Capital Allocation, Portfolio Enhancement and Performance Measurement : A Unified Approach *)

Transcription

1 Caital Alloation, Portfolio Enhanement and Performane Measurement : A Unified Aroah *) Winfried G. Hallerbah ** ) Aril 30, 2003 * ) I d like to thank artiiants of the EURO Working Grou on Finanial Modeling (Haarlem), and Quantitative Methods in Finane (Sydney) onferenes for their valuable feedbak. Of ourse, the usual dislaimer fully alies. ** ) Det. of Finane, Erasmus University Rotterdam, POB 1738, NL-3000 DR Rotterdam, The Netherlands, hone , fasimile , hallerbah@few.eur.nl, home age: htt:// see also htt://

2 Abstrat Risk analysis, eonomi aital alloation and erformane evaluation are ruial stes in the roess of enterrise-wide risk management. Caital-at-Risk (CaR) lays a entral role sine it determines the amount of eonomi aital that is required to suort firm-wide onsolidated risks and it is the key ingredient of risk-adjusted return (RAROC) measures. The existing literature, however, offers various definitions of RAROC. In addition most aroahes assume a joint-ellitial world. Eseially in the ontext of redit risk, where loss distributions are skewed, this is not realisti. Moreover this leads to biases in estimating the risk ontributions of ortfolio omonents and in determining the subsequent alloation of eonomi aital. In this aer we study aital alloation and risk-adjusted erformane measurement (RAPM) in a oherent and non-arametri framework. Our results an readily be used in a simulation ontext and serve as a benhmark to evaluate the orresonding CreditMetris, CreditRisk+ and KMV aroahes. We first disuss the alloation of eonomi aital over business units or ortfolio omonents aording to their risk ontributions. We then show that the relevant RAROC measure, based on relative risk-return ontributions, atually emerges from the solution to a suitable CaRonstrained ortfolio otimization roblem. This imlied RAROC is imortant as a deision measure for shaing ortfolio omosition ex ante fato; as a erformane measure it serves to evaluate and to attribute ortfolio erformane ex ost fato. However, different deision roblems imly different RAROC measures. The relevant definition of a RAROC measure deends on the seifi deision ontext at hand and, onsequently, no generally valid reies an exist. Hene we roose a unified aroah to ortfolio otimization, eonomi aital alloation and RAPM. Key words: Caital-at-Risk, risk-adjusted erformane evaluation, RAROC, ortfolio otimization, non-arametri methods JEL lassifiation: C13, C14, C15, D81, G11, G20 2

3 1. Introdution Market risks, redit risk and oerational risks are the main risk ategories faed by finanial institutions. To an inreasing degree the allowed exosures from these risks are subjet to regulation (notably by the Basle II roosals). Under the denominator of Enterrise-Wide Risk Management (heneforth EWRM) these risks are analyzed in a oherent way. This is a hallenging task. 1 Finanial institutions hold reserves and rovisions in order to over exeted losses inurred in the normal ourse of business. In order to rovide a ushion against unexeted losses they must hold some amount of aital. The minimum amount of aital required by BIS regulations is termed regulatory aital. Finanial firms also seify internal aital requirements in order to ensure solveny. The minimum amount of internal aital is termed eonomi aital. Defined as a one-sided onfidene interval on otential ortfolio losses over a seifi horizon, Value-at-Risk (VaR) serves the role of setting the aital requirement for market risks. Beause of the frequent ortfolio revisions the VaR horizon is hosen fairly short, ranging from one to twenty trading days. In the ontext of redit risk VaR is often denoted as Credit- VaR; in the general ontext of enterrise-wide risk the VaR measure is termed Caitalat-Risk (CaR). These metris serve to set the amount of eonomi aital. Comared to VaR the fous is more on solveny than on liquidity so that the horizon is longer, tyially one year. 2 Also, sine the ontinuity (non-default) of the firm is at stake the onfidene level is set fairly high, tyially 99% or even 99.5%. 3 For simliity we heneforth gather the onets of VaR, Credit-VaR and CaR under the generi term of CaR. Probabilisti analyses of otential ortfolio losses date bak more than a entury. An analysis of CaR avant la lettre is rovided by Edgeworth [1888] who invoked the entral limit theorem and used quantiles of the normal distribution to analyze otential bank losses and to evaluate bank solveny. Motivated by the BIS roosals and the EU s Caital Adequay Diretives, the release of RiskMetris by J.P.Morgan [1994] in Otober 1994 surred the develoment of the VaR onet. Nowadays a wide variety of analytial and simulation-based estimation methods is available for market risk and redit risk analysis. 4, 5 For a ritial overview of reent develoments, see for examle the seial issue of the Journal of Banking & Finane [2002]. Extending the VaR onet from a trading environment to a redit risk and more general, an EWRM ontext raises some interesting roblems. EWRM entails several stes, viz. risk analysis, ortfolio otimization, eonomi aital alloation, and risk-adjusted erformane (RAP) evaluation. In the first ste the market and redit risks must be analyzed in a onsistent way, reognizing the interdeendeny of these risks. The arametri assumtion of symmetrial (viz. ellitial) distributions may be 1 See for examle Bookstaber [1997] and Cumming & Hirtle [2001]. 2 Horizon issues are disussed in Kuie [1999] and Shen [2001]. 3 Suh high onfidene level renders model validation by baktesting virtually imossible, eseially when ombined with a long horizon. See also footnote 2. 4 The hoie for a seifi estimation method deends on both the degree of non-linearity of the instruments omrised in the ortfolio and the willingness to make restritive assumtions on the underlying statistial distributions. See for examle Duffie & Pan [1997] and Jorion [2001] for an overview. 5 Kuie [2001] disusses estimating Credit-VaR vis à vis different aliations. 3

4 analytially onvenient in CaR analyses 6, but it is not aroriate. The deomosition of ortfolio CaR with reset to individual ativities is then troublesome and may distort the alloation of eonomi aital over ortfolio omonents, ortfolio otimization, and the RAP analysis. In CreditSuisse s CreditRisk+ [1997], for examle, the alloation is erformed roortionally to standard deviation and this aroah breaks down when the imlied ellitiity assumtion is violated. 7 In KMV s [1997] ortfolio otimization roedure a onventional Share [1966,1994] ratio is used, whih is also biased when returns are not ellitial. Also the erformane measurement on the basis of onventional RAROC measures is troublesome. Firstly beause these measures may be based on an unsuitably arametrially estimated CaR. Seondly beause throughout the literature RAROC measures are defined and not derived. 8 Hene there exists substantial ambiguity in ex athedra roosed RAP measures. We here argue that the relevant risk-adjusted erformane measure is imlied by the underlying otimization roblem. Hene the relevant definition of a RAROC measure deends on the seifi deision ontext at hand. Consequently no generally valid reies an exist. In this aer we study ortfolio otimization, aital alloation and risk-adjusted erformane measurement (RAPM) in a oherent and non-arametri framework. Our results an readily be used in a simulation ontext and serve as a benhmark to evaluate the orresonding CreditMetris, CreditRisk+ and KMV aroahes. 9 The outline of the aer is as follows. In setion 2 we briefly review the onets of eonomi aital, CaR, and RAROC, and we disuss the alloation of eonomi aital over business units or ortfolio omonents aording to their risk ontributions. In setion 3 we analyze ortfolio otimality and RAPM in some simlified deision ontexts. We show that the imlied ortfolio otimality onditions guide the hoie of the aroriate RAROC metri. More seifially we show that the relevant RAROC measure, based on relative risk-return ontributions, atually emerges from the solution to the suitable CaR-onstrained ortfolio otimization roblem. This imlied RAROC is imortant as a deision measure for shaing ortfolio omosition ex ante fato; as a erformane measure it serves to evaluate and to attribute ortfolio erformane ex ost fato. Hene we roose a unified aroah to ortfolio otimization, eonomi aital alloation and RAPM. In ratie, there are restritions on ortfolio revisions and flexibility in ortfolio omosition is limited. Attention thus shifts from fully-fledged ortfolio otimization to ortfolio enhanement. In this ontext our results an be used (i) to estimate the risk-return trade-off that is imlied by a given sub-otimal ortfolio, (ii) to gauge the degree of its sub-otimality, and (iii) to imrove the ortfolio in aordane with the estimated riskreturn trade-off. Setion 4 onludes the aer and resents lines for further researh. The Aendix ontains tehnial details. 6 See for examle Saita [1999], Stoughton & Zehner [1999,2000] and Dowd [1998,1999,2000] who assume normality throughout. In resonse, Tashe & Tibiletti [2001] relax this assumtion by investigating suitable aroximations. In Hallerbah [2003] we also resent a non-arametri aroah. 7 See Hallerbah [2003]. For orret deomosition roedures in CreditRisk+ see Haaf & Tashe [2002] and Kurth & Tashe [2003]. 8 See Bessis [2002] and Matten [2000], e.g. 9 For a omarison of these Credit-VaR models we refer to Crouhy, Galai & Mark [2000]. 4

5 2. Preliminaries In this setion we introdue notation and resent some useful results. We first disuss the theoretial onets of overall CaR, marginal CaR and omonent CaR. 10 Sine we want to disuss CaR and RAROC in the most general ontext, the only (and very weak) assumtion we make is that all relevant return distributions have finite first moments. We then summarize the onet of RAROC and outline its role in RAPM. defining overall CaR Consider a ortfolio with urrent value V, onsisting of N omonents. In the broad ontext of EWRM the ortfolio is the overall firm and the omonents reresent the artitioning of the firm s business ativities aording to searate ativities, organized ativities or business units, e.g. In the ontext of redit risk the omonents are the individual redits or loans omrised in the ortfolio. Given the urrent values { V i } i of the dollar ositions in eah of the omonent ativities, the hange in ortfolio value over a holding eriod t equals: = with : i = V (2.1) V V r Vi ri i V i where r and r i denote the t return on the ortfolio and ativity i, resetively. A tilde marks a stohasti variable. 11 All returns denote total returns and reflet both hanges in market value ( rie return ) and ash flows ( ash return ) during the eriod. 12 The ortfolio omosition is assumed onstant over the eriod t. Given the ortfolio, its exeted dollar return is: (2.2) V Vr Vr i i i = where r i is the exeted erentage return on ativity i. Given a onfidene level and an evaluation horizon of quantile erentage return t, we define the quantile dollar return V and the r (given V ) that satisfy the seified onfidene level: (2.3) { } { } = : Pr V = r V = 1 r : Pr r r 1 10 The following results (and notably the orresondene between CaR ontributions and onditional exetations) have been first derived by Hallerbah [2003] and Tashe [1999]. V on the robability sae (Ω,F, Pr( )), where the σ-field F 11 We define the variates { V } i and ontains subsets of the samle sae Ω. 12 When a market value is not available, for examle for non-traded redits, the mark-to-market valuation is relaed with mark-to-model valuation. This introdues model risk; f. Kato & Yoshiba [2000]. 5

6 Overall ortfolio CaR is now given by CaR V = r V sine CaR is defined in terms of losses. 13 When V is initially given, fous an be on r instead of on V. defining marginal exeted return and marginal CaR When studying ortfolio otimality in setion 3 we need information about the marginal exeted returns and the marginal CaRs of the individual seurities omrised in the ortfolio. From (2.2) it follows that marginal exeted return of seurity i is: (2.4) ( rv ) = ri i V i The marginal CaR MCaR i is the hange in ortfolio CaR resulting from a marginal hange in the dollar osition in omonent ativity i : (2.5) MCaR i CaR i V i Note that eq.(2.5) also alies to an ativity N that has not yet been inluded in the firm s ortfolio. The initial ortfolio then omrises N 1 omonent ativities and we onsider this (N 1)-element ortfolio as an N-element ortfolio where V N = 0 initially. To evaluate marginal CaR we start from eq.(2.1) whih identifies the ortfolio dollar return as a onvex ombination of the dollar returns on the individual omonents. Beause of the ortfolio artitioning and by the very definition of onditional exetations we have: (2.6) V = E{ r V V } = VE i { r i } i Note that the onditional exetation E { ri } is a random variable. 14 Also note that the ortfolio dollar return is a linearly homogeneous funtion of the ositions { V i } i. Sine this funtion is ontinuous and analyti we an aly Euler s theorem: (2.7) V = V = V r i i i i Vi i 13 We obtain hanges in dollar values by ombining returns with mark-to-market values. Whether fousing on returns or on dollar ositions, transations with zero initial value (suh as redit swas) have to be deomosed into non-zero long and short ositions (maing). The firm s overall ortfolio will have strit ositive initial value. 14 Hene E { r } is to be interreted as the exetation of r ~ i onditional to the σ-field F relative to whih i is defined. For details, see Sanos [1986]. 6

7 Substituting eq.(2.7) in (2.6) and onditioning on V = yields: (2.8) CaR { = V = VE i V = V = VE i r i } i Vi where we have added a minus sign sine CaR is defined in terms of losses. Sine the ortfolio return now takes the artiular value V the onditional exetations beome deterministi. Under mild regularity onditions we an interhange the exetations integral and the derivative in the seond equality of eq.(2.8). Hene, the onditional exetation reresents / Vi : minus the marginal CaR of i, MCaR i. From the third equality it then follows that: (2.9) MCaRi = E{ r i } Following omlementary reasoning, Tashe [1999] indeendently derived the same result in a VaR ontext. For formal roofs we refer to Tashe [1999] and Gouriéroux et al. [2000]. The intuition behind eq.(2.9) is lear. When there is a ositive (negative) interdeendene between i and then large negative ortfolio returns will on average be assoiated with large negative (ositive) omonent returns. Inreasing (dereasing) the size of the ativity osition V i will then lower the ortfolio value even more, thus inreasing the ortfolio s CaR. i defining and relating omonent CaR Sine: (2.10) { } CaR = V E r V = V MCaR i i i i i i eah term i measures the total ontribution of asset i to the overall ortfolio VaR. Hene Vi MCaRi = CCaRi is the omonent CaR of ativity i. These omonent CaRs an uniquely be attributed to eah of the individual omonents of that ortfolio and aggregate linearly into the total diversified ortfolio CaR: (2.11) CaR CCaRi = Vi MCaRi i i Note that beause of return interdeendenies and diversifiation effets the omonents stand-alone CaRs do not add u to the diversified ortfolio CaR. 15, The break-down of VaR aording to ortfolio omonents or market risk fators as suggested by Fong & Vasiek [1997], for examle, suffers from this shortoming and is hene not useful. For a break-down of VaR aording to an underlying fator model, see Hallerbah & Menkveld [2003]. For an evaluation of different deomosition methods we refer to Koyluoglu & Stoker [2002]. 7

8 In Hallerbah [2003] we show how marginal and omonent CaRs an be estimated in a non-arametri ontext. Eq.(2.11) is a owerful result. It does not deend on any distributional assumtions but revails sine the ortfolio oerator is linear. Without loss of generalization the omonent ativities may be maed in a non-linear fashion onto standardized ositions or underlying state variables (suh as default roesses or reovery rates, as in J.P.Morgan s [1997] CreditMetris ). eonomi aital, RAPM and RAROC Sine eonomi aital is neessary to over otential losses from the firm s ativities ositions { V i } i, the RAP is measured by relating generated inome to eonomi aital. The resulting RAP metri, termed RAROC, was first roosed by Bank of Ameria. 17 It takes the form: (2.12) adjusted inome RAROC = CaR The denominator is risk-adjusted or eonomi aital. On the aggregate level this is the firm s equity and the CaR s onfidene level is in line with the firm s default robability. In the numerator inome is revenues minus osts minus exeted losses. The adjustment for exeted loss is generally onsidered as a risk orretion (although it is a rovision for exeted losses, whih by definition does not reresent risk). For this reason, (2.12) is sometimes alled a RARORAC (risk-adjusted return on riskadjusted aital) measure. Various seifi definitions exist (suh as RAROC, RARORAC and RORAC), but most variations are due to the seifiation of the numerator. The numerator indeed raises some questions. Should finaning (oortunity) osts be taken into aount? The numerator then reresents the eonomi rofit. More imortantly we ask ourselves why the fous is on ash inome? Return an also be generated from aital gains or losses? Defining inome on a mark-to-market basis an orret for this. In ex ost aliations, how is the numerator measured? Sine risk is involved we would like to seify the numerator as an exeted return. In ex ost aliations the fair game assumtion ould be invoked to estimate the exeted value by means of an historial average. The denominator raises the issue of how to alloate the amount of eonomi aital that relates to the overall diversified firm ortfolio over the different sub-levels within the firm (ranging from business units, via geograhial loations to trading desks, individual traders and ultimately to individual business transations). These ambiguities all for a fool roof definition of RAROC. However, in the next setion we argue that the relevant definition of the RAP measure deends on the deision ontext, omrising the ursued objetive(s) and the imosed onstraints. 16 The sum of the stand-alone VaRs/CaRs an be larger than the ortfolio VaR/CaR but also smaller. The latter henomenon indiates that VaR is not sub-additive. Hene, VaR is not a oherent risk measure; see Artzner et al. [1999]. Denault [2001] translates the formulated oherene requirements to aital alloation measures. 17 See Zaik, Walter & Kelling [1996]. For general exosés we refer to Bessis [2002], Matten [2000] or Smithson & Hayt [2001]. 8

9 3. Portfolio otimization, RAROC and RAPM In this setion we show how the relevant definition of risk ontributions and various RAROC measures are imlied by a ortfolio otimization model. Eah RAROC measure is relevant within the seifi underlying otimization ontext. We assume that the firm strives to imize the exeted return on its ativities ortfolio subjet to a onstraint on the required eonomi aital. Eonomi aital is measured by the firm s CaR over a given horizon. We lassify the ossible models using two dimensions. The first dimension is defined by the sale of oerations. The aital invested in the firm ativities (or the budget available for the firm s business ventures) V may either be fixed or free. In the former ase the available aital is restrited (arallelling a standard ortfolio investment roblem); in the latter ase the firm may inrease (derease) the sale of its ativities by raising more (less) aital. The seond dimension is defined by the tye of CaR onstraint. 18 This onstraint on eonomi aital may be formulated in either absolute or relative terms. The absolute CaR onstraint is given by the ortfolio dollar CaR level CaR that should not be, exeeded. The relative onstraint is defined in erentage terms r of V (see eq.(2.3)). The ossible ombinations are summarized in Exhibit 1. Exhibit 1: A simle tyology of CaR-onstrained otimization roblems CaR restrition firm s aital absolute ($) relative (%) V free I: V = i s.t. CaR IV: V = i s.t. V r, V fixed II: V = rv s.t. CaR V Vi III: V = rv s.t., V r V Vi In all ases we may wish to restrit short ositions, V 0, i. In that ase Kuhn- Tuker onditions will aly and only ositive ositions are onsidered. 19 When V is i 18 There is some debate whether CaR should be disounted over the horizon or not. When CaR should over otential losses at the end of the horizon the disounting argument is lear. When also intermediate losses should be overed the ase is not lear. In the following we refrain from disounting CaR, but the neessary adjustment is obvious. 19 Throughout the aer we assume that seond order onditions are satisfied. Hene we assume that the feasible CaR region is onvex. 9

10 , fixed, CaR r V so the otimization roblems II and III are equivalent. When the CaR restrition is formulated in relative terms and V is not fixed, as in IV, the roblem beomes indeterminate and an only be solved for { V i } given some level of V. Atually, IV is not realisti under our simle assumtions sine in ratie there will be some limit to the firm s ativities anyhow. Should we allow for a trade-off between exeted return and CaR, or when we would add other restritions, IV would beome a relevant starting oint. But for now we are left with two different ases: on the one hand we have roblem I, and on the other roblems II and III. Another aset that roves to be imortant is whether riskfree ativities (riskfree borrowing and/or lending) are available to the firm. In setion 3.1 we assume that all ortfolio omonents are risky, so there does not exist a riskfree rate. In setion 3.2 we dro this assumtion and allow for riskfree investment oortunities. 3.1 Portfolio otimization without riskfree rate roblem I In situation I the firm strives to imize the exeted dollar return over the hosen time horizon subjet to an absolute CaR onstraint. The risk and hene the eonomi aital of the firm s ativities is restrited by the imum admissible CaR level CaR. The otimization roblem beomes hoosing { V } suh that: i i (3.1) { Vi } V st.. CaR We an safely assume that there exist suffiient rofitable business ventures so that the CaR onstraint is binding. Hene the imum allowed amount of eonomi aital will be emloyed. Forming the Lagrangian and taking the artial derivatives to V i leads to the following first order onditions (FOCs heneforth): (3.2) r λ MCaR = 0 i * i i together with the original CaR onstraint. λ is the Lagrange multilier and an asterisk refers to the otimum. Multilying with V i and summing over i * yields, in ombination with (3.2): (3.3) * j * i * = = i, j * CCaR CCaR CaR i* j* * rovided that MCaRi* 0, i *. Eq.(3.3) is the ortfolio otimality ondition. When the ortfolio is otimal the ratio of marginal (total) return ontribution and marginal (total) CaR ontribution is onstant over all ativities in *. To allow for zero marginal CaRs we rewrite (3.3) as: 10

11 (3.4) * CCaRi* Vi = * i * CaR * Otimal alloation of aital is ahieved when (3.3) (or (3.4)) is satisfied. Note that for eah ativity its exeted dollar return should be related to its total ontribution to the diversified ortfolio CaR. In the last term of (3.3) we reognize the familiar firm-wide RARORAC (or RAROC). The numerator is the exeted dollar return on the ativity ortfolio. By definition this return (i) is net of the exeted loss and (ii) inludes rie returns (aital gains/losses). In the onventional definition, only the ash return (i.e. inome) is onsidered (ontrasting (ii)) and subtrating the exeted loss is meant to yield the risk-adjusted return. Sine the exeted loss is exeted by definition, this is not a risk orretion at all. The first term of (3.3) indiates how to araise the ex ante erformane of ativity i: by relating its exeted dollar return to its ontribution to overall eonomi aital. Now suose that given some ortfolio we find that: i j (3.5) > > i, j CCaR CCaR CCaR i j Obviously is not otimal. This imlies that an be enhaned by inreasing the osition in ativity i and dereasing the osition in j. Ex ante erformane analysis is thus relevant for evaluating the otimality of some (initial) ortfolio and deriving ortfolio revision reies. Under a fair game assumtion ativity i s ex ost erformane an be gauged by relating its average realized dollar return to its ontribution to overall eonomi aital. In ratie the resrition is to imize the firm s RAROC. As the Aendix shows, imizing RAROC yields the same FOCs eqs.(3.3) and (3.4). Obviously the unonstrained imization of onventional RAROC assumes the underlying otimization roblem (3.1). Conversely, imizing RAROC an be justified on the basis of (3.1). But suose now that the firm s total aital V is fixed, or that riskfree ventures exist. Obviously the ortfolio otimality onditions will hange and hene the imlied risk-adjusted erformane measure. This is investigated below. roblems II and III In situations II and III the total available aital V is fixed and the firm strives to imize the exeted dollar return over the hosen time horizon subjet to an absolute or relative CaR onstraint. The otimization roblem now beomes hoosing { V i } i suh that: (3.6) { V } i V = r V, st.. CaR = V r 11

12 V Vi From the Lagrangian the FOCs are: (3.7) r λ MCaR θ = 0 i * i i together with the original onstraints. λ and θ are the Lagrange multiliers of the CaR and the aital onstraints, resetively. Multilying with V i and summing over i * yields, in ombination with (3.7): (3.8) * * j θ j * θ * * * i θv V V i = = i, j * CCaR CCaR CaR i* j* * with CaR * = CaR, rovided that MCaR * 0, i *. This imlied ortfolio otimality ondition is equivalent to: i (3.9) ri θ r θ r θ = = i, j * MCaR MCaR r j * i* j* * When the ativity ortfolio is otimal the ratio of marginal (total) adjusted return ontribution and marginal (total) CaR ontribution is onstant over all ativities in *. Exeted returns are adjusted with a fator θ (the shadow rie of relaxing the aital onstraint) indiating the erentage oortunity ost of obtaining additional funds. To allow for zero marginal CaRs we rewrite (3.8) as: (3.10) CCaR V = V i * * * i* i θ i * * CaR θ * Otimal alloation of the available restrited aital is now ahieved when (3.8) (or (3.9) or (3.10)) is satisfied. The last term of (3.8) (or (3.9) in erentage terms) is the relevant imlied riskadjusted erformane metri. The numerator is the exeted adjusted return on the ativity ortfolio. Again, this return (i) is net of the exeted loss and (ii) inludes rie returns (aital gains/losses). Moreover it is (iii) adjusted for the imlied shadow ost θ of obtaining additional funds. The first term of (3.8) indiates how to araise the erformane of ativity i: by relating its exeted or average adjusted dollar return to its ontribution to overall eonomi aital. From an ex ante ersetive, deviations for the FOCs an be used to guide ortfolio revisions in order to enhane the subotimal ortfolio. When the adjustment sub (iii) is ignored, otimal alloation is not guaranteed. Likewise, ex ost erformane analysis is distorted. Conventional RAROC analysis on the basis of (3.3) for the ortfolio, or the individual ativities omrised therein, will fail in this ase. As shown in the Aendix, the unonstrained imization of onventional RAROC is at odds with the underlying otimization roblem (3.6). 12

13 Let θ now be the exliitly seified erentage ost of inreasing the aital base. The otimization roblem beomes rv i i θ Vi V subjet to CaR (with V no longer restrited). This alternative roblem resembles situation I where V is not fixed, but has the same FOCs (3.8) and (3.10) as above. Alternatively, the total finaning osts over V (not restrited) an be taken into aount, leading to ( ri θ ) Vi subjet to the CaR onstraint. Again this results in the same FOCs (3.8) and (3.10). 3.2 Portfolio otimization allowing for riskfree ativities We now assume that riskfree borrowing and lending oortunities exist for the firm. Denoting ativity 1 as riskfree, its return is the riskfree rate r f. In general, the dollar return on the firm s total ativity ortfolio is: N (3.11) V = Vr 1 f + Vr i i = Vr 1 f + q with V q = Vqr q i= 2 where V q is the risky art q of ortfolio, satisfying V q N Vr. When V is fixed, we have V1 V Vr i= 2 ii as a funtion of the risky ventures. Defining the weight w of the risky ativities in the total ortfolio, the exess total ortfolio return is: (3.12) r rf = w ( rq rf ) = with w V / V Portfolio s exess dollar return CaR is: (3.13) ( ) CaR V r r = CaR + V r f f f It finally readily follows that the exess dollar return CaRs of and q are equal: 1 (3.14) ( ) ( ) CaR = V r r = V w r r CaR w f f q q f qf We now revisit the four roblems in Exhibit 1. q N i= 2 i i roblem I In this situation the firm strives to imize the exeted dollar return over the unrestrited aital V subjet to the absolute CaR onstraint: { } (3.15) { } q ( 1 f ) Pr CaR = Pr CaR + V r = 1 13

14 where we have used (3.11). Hene: (3.16) q = q = + 1 f CaR V CaR V r Eq.(3.16) shows that this otimization ase is not interesting. The absolute CaR restrition on an be satisfied with any risky ortfolio q simly by adding suffiient riskfree investment V 1 > 0. roblems II and III When onsidering V fixed, the otimization roblem is given by eq.(3.6) with FOCs (3.7). For the riskfree ativity i=1 we have MCaR1 = rf, so the FOC beomes: (3.17) θ = ( + λ) 1 rf Substituting in (3.7) yields: (3.18) i f λ ( i f ) r r = MCaR + r i * * Multilying with V i and summing over i * gives: (3.19) V ( r* rf ) = λ ( CaR* + Vrf ) The LHS of (3.19) is the exeted exess dollar return (i.e. dollar risk remium) on ortfolio, and the term in arentheses on the RHS is s exess dollar return CaR: (3.20) CaR ( f CaR + Vrf = V r rf ) Eqs.(3.18) and (3.19) translate into: (3.21) * * * * * i f i j f j * f r V r V r V = = i q* CCaR CCaR CaR if * jf * f * with CaR f* f = CaR in exess return form, rovided that MCaR * 0, i *. This is the imlied ortfolio otimality ondition, equivalent to: if (3.22) r r r r r r r r = = = i q* MCaR r MCaR r r r r r i f j f * f q* f i* + f j* + f * + f q* + f The last equality follows from (3.12) and (3.14).The notable differene with (3.9) is that the denominators are the exess return CaR ontributions. Dowd [2000,.221] disqualifies onventional RAROC sine it an beome infinitely large by investing all 14

15 aital in riskfree ventures. But we see that the relevant RAROC measure in this limit ase beomes indeterminate. To allow for zero marginal CaRs we rewrite (3.22) as: (3.23) MCaR + r r r = r r i * i* f i f * f CaR* + V* r f Multilying both sides of (3.23) with V translates the exression into dollar terms as in (3.21).Note that (3.23) orresonds to (3.4) ast in exess return form. The FOC eq.(3.22) is idential to the FOC for the mean-var ortfolio seletion roblem as derived in Grootveld & Hallerbah [2003]. It reveals linear twofund searation, i.e. the otimal alloation within the risky ortfolio q is indeendent of the total ortfolio. For any value of the imum admissible CaR the orresonding otimal ortfolio alloation onsists of a linear ombination of the risk free investment and only one single risky ortfolio q. 20 Note that this searation roerty is fundamentally different from that imlied by the imization of exeted utility; for the latter otimization ase the results of Cass & Stiglitz [1970] are definitive. The otimal alloation of the available aital is ahieved when (3.21) or (3.22) is satisfied, and these onditions aly for ortfolio q. Given ortfolio q and the CaR onstraint, ortfolio readily follows. The last term of (3.21) (or (3.22) in erentage terms) is the relevant imlied risk-adjusted erformane metri for this ase. The numerator is the (dollar) risk remium on the ativity ortfolio. 21 The first term of (3.21) or (3.22) shows how to araise the erformane of ativity i: by relating its risk remium (or average exess return) to its ontribution to overall exess eonomi aital CaRf = CaRqf. Again, using a onventional RAROC measure will loud the erformane analysis. An interesting aset of this artiular roblem is that it simly is otimization roblem I as studied in setion 3.1, but now fully ast in exess returns. Hene the result from the Aendix alies: unonstrained imization of the imlied adjusted RAROC measure (the last term of (3.21) or (3.22)): (3.24) rfv r rf = CaR r r ( ) f f yields the same FOCs as in the two LHS terms of (3.21) or (3.22): (3.25) * * * * i f i j f j r V r V = i * CCaR CCaR if * jf * 20 See also Arza & Bawa [1977] and Tashe [1999] on this oint. 21 Crouhy, Turnbull & Wakeman [1999] define an adjusted RAROC measure in the form of RAROC minus the riskfree rate, divided by CAPM beta. This is to aount for systemati (ried) risk instead of total firm risk. However, they do not derive the metri nor indiate how it ould be derived. 15

16 Note that the imand in (3.24) is in sirit similar to the Share ratio 22 : it measures the exeted exess return er unit of risk where risk is here defined in terms of the CaR of the exess returns. Without restriting referenes, mean-variane analysis is valid when ortfolio returns are ellitially distributed with finite variane. 23 Suh ellitial distributions are fully defined by the loation and sale arameters and do not exhibit skewness. 24 In an ellitial world the exess CaR of ortfolio is simly defined by the first two statistial moments of the ortfolio return distribution: (3.26) CaRf = rf = rf + k() σ f (with σ f < ) whereσ f is the standard deviation of the exess return on ortfolio and k () is the roortionality fator belonging to the CaR onfidene level. For examle when 1 ortfolio returns are normally distributed we have k () = N () where N( ) is the standard normal distribution funtion. In addition, in an ellitial world CaR satisfies the sub-additivity roerty; f. Embrehts et al. [2002]. Inororating (3.26) in (3.24) and some rearranging yields: (3.27) 1 f σ f f r r r r = 1 + k ( ) rf + k() σ f r rf σ f given. So in an ellitial world, the otimization roblem (3.24) entails finding the ortfolio that imizes its Share ratio (the last term in (3.27)). Hene in an ellitial world the two ortfolio otimization roblems are omletely equivalent. This ontradits Cambell, Huisman & Koedijk [2001]. However, as we exet that the underlying distributions in a RAPM ontext are asymmetri, this disqualifies the onvenient ellitial arametri assumtion. For onveniene, the relevant RAROC measures imlied in eah of the deision situations is summarized in Exhibit 2. (Reall that IV is indeterminate.) 22 See Share [1966,1994]. 23 See Owen & Rabinovith [1983]. 24 The general lass of (both finite and infinite variane) ellitial distributions inludes the Student t distribution, symmetri stable (Pareto-Lévy) distributions with harateristi exonent smaller than two and the normal distribution. Also non-normal variane mixtures of multivariate normal distributions belong to the ellitial lass, see for instane Chmielewski [1981] and Fang, Kotz & Ng [1990]. 16

17 Exhibit 2: Adjusted RAROC measures imlied in eah of the deision situations (see in Exhibit 1) of CaR-onstrained otimization. riskfree ventures firm s aital no yes ( r f ) V free I: CaR * * I: NA V fixed II, III: θv * * CaR * II, III: * * rfv CaR f * 4. Conlusions We argue that without exliitizing the relevant EWRM deision ontext (in the form of learly and unambiguously stiulating the objetives and onstraints), it is not feasible to: ex ante otimize the firm s ativities ortfolio; ex ante alloate eonomi aital over ativities omrised in the firm-wide ortfolio aording to their risk ontributions; ex ante evaluate the quality of the ativities ortfolio in the light of ursued objetives and imosed onstraints; ex ost evaluate overall ortfolio erformane; ex ost attribute erformane to individual ativities. We illustrated our argument with various simle otimization examles. Even though the resented deision ontexts are simlified, our results learly show that alying RAPM on the basis of onventional RAROC measures (as resented in the literature) more often than not may lead to erroneous onlusions and ations. For deriving relevant RAP measures we suggest the following reie. Firstly, identify the relevant objetive(s) and onstraints. Seondly, derive the imlied ortfolio otimality onditions from the aroriate otimization roblem. Finally, aly the imlied relevant RAP measure for ex ante ortfolio enhanement and ex ost erformane evaluation and attribution. A hallenging route for further researh is to unover more omlex tyologial deision ontexts as we may enounter them in ratie and to reveal the imlied and hene adequate RAP metris. 17

18 Aendix Let A(x) and B(x) be analyti funtions of vetor x=[x i ], homogeneous of degrees g and h, resetively. Consider the following onstrained otimization roblem: (A.1) A( x ) st.. B( x ) B { x} where B is a (negative) number. The FOCs are: (A.2) A( x) + λ B( x ) =0 together with the original onstraint, where λ is the Lagrange multilier and is the gradient oerator. We assume seond order onditions are satisfied. Premultilying (A.2) with x, solving for λ and substituting bak yields the ortfolio otimality ondition: (A.3) B( x* ) A( x* ) = A( x* ) h B( x* ) g where asterisks denote the otimum. Now onsider the first unonstrained roblem, stiulating a fixed trade-off between A(x) and B(x) governed by the arameter γ > 0 : (A.4) A( x) +γ B( x ) { x} This yields the same FOCs (A.2) and ortfolio otimality ondition (A.3) Next onsider the seond unonstrained roblem: (A.4) A( x) B( x) 0 { x} B( x) The FOC is: (A.5) A( x* ) A( x* ) B( x* ) =0 B( x* ) whih translates into (A.3) when g = h= 1 (i.e. linear homogeneity). The third unonstrained roblem: (A.6) A( x) θ B( x) 0, θ 0 onstant { x} B( x) has FOCs: 18

19 (A.7) A( x* ) θ A( x* ) B( x* ) =0 B( x* ) whih is inomatible with (A.3), even when g = h= 1. Now let the exeted ortfolio return orresond to A(x) above, and the ortfolio CaR CaR (defined in terms of losses) orresond to -B(x). Sine V and CaR are linearly homogeneous in the ativities { V i } i, imizing V subjet to only a onstraint on CaR is equivalent to imizing RAROC CaR unonstrained. Moreover, sine RAROC is homogeneous of degree zero, this measure an be imized without taking into aount any restrition on V. The solution an simly be saled to satisfy this restrition. However, from the third roblem (A.6) we have: (A.8) CaR θv CaR 19

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