Debt Portfolio Optimization Eric Ralaimiadana

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1 Deb Porfolio Opimizaion Eric Ralaimiadana 27 May 2016 Inroducion CADES remi consiss in he defeasance of he Social Securiy and Healh public deb To achieve is ask, he insiuion has been assigned a number of resources, based on axaion The firs sep of he «defeasance» aciviy is o borrow money from capial markes and pass i on o he Social Securiy accoun o clear accumulaed deficis The second sep is o repay deb : providing bondholders wih coupon paymens and reurning heir capial in he end The purpose of he paper is o show how from he asse and liabiliy managemen sandpoin we can help design boh sraegies of bond issuance and repaymen 27 May

2 Deb ransfer mechanism 27 May Resource breakdown 13.1% 1.1% 16.3% 52.4% 17.1% Earned income Subsiuion income Capial income FRR Oher 27 May

3 Modelling he economy The iniial value of he conribuion revenue once assessed, is evoluion can be modelled by a real growh rae, and a rae of inflaion. e g, real growh rae, inf laion rae and k he value growh rae in coninuous - ime is he sum g The economy is also driven by he shor-erm ineres rae. We assume is dynamic follows a Ornsein-Uhlenbeck process, along wih he real growh rae and he inflaion rae. We specify a Vasicek formulaion of he zero-coupon rae curve. For insance he shor-erm ineres rae evoluion has he following differenial equaion : dr ab r d r dwr Each of he hree variables conains a source of risk modelled by a Wiener process ; he hree Brownian moions are linked one-anoher by heir cross-correlaions. Afer some ransformaion one ges a sysem of hree equaions wih uncorrelaed sources of risk. For insance he inflaion rae dynamic will be modelled by he following equaion: 2 d cd d r, dwr 1 r, dz 27 May Modelling he balance-shee The asse A has he following dynamic da A k d Knowing a given liabiliy amoun, We compue he disposable as he resources/expenses balance. From here, he ne balance is aained by subracing payables. When posiive we can furher reduce he exising deb wih buy-backs, oherwise a financing requiremen is me wih an issuance program. The value of he ne deb annual variaion will flucuae wih he cheap/dearness of he raes environmen. Our goal is o opimize he yearly ne deb reducion, or amorizaion. We describe he opimizaion problem in boh saic and dynamic frameworks 27 May

4 Opimizaion in a saic framework Along wih Marellini and Milhau (2009), le us define G he asse-liabiliy margin and Q he funding raio, wih A G and A Q The insiuion s goal is o maximize he margin uiliy expecaion a a given horizon T, E[u(G T )], upon he deb porfolio allocaion vecor X, The opimal soluion found by he auhors is hen (in he absence of a riskless ineres rae) X wih 1 e b Q0 1 eb C 1 Q e 0 b 1 Q0 e b 1 Q 1 C e b e b One recognizes he hree funds from he separaion heorem : he minimum variance porfolio, he angen porfolio, and he asse hedge porfolio May Opimizaion in a dynamic framework Opimizing he asse-liabiliy margin can read u max E GT G 0 X The opimizaion program can be wrien as T s 0k s ds max E u A0e X G0 X 0 Under he following consrains wih he ime price index, k he nominal revenue rae of growh obained by compounding he rae of growh in real erms g(.) and he inflaion rae (.), r he money marke rae, 27 May

5 Opimizaion in a dynamic framework (cd) he emporal variaion of he cos of deb is expressed in vecor form as X. For if we name he deb porion comprising nominal bonds, he one wih inflaion-linked noes, he las one wih noes running a he money marke rae, hen he column vecor. reads 1 2 R d d rd 3 d where R() is he cos of he nominal bond 27 May Oupus 100% index-linked deb 100% floaing rae deb Performance increase CADES deb december 2015 Risk improvemen 100% fixed rae deb Var5% of Repaymen Capaciy (wih 5% chances of saying underneah) Expeced Repaymen Capaciy (bn Eur) 27 May

6 Roues of enhancemen Bring he deb ransfer phenomenon ino he model, i.e. change is saus from exogenous o endogenous Change he emporal sep of he model from a yearly frequency o an infra-annual frequency (quarerly e.g.) Enhance he deb porfolio represenaion wih a much deailed approach of he shor-erm noes porfolio 27 May Deb ransfer Deb ransfer is similar o a jump in he process of deb amorizaion, oherwise a raher smooh process. Usually, wo hings maer : ime (frequency) and magniude Observed facs each us ha deficis accumulaion looks like a recurring process. This process bears no obviously perceivable consan frequency, and no consan magniude eiher he magniude meaning he size of he cumulaive deficis a he ime public policy decides i mus be sopped and ransferred o a defeasance agency. 27 May

7 Deb ransfer : exisence of a hreshold? Deb ransfer risk : waiing ime and hreshold 1,2 1,5% 1 1,0% 0,8 0,5% 0,0% 0,6 Social Securiy accoun accumulaed ne balance -0,5% 0,4-1,0% 0,2-1,5% ,0% Transfer indicaor Cumulaive Weigh relaive o GDP 27 May Deb ransfer : aemp o modelize he public policy decision rule The governing body in charge of social affairs budge facing a new defici on a given year has a choice beween : le he deficis build up, and have he Social Securiy reasurer finance he defici by shor-erm borrowing clear he new defici by ax levy, or compensae by a cu in spending somewhere else urn he accumulaed deficis ino a new deb and ransfer i o he defeasance agency 27 May

8 Deb ransfer : aemp o modelize he public policy decision rule 27 May Deb ransfer : aemp o modelize he public policy decision rule 27 May

9 Deb ransfer : he jump opion e a ime 2,S 2 be he new defici, C 2 he global cos of public choice regarding deficis as of ime 2, p and q he marginal probabiliy values for deciding o pospone deb ransfer or clear S 2 defici by ax levy, respecively, We can wrie he price of he risk of a deb jump seen from ime 1, as he cos share associaed wih soluion n 3 R 1 1 r 1 1 p qecoss 2 D3 2 E E 1 C 27 May Deb modelizaion A general formulaion of he deb process sochasic equaion could hen read d H dj where represens he amorizaion process H is he magniude of he jump J he jump process, defined by dj 1 p qd 27 May

10 Quesions Answers Thank you 27 May Quesions Commens The views expressed in his [publicaion/presenaion] are hose of invied conribuors and no necessarily hose of he IFoA. The IFoA do no endorse any of he views saed, nor any claims or represenaions made in his [publicaion/presenaion] and accep no responsibiliy or liabiliy o any person for loss or damage suffered as a consequence of heir placing reliance upon any view, claim or represenaion made in his [publicaion/presenaion]. The informaion and expressions of opinion conained in his publicaion are no inended o be a comprehensive sudy, nor o provide acuarial advice or advice of any naure and should no be reaed as a subsiue for specific advice concerning individual siuaions. On no accoun may any par of his [publicaion/presenaion] be reproduced wihou he wrien permission of he IFoA [or auhors, in he case of non-ifoa research]. 27 May

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Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion