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Asset-Liability Management Decisions in Private Banking

February 2007

An EDHEC Risk and Asset Management Research Centre Publication

Sponsored by

2 EDHEC RISK AND ASSET MANAGEMENT RESEARCH CENTRE

Published in France, March 2007. Copyright© EDHEC 2007The ideas and opinions expressed in this paper are the sole responsibility of the authors.

About the Authors ................................................................................................................................................4 Foreword..................................................................................................................................................................5

Executive Summary .....................................................................................................................................8

Résumé.........................................................................................................................................................11 1. Introduction .....................................................................................................................................................16

2. Asset-Liability Management as a Truly Client-Driven Approach to Private Banking ..................18 2.1. Sources of Added-Value in Wealth Management ............................................................................................................18 2.2. A Typology of Clients’ Profiles ..................................................................................................................................................18

3. A Brief History of ALM Techniques ...........................................................................................................20 3.1. Cash-Flow Matching and Immunization .............................................................................................................................20 3.2. Surplus Optimization ....................................................................................................................................................................21 3.3. LDI Solutions ....................................................................................................................................................................................22 3.3.1.StaticLDISolutions................................................................................................................................................................................................................23 3.3.2.DynamicLDISolutions.........................................................................................................................................................................................................23

3.4. Overview ............................................................................................................................................................................................24

4. Illustrations of the Usefulness of an ALM Approach to PWM ...........................................................25 4.1. Pension-Related Objective .........................................................................................................................................................26 4.1.1.Cash-FlowMatchingStrategy..........................................................................................................................................................................................26 4.1.2.SurplusOptimizationStrategies.......................................................................................................................................................................................26 4.1.3.DynamicLDIStrategies........................................................................................................................................................................................................28 4.1.4.AVariant.....................................................................................................................................................................................................................................30

4.2. Expenditure-Related Objective: the Case of Real Estate ..............................................................................................32 4.3. Bequest-Related Objective .........................................................................................................................................................33 4.3.1.TheBaseCase...........................................................................................................................................................................................................................33 4.3.2.IntroducingConstraints.......................................................................................................................................................................................................34 4.3.3.AVariantwithSignificantLump-SumPaymentsExpected.................................................................................................................................34

5. Conclusion........................................................................................................................................................37

6. Mathematical Appendix ...............................................................................................................................38 6.1. Stochastic Model for the Value of Asset and Liabilities ................................................................................................38

6.2. Objective and Investment Policy .............................................................................................................................................39

6.3. Solution using the Dynamic Programming Approach ...................................................................................................40 6.3.1.GeneralSolution.....................................................................................................................................................................................................................40

6.4. From Static to Dynamic Portfolio Management ...............................................................................................................41

References ............................................................................................................................................................43

About the EDHEC Risk and Asset Management Research Centre ..........................................................45

About Pictet & Cie ..............................................................................................................................................47

Table of Contents

Asset-Liability Management Decisions in Private Banking 3

Noël Amenc PhD isProfessorofFinanceandDirectorofResearchandDevelopmentattheEDHECGraduateSchoolofBusiness,whereheheadstheRiskandAssetManagementResearchCentre.HehasaMastersinEconomicsandaPhDinFinanceandhasconductedactiveresearchinthefieldsofquantitativeequitymanagement,portfolioperformanceanalysis and active asset allocation, resulting in numerous academic and practitionerarticlesandbooks.HeisanAssociateEditoroftheJournal of Alternative Investments andamemberofthescientificadvisorycounciloftheAMF(Frenchfinancialregulatoryauthority).

Lionel Martellini PhD is a Professorof Finance at the EDHECGraduateSchool ofBusinessandtheScientificDirectoroftheEDHECRiskandAssetManagementResearchCentre.HeholdsgraduatedegreesinEconomics,StatisticsandMathematics,aswellasaPhDinFinancefromtheUniversityofCaliforniaatBerkeley.LionelisamemberoftheeditorialboardoftheJournal of Portfolio ManagementandtheJournal of Alternative Investments.Anexpertinquantitativeassetmanagementandderivativesvaluation,Lionelhaspublishedwidelyinacademicandpractitionerjournals,andhasco-authored reference textbooks on Alternative Investment Strategies and Fixed-IncomeSecurities.

Volker Ziemann isaResearchEngineerattheEDHECRiskandAssetManagementResearchCentre.HeholdsaMaster’sDegreeinEconomicsfromHumboldt-UniversityinBerlinandaMaster’sDegreeinStatisticsfromENSAEinParis,andiscurrentlyaPhDstudentinfinanceattheUniversityofAix-en-Provence.


About the Authors

High net worth individuals (HNWIs) havenumerous characteristics, in terms of assetsunder management and the sophisticationof their requirements, that they share withinstitutionalinvestors.Thisisafactthathaslongbeen recognisedby themarketingdepartmentsof asset management companies and privatebanks,who typicallyhave special considerationfor these profiles in their marketing and salessegmentation. We can therefore consider, withstrong justification, that a similar approachwould be appropriate for the investmentmanagement techniques employed for HNWIsandinstitutionalinvestors.Thisisthelogicthatwe have applied in the present research paper,which is drawn from EDHEC’s ALM and AssetManagementresearchprogramme.

This programme aims to apply recent researchin asset-liability management for institutionalinvestors and to improve asset managementtechniques,andinparticularstrategicallocationtools, to positively impact the performance ofALM programmes. Recent EDHEC publicationsin this field include ‘Assessing the Impacts ofIFRS and Solvency II Rules on the FinancialManagementofEuropeanInsuranceCompanies,’a major study which was jointly produced bythe EDHEC Financial Analysis and AccountingResearchCentreand theEDHECRiskandAssetManagement Research Centre; an academicanalysis of Liability-Driven Investing by LionelMartellini,‘TheTheoryofLDI,’whichwaspublishedintheMay2006issueofLife and Pensions;andapaperentitled‘TheBenefitsofHedgeFundsinAssetLiabilityManagement’,byLionelMartelliniand Volker Ziemann, which appeared in theAlternative Investment Quarterlyin2005.

The current paper discusses the sources ofadded-value in private wealth management,and argues through a series of illustrationsthat asset-liability management is the naturalapproach for the design of truly client-drivenservicesinprivatebanking.

Wewouldliketoextendoursincerethankstoourpartners atPictet&Cie,who lent considerablesupporttothisproject.Wehopeyouwillfindthestudybothinterestingandinformative.

NoëlAmenc,PhD,Director of the EDHEC Risk and Asset Management Research Centre

Foreword

Asset-Liability Management Decisions in Private Banking �

Executive Summary


The private wealth management industry hasnowbecomea very significant industrydue tocontinuing strong economic growth in specificregions of the world. This increase is currentlydriving a larger wealth management marketcreating greater opportunities for wealthadvisorstoleveragenewtechnologywithaviewto acquiring new clients and boosting profits.Asaresult,competitionamongwealthadvisoryfirms is increasing to find ways to improveexisting client relationships and provide newtools to improve advisor efficiency. Currentprivatebankingtoolsaretypicallytaxandestateplanning geared towards one specific countryand financial simulation software, relying onsingle period mean-variance optimization ofthe asset portfolio. These tools suffer fromsignificant limitations and cannot satisfy theneedsofasophisticatedclientele.

While some industry players have recentlydeveloped planning tools that model assets ina multi-period stochastic framework, asset-liabilitymatchingforindividualsremainsanareaforexploration.ThispaperadaptsAsset-LiabilityManagement (ALM) techniques developed forinstitutional investors tothecontextofprivatebankingcustomers.Asset-LiabilityManagement(ALM) denotes the adaptation of the portfoliomanagement process in order to handle thepresence of various constraints relating to thecommitments of an investor’s liabilities. Weargue that portfolio optimization techniquesused by institutional investors, e.g., pensionfunds, could usefully be transposed to thecontextofprivatewealthmanagementbecausethey have been engineered precisely to allowfor the incorporation of an investor’s specificconstraints, objectives and horizon in theportfolioconstructionprocess.Taking investor’sliability constraints and specific objectives intoaccountactuallyhasadramaticimpactonassetallocation decisions. For example, clients whowish to maintain a given level of expenses fortheirretirementyearswillexpecttheinvestmentprocessperformedontheircurrentwealthtobeable to generate cash-flows sufficient to meettheirconsumptionneeds,whichjustifiesafocus

oninflationhedgingthatisnottypicallyinvolvedinastandardassetmanagementsolution.

As an illustration, we consider the situation ofan investor who wishes to invest fixed annualcontributions(€x)forafutureexpenditure,e.g.,the purchase of a house in 5 years, for whichthe current value is normalized at €100. Weintroduceanexplicitmodelforthedynamicsofreal estate prices and the exhibit below showsthe impact of real estate price uncertainty onthe value of the €100 payment scheduled tobepaidin5yearsfromnow.Aswecansee,realestate price risk is significant, with a nominalamount to be secured equal to €156.59 onaverageanda€27.18standarddeviation.

In practical terms, the goal is to generate alump sum payment at horizon date (5 years).It is not possible in general to find a perfectliability-matching portfolio. The existence of aperfect liability-matching portfolio is actuallyonly ensured on the following two conditions:the investor must be able to borrow againstfuture income and invest the present value ofthefuturecontributionsattheinitialdate;andtheremustbeaninvestmentvehicle(e.g.,REITS)with a payoff which is directly related to realestate price uncertainty. We test two differentsituations:anopportunitysetcontainingstocks,bondsandTIPSandanopportunitysetcontainingstocks,bonds,TIPSandrealestate(modelledas

Distribution of house prices at final date; mean value = 156.59;standarddeviation=27.18.

Executive Summary


an investment that will pay the compoundedreturn on real estate). To generate comparableportfolios, we looked at the improvement insurplus volatility for a given level of expectedsurplus.

Thegraphshowstheefficient frontier inbothcases, while risk-return indicators are reportedin the table. As expected, the presence ofassets allowing investors to span real estatepriceuncertaintyprovestobeakeyelementinimproving theefficient frontiersobtained froman ALM perspective. Looking for example atportfolioDandD’ inthetable,weseethatforthesamelevelofexpectedsurplus(12.60inboth

cases),thesurplusvolatilityattheoptimallevelreaches 21.95 when the opportunity set doesnot contain a real estate asset,while itmerelyamounts to 4.25, a dramatic risk reduction,whentherealestateassetisincluded.Againthissignals the relevance of an ALM approach toprivatewealthmanagement:itisonlybytryingto fit the client liability constraints that trulyoptimalsolutionscanbeproposed.

Inthesamevein,wealsoconsideranumberofother illustrations that are typical of standardprivatewealthmanagementproblemsandshowthat optimal solutions are strongly affected bythepresenceofliabilityconstraints.Inparticular,we focus on various pension-related objectivesandconsideranindividualwhoiseitheralreadyretiredorstillemployed,andwhoseekstoensureastreamofinflation-protectedfixedpayments,

based either on a lump-sum contribution or aseriesofannualcontributions.Wealsointroduceavarietyofbequest-relatedobjectives.

In conclusion, we argue that it is not theperformanceofaparticular fundnorthatofagivenassetclass(includingcommoditiesorhedgefunds)thatwillbethedeterminingfactorintheability of private wealth management to meet

Executive Summary

ALMEfficientFrontierswithoutRealEstate (A,B,C,D,E,F)andwithRealEstate(A’,B’,C’,D’,E’,F’)

Portfolio

Weights

StocksBondsTIPSRealEstate

Expectedsurplus

Volatilityofsurplus

Prob(S<0)Expectedshortfall

Necessarynominal

contributionp.a.

Relativecontributionsavingp.a.

A 0% 0% 100% - 2.46 14.64 41.�% 11.40 (12.�%) 19.07 4.7%

B 10% 90% 0% - �.84 13.96 31.�% 10.32 (11.3%) 17.8� 10.8%

C 33% 67% 0% - 9.22 17.16 28.9% 11.03 (12.1%) 16.70 16.�%

D ��% 4�% 0% - 12.60 21.9� 28.�% 12.93 (14.2%) 1�.62 21.9%

E 78% 22% 0% - 1�.98 27.�1 28.0% 1�.92 (17.�%) 14.60 27.0%

F 100% 0% 0% - 19.36 33.46 27.9% 19.09 (21.0%) 13.63 31.8%

A’ 0% 0% 100% 0% 2.46 14.64 41.�% 11.40 (12.�%) 19.07 4.7%

B’ 0% 0% 68% 32% �.84 10.87 28.6% 7.44 (8.2%) 17.8� 10.7%

C’ 0% 0% 37% 63% 9.22 7.27 10.6% 4.�7 (�.0%) 16.70 16.�%

D’ 0% 0% �% 9�% 12.60 4.2� 0.6% 2.07 (2.3%) 1�.61 21.9%

E’ 46% 0% 0% �4% 1�.98 16.43 16.7% 7.36 (8.1%) 14.�9 27.0%

F’ 100% 0% 0% 0% 19.36 33.46 27.9% 19.09 (21.0%) 13.63 31.8%

Allocationstrategiesandrisk-returnindicators;allvaluesaregivenaspresentvaluesatinitialdatebasedona€20annualcontributionfor5years,thepresentvalueofwhichamountsto€90.91givenourchoiceofparametervalues.Expectedshortfallisexpressedasapercentageofthisvalue.Therelativecontributionsavingcorrespondstotheincrease(inpercentage)ininitialinvestmentthatshouldhavetakenplacewithagivenstrategysoastogenerateanexpectedsurplusequaltozero.


investors’expectations.Whatwillprovetobethedecisive factor is the private wealth manager’sability to design an asset allocation solutionthatisafunctionofthekindsofparticularrisksto which the investor is exposed, as opposedto the market as a whole. Hence, an absolutereturnfund,oftenperceivedasanaturalchoicein the context of private wealth management,would not be a satisfactory response to theneedsofaclientfacinglong-terminflationrisk,where the concern is capital preservation inreal, as opposed to nominal, terms. Similarly, aclientwhoseobjectivewouldbe related to theacquisitionofapropertywouldacceptlowandeven negative returns in situations when realestatepricessignificantlydecrease,butwillnotsatisfy himself or herself with relatively highreturnsifsuchhighreturnsarenotsufficienttomeet a dramatic increase in real estate prices.In such circumstances, a long-term investmentinstocksandbondswithaperformanceweaklycorrelatedwith real estatepriceswouldnotbetherightinvestmentsolution.

In other words, the success or failure of thesatisfactionof theclient’s long-termobjectivesisfundamentallydependentonanALMexercisethat aims to determine the proper strategicinter-classes allocation as a function of theclient’s specific objectives and constraints.Assetmanagementshouldonlycomenextasaresponse to the implementation constraints oftheALMdecisions.Ontheonehand,itismeanttodeliver/enhancetheriskandreturnparameterssupportingtheALManalysisforeachassetclass.On the other hand, it can also allow for themanagementofshort-termconstraints,suchascapitalpreservationatagivenconfidencelevel,whicharenotnecessarilytakenintoaccountbyanALMoptimizationexercise,whichbynaturefocusesonlong-termobjectives.

Executive Summary


Résumé


Grâce à une croissance économique soutenuedans plusieurs régions du monde, l’industriede la gestion privée s’est octroyée une placeconsidérabledanslepaysagefinanciermondial.Cetteaccélération sertactuellementdemoteurdans un marché croissant de gestion depatrimoine, créant ainsi la possibilité pour lesconseillersdecedomained’attirerdenouveauxclients et d’augmenter leurs bénéfices. Enconséquence, la concurrence entre les sociétésde conseil en gestion de patrimoine est enconstante progression dans le but de trouverdes moyens d’améliorer les relations clientsexistantesetdeseprocurerdenouveauxoutilsafin d’améliorer l’efficacité de leurs conseils.Les expertises actuelles en gestion privée sonttypiquementcellesdelafiscalitéetdelagestiondeshéritagespropresàunpaysparticulier,ainsique des progiciels de simulation financière,souvent basés sur une optimisation moyenne-varianced’unportefeuilled’actifsdansuncadrestatique. Ces outils souffrent de limitationsimportantesetnepeuventrépondreauxbesoinsd’uneclientèlesophistiquée.

Siquelquesacteursdel’industrieontrécemmentdéveloppédesoutilsprévisionnelsquimodélisentles actifs dans un cadre stochastique multi-périodes, la gestion actif-passif pour lesparticuliers reste un domaine à explorer. Cedocumentadaptelestechniquesdegestionactif-passif (GAP ou ALM en anglais), développéespourlesinvestisseursinstitutionnels,aucontextedes clients privés. L’Asset-Liability Management(ALM) désigne l’adaptation du processus degestion de portefeuille afin de prendre encomptelaprésencedediversescontraintesliéesaux engagementsque représente lepassif d’uninvestisseur. Nous pensons qu’il est intéressantde transférer les techniques d’optimisationde portefeuille utilisées par les investisseursinstitutionnels,parexemplelesfondsdepension,aucontextedelagestionprivée,parcequecelles-ciontprécisémentétéconçuesafindepermettrel’intégrationdescontraintes,desobjectifsetdeshorizons de l’investisseur dans le processus deconstructiondeportefeuille.Enfait, lapriseencomptedescontraintesdepassifetdesobjectifs

précisdel’investisseuraprioriimpactedefaçonsignificative les décisions d’allocation d’actifs.Parexemple,lesclientsquisouhaitentgarderunniveau donné de dépenses durant leurs annéesde retraite s’attendront à ce que le processusd’investissement appliqué à leur patrimoineactuel puisse générer des flux de trésoreriesuffisants pour satisfaire à leurs besoins deconsommation,cequijustifiel’intégrationd’unecouverturepar rapport à l’inflation,quine faitpastypiquementpartied’unesolutiondegestiond’actifsstandard.

Afin d’illustrer ce concept, nous examinons lasituation d’un investisseur qui souhaite allouerdes contributions annuelles fixes (x€) pourune dépense future, par exemple l’achat d’unemaisondans5ans,lavaleuractuelledecelle-ciétantnormaliséeà100€.Nous introduisonsunmodèleexplicitepourladynamiquedesprixdel’immobilier et le graphique ci-dessous montrel’impactdel’incertitudedesprixdel’immobiliersurlavaleurduversementde100€prévupourdans5ans.Commenouspouvonsleconstater,lerisquedeprixdel’immobilierestimportant,avecunevaleurnominalede156,59€enmoyenneàobteniretunécarttypede27,18€.

En termes pratiques, le but est de générer leversementd’unesommeforfaitaire indexéeauxprixde l’immobilierà ladated’horizon (5ans).Il n’est pas toujours possible de trouver unportefeuille parfaitement adossé au passif.

Distribution des prix de maison à la date finale ; valeur moyenne =156,59;écarttype=27,18.


En effet, dans cet exemple l’existence d’unportefeuille parfaitement adossé au passifdépendrait des deux conditions suivantes :l’investisseurpeutempruntersurlabasedessesrevenusfutursetpeutinvestirlavaleuractuellede ses futures contributions à la date initiale ;et il existe un support d’investissement (parexempleREITS)avecunrendementdirectementlié à l’incertitude du prix de l’immobilier. Noustestons deux situations différentes, un exerciced’allocation avec un menu de classes d’actifscontenant des actions, des obligations et desobligationsd’Etatindexéessurl’inflation(TIPS),etunexerciced’allocationavecunmenudeclassesd’actifs contenant des actions, des obligations,des TIPS et de l’immobilier (modélisé commeun investissement qui réalisera le rendementcomposé de l’immobilier). Afin de générer desportefeuilles comparables, nous avons regardél’améliorationdelavolatilitédel’excédentpourunniveaudonnéd’excédentescompté.

Le graphique montre la frontière efficientedanslesdeuxcas,etlesindicateursderisqueetde rendement sont renseignés dans le tableauci-contre. Comme on aurait pu s’y attendre, laprésenced’actifspermettantauxinvestisseursdecouvrir l’incertitudedesprixde l’immobilierestunélémentclédansl’améliorationdesfrontièresefficientesobtenuesdansuneoptiqueALM.Enregardantpar exemple les portefeuillesD etD’dans le tableau, nous constatons que pour unmêmeniveaud’excédent escompté (12,60dans

lesdeuxcas),lavolatilitédel’excédentauniveauoptimalatteint21,95quandl’immobiliernefaitpas partie du menu des classes d’actifs, alorsqu’elleatteint4,25,uneréductionderisquetrèsimportante,quandl’actifimmobilierestcompris.Ceci témoigne à nouveau de la pertinenced’uneapprocheALMdanslagestionprivée:cen’est qu’en essayant de garantir l’adéquationdes contraintes de passif du client que dessolutions véritablement optimales peuvent êtreproposées.

Dans la même lignée, nous développonsplusieurs autres expériences qui sont typiquesdes problématiques de gestion privée et nousmontrons que les solutions optimales sontfortement impactées par la présence descontraintes de passif. Nous nous concentronsnotamment sur différents objectifs liés à laretraite,etnousconsidéronslecasd’unindividuquiestdéjàretraitéoubientoujourssalarié,etqui cherche à garantir un flux de versementsfixes protégés contre l’inflation, à partir soitd’une contribution forfaitaire soit d’une sériede contributions annuelles. Nous introduisonségalementdiversobjectifsrelatifsàdeslegs.

En conclusion, nous avançons l’idée qu’iln’est pas tant la performance d’un fonds enparticuliernimêmed’uneclassed’actifsdonnée(y compris les matières premières ou les hedgefunds) qui sera le facteur déterminant dans lacapacité de la gestion privée à répondre auxattentes des investisseurs. Ce qui sera décisifest lacapacitédugérantprivéàconcevoirunesolution d’allocation d’actifs en fonction desrisquesprécisauxquelsl’investisseur,plutôtquelemarchédanssonensemble,estexposé.Ainsi,un fonds de rendement absolu, souvent perçucomme un choix naturel dans le contexte dela gestion privée, ne fournira pas une réponsesatisfaisante aux besoins d’un client qui doitfaire face à un risque d’inflation sur le longterme, auquel cas le souci sera la préservationducapitalentermesréelsplutôtquenominaux.De même, un client dont l’objectif est lié àl’acquisition d’une propriété accepterait desrendements bas ou même négatifs dans des

Résumé

Frontières efficientes ALM sans immobilier (A, B, C, D, E, F) et avecimmobilier(A’,B’,C’,D’,E’,F’)

situations où les prix de l’immobilier sont ennettediminution,maisnesecontenterapasderendements relativement élevés si ceux-ci nelui permettent pas de faire face à des haussessensiblesdeprixde l’immobilier.Dansde tellescirconstances, un investissement sur le longterme dans des actions et des obligations avecune performance faiblement corrélée avec lesprix de l’immobilier ne serait pas la bonnesolutiond’investissement.

En d’autres termes, la capacité de répondreaux objectifs à long terme du client dépendfondamentalementdel’exerciced’ALMquiviseàdéterminerlabonneallocationstratégiqueentrelesclassesenfonctiondesobjectifsetcontraintesspécifiques du client. La gestion d’actifs doitseulementsuivreenréponseauxcontraintesdemiseenœuvredesdécisionsd’ALM.D’unepart,cela doit permettre d’améliorer des paramètresde risque et de rendement soutenant l’analyseALMpourchaqueclassed’actifs.D’autrepart,leprocessus de gestion d’actifs peut permettre lagestiondecontraintesàcourtterme,tellequelapréservationducapitalàun seuildeconfiancedonné, qui ne sont pas forcément prises encompteparunexerciced’optimisationd’ALM,cederniersefocalisantparnaturesurlesobjectifsàlongterme.

Résumé

Portefeuille

AllocationObligations

d’état indexées sur l’inflation

Actions Obligations (TIPS) Immobilier

Excédent escompté

Volatilité de l’excédent

Prob(S<0)Déficit

escompté

Contribution nominale

nécessaire p.a.

Economie de contribution relative p.a.

A 0% 0% 100% - 2,46 14,64 41,�% 11,40 (12,�%) 19,07 4,7%

B 10% 90% 0% - �,84 13,96 31,�% 10,32 (11,3%) 17,8� 10,8%

C 33% 67% 0% - 9,22 17,16 28,9% 11,03 (12,1%) 16,70 16,�%

D ��% 4�% 0% - 12,60 21,9� 28,�% 12,93 (14,2%) 1�,62 21,9%

E 78% 22% 0% - 1�,98 27,�1 28,0% 1�,92 (17,�%) 14,60 27,0%

F 100% 0% 0% - 19,36 33,46 27,9% 19,09 (21,0%) 13,63 31,8%

A’ 0% 0% 100% 0% 2,46 14,64 41,�% 11,40 (12,�%) 19,07 4,7%

B’ 0% 0% 68% 32% �,84 10,87 28,6% 7,44 (8,2%) 17,8� 10,7%

C’ 0% 0% 37% 63% 9,22 7,27 10,6% 4,�7 (�,0%) 16,70 16,�%

D’ 0% 0% �% 9�% 12,60 4,2� 0,6% 2,07 (2,3%) 1�,61 21,9%

E’ 46% 0% 0% �4% 1�,98 16,43 16,7% 7,36 (8,1%) 14,�9 27,0%

F’ 100% 0% 0% 0% 19,36 33,46 27,9% 19,09 (21,0%) 13,63 31,8%

Stratégiesd’allocationetindicateursderisqueetderendement;touteslesvaleurssontprésentéescommedesvaleursactuellesàladateinitialeàpartird’unecontributionannuellede20€pendant5ans.Lavaleuractuelledecelle-ciestégaleà90,91€étantdonnénotrechoixdevaleursdeparamètres.Ledéficitescomptéestexprimécommeunpourcentagedecettevaleur.L’économiedecontributionrelativecorrespondàl’augmentation(enpourcentage)del’investissementinitialquiauraitdûêtreappliquéeafindegénérerunexcédentescomptéégalàzéro.


Asset-Liability Management Decisions in Private Banking 1�

Asset-Liability Management Decisions

in Private Banking

The private wealth management industry hasnow become a very significant industry due tocontinuing strong economic growth in specificregionsoftheworld.Accordingtoarecentsurvey,thewealthofhighnetworthindividuals(HNWIs),peoplewithnetfinancialassetsofatleastUS$1million excluding their primary residence andconsumables,climbedtoUS$33.3trillionin2005,whichrepresentsanannualrateof8.0%overthelast decade1. According to the same survey, thenumberofHNWIsgrewby6.5percentover2004,to 8.7 million, and the number of Ultra-HNWIs—thosewhohavefinancialassetsofmorethanUS$30 million — grew by 10.2%, to 85,400 in2005.

This increase is currently driving a largerwealth management market creating greateropportunitiesforwealthadvisorstoleveragenewtechnology to acquire new clients and increaseprofits. As a result, competition among wealthadvisoryfirmsisincreasingtofindwaystoimproveexistingclientrelationshipsandprovidenewtoolstoimproveadvisoreffectiveness.Currentprivatebankingtoolsaretypicallytaxandestateplanninggearedtowardsonespecificcountryandfinancialsimulation software, relying on single periodmean-varianceoptimizationofanassetportfolio.These tools suffer from significant limitationsand cannot satisfy the needs of a sophisticatedclientele.

Firstly,singlecountrytaxplanningtoolsareoflittlerelevancetohighnetworthindividualsoperatingoffshore or across multiple tax jurisdictions.Secondly, financial simulation software relyingon single period mean-variance optimization ofasset portfolios cannot yield a proper strategicallocation for at least two reasons. On the onehand,optimizationparameters(especiallyexpectedreturns)aredefinedasconstants,apracticewhichiscontradictedbyempiricalobservationanddoesnotallowforthelengthoftheinvestmenthorizon.Ontheotherhand,andmostimportantlyperhaps,liabilityconstraintsandriskfactorsaffectingthem,such as inflation-risk on targeted spending, areneithermodelednorexplicitlytakenintoaccountintheportfolioconstructionprocess.

The process involved in dealing with a privateclient typically leads to a detailed analysis ofthe client’s objectives, constraints, as well asrisk-aversion parameters (sometimes on thebasisofrathersophisticatedapproaches).Yet ititsstrikingthatoncethis informationhasbeencollected,andsometimesformalized,verylittleisdoneintermsofcustomizingaportfoliosolutiontothebenefitofthespecificneedsoftheclient.Typically, the approach consists in providingseveral profiles expressed in terms of volatilityor drawdown levels, with in some instances adistinctioninhowthecapitalwilleventuallybeaccessed (annuitiesor lumpsumpayment),buttheclient’sobjectives,constraintsandassociatedspecific risk factors are simply not taken intoaccountinthedesignoftheoptimalallocation.

While some industry players have recentlydeveloped planning tools that model assets ina multi-period stochastic framework, asset-liability matching for individuals remains anareaforexploration.Theobjectiveofthispaperis to adapt Asset-Liability Management (ALM)techniquesdevelopedfor institutional investorsto the context of private banking customers.Asset-Liability Management denotes theadaptationoftheportfoliomanagementprocessin order to handle the presence of variousconstraints relating to the commitments thatrepresenttheliabilitiesofaninvestor. Itshouldbeemphasizedatthisstagethatthedefinitionof“liabilities”weuseinwhatfollowsisratherbroadand encompasses any commitment, whetherexternalorself-imposed,thataprivateinvestoris facing. For example, an investor committedto a real estate acquisition will perceive suchan expense as a future commitment for whichmoneyshallbeavailablewhenneeded.Similarly,clientswhodesire tomaintain a given level ofexpenses for their retirement years will expectthe investment process performed on theircurrentwealthtobeabletogeneratesufficientcash-flowstomeettheirneeds.Inwhatfollows,wearguethatportfoliooptimizationtechniquesused by institutional investors, e.g., pensionfunds, could usefully be transposed to thecontextofprivatewealthmanagementbecause

1. Introduction


1-MerrillLynch&CapgeminiWorldWealthReport2006availableatwww.us.capgemini.com/worldwealthreport06

they have been engineered precisely to allowfor the incorporation of an investor’s specificconstraints,objectivesandhorizon(allofwhichcanbebroadlysummarizedintermsof“liabilityconstraints”) in the portfolio constructionprocess.

The rest of the paper is organized as follows.In section 2, we discuss the sources of added-valueinprivatewealthmanagement,andarguethat asset-liability management is the naturalapproach for the design of truly client-drivenservices in private banking. In section 3, weprovideabriefhistoryofALMtechniques,withaspecificemphasisonthebenefitsandweaknessesofcompetingapproaches,bothfromapracticaland a conceptual standpoint. In section 4, wepresentaseriesofillustrationsoftheusefulnessof asset-liability management techniques in aprivate banking context. A conclusion can befound in section 5, while technical details arerelegatedtoadedicatedappendix.

1. Introduction


2.1. Sources of Added-Value in Wealth Management

It has often been argued that the proximityto clients is themain “raisond’être” anda keysource of competitive advantage for privatewealthmanagement.Buildingonthisproximity,private bankers should be ideally placed tobetteraccountfortheirclients’specificliabilityconstraints when engineering an investmentsolutionforthem.Inotherwords,asset-liabilitymanagementisthetruesourceofadded-valueinprivatewealthmanagement.

Mostprivatebankersactuallyimplicitlypromotean ALM approach to wealth management.In particular, they claim to account for theclient’s goals and constraints. The technicaltools involved,however,areoftennon-existentor ill-adapted. As a result, current practice inaddressingclients’needsismostlyafailure,withonly a very limited fraction of private bankersactuallydesigningportfoliosconsistentwiththeclients’needs.Whiletheclientisroutinelyaskedallkindsofquestionsregardingcurrentsituation,goals,preferences,constraints,etc.,theresultingservice and product offering mostly boil downto a rather basic classification in terms of riskprofiles.

In principle, several situations exist,correspondingtovaryinglevelsofsophisticationandconsiderationofclients’needs.

Thefirstcaseiswhenprivatebankerssimplydonot use any portfolio construction tool. Sincethe solutions they then offer do not take intoaccount clients’ objectives, risk-aversion orconstraints, this is simply not acceptable. Aslightlymoresatisfyingsituationinvolvesprivatewealthmanagementperceivedasapureasset-management exercise. The solution consists ofthe optimal design of different portfolios withdifferent risk profiles, where the clients’ goalsand constraints are not taken into account. Athirdsituationinvolvestestingfortheimpactofassetallocationdecisionsintermsofcompliancewith respect to theclients’ liability constraints.

Forexample,someprivatebankersuseamodeltotestprobabilityofashortfallathorizon.Theoptimal asset portfolio, however, is designedindependently of clients’ needs. Finally, thelast, fully satisfactory situation, involves theincorporation of the client’s full profile inportfolio construction. Only this can ensurethat clients’needsareproperlyaddressed.Thisrequires the development of proper portfolioconstruction tools similar to the ones used ininstitutionalmoneymanagement.Explaininghowasset-liability management techniques used inthecontextofinstitutionalmoneymanagementcan/should be transposed to private wealthmanagementispreciselythefocusofthispaper.

2.2. A Typology of Clients’ Profiles

Broadly speaking, there are at least fourdimensionsinaclientprofile.

•Objectiveprofile•Time-horizonprofile•Constraintsandrisk-aversionprofiles•Contributionprofile

Each of these dimensions is related to thedefinition of a client’s liabilities. The firstdimension, the objective profile, is relatedto the particular type of liability a client isfacing. Examples are pensionneeds, real estateacquisition,payingforchildren’seducation,etc.Theseconddimension,thetime-horizonprofile,is of significance since it can be shown that,unlessunderveryspecificassumptions,optimalportfolioallocationsdependontherisk-horizon(see Merton (1971) for a general theory ofdynamicassetallocationdecisions).Itisoftenthecasethattheactualhorizonis long,sometimeswith intermediate, short-term constraints orgoals. The third dimension, the constraints andrisk-aversionprofiles,correspondstoanecessaryenlargement of typical clientele segmentation,whichoftenboilsdowntosubjectiveclassificationintermsofrisktolerance.Abetterunderstandingcan be obtained from the perspective in termsof risk constraints. The fourth dimension, the

2. Asset-Liability Management as a Truly Client-Driven Approach to Private Banking


contribution profile, is as important in privatewealth management as it is in institutionalmoneymanagement.Justasapensionfundwithahealthysponsorcompanyshouldnotholdthesame allocation as another pension fund withthesamefundingstatusbutamorefinanciallyconstrained sponsor company, two clients withthe same objectives but different contributionschedulesshouldnotbeproposedforthesameportfolios. In other words, the risk level in aportfolio allocation does not solely dependon the (private or institutional) investor’s risk-aversion but also on the margin for error (seefor example Martellini (2006b) for an explicitmodel of asset-liability management whereoptimalportfolioallocationexhibits anexplicitdependence in the margin for error measuredintermsofdistancewithrespecttoaminimumfunding ratio requirement). Inparticular, aswewill see, a key distinction exists between fixedandflexiblecontributionschedules.

Inwhatfollows,wepresentanexampleofpossiblesegmentationof aprivatewealthmanagementclientele.1) Objective profiles a/Pension-relatedobjectives •Clientisretiredandhasalready contributed •Clientisnotretiredyetandwill contributeinthefuture b/Expenditure-relatedobjectives • Accumulateforfutureexpenditure basedonasinglecontribution • Accumulateforfutureexpenditure basedonascheduleofcontributions

2) Time-horizon profiles: 0-3 years, 3-5 years, 5-10 years, 10-20 years, 20-40 years

3) Constraints and risk-aversion profiles a/Short-termconstraints:noannual protection,annualcapitalprotectionuptoa 100%,95%,90%,85%confidencelevel b/Long-termconstraints:noprotectionat horizon,fullprotectionathorizoninrealor nominalterms,probabilityofnotreachingthe goalkeptbelow10%,15%,20%,25%,50%

4) Contribution profiles a/Fixedcontributionschedule b/Uncertaincontributionschedule •Presenceofascheduleofminimum contributions • Absenceofascheduleofminimum contributions

A simplified version could be proposed to adifferent, less affluent clientele with less roomforflexibility.1) Objective profiles a/Pension-relatedobjectives •Clientisretiredandhasalready contributed •Clientisnotretiredyetandwill contributeinthefuture b/Expenditure-relatedobjectives •Accumulateforfutureexpenditure basedonasinglecontribution •Accumulateforfutureexpenditure basedonascheduleofcontributions

2) Time-horizon profiles: 0-3 years, 3-10 years, 10-30 years

3) Constraints and risk-aversion profiles a/Short-termconstraints:noannual protection,annualcapitalprotectionuptoa 95%confidencelevel b/Long-termconstraints:noprotection athorizon,fullprotectionathorizoninreal ornominalterms,probabilityofnotreaching thegoalkeptbelow10%

4) Contribution profiles a/Fixedcontributionschedule

Inwhatfollows,weprovideanintroductiontothesophisticatedALMtechniquesthathavebeenusedininstitutionalmoneymanagementtodesignanofferingofproductsandservicesthatreallyaccountforinvestors’goalsandconstraints.Wethenprovideillustrationsoftheirusefulnessinprivatewealthmanagement.

2. Asset-Liability Management as a Truly Client-Driven Approach to Private Banking


Recentdifficultieshavedrawnattentiontotheriskmanagementpracticesof institutional investorsin general and defined benefit pension plansin particular.Whathas been labeled “a perfectstorm”ofadversemarketconditionsattheturnofthemillenniumhasdevastatedmanycorporatedefined benefit pension plans. Negative equitymarketreturnshaveerodedplanassetsatthesametime as declining interest rates have increasedmark-to-market value of benefit obligationsandcontributions.Inextremecases,thishasleftcorporate pension plans with funding gaps aslargeasorlargerthanthemarketcapitalizationoftheplansponsor.That institutional investorsingeneralandpensionfundsinparticularhavebeensodramaticallyaffectedbyrecentmarketdownturns has emphasized the weakness ofriskmanagementpractices. In particular, it hasbeen argued that the kinds of asset allocationstrategies implemented in practice, which usedto be heavily skewed towards equities in theabsenceofanyprotectionwithrespecttotheirdownsiderisk,werenotconsistentwithasoundliabilityriskmanagementprocess.

In this context, a renewed interest in asset-liability management techniques has surfacedin institutional money management. Newapproaches that are referred to as liabilitydriven investment (“LDI”) solutions have alsobeen introduced following recent changes inaccounting standards and regulations thathave led to an increased focus on liability riskmanagement. In what follows, we will providea brief review of standard asset allocationtechniquesusedinALM,whichcanbeclassifiedintoseveralcategories.

3.1. Cash-Flow Matching and Immunization

A first approach called cash-flow matchinginvolvesensuringaperfectstaticmatchbetweenthe cash flows from the portfolio of assetsand the commitments in the liabilities. Let usassume for example that apension fundhas acommitment to pay out a monthly pension to

a retired person. Leaving aside the complexityrelatingtotheuncertainlifeexpectancyoftheretiree,thestructureoftheliabilitiesisdefinedsimply as a series of cash outflows to be paid,therealvalueofwhichisknowntoday,butforwhich the nominal value is typically matchedwith an inflation index. It is possible in theorytoconstructaportfolioofassetswhose futurecash flowswillbe identical to this structureofcommitments.Todoso,assumingthatsecuritiesofthatkindexistonthemarket,wouldinvolvepurchasing inflation-linked zero-coupon bondswith amaturity corresponding to thedates onwhichthemonthlypensioninstallmentsarepaidout,withamountsthatareproportional totheamountofrealcommitments.Thetechniquecanalso be implemented in a synthetic way usinginterestratesandinflationswaps.

Thistechnique,whichprovidestheadvantageofsimplicityandallows, intheory, forperfectriskmanagement, nevertheless presents a numberof limitations. First of all, it will generally beimpossible to find inflation-linked securitieswhose maturity corresponds exactly to theliabilitycommitments.Moreover,mostof thosesecurities pay out coupons, which leads to theproblemofreinvestingthecoupons.Totheextentthatperfectmatchingisnotpossible,thereisatechniquecalledimmunization,whichallowstheresidualinterestrateriskcreatedbytheimperfectmatch between the assets and liabilities to bemanagedinadynamicway.Thisinterestrateriskmanagementtechniquecanbeextendedbeyonda simple duration-based approach to fairlygeneralcontexts,includingforexamplehedginglarger changes in interest rates (through theintroductionofaconvexityadjustment),hedgingnon-parallel shifts in the yield curve (see forexampleFabozzi,MartelliniandPriaulet(2005)),or simultaneous management of interest rateriskandinflationrisk(SiegelandWaring(2004)).Itshouldbenoted,however,thatthistechniqueisdifficulttoadapttohedgingnon-linearrisksrelated to the presence of options hidden inthe liability structures, and/or tohedgingnon-interestraterelatedrisksinliabilitystructures.

3. A Brief History of ALM Techniques


Another,probablymoreimportant,disadvantageofthecash-flowmatchingtechnique(oroftheapproximate matching version represented bytheimmunizationapproach)isthatitrepresentsapositioningthatisextremeandnotnecessarilyoptimalfortheinvestorintherisk/returnspace.Infactitcanbesaidthatthecash-flowmatchingapproach in asset-liability management is theequivalent of investing in the risk-free assetin an asset management context. It allows forperfect management of the risks, namely acapital guarantee in the passive managementframework, and a guarantee that the liabilityconstraintsarerespectedintheALMframework.However, the lack of return, related to theabsenceofriskpremia,makesthisapproachverycostly, which leads to an unattractive level ofcontributiontotheassets.

3.2. Surplus Optimization

In a concern to improve the profitability ofthe assets, and therefore reduce the level ofcontributions, it isnecessary to introduceassetclasses(stocks,governmentbondsandcorporatebonds) that are not perfectly correlated withthe liabilities into the strategic allocation.It will then involve finding the best possiblecompromise between the risk (relative to theliability constraints) thereby taken on, and theexcessreturnthattheinvestorcanhopetoobtainthroughtheexposuretorewardedriskfactors.

Different techniquesare thenused tooptimizethe surplus, i.e., the excess value of the assetscomparedtotheliabilities,inarisk/returnspace.In particular, it is useful to turn to stochasticmodels that allow for a representation of theuncertaintyrelatingtoasetofriskfactorsthatimpactupontheliabilities.Thesecanbefinancialrisks (inflation, interest rate, stocks) or non-financialrisks(demographiconesinparticular).

Twokeystepsareinvolvedinsurplusoptimization.The first step consists in using a mathematicalmodel for generating stochastic scenarios forall risk factors affecting assets and liabilities

(typically, interest rates, inflation, stock prices,real estate, etc.). Models are chosen so as torepresent actual as well as possible behaviorsandparametersarechosensoastobeconsistentwithlong-termestimates.Thenextstepinvolvesusinganoptimizationtechniquetofindthesetofoptimalportfolios.

In terms of stochastic scenario simulation, onetypically distinguishes between three mainrisk factors affecting asset and liability values:interest rate risk (or, more accurately, interestraterisks,sincethereismorethanoneriskfactoraffecting changes in the shape of the yieldcurve),inflationrisk,andstockpricerisk.WhenrealestateisusedasanALMassetclass,thenanadditionalmodelforthedynamicsofrealestatepricesshouldbeadded.Intheillustrationsthatfollowina latersection,wehaveusedasetofstandardstochasticmodelsfortheseriskfactors,including as key features a two-factor mean-reverting process for real interest rates, a one-factormean-revertingprocessforinflationratesandaMarkovregimeswitchingmodelforexcessreturn on equity (excess return)2. Our model isborrowed from Ahlgrim, D’Arcy and Gorvett(2004)andcanbewrittenas3:

( )( )( )( ) x

ts

xs

xtttt

ttt

ltltllt

rtrttrt

dWdtbdtrSdS

dWdtbad

dWdtlbadl

dWdtrladr

σπ

σππ

σ

σ

ππππ

+++=

+−=

+−=

+−=

Here rt (respectively, πt) is the real short-termrate (respectively, inflation rate) at date t, ar(respectively,aπ)isthespeedofmeanreversionoftheshort-termrate(respectively,inflationrate),lt(respectively,bπ)isthelong-termmeanvalueof the short-term rate (respectively, inflationrate),andσr(respectively,σπ)isthevolatilityofthe short-rate (respectively, inflation rate). Thismodel assumes a particular two-factor processfor the real rate soas toaccount for thenon-perfect correlation between bonds of differentmaturities. In particular, it assumes that thelong-termmeanvalueltoftheshort-termrateisalsostochasticallytime-varying,withaspeedof



2-Amean-revertingmodelforrealestatepriceshasalsobeenusedfortheillustrationswhererealestatewasintroduced.

3-OthercompetingmodelscanofcoursebeusedinALMsimulationsandoptimization,buttheyaremostlyconsistentinspiritwiththisparticularmodel,whichwehavechosenbecauseitrepresentsastandardexampleofastate-of-the-artALMmodelwhichismadeavailableforpublicusebytheCasualtyActuarialSociety(CAS)andtheSocietyofActuaries(SOA)(seereferencelistforexactreferencesofthepaperandawebsitewherethepapercanbedownloaded).

meanreversiondenotedbyal,along-termmeanvaluedenotedbyblandavolatilitydenotedbyσl. By contrast the long-term mean value ofthe inflation rate is assumed to be a constant.Here rW , lW and πW arethree(correlated)standard Brownian motions representinguncertainty driving the three risk-factors.Besides, a Markov-regime switching model isassumedforequityreturns,withbxasthe(state-dependent) excess expected return (over thenominal rate ( )ttr π+ ) and σx as the (state-dependent)stockvolatility.Here sW isastandardBrownian motion representing uncertaintydrivingstockreturns,and iscorrelatedto rW ,

lW and πW . The introduction of a Markovregime-switching model is motivated by thedesire to fit important empirical characteristicsof equity returns, such as the presence of fat-tails and stochastic volatility with volatilityclusteringeffects.Thebasicideaisthatreturnsarenotdrawnfromasinglenormaldistribution;rather there are two distributions at workgenerating the returns observed. The equityreturnsdistributionisassumedtojumpbetweentwopossiblestates,usuallyreferredtoasregimes,denoted by x=1 and x=2 and interpreted as alow and a high volatility regimes. A transitionmatrix controls the probability of movingbetweenstates.

In terms of optimization, the objective can betominimizethevolatilityofthesurplus/deficit;it can also involve other risk measures such astheexpectedshortfall(averagevalueofadeficitconditionalonadeficit),ortheprobabilityofan(extreme)deficit.Theperformance,ontheotherhand,istypicallymeasuredintermsofexpectedsurplus, or necessary contributions. Differentchoicesintermsofoptimizationmodelarealsoavailable,withpossibleoptionsinvolvingsimplestatic optimization or dynamic optimizationwith time- and state-dependent solutions (seefor example Ziemba and Mulvey (1998), aswell as references therein for more details onoptimizationmodelsusedinALM).

3.3. LDI Solutions

Surplus optimization typically allows forhigher returns (on average), and hence lowercontributions(onaverage),sinceit leadstotheintroductionofriskyassetclasses,withtheaccessto associated risk premia. On the other hand,it introduces a significant source of risk sinceassetclassespoorlycorrelatedwithliabilitiesareintroduced.

Inanattempttomitigatetheserisks,andenhanceliability risk management, a new approach(known as liability-driven investment, or LDI)has recently been proposed; it is based on theintroductionofaliability-matching(orliability-hedging)portfoliointhemenuofassetclasses.Itthusbuildsonthetraditionalapproachofcash-flow matching and immunization, focused onriskmanagement,towhichitaddsacomponentdedicatedtoperformance.

It should be noted that when the liabilitymatching portfolio is available in the menuof asset classes, the minimum risk solution ofsurplus optimization corresponds to the cash-flowmatchingstrategy,whichisthusrecoveredasaspecificcase.Inprinciple,oneshouldagaindistinguishbetween:• Cash-flow matching: a perfect match ispossiblebetweenasset and liability cash-flows,usingcashinstruments(nominalandrealbonds)andpossiblydedicatedderivatives(interestrateandinflationswaps)(seeExhibit1).



Exhibit1:Surplusoptimizationwithoutaliability-matchingportfolio

•Cash-flowhedging (immunization): a perfectmatchisnotpossibleandduration(orextendedduration) hedging techniques are implementedsoastominimizemismatchrisk(seeExhibit2).

Fromthepreviouscomments,itmightseemthatso-calledLDIsolutionsaremerelyaspecificcaseofsurplusoptimizationtechniques,inacontextwherealiability-matching(orliability-hedging)portfolioisavailableinthemenuofassetclasses.There is a somewhat subtle difference, though,betweenLDIsolutionsandsurplusoptimizationwithaliability-matchingportfolio.LDIsolutionsadvocateanapproachtoALMthatisexpressedintermsofallocationtothreebuildingblocks(cash,liability-matching portfolio, and performanceportfolio),asopposedtoallocationtostandardasset classes, asdone in thecontextof surplusoptimizationtechniques.Assuch,itisconsistentwithanextendedversionof the standard fundseparationtheoremthat iswell-known inassetmanagement(seenextsectionandtheappendix,orMartellini(2006ab)).

3.3.1. Static LDI SolutionsThis is the standard approach that has rapidlygained interest from pension funds, insurancecompanies,andinvestmentconsultantsalike.Asrecalledbefore,whiletheycanvarysignificantlyacross providers, LDI solutions typically involvea hedge of the duration and convexity risksvia several standard building blocks, whilekeepingsomeassetsfreeforinvestinginhigheryielding asset classes. These solutions may or

may not involve leverage, depending on theinstitutional investor’s risk aversion. Whenno leverage is used, a fraction of the assets(known as the liability-matching portfolio) isallocated to risk management, while anotherfractionoftheassetsisallocatedtoperformancegeneration.Onemayactuallyviewthisapproachas a combination of two strategies, involvinginvesting in immunization strategies (for riskmanagement) as well as investing in standardasset management solutions (for performancegeneration). As explained above, this approachstands in contrast to more traditional surplusoptimization methods (in particular when adedicated liability-matching portfolio is notintroduced),wherebothobjectives(liabilityriskmanagement and performance generation) arepursuedsimultaneouslyinanattempttoachievethe portfolio with the highest possible relativerisk/relativereturnratio.

3.3.2. Dynamic LDI SolutionsThe implementation of LDI solutions cruciallydependsontheinvestor’sriskaversion.Highriskaversion leads to a predominant investment intheliability-hedgingportfolio,whichimplieslowextremefundingrisk(zeroriskincompletemarketcase)aswellaslowperformance(andthereforehigh necessary contributions), while low riskaversionleadstopredominantinvestmentintheperformance-seeking portfolio, which implieshigh funding risk as well as higher expectedperformance,andhencelowercontributions.

Another way to approach the trade-offbetweenriskmanagementontheonehandandperformance generation on the other consistsin implementing a dynamic, as opposed tostatic,allocationbetweentheliability-matchingportfolioandtheperformance-seekingportfolio.Suchdynamicallocationmethods,whichattempttodeliverthebestofbothworlds(downsideriskprotection and access to upside potential), areinspired by the portfolio insurance techniques,whichareextended toanALMframework (seeinparticularLeibowitzandWeinberger(1982ab)for the contingent optimisation technique, aswellasAmenc,MalaiseandMartellini(2004)or



Exhibit2:Surplusoptimizationwithaliability-matchingportfolio

Martellini (2006b) forageneralisation in termsofadynamiccore-satellitemanagement).

One interesting formofdynamic LDI strategiesrecommends that the fraction of wealth Atallocated to the optimal growth portfolio isequal toaconstantmultiplemof thecushion,i.e.,thedifferencebetweentheassetvalueandthefloordefinedasAt-kLt,wherekrepresentsa regulatoryor self-imposedminimumfundingratiorequirement.Thisisreminiscentofconstantproportion portfolio insurance strategies (CPPIstrategies), which the present setup extends toarelativeriskmanagementcontext.WhileCPPIstrategiesaredesignedtopreventfinalterminal

wealth from falling below a specific threshold,extended CPPI strategies (a.k.a. contingentimmunizationstrategies)aredesignedtoprotectasset value from falling below a pre-specifiedfraction of some benchmark value, here theliability value (see the Appendix for moredetails).

3.4. Overview

ThefollowingexhibitpresentsanoverviewofALMtechniquesandthecorrespondingtechniquesinassetmanagement.



Exhibit3:OverviewofALMtechniquesandthecorrespondingtechniquesinassetmanagement.

Risk/Return Profile Asset Management(absolute risk)

Asset-Liability Management(relative risk)

Zero risk – no access to risk premiaInvestment in

risk-free asset

Cash-flow matching

and/or immunization

Optimal risk-return trade-offOptimally diversified portfolio of

risky assetsOptimisation of the surplus

Fund separation theoremCapital market line

(static mix of cash and optimal performance-seeking risky portfolio)

LDI solution

(static mix of cash, liability matching portfolio and optimal risky portfolio)

Dynamic and skewed risk management

(non-linear payoffs)

Portfolio insurance

(dynamic mix of risk-free asset and optimal risky portfolio)

Dynamic LDI

(also known as contingent immunization)

4. Illustrations of the Usefulness of an ALM Approach to PWM


Inwhatfollows,wepresentasetofexamplesoftheuse of asset-liability management techniques inprivatebanking.Ourexamplesaredrawnfromthesimplifiedtypologyofclientprofilesdocumentedinsection2.Weusethestandardmodelintroducedinsection3forgeneratingstochasticscenariosforriskfactorsaffectingassetandliabilityvalues;andwegenerateasetof1,000scenariosforinterestrates, inflation rateandequityprices,aswellasrealestateprices,whenneeded.

Inordertoalleviateapossibleconcernovertheimpact of arbitrary parameter values, we takeparameter values that are identical to those inAhlgrim,D’ArcyandGorvett(2004),whocalibratethe model with respect to long time-series.Otherchoicesofparametervaluescanofcoursebe adopted and their implementation would bestraightforward.

TheparametervaluesaregiveninExhibit4.

Real interest Parameter valueMean reversion speed for short rate process 1

Volatility of short rate process 0.01

Mean reversion speed for long-term mean value 0.1

Volatility of long-term mean value 0.016�

Long-term mean reversion level for long-term mean value 0.028

Correlation between short-rate and long-term mean value 0.�

InflationMean reversion speed for inflation process 0.4

Volatility of inflation process 0.04

Long-term mean reversion level for inflation 0.048

Correlation between inflation and short-term interest rate -0.3

Equity model – Regime switching(Monthly) mean equity excess return in state 1 0.008

(Monthly) volatility of equity return in state 1 0.039

(Monthly) mean equity excess return in state 2 -0.011

(Monthly) volatility of equity return in state 2 0.113

Equity model - Regime switching probabilitiesProbability of staying in state 1 0.989

Probability of switching from state 1 to state 2 0.011

Probability of staying in state 2 0.941

Probability of switching from state 2 to state 1 0.0�9

Real estateReal estate yield reversion speed 1.2

Real estate quarterly yield reversion level 0.023

Real estate yield volatility 0.013Exhibit4:Parametervalues–borrowedfromAhlgrim,D’ArcyandGorvett(2004)

For simplicity of exposure, we have chosento focus on static allocation strategies. Whileappealingfromaconceptualstandpoint,generaltime- and state-dependent portfolio strategiestend to generate a source of confusion forprivateclients,whomayperceivesuchdynamicallocation strategies as attempts to implementtactical asset allocation decisions. In whatfollows,wehavetestedfortheimplementationofextendedCPPIALMstrategiesasachoiceofpragmatic,rule-basedtechniquesallowingustobetter understand the benefits of introducingtime-varyingallocations.Forthesakeofbrevity,theresultsrelatedtothedynamicLDIstrategiesare only reported for a single illustration, thefirstone.Thebenefitstobeexpectedfromsuchstrategies would be qualitatively equivalent inthe context of the other illustrations discussedbelow.

In all cases, we report standard risk-returnindicatorssuchasexpectedsurplus,volatilityofthe surplus, probability of a deficit, as well asexpectedshortfall(expectedvalueofthedeficitconditionalonhavingadeficit).

4.1. Pension-Related Objective

As a first illustration, we focus on a pensionobjectiveandconsidera65-year-old individualwhoisalreadyretired.His/hergoal istoensureastreamofinflation-protectedfixedpayments,whichwenormalizedat€100, for thenext20years (i.e., fromage65 toage85)4. Toachievethis goal the individual is prepared to invest afixedamountofmoney.

Wetestthreedifferentstrategies:•Cash-flowmatchingstrategies•Surplusoptimizationstrategies•DynamicLDIstrategies

4.1.1. Cash-Flow Matching StrategyOne natural solution for meeting the client’sobjective consists in buying equal amountsof zero-coupon inflation-protected securities(TIPS) with maturities ranging from 1 year to

20 years, assuming they exist (alternatively, anOTC interest rate and inflation swaps can beused to complement existing cash instrumentssoastogenerateaperfectmatchwithliabilities,here a stream of 20 annual €100 payments).This equally-weighted portfolio of TIPS isthe practical implementation of the liabilitymatching portfolio introduced at a conceptuallevelinsection3.

Usingtheaforementionedstochasticmodelandassociatedparametervalues,wegeneraterandompathsforthepriceof20zero-couponTIPSwithmaturities matching expected payment dates.Wefindthepresentvalueof liability-matchingportfolio, denoted as L(0), and we obtain L(0)= 1777.15. As we can see, the performance ispoor and the burden of contributions is veryhigh:theamountofmoneyneededtogenerate20annual€100payments isnotmuchsmallerthan20x100.Thisisduetothefactthatratesaretypicallyvery low. Theclientneedsaveryhighcurrent contribution to sustain his/her futureconsumptionneeds.

On the other hand, one key advantage ofthis approach, which represents an extremepositioning in the risk-return space, is that thedistribution of surplus at date 20 is triviallyequal to 0. There is no possible deficit (norsurplus),becausethepresentvalueofthefutureliabilitypaymentshasbeeninvestedinaperfectreplicatingportfoliostrategy.

In this context, it is reasonable, unless in thepresence of an extremely (infinitely) highrisk aversion, to add risky asset classes toenhancethereturnanddecreasethepressureoncontributions,atthecostsofintroducingariskofmismatchbetweenassetsand liabilities.Thisiswhatweturntonext.

4.1.2. Surplus Optimization StrategiesWenowgeneratestochasticscenariosfornominalbonds and stocks also. We then start with thesame initial amount L(0), and find the bestfixed-mixstrategythatconsistsofinvestmentinstocks,bondsanda liability-matchingportfolio



4-Wethusassumeawaythecomplexityrelatedtomortalityrisk,whichcanbedealtwiththroughanannuitycontractprovidedforbyaninsurancecompany.

(regarded as a whole) so as to generate anefficient frontier in a surplus space based onoptimizing the trade-off between expectedsurplusandvarianceofthesurplus(boldlineinExhibit5).Ofcourse,ashighlightedinsection3,theminimumriskportfoliocorrespondsto100%investment in the liability-matching portfolio(correspondingtopointAinExhibit5).Formally,weassumethattheassetportfolioisliquidatedeachyear, a liabilitypayment ismade,and theremaining wealth is invested in an optimalportfolio; in scenarios such that the remainingwealth is not sufficient to make the promisedliability payment, we assume that borrowingattherisk-freerateisperformedsoastomakeupforthedifference.Weestimateprobabilitiesof not meeting the objectives (probability of adeficit),whicharereportedinExhibit6,andalsoplotthedistributionofthesurplusatdate20forafewpointsontheefficientfrontier(seeExhibit7). As can be seen in Exhibit 6, increasing theallocationstostocksandnominalbonds,whichhavealong-termperformancehigherthanthatof inflation-protected bonds but are not asgood a match with respect to liabilities, leadsto a higher value of the expected surplus, andtherefore to average contribution savings, butalsotoanincreasedvolatilityofthesurplusandanincreasedprobabilityofthedeficit.

For comparison purposes, we also perform thesame exercise and design the efficient frontierwhen the liability-matching portfolio is notavailable in the menu of asset classes (see thefinelineinExhibit5).Theimprovementinducedby the introduction of a liability-matchingportfolio is spectacular, as can be seen by asimple comparison between point A and A’ orBandB’.RegardingpointBandB’forinstance,onecanseethatforthesamelevelofexpectedsurplus(€376.78),thevolatilityofthesurplusisincreasedbymorethan50%whentheliability-matching portfolio is not available (€640.24versus€423.65).Theriskreductionbenefitsarealsospectacularwhenriskismeasuredintermsofprobabilityofadeficitorexpectedshortfall.Intuitively, such a dramatic improvement ininvestor’s welfare is related to the fact that it

isonlythroughthecompletionofthemenuofasset classes that arises from the introductionofadedicated liability-matchingportfolio thatthe investor’s specificobjectiveandconstraintsaswellasrelatedriskexposuresarefullytakenintoaccount.

Of course, the difference between optimalportfoliosinthepresenceandintheabsenceofaliability-matchingportfoliodecreaseswiththeinvestor’s risk-aversion: risk-seeking investorsdo not seek to enjoy the benefits of liabilityprotectionandmostlyinvestinstocksandbondsanyway.

This ALM optimization exercise consists infindingtheportfoliosthatareoptimalfromthestandpoint of protecting investors’ liabilities. Apure asset management (AM) exercise, on theotherhand,focusesondesigningtheportfolioswiththeoptimalrisk-returntrade-off.Ofcourse,nothingguaranteesthatAMefficientportfolioswill be efficient fromanALMperspective (andvice-versa);inparticular,thefocusisonnominalreturn from an AM perspective, while it is onrealreturnfromanALMperspective.TotestfortheALMperformanceofAMefficientportfolios,we have conducted the following experiment.Wefirstfindthestandard(Markowitzefficient)frontier based on horizon returns, i.e., theportfolios that achieve the lowest level of



Exhibit5:Efficientfrontierinamean-variancesurplusspace

volatility(acrossscenariosathorizon)foragivenexpected return (across scenarios at horizon).We thenplot theseportfolios (fine line) in the(expectedsurplus-volatilityofthesurplus)ALMspace(seeExhibit8).From Exhibit 8, we can see that a portfolioefficientinanAMsenseisindeednotnecessarilyefficientinanALMsense,andvice-versa.Hence,not taking into account liability constraintsleads to potentially severe inefficiencies fromtheinvestor’sstandpoint.

Wenowturntodynamicportfoliostrategies.

4.1.3. Dynamic LDI StrategiesIn testing the implementation of the dynamicLDIstrategies,theperformanceportfolioistakentobethestock-bondportfoliowiththehighestSharpe ratio (with our choice of parametervalues, and a 4% risk-free rate, we obtain thefollowingportfolio:28.5%instocksand71.5%inbonds),whiletheliability-matchingportfolioistheaforementionedportfolio invested inthe20 zero-coupon TIPSwithmaturitiesmatchingexpectedpaymentdates.

We consider the extended CPPI strategyintroducedinsection3.Weconsider6variantsof the strategy, with the level of protectionk=90%,ork=95%,andthemultipliervaluem=2,3and4.Theresultsarereportedinexhibits9to12, where we present the performance of thevariousdynamicstrategiesandcomparethemto



Exhibit7:Distributionoffinalsurplus/deficit

Exhibit8:AMandALMefficientfrontiersinamean-variancesurplusspace

Weights

StocksBondsLiab-PF

Expectedsurplus

Volatilityofsurplus

Prob(S<0) Expectedshortfall

Necessarynominal

contributionp.a.


A 0% 0% 100% 0.00 0.00 0.00% 0.00 (0.00%) 1777.1� 0.00%

B 19% 22% �9% 376.78 423.6� 1�.�0% 204.4� (11.�0%) 1��6.99 12.39%

C 43% 41% 16% 1130.33 1478.91 18.70% 427.92 (24.08%) 1314.07 26.06%

D �2% 4�% 3% 1�07.11 2093.77 19.60% �02.64 (28.28%) 1233.�8 30.�9%

A’ 10% 90% 0% 0.00 �08.92 36.60% 432.34 (24.33%) 1777.1� 0.00%

B’ 20% 80% 0% 376.78 640.24 2�.20% 38�.97 (21.72%) 1��6.99 12.39%

C’ 43% �7% 0% 1130.33 1�00.11 19.30% 4�7.69 (2�.7�%) 1314.07 26.06%

D’ �1% 49% 0% 1�07.11 2094.78 20.00% 499.21 (28.09%) 1233.�8 30.�9%

Exhibit6:Allocationstrategiesandrisk-return indicators;allvaluesaregivenaspresentvaluesat initialdate (basedonaL(0)=1777.15 initialinvestment);lossesrelativetoL(0)arereportedinparenthesesforexpectedshortfall;therelativecontributionsavingcorrespondstotheincrease(inpercentage)ininitialinvestmentthatshouldhavetakenplacewithagivenstrategysoastogenerateanexpectedsurplusequaltozero.

theperformanceoftheirstaticcounterpart.Thestaticcounterpartofagivendynamicportfoliostrategy is defined as the strategy involvingconstant (fixed-mix) allocation to theportfoliowith the highest Sharpe ratio and liability-matching portfolio that matches the initialallocationofthecorrespondingdynamicstrategy.Forexample,whenk=95%andm=4,theinitialallocation to the liability-matching portfolio(respectivelythehighestSharperatioportfolio)isgivenby1-(1-k)m=80%(respectively,20%).ThestaticcounterpartoftheextendedCPPIstrategywithparametersk=95%andm=4isthereforeafixed-mix strategy with a constant 80%-20%allocation to liability matching portfolio andperformance-seekingportfolio.

As can be seen in Exhibit 9 and Exhibit 10,most dynamic strategies allow for significantlylower expected shortfall numbers as well ashigher expected surplus (and hence highercontribution savings) when compared to theirstatic counterparts. On the other hand, theytend to generate higher volatility. Also, theprobability of a deficit is rather large withdynamicstrategies,whichaimtoavoidalldeficitbeyond the minimum threshold (90% or 95%),as opposed to minimizing the probability offacing such a relatively low deficit. In essence,dynamic ALM strategies generate asymmetricsurplusdistributions,asconfirmedbyExhibits11and12,where thevarious surplusdistributionsare presented. We also note, as expected, that



DynamicCPPIExpectedsurplus

Volatilityofsurplus


Necessarynominalcontributionp.a.


m=2 k=0.90 121.97 188.42 2�.20% 66.4� (3.74%) 1694.19 4.67%

m=3 k=0.90 184.7� 326.33 30.20% 97.21 (�.47%) 16�8.70 6.66%

m=4 k=0.90 203.97 388.70 36.60% 119.11 (6.70%) 1646.97 7.33%

Static CPPIExpected surplus

Volatility of surplus

Prob(S<0)Expected shortfall

Necessary nominal contribution p.a.

Relative contribution saving p.a.

m=2 k=0.90 99.39 110.72 14.90% 80.80 (4.��%) 1706.70 3.96%

m=3 k=0.90 1�3.12 174.9� 1�.90% 119.84 (6.74%) 1673.88 �.81%

m=4 k=0.90 209.74 24�.63 16.80% 1�8.93 (8.94%) 1642.37 7.�8%

Exhibit9:risk-returnindicatorsforextendedCPPIstrategiesforalevelofguaranteek=90%,aswellasfortheirstaticcounterpart;allvaluesaregivenaspresentvaluesatinitialdate(basedonaL(0)=1777.15initialinvestment);lossesrelativetoL(0)arereportedinparenthesesforexpectedshortfall;therelativecontributionsavingcorrespondstotheincrease(inpercentage)ininitialinvestmentthatshouldhavetakenplacewithagivenstrategysoastogenerateanexpectedsurplusequaltozero.

DynamicCPPI Expectedsurplus

Volatilityofsurplus Prob(S<0) Expectedshortfall Necessarynominal

contributionp.a.


m=2 k=0.9� �8.48 88.18 2�.10% 32.7� (1.84%) 1734.�1 2.40%

m=3 k=0.9� 94.38 17�.82 29.90% 48.12 (2.71%) 1711.43 3.70%

m=4 k=0.9� 11�.40 240.19 36.80% �8.48 (3.29%) 1698.24 4.44%

StaticCPPI Expectedsurplus

Volatilityofsurplus Prob(S<0) Expectedshortfall Necessarynominal

contributionp.a.


m=2 k=0.9� 48.38 �2.�8 14.10% 40.43 (2.27%) 1741.04 2.03%

m=3 k=0.9� 73.�� 80.93 14.30% 61.46 (3.46%) 1723.66 3.01%

m=4 k=0.9� 99.39 110.72 14.90% 80.80 (4.��%) 1706.70 3.96%

Exhibit10:risk-returnindicatorsforextendedCPPIstrategiesforalevelofguaranteek=95%,aswellasfortheirstaticcounterpart;allvaluesaregivenaspresentvaluesatinitialdate(basedonaL(0)=1777.15initialinvestment);lossesrelativetoL(0)arereportedinparenthesesforexpectedshortfall;therelativecontributionsavingcorrespondstotheincrease(inpercentage)ininitialinvestmentthatshouldhavetakenplacewithagivenstrategysoastogenerateanexpectedsurplusequaltozero.

increasingtheguaranteedlevelkanddecreasingthemultipliervaluemleadtomoreconservativestrategies, with less potential for surplusperformanceandlowerrisk.

Overall, the results reported in exhibits 7 to10 show the very significant risk managementbenefitsthatarisefromdynamicstrategies.

4.1.4. A VariantWenowconsideraslightvariantofthepensionrelatedobjective,wheretheclientisassumedtobe a 45-year-old individual who is not retiredyetandplanstoretireatage65.Thegoalistoensure at age 65 a single lump-sum paymentnormalizedat€100plusinflationforretirement.Toachievethisgoaltheindividualispreparedtocontributeanamountx(outofhisyearlysalary)fortheremaining20yearsofhisworkinglife.

Exhibit 13 shows the impact of inflation riskon the value of the €100 payment scheduled

to be paid in 20 years from now. As we cansee, inflation risk is significant,withanominalamount to be secured for retirement equalto €247.39 on average and a 94.50 standarddeviation.The main difference with the previous case isthattheinvestormaynotbeabletoimplement

aperfectliability-matchingportfoliounlesshe/she isallowed toborrowagainsthis/her futureincome.

Where borrowing is possible, the strategy is asfollows:• Borrow xB(0,1)+xB(0,2)+…+xB(0,20), whereB(s,t) isthepriceatdatesofaunitfacevaluepure discount nominal bond that matures attimet.•Investthisamountinazero-couponinflationprotectedbondwitha20-yearmaturity.The optimal value for x is given by: x =100P(0,20)/(B(0,1)+B(0,2)+…+B(0,20)), whereP(s,t) is thepriceatdatesofaunitfacevaluepurediscountrealbondthatmaturesattimet.Withourchoiceofparametervalues,xturnsouttobeequal to6.07.This is theamountneededto allow for a perfect ALM match. In practice,it ishowevergenerallynot feasible/practical toborrowagainstfutureincome,anditisthereforeimpossibletogenerateaperfectALMmatchdueto uncertainty over investment conditions forfuturecontributions.



Exhibit13:Distributionofliabilitiesatfinaldate;meanvalue=247.39,standarddeviation=94.50.

Exhibit 11: Distribution of final surplus/deficit for extended CPPIstrategiesfora90%guaranteelevel

Exhibit 12: Distribution of final surplus/deficit for extended CPPIstrategiesfora95%guaranteelevel

Asanattempttoestimatetheoptimalallocationstrategies in this context, we perform thefollowingnumerical exercise.We first generaterandom paths for stock, bond and TIPS priceswith parameters consistent with long-termestimates, where bond and TIPS are regardedas indices(modelledasconstantmaturityzero-couponsecurities).Wethentakex=100P(0,20)/(B(0,1)+B(0,2)+…+B(0,20))= 6.07, as explainedbefore, and find the set of optimal portfoliosthat will minimize the volatility of a deficit/surplus,definedasassetvalueatdate20minusliability valueon retirementdate (i.e., 100plus

20yearsworthofinflation),foragivenlevelofsurplusexpectedvalue.Foreachportfolioontheefficientfrontier,wethenfindthevaluex’<xthatisneededtogenerateazeroexpectedsurplusatretirementdate.Therelativecontributionsavingisgivenby(oneminus)theratioofthepresentvalue of the 20 annual x’ payments over the20 annual x payments. The optimal portfolioallocations and risk-return indicators are givenin Exhibit 14, while the efficient frontier inthe ALM space of expected value and surplusvolatilityappearsinExhibit15.



Exhibit15:Efficientfrontierinamean-variancesurplusspace Exhibit16:AMandALMefficientfrontiersinamean-variancesurplusspace

PortfolioWeights

StocksTIPSBonds

Expectedsurplus

Volatilityofsurplus Prob(S<0) Expected

shortfall

Necessarynominal

contributionp.a.


LMP - - - 0.00 0.00 0.0% 0 (00.0%) 6.07 -

A 0% 0% 100% 9.18 27.10 34.8% 17.6� (22.7%) �.44 10.4%

B 9% 0% 91% 23.61 39.11 24.0% 17.62 (22.6%) 4.7� 21.8%

C 19% 0% 81% 39.64 �9.93 19.6% 18.64 (23.9%) 4.16 31.�%

D 30% 0% 70% ��.67 83.03 18.2% 19.42 (24.9%) 3.70 39.1%

E 40% 0% 60% 71.70 106.94 17.9% 20.39 (26.2%) 3.33 4�.2%

F �0% 11% 39% 87.73 131.20 18.1% 22.38 (28.7%) 3.03 �0.1%

G 60% 24% 16% 103.76 1��.64 17.8% 2�.92 (33.3%) 2.78 �4.3%

H 70% 30% 0% 119.80 180.17 17.3% 29.69 (38.1%) 2.�7 �7.8%

I 80% 20% 0% 13�.83 204.84 17.3% 31.60 (40.6%) 2.38 60.8%

K 90% 10% 0% 1�1.86 229.60 16.9% 34.79 (44.7%) 2.22 63.4%

L 100% 0% 0% 167.89 2�4.43 17.�% 36.31 (46.6%) 2.09 6�.7%

Exhibit14:Allocationstrategiesandrisk-returnindicators;allvaluesaregivenaspresentvaluesatinitialdatebasedona€6.07annualcontributionfor20years,whosepresentvalue6.07x(B(0,1)+B(0,2)+…+B(0,20))amountsto €77.87givenourchoiceofparametervalues.Expectedshortfall isexpressedasapercentageofthisvalue.Therelativecontributionsavingcorrespondstotheincrease(inpercentage)ininitialinvestmentthatshouldhavetakenplacewithagivenstrategysoastogenerateanexpectedsurplusequaltozero.

Asbefore,wecanseethataportfolioefficientinanAMsenseisindeednotnecessarilyefficientinanALMsense,andvice-versa(seeexhibit16),which suggests that omitting to take liabilityconstraintsintoaccountinthedesignofoptimalportfolio solutions leads to potentially severeefficiencylossesfromtheinvestor’sstandpoint.

4.2. Expenditure-Related Objective: the Case of Real Estate

We now consider an investor who wishes toinvestfixedannualcontributions(€x)forfutureexpenditure,e.g.,tobuyahousein5years,thecurrentvalueofwhichisnormalizedat€100(wemay alternatively interpret this as the requireddownpayment).Forsimplicity,onecouldassumethathousepricesincreasewithinflationandusethe stochastic model for inflation to generateadistributionoffuturehouseprices.Ofcourse,because real estate prices are only imperfectlycorrelated with a broad-based consumer priceindex,itismoreaccuratetointroduceanexplicitmodel for the dynamics of real estate prices,whichiswhatwedohere.

Exhibit17showstheimpactofrealestatepriceuncertaintyonthevalueofthe€100paymentscheduledtobepaidin5yearsfromnow.Aswecansee,realestatepriceriskissignificant,withanominalamounttobesecuredequalto€156.59onaverageanda€27.18standarddeviation.

In practical terms, the goal is to generate alump sum payment at horizon date (5 years).As in the previous example, it is not possiblein general to find a perfect liability-matchingportfolio. The existence of a perfect liability-matching portfolio is actually only ensured onthefollowingtwoconditions:• Investors can borrow against future incomeand can invest at the initial date the presentvalueofthefuturecontributions.

•Thereexistsaninvestmentvehicle(e.g.,REITS)whose payoff is directly related to real estatepriceuncertainty.

In what follows, we test two differentsituations:• The opportunity set contains stocks, bondsandTIPS• The opportunity set contains stocks, bonds,TIPS,plusrealestate(modelledasaninvestmentthat will pay the compounded return on realestate)

To generate comparable portfolios, we havelookedat the improvement in surplusvolatilityforagivenlevelofexpectedsurplus.



Exhibit 17: Distribution of house prices at final date; mean value =156.59,standarddeviation=27.18.

Exhibit18:ALMEfficientFrontierswithoutRealEstate(A,B,C,D,E,F)andwithRealEstate(A’,B’,C’,D’,E’,F’)

Exhibit 18 shows the efficient frontier in bothcases, while risk-return indicators are reportedin Exhibit 19. As was expected, the presenceof assets allowing investors to span real estatepriceuncertaintyprovestobeakeyelementinimproving theefficient frontiersobtained froman ALM perspective. Looking for example atportfolioDandD’fromExhibit19,weseethatforthesamelevelofexpectedsurplus(12.60inbothcases),thesurplusvolatilityattheoptimallevelreaches21.95whentheopportunitysetdoesnot contain a real estate asset,while itmerelyamounts to 4.25, a dramatic risk reduction,whentherealestateassetisincluded.Againthissignals the relevance of an ALM approach toprivatewealthmanagement:itisonlybytryingto fit the client liability constraints that trulyoptimalsolutionscanbeproposed.

4.3. Bequest-Related Objective

We now consider a wealthy 65-year-oldindividual who is already retired. He/she hassignificant wealth (say 100 million euros) andwishestomaintainastandardofliving(annualexpenses say at 2 million euros plus inflation)withanadditionalbequestmotivein20years5.

The analysis aims to find the optimal policyso as to generate the highest possible bequestlevelwithagivenprobabilitydenotedby α.Wefirst discuss this situation as a base case, andsubsequentlyturntodifferentvariants.

4.3.1. The Base CaseExhibit20showstheoptimalallocationstrategy,as well as related risk-return indicators, forvarious values of the confidence level α,while Exhibit 21 shows the distribution of thediscountedvalueoffinalbequestalsoforthesedifferentvalues.



5-Weagainassumeawaythecomplexityrelatedtomortalityrisk,whichcanbedealtwiththroughanannuitycontractprovidedforbyaninsurancecompany.

Portfolio

Weights

StocksBondsTIPSRealEstate

Expectedsurplus

Volatilityofsurplus


Necessarynominal

contributionp.a.


A 0% 0% 100% - 2.46 14.64 41.�% 11.40 (12.�%) 19.07 4.7%

B 10% 90% 0% - �.84 13.96 31.�% 10.32 (11.3%) 17.8� 10.8%

C 33% 67% 0% - 9.22 17.16 28.9% 11.03 (12.1%) 16.70 16.�%

D ��% 4�% 0% - 12.60 21.9� 28.�% 12.93 (14.2%) 1�.62 21.9%

E 78% 22% 0% - 1�.98 27.�1 28.0% 1�.92 (17.�%) 14.60 27.0%

F 100% 0% 0% - 19.36 33.46 27.9% 19.09 (21.0%) 13.63 31.8%

A’ 0% 0% 100% 0% 2.46 14.64 41.�% 11.40 (12.�%) 19.07 4.7%

B’ 0% 0% 68% 32% �.84 10.87 28.6% 7.44 (8.2%) 17.8� 10.7%

C’ 0% 0% 37% 63% 9.22 7.27 10.6% 4.�7 (�.0%) 16.70 16.�%

D’ 0% 0% �% 9�% 12.60 4.2� 0.6% 2.07 (2.3%) 1�.61 21.9%

E’ 46% 0% 0% �4% 1�.98 16.43 16.7% 7.36 (8.1%) 14.�9 27.0%

F’ 100% 0% 0% 0% 19.36 33.46 27.9% 19.09 (21.0%) 13.63 31.8%

Exhibit19:Allocationstrategiesandrisk-returnindicators;allvaluesaregivenaspresentvaluesatinitialdatebasedona€20annualcontributionfor5years,thepresentvalueofwhichamountsto€90.91givenourchoiceofparametervalues.Expectedshortfallisexpressedasapercentageofthisvalue.Therelativecontributionsavingcorrespondstotheincrease(inpercentage)ininitialinvestmentthatshouldhavetakenplacewithagivenstrategysoastogenerateanexpectedsurplusequaltozero.

Targetpercentile

WeightsExpectedbequest

Volatilityofbequest

Bequestpercentiles

Stocks Bonds LMP 5 10 20 Median 75 95

No

constraints

alpha= �% 11% 49% 40% 90.�8 30.36 48.98 �6.01 64.79 86.7� 107.�6 14�.93

alpha=10% 2�% 38% 37% 118.43 �8.20 46.78 �9.�2 71.76 10�.84 147.39 226.�0

alpha=20% �3% 23% 24% 194.64 163.44 34.�8 �1.1� 76.3� 149.39 2�4.62 �18.�0

Exhibit20:Allocationstrategiesandrisk-returnindicatorsasafunctionoftheconfidencelevelα

4.3.2. Introducing ConstraintsWealsoconsidertwovariantsinwhich:•Halfoftheclientwealth(100million)isheldasstockinhis/herownprivatecompany,whichwill be sold in 5 years from now; in this case,we impose a 50% lower constraint on equityallocation)6.• The value of existing property is accountedfor (e.g., theclienthasae10millionworthofpropertyvalueinadditiontothee100million).

TheseresultscanbefoundinExhibits22and23.



6-Inotherworlds,weassumeawayspecificriskinprivateequityreturnwhenoptimizingtheportfolio,andmodelthe2100millionasifitwasinvestedintheequityindex.

Exhibit23:Distributionofthediscountedvalueoffinalbequestasafunctionoftheconfidencelevelα,withadditionale10millioninitiallyheldinrealestate(left)andwithaminimumof50%investedinequity(right)

AswecanseethroughacomparisonwithExhibit20, the presence of constraints related to the

client’s situation will impact upon the optimalportfoliostrategy.

Exhibit21:Distributionofthediscountedvalueoffinalbequestasafunctionoftheconfidencelevelα

Exhibit22:Allocationstrategiesandrisk-returnindicatorsasafunctionoftheconfidencelevelα,includingallocationconstraints(realestateorequity)

TargetPercentile

WeightsStocksBondsLMP

Expectedbequest

Volatilityofbequest

Bequestpercentiles

5 10 20 Median 75 95

Min �0%

in stocks

alpha= �% �0% 40% 10% 181.3� 142.68 37.23 �2.0� 73.�� 142.62 239.44 462.09

alpha=10% �0% 1�% 3�% 187.21 1�2.�9 3�.82 �3.11 7�.00 147.21 242.09 490.4�

alpha=20% �3% 23% 24% 194.20 162.69 34.�9 �1.10 76.33 149.1� 2�4.2� �16.30

Additional 10m

in real estate

property at T0

alpha= �% 22% 36% 42% 1�2.14 �9.32 79.21 88.47 103.63 141.31 182.29 260.71

alpha=10% 24% 44% 32% 1��.60 62.90 78.76 90.32 104.�1 143.97 186.�0 277.�0

alpha=20% �2% 23% 2�% 241.94 179.88 62.03 83.67 111.22 192.13 309.43 �94.38

4.3.3. A Variant with Significant Lump-Sum Payments ExpectedWefinallyconsidera65-year-oldindividualwhois already retired. He/she has significant wealth(say e100 million) and wishes to maintain a

standard of living (annual expenses say at 2millioneurosplusinflation),plustwosignificantexpenses(10millionin5yearsand10millionin10years),e.g.,tobuyaprivatejetorayacht,withanadditionalbequestmotivein20years.



Exhibit24:Distributionofthediscountedvalueoffinalbequestasafunctionoftheconfidencelevelα

Exhibit26:Distributionofthediscountedvalueoffinalbequestasafunctionoftheconfidencelevel αandexpectedbequestlevel

Exhibit25:Allocationstrategiesandrisk-returnindicatorsasafunctionoftheconfidencelevelα

Target

Percentile

Weights

Stocks Bonds LMP

Expected

Bequest

Volatility

of Bequest

Bequest Percentiles

� 10 20 median 7� 9�

alpha= �% 17% 47% 36% 74.�7 31.11 3�.30 41.26 48.70 69.�2 91.33 133.3�

alpha=10% 17% 40% 43% 7�.88 32.27 34.43 41.84 49.69 70.01 93.46 13�.48

alpha=20% 48% 31% 21% 141.03 118.82 19.73 32.68 �4.10 110.9� 187.23 363.20

The analysis aims at finding the optimalpolicysoasto:• Generate the optimal distribution ofbequestforagivenlevelofannualexpenses(Exhibits24and25).•Generate theoptimaldistributionof levelof annual expenses for a given level ofbequest(Exhibits26and27).



Bequest

level

Target

percentile

Weights

Stocks Bonds LMP

Expected

annual

expenses

Volatility

of annual

expenses

Annual expenses percentiles

� 10 20 Median 7� 9�

7� alpha= �% 17% 47% 36% 1.�3 1.�8 -1.20 -0.60 0.33 1.73 2.64 3.72

alpha=10% 17% 40% 43% 1.�� 1.62 -1.11 -0.67 0.31 1.76 2.67 3.81

alpha=20% 48% 31% 21% 3.28 3.23 -2.06 -0.63 0.81 3.44 �.43 8.01

100 alpha= �% 17% 47% 36% 0.26 2.04 -3.62 -2.30 -1.10 0.�6 1.70 2.98

alpha=10% 17% 40% 43% 0.30 2.08 -3.63 -2.34 -1.02 0.6� 1.74 3.08

alpha=20% 48% 31% 21% 2.1� 3.70 -3.98 -2.17 -0.61 2.47 4.66 7.38

1�0 alpha= �% 17% 47% 36% -2.3� 3.07 -8.18 -6.26 -4.�1 -1.86 -0.18 1.�0

alpha=10% 17% 40% 43% -2.31 3.14 -8.29 -6.38 -4.�4 -1.78 -0.08 1.�9

alpha=20% 48% 31% 21% -0.13 4.80 -8.33 -�.63 -3.44 0.37 3.10 6.40

Exhibit27:Allocationstrategiesandrisk-returnindicatorsasafunctionoftheconfidencelevelα andexpectedbequestlevel

This paper has provided ample evidence thatasset-liability management is an essentialimprovement in private wealth managementthat allows private bankers to provide theirclients with investment solutions and assetallocation advice that truly meet their needs.We have also provided a series of illustrationsthat showthat someof themost sophisticatedALM techniques used in institutional moneymanagementcansatisfactorilybe implementedinprivatewealthmanagement.

Ultimately,wearguethatitisnottheperformanceof a particular fund nor that of a given assetclass (including commodities or hedge funds)thatwillbethedeterminingfactorintheabilityofprivatewealthmanagementtomeetinvestors’expectations.Whatwillprovetobethedecisivefactor is the private wealth manager’s abilityto design an asset allocation solution that isa function of the kinds of particular risks towhichtheinvestorisexposed,asopposedtothemarket as a whole. Hence, an absolute returnfund,oftenperceivedasanaturalchoiceinthecontext of private wealth management, shallnot be a satisfactory response to the needsof a client facing long-term inflation risk,where the concern is capital preservation inreal, as opposed to nominal, terms. Similarly, aclientwhoseobjectivewouldbe related to theacquisitionofapropertywouldacceptlowandeven negative returns in situations when realestatepricessignificantlydecrease,butwillnotsatisfy himself or herself with relatively highreturnsifsuchhighreturnsarenotsufficienttomeet a dramatic increase in real estate prices.In such circumstances, a long-term investmentinstocksandbondswithaperformanceweaklycorrelatedwith real estatepriceswouldnotbetherightinvestmentsolution.

In other words, the success or failure of thesatisfactionof theclient’s long-termobjectivesisfundamentallydependentonanALMexercisethat aims to determine the proper strategicinter-classes allocation as a function of theclient’s specific objectives and constraints.Assetmanagementshouldonlycomenextasa

response to the implementation constraints oftheALMdecisions.Ontheonehand,itismeanttodeliver/enhancetheriskandreturnparameterssupportingtheALManalysisforeachassetclass.On the other hand, it can also allow for themanagementofshort-termconstraints,suchascapitalpreservationatagivenconfidencelevel,whicharenotnecessarilytakenintoaccountbyanALMoptimizationexercise,whichbynaturefocusesonlong-termobjectives.

5. Conclusion


Inthisappendix,wepresentageneralcontinuous-timemodel of asset allocationdecisions in thepresence of liability constraints. This materialis borrowed from Martellini (2006ab). Froman academic standpoint, several authors haveattempted to cast the ALM problem in acontinuous-timeframework,andextendMerton’sintertemporal selection analysis (see Merton(1969, 1971)) to account for the presence ofliabilityconstraintsintheassetallocationpolicy.Afirststepintheapplicationofoptimalportfolioselectiontheorytotheproblemofpensionfundshasbeen takenbyMerton (1990)himself,whostudies the allocation decision of a universitythatmanagesanendowmentfund. Inasimilarvein, Boulier et al. (1995) have formulated acontinuous-time dynamic programming modelofpensionfundmanagement.Itcontainsallofthebasicelementsformodelingdynamicpensionfund behavior, and can be solved by means ofanalyticalmethods7.Rudolf andZiemba (1994)extend these results to the case of a time-varying opportunity set, where state variablesare interpreted as currency rates that affectthe value of the pension’s asset portfolio. Alsorelated is a paper by Sundaresan and Zapatero(1997), which is specifically aimed at assetallocation and retirement decisions in the caseofapensionfund.

Thiscontinuous-timestochasticcontrolapproachto ALM is appealing because it enjoys thedesirablepropertyof tractabilityandsimplicity,allowingone to fully and explicitlyunderstandthe various mechanisms affecting the optimalallocation strategy8.Ourpaper fallswithin thisstrand of the literature, which it complementsinavarietyofways.Inparticular,wearguethatthe value of the liability portfolio is a naturalnumeraire in the investor’s objective function,andwerelatethesolutiontothisproblem,whichinvolves a three fund separation theorem, tosome of the recent liability-driven investmentsolutions offered by several investment banksandassetmanagementfirms.

6.1. Stochastic Model for the Value of Asset and Liabilities

Let [0,T] denote the (finite) time span of theeconomy, where uncertainty is describedthroughastandardprobabilityspace(Ω,A,P)andendowed with a filtration { }0; ≥tFt

, whereAF ⊂∞ and 0F istrivial,representingtheP-

augmentationofthefiltrationgeneratedbythen-dimensionalBrownianmotion ( )nWW ,...,1 .Weconsidernriskyassets(orassetclasses),thepricesofwhicharegivenby:

nidWdtPdP

n

j

jtiji

it

it ,...,1 ,

1

=⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑

=

σμ

We shall sometimes use the shorthand vectornotation for the expected return (column)vector ( )'

,....,1 nii == μμ and matrix notation

( )njiij ,....,1, =

= σσ fortheassetreturnvariance-covariancematrix,assumedtobenon-singular.1=(1,…,1)’ denotesann-dimensional vectorofonesand ( ) nj

jWW ,....,1'

== denotesthevectorof Brownian motions. A risk-free asset, the 0th

asset, isalsotradedintheeconomy.Thereturnon that asset, typically a default free bond, isgivenby

€

dPt0 = Pt

0rdt ,whereristherisk-freerateintheeconomy.Weassumethroughoutthepaperthatallparametervaluesareconstant9.

We also introduce a separate process thatrepresents in a reduced-form manner thedynamicsofthepresentvalueoftheliabilities:

⎟⎟⎠

⎞⎜⎜⎝

⎛++= ∑

=

εεσσμ tL

jt

n

jjLLtt dWdWdtLdL ,

1,

where ( )εtW is a standard Brownian motion,

uncorrelated with W, that can be regarded astheprojectionresidualofliabilityriskontoassetpriceriskandrepresentsthesourceofuncertaintythat is specific to liability risk,emanatingfromvariousfactorssuchasuncertaintyinthegrowthof the work force, uncertainty in mortalityand retirement rates, etc. Lμ represents theexpectedreturnontheliabilityportfolio; εσ ,L

6. Mathematical Appendix


7-ArelatedreferenceisSiegmannandLucas(2002)whoextendtheapproachtakenbyBoulieretal.(1995)byconsideringCARAandCRRApreferences,asopposedtoasimplequadraticlossfunction.

8-AsecondstrandoftheliteraturehasthereforefocusedondevelopingmorecomprehensivemodelsofuncertaintyinanALMcontext.ThishasledtothedevelopmentofastochasticprogrammingapproachtoALM,includingKallbergetal.(1982),KusyandZiemba(1986),orMulveyandVladimirou(1992).Agoodnumberofapplicationsinasset-liabilitymanagementareprovidedinZiembaandMulvey(1998)andZiemba(2003).

9-Moregenerally,onecanmakeexpectedreturnandvolatilitiesoftheriskyassets,aswellastherisk-freerate,dependuponamulti-dimensionalstatevariableX.Thesestatevariablescanbethoughtofasvarioussourcesofuncertaintyimpactinguponthevalueofassetsandliabilities.Inparticular,onemayconsidertheimpactofstochasticinterestrateontheoptimalpolicy.

represents pure liability volatility, while thevector ( )'

,....,1, njjLL == σσ canbe regardedas

measuring the magnitude of financial risks inliabilitystreams.

Theintegrationoftheabovestochasticdifferentialequationgives:

€

LT = Ltη t ,T( )ηL t ,T( ) ,with:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )⎭⎬⎫

⎩⎨⎧

+−≡

⎭⎬⎫

⎩⎨⎧

+⎟⎠

⎞⎜⎝

⎛ −≡

∫ ∫

∫ ∫T

t

T

tsLLL

T

t

T

tsLLLL

dWsdssTt

dWsdssssTt

εεε σση

σσσμη

,2,

''

21

exp,

21

exp,

When 0, =εσ L , then we are in a completemarket situation where all liability uncertaintyisspannedbyexistingsecurities.Becauseofthepresence of non-financial risks (e.g., actuarialrisks), suchasituationneveroccurs inpractice,and the correlation between the liability andtheliability-hedgingportfolio(i.e.,theportfoliowiththehighestcorrelationwithliabilityvalues)is always strictly lower than one. In generaltherefore, 0, ≠εσ L andthepresenceofliabilityriskthatisnotspannedbyassetpricesinducesaspecificformofmarketincompleteness.

6.2. Objective and Investment Policy

The investmentpolicy isa (column)predictableprocess vector ( )( )

01' ,...,

≥= tnttt www that

represents allocations to risky assets, with theremainder invested in the risk-free asset. Wedefineby w

tA theassetprocess,i.e.,thewealthattimetofaninvestorfollowingthestrategywstartingwithaninitialwealth 0A .

Wehavethat:

( ) ⎥⎦

⎤⎢⎣

⎡+−=

t

t

t

twt

wt P

dPw

BdB

wAdA ''.1 1

or:

( )( )[ ]tw

tw

t dWwdtrwrAdA σμ '' +−+= 1

Wenowintroduceonevariableofinterest,whichwillbeusedasastatevariableinthismodel—thefundingratio—definedastheratioofassetsto liabilities: t

wtt LAF = . Apension trust is

saidtohaveasurpluswhenthefundingratioisgreaterthan100%,fullyfundedwhenitisequalto100%,andunderfundedwhenitislessthan100%.

Inanasset-liabilitymanagementcontext,whatmattersisnotthevalueoftheassetsperse,buthow the asset value compares to the value ofliabilities.Thisisalsothereasonwhyitisnaturalto assume that the (institutional) investor’sobjective is written in terms of relative wealth(relative to liabilities), as opposed to absolutewealth: ( )[ ]T

wFUE 0max 10.

Using Itô’s lemma, we can also derive thestochasticprocessfollowedbythefundingratioundertheassumptionofastrategyw:

( )2

322

11t

t

wt

tw

tt

tt

wtw

ttt

wt

t dLLA

dLdAL

dLLA

dALL

AddF +−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

whichyields:

( )( )( ) ( ) ( ) ( )( )dtdtdtwdWdWdtdWwdtrwrFdF

LLLLtLtLLtt

t 2,

',

' ''' εε

ε σσσσσσσμσμ ++−++−+−+= 1

or

( ) ( )( ) ( ) εεε σσσσσμσσσμ tLtLLLLLL

t

t dWdWwdtrwdtrFdF

,'2

,' '' −−+−−+++−= 1

Forlateruse,letusdefinethefollowingquantitiesasthemeanreturnandvolatilityofthefundingratioportfolio,subjecttoaportfoliostrategyw:

( ) ( )( )

( )( )( )21

2,

'''

2,

'

''

1'

ε

ε

σσσσσσ

σσμσσσμμ

LLLw

F

LLLLLw

F

ww

rwr

+−−≡

−−+++−≡

( )( )( ) ( ) ( ) ( )( )dtdtdtwdWdWdtdWwdtrwrFdF

LLLLtLtLLtt

t 2,

',

' ''' εε

ε σσσσσσσμσμ ++−++−+−+= 1

( ) ( )( ) ( ) εεε σσσσσμσσσμ tLtLLLLLL

t

t dWdWwdtrwdtrFdF

,'2

,' '' −−+−−+++−= 1



10-Anotherrelevantvariableisthesurplus,definedasthedifference,asopposedtotheratio,ofassetsandliabilities.



6.3. Solution using the Dynamic Programming Approach

Define the indirect or derived utility processat time t: ( )[ ]Ttwt FUEJ max= , where

[ ]•tE denotes theexpectationconditionaloninformationavailableattimet,suchasdescribedby the filtration generated by the asset pricesthat aredrivenbynBrownianmotionand the(n+1)th Brownian motion driving pure liabilityuncertainty.

6.3.1. General SolutionFor a Markovian control process ( )

0≥ttw anda function ( ) 2,1, CFt t ∈ϕ the infinitesimalgenerator of the funding ratio process is:

( ) ( )2221, w

FFFw

FFttw FFFtA σϕμϕϕϕ ++=

wherethederivativeofafunctionfwithrespecttovariablexisdenotedasfx.

Given the objective function, the appropriateHamilton-Jacobi-Bellman equation associatedwiththisproblemis: ( ){ } 0,sup =t

w

w

FtJA ,

subjectto ( ) ( )tT FUFTJ =, .

Optimizingwithrespecttowyields:

( ) ( ) ( ) 0*

2

221* =

∂

∂+

∂

∂w

wFw

wF

wF

FF

wF

F

σϕ

μϕ

or:

( )( ) ( )( ) 0'*'2 =−+−− LFFLF wFrF σσσσϕσσμϕ 1

withsolution:

( ) ( ) ( )( ) ( )( )

( ) LtFF

tFLt FtF

FtrFtww σσσσ

ϕ

ϕσσμσσ 11** '

,

,', −− +−−−== 1

( ) ( ) ( )( ) ( )( )

( ) LtFF

tFLt FtF

FtrFtww σσσσ

ϕ

ϕσσμσσ 11** '

,

,', −− +−−−== 1

or:

( ) ( ) ( ) ( )( )

( )( )

( ) LtFF

tF

tFF

tFt FtF

FtFtF

FtrFtww σσ

ϕ

ϕ

ϕ

ϕμσσ 11** '

,

,1

,

,', −−

⎟⎟⎠

⎞⎜⎜⎝

⎛++−−== 1

( ) ( ) ( ) ( )( )

( )( )

( ) LtFF

tF

tFF

tFt FtF

FtFtF

FtrFtww σσ

ϕ

ϕ

ϕ

ϕμσσ 11** '

,

,1

,

,', −−

⎟⎟⎠

⎞⎜⎜⎝

⎛++−−== 1

Wethusobtainathreefundseparationtheorem,wheretheoptimalportfoliostrategyconsistsofholdingtwofunds,onewithweights:

( ) ( )( ) ( )11'

1rr

wM−

−=

−

−

μσσ

μσσ1

1

'

'

andanotheronewithweights:

( )( ) L

LLw

σσ

σσ1

1

'

'−

−

=1'

therestbeinginvestedintherisk-freeasset.

Thefirstportfolioisthestandardmean-varianceefficient portfolio. Note that the amountinvestedinthatportfolioisdirectlyproportionaltotheinvestor’sArrow-Prattcoefficientofrisk-tolerance

FF

F

F ϕ

ϕ− (theinverseoftherelative

risk aversion). Thismakes sense: thehigher theinvestor’s(funding)risktolerance,thehighertheallocationtothatportfoliowillbe.

Inordertobetterunderstandthenatureofthesecond portfolio, it is useful to remark that itisaportfoliothatminimizesthe localvolatility

wFσ ofthefundingratio.Toseethis,recallthat

theexpressionforthelocalvarianceisgivenby( )( )( )2

12,

''' '' 2 εσσσσσσ LLLw

F ww +−−= ,whichreachesa minimum for ( ) Lw σσ 1* ' −= , with theminimumbeing 2

,εσ L .Assuch,itappearsastheequivalentoftheminimumvarianceportfolioinarelativereturn-relativeriskspace,andalsoastheequivalentoftherisk-freeassetinacompletemarket situation where liability risk is entirelyspanned by existing securities ( 02

, =εσ L ).Alternatively,thisportfoliocanbeshowntohavethehighestcorrelationwiththeliabilities.Assuch,itcanbecalleda liability-hedgingportfolio, inthespiritofMerton(1971)intertemporalhedgingdemands. Indeed, if we want to maximize thecovariance Lw σσ' betweentheassetportfolioandtheliabilityportfolioL,undertheconstraintthat wwA ''2 σσσ = , we obtain the followingLagrangian:

( )2''' AL wwwL σσσλσσ −−=

Differentiating with respect to w yields:

ww

LL σλσσσ '2−=

∂

∂ ,

with a strictly negative second derivativefunction.Settingthefirstderivativeequaltozeroforthehighestcovarianceportfolioleadstothefollowingportfolio,whichisindeedproportionaltotheliabilityhedgingportfolio:

( ) ( ) LLw σσλ

σσσσλ

11 '2

1'

21 −− == .

6.3.2. Specific Solution in Case of CRRA Utility and Constant Parameter ValuesLetusnowconsideraspecificutilityfunctionoftheCRRAtype:

( ) ( )γ

γ

−=

−

1

1T

T

FFU

We try a solution to the non-linear Cauchyproblem:

( ) 022

21

**

=++ wFFF

wFFt FF σϕμϕϕ

whichisseparableinFandcanbewrittenas:

( ) ( )( )γ

ϕγ

−=

−

1

,,

1t

t

FTtgFt

with:

( ) ( ) ( )( ) ( ) ( )( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+−−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−= 2

,,2, 2

21'

11

21

exp, εεε σγγ

σθσγσθσθγ LLLLLLtTTtg

( ) ( ) ( )( ) ( ) ( )( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+−−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−= 2

,,2, 2

21'

11

21

exp, εεε σγγ

σθσγσθσθγ LLLLLLtTTtg

where Lσσ − is defined as the matrix thegeneral term of which is equal to that of σ outsidethediagonalandisequalto Lii σσ − ,alsowrittenas Li σσ −2 ,onthediagonal.

Giventhatγ

1=−

FF

F

FJ

J,wefinallyobtain:

( ) ( ) ( ) ( ) Lt rFtww σσγ

μσσγ

11** '1

1'1

, −−

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−== 1

Asiswellknown,itshouldbenotedthatwhen1=γ , i.e., inthecaseofthe log investor, the

intertemporal hedging demand is zero (myopicinvestor).Ingeneral,again,theoptimalstrategyconsists of holding two funds, in addition tothe risk-freeasset, the standardmean-varianceportfolioandtheliabilityhedgingportfolio,andtheproportionsinvestedinthesetwofundsareconstantintime.

6.4. From Static to Dynamic Portfolio Management

Itcanbedesirablefromaninvestor’sstandpointto set a strict constraint on the potentialunderperformanceoftheportfoliowithrespecttotheliabilitybenchmark.Therecanbetwotypesofconstraints,explicitorimplicit. Inaprogramwithexplicitconstraints,marginalindirectutilityfromwealthdiscontinuouslyjumpstoinfinity:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

⎟⎠⎞

⎜⎝⎛

−

≤≤ γ

γ

1

1

,

T

T

tTstw

LA

EMaxs

suchthat TT kLA ≥ almostcertainly.Ontheotherhand,inaprogramwithimplicitconstraints,marginalutilitygoessmoothlytoinfinity:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

⎟⎠⎞⎜

⎝⎛ −

−

≤≤ γ

γ

1

1

,

kLA

EMax T

T

tTstws

Itcanbeshown(Martellini(2006b))inacompletemarket setting that the solution to a programwith implicit constraints yields the followingtime-dependentsolution11:

( ) ( ) ( ) ( ) Btt

t Fk

rFk

Fww σσγ

μσσγ

11** '11

1'11 −−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−== 1

( ) ( ) ( ) ( ) Btt

t Fk

rFk

Fww σσγ

μσσγ

11** '11

1'11 −−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−== 1



11-SeeMartellini(2006b),whousesinacompletemarketsettingtheconvexduality(ormartingale)techniqueforsolvingtheoptimalportfolioallocationprobleminthepresenceofminimumfundingratioconstraints.



Consider the fractionofwealthAtallocated totheperformance-seekingportfolio.Itisgivenby:

( ) ( ) ( ) ( )( )ttt

tt kLA

rF

kAA

r−

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

− −−

γμσσ

γμσσ 11'11' 11 ''

Hence it appears that the fraction of wealthallocatedtothesatellite isequal toaconstantmultiple m of the cushion, i.e., the differencebetweentheassetvalueandthefloordefinedasAt-kLt.Thisstrategyisreminiscentofconstantproportionportfolio insurance(CPPI)strategies,which itextendstoarelativeriskmanagementcontext. While CPPI strategies are designed topreventfinalterminalwealthfromfallingbelowaspecificthreshold,extendedCPPIstrategies(ordynamic core-satellite strategies) are designedtoprotectassetvaluefromfallingbelowapre-specifiedfractionofthebenchmarkvalue,heregivenbytheliabilityportfolio.

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•Ziemba,W.andJ.Mulvey (Eds.) (1998),Worldwide Asset and Liability Modeling,Cambridge,UK:CambridgeUniversityPress.

EDHECisoneofthetopfivebusinessschoolsinFranceandwasranked7thintheFinancialTimesMastersinManagementRankings2006owingtothehighqualityofitsacademicstaff(over100permanent lecturers from France and abroad)anditsprivilegedrelationshipwithprofessionalsthattheschoolhasbeendevelopingsinceitwasestablishedin1906.EDHECBusinessSchoolhasdecided todrawon its extensive knowledgeoftheprofessionalenvironmentandhasthereforeconcentrateditsresearchonthemesthatsatisfytheneedsofprofessionals.EDHECisoneofthefewbusinessschoolsinEuropetohavereceivedthe triple international accreditation: AACSB(USGlobal), Equis (Europe-Global) and AMBA(UK-Global). EDHEC pursues an active researchpolicyinthefieldoffinance.Its“RiskandAssetManagement Research Centre” carries outnumerous researchprogrammes in theareasofassetallocationandriskmanagementinboththetraditionalandalternativeinvestmentuniverses.

The choice of asset allocationTheEDHECRiskandAssetManagementResearchCentrestructuresallofitsresearchworkaroundasset allocation. This issue corresponds to agenuine expectation from the market. On theonehand,theprevailingstockmarketsituationin recent years has shown the limitations ofactivemanagementbasedsolelyonstockpickingas a source of performance. On the other, theappearance of new asset classes (hedge funds,private equity), with risk profiles that are verydifferentfromthoseofthetraditionalinvestmentuniverse,constitutesanewopportunityinbothconceptualandoperationalterms.Thisstrategicchoice is applied toall of the centre’s researchprogrammes,whethertheyinvolveproposingnewmethodsofstrategicallocation,whichintegratethealternativeclass;measuringtheperformanceof funds while taking the tactical allocationdimension of the alphas into account; takingextreme risks into account in the allocation;or studying the usefulness of derivatives inconstructingtheportfolio.

An applied research approach Inadesiretoensurethattheresearchitcarriesout is truly applicable in practice, EDHEC hasimplemented a dual validation system for theworkoftheEDHECRiskandAssetManagementResearchCentre.Allresearchworkmustbepartof a research programme, the relevance andgoals of which have been validated from bothan academic and a business viewpoint by thecentre’s advisory board, which is made up ofboth internationally recognised researchersandthecentre’sbusinesspartners.

Todate,thecentrehasimplementedsixresearchprogrammes:•Multi-style/multi-classallocation•Performanceandstyleanalysis•Indicesandbenchmarking•Assetallocationandextremerisks•Assetallocationandderivativeinstruments•ALMandassetmanagement

Research for businessTooptimiseexchangesbetweentheacademicandbusinessworlds,theresearchcentremaintainsawebsitedevotedtoassetmanagement researchfortheindustry:www.edhec-risk.com,circulatesamonthlynewslettertoover75,000practitioners,conducts regular industry surveys andconsultations,andorganisesannualconferencesforthebenefitofinstitutionalinvestorsandassetmanagers.Thecentre’sactivitieshavealsogivenrisetothebusinessoffshootsEDHECInvestment

About the EDHEC Risk and Asset Management Research Centre


Research, which supports institutional investorsand asset managers in the implementation ofthe centre’s research results and proposes assetallocation services in the context of a ‘core-satellite’ approach encompassing alternativeinvestments and EDHEC Asset ManagementEducation, which helps investment professionalsto upgrade their skills with advanced risk andassetmanagementtrainingacrosstraditionalandalternativeclasses.

About the EDHEC Risk and Asset Management Research Centre



Founded in1805 inGeneva,Pictet&Cie is todaySwitzerland’s leading private bankers and oneof the leading independent asset managers inEurope.HeadquarteredinGeneva,ithasofficesin:Florence,Frankfurt,Hong-Kong,Lausanne,London,Luxembourg, Madrid, Milan, Montreal, Nassau,Paris, Rome, Singapore, Tokyo, Turin and Zurich.With 2,300 employees, including 500 investmentprofessionals, as well as 100 financial analystsand economics, Pictet’s investments spread overmorethan80countries.ThePictetGrouphasCHF370bnunderadministration(overCHF230bnundermanagement)(31December2006figures).

The Bank is a partnership owned and managedby eight Partner-Managers who pledge theirentirepersonalassetsagainstthebank’sliabilities.Since its inception, Pictet has specialised inwealthmanagement servicesand related services:private banking, institutional asset management(Pictet Asset Management), management andadministrationofinvestmentfunds(PictetFunds),Global Custody and family office services to adiscerningprivateandinstitutionalclienteleworld-wide.

Pictet Asset Management (PAM) includes all theoperating subsidiaries and divisions of the Groupthat carry out institutional asset management.ThroughPAM,Pictet isSwitzerland’sthirdbiggestinstitutional asset manager after UBS and CreditSuisse with CHF 121bn under management. Thefoundationsofthisareaofbusinesswerelaidin1967,andintheearly1980saseparateinstitutionalentitywassetup,makingtheBankoneofthepioneersofinstitutionalmanagementinSwitzerland.

Pictet & Cie has always been committed to themaxim«Innovationisourtradition»,and,inkeepingwith this, theBankwasoneof the first to investin newly industrialising countries through anEmergingMarketsfund,whichitsetupin1990.Thedevelopmentofseveralindicesthatallowinvestorsto compare the performance of occupationalpensionfunds(includingCHFbondindicesin1983andtheBVGIndexin1985)isfurtherevidenceofthisspiritofinnovation.

Pictet isalsoSwitzerland’snumberthreeforfunddistribution, with a volume of CHF 53 billion,

and was a trailblazer for the biotech, water andgenericssectors,beingthefirsttomarkettheseasseparate investment products. Today, the productrange includes over 70 funds. In addition toemergingmarketsandthemedportfolios,PictetisanacknowledgedspecialistinglobalandEuropeanequitiesandsmallcaps,andisalsowellknownforits top-class fixed income products. Furthermore,Pictet now has some CHF 15 billion invested inhedge funds, and can call on nearly 20 years ofexpertiseintheselectionoffundsofhedgefunds.

Pictet was the first bank in Continental Europeto launch a Family Office in 1998 and has beenconsistentlybeenrankedasoneofthebestGlobalCustodiansworldwideforthequalityoftheserviceoverthepast10years.

This specialization and independence enablePictet to avoid conflicts of interest and to makerecommendations with optimal objectivity, freefrom pressures that can occur in major financialgroups. Pictet’s diversified client base includesprivate clients, (e.g. individuals, family trusts andfoundations and independent asset managers),aswellas institutionalclients (e.g.pensionfunds,companies, insurance firms and state entities,distributionpartnersforPictet’sinvestmentfunds).

The partnership status ensures that a long-termviewistakensincethebankispasseddownfromone generation to the next. The bank’s legalstructure, stilloperatingonahumanscale,placesperson-to-person relations at the centre of thedealings.Moreover,theverylowstaffturnoverofPictet enables thebank to foster aunique, long-termrelationshipwitheachandeveryclient.

Pictet remains true to its traditional values andintends to continue to expand in a natural way,preservingbothitsindependenceanditslegalstatusasapartnership,wherebythepartnersstandentirelyandpersonallyliablefortheBank’scommitments.

Ivan Pictet, who today represents the eighthgeneration of the Pictet family, has been seniorpartner since 1 July 2005 and recently reiteratedthat“thebedrockofourbusinesshasalwaysbeenwealthmanagement,andthiswillremainsointhefutureaswell.»

About Pictet & Cie

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�0 EDHEC RISK AND ASSET MANAGEMENT RESEARCH CENTRE

EDHEC RISK AND ASSET MANAGEMENT RESEARCH CENTRE

393-400 promenade des Anglais06202 Nice Cedex 3Tel.: +33 (0)4 93 18 78 24Fax: +33 (0)4 93 18 78 40e-mail: [email protected]: www.edhec-risk.com

PICTET & CIERoute des Acacias 60 1211 Geneva 73SwitzerlandTel: +41 (0) 58 323 23 23Web: www.pictet.com

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