous ime», preprin [6] L.E.O.Svensson, I.M.Werner (1993), «Non-raded asses in incomplee markes», European Economic Review [7] H.K.Koo (1998), «Consumpion and porfolio selecion wih labour income : a coninuous ime approach», Mahemaical Finance [8] 0.Vasicek (1977), «An equilibrium characerizaion of he erm srucure», Journal of Financial Economics [9] F.Black, A.Perold (1992), «Theory of Consan Proporions Porfolio Insurance», Journal of Economic Dynamics and Conrol [10] R.Jarrow, Y.Yildirim (2002), «Pricing reasury inflaion proeced securiies and relaed derivaives using an HJM model», Working Paper, Cornell Universiy [11] Rober C.Meron (1971), «Opimal consumpion and porfolio rules in a coninuous-ime modem», Journal of Economic Theory [12] I.Toder (Sep 2004), «Modelizaion of inflaion-linked deb», Memorandum of DEA «Probabiliés e Finances» Pierre e Marie Curie PARIS VI Universiy, CADES 22/22