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

Transcription

1 Catal Alloaton, Portfolo Enhanement and Performane Measurement : A Unfed Aroah Wnfred G. Hallerbah * ) November 15, 2001 reared for : EURO WGFM 2001, Haarlem NL QMF2001, Sydney AUS * ) Det. of Fnane, Erasmus Unversty Rotterdam, POB 1738, NL-3000 DR Rotterdam, The Netherlands, hone , fasmle , hallerbah@few.eur.nl, home age: htt:// see also htt://

2 Abstrat Rsk analyss, eonom atal alloaton and erformane evaluaton are rual stes n the roess of enterrse-wde rsk management. Catal-at-Rsk (CaR) lays a entral role sne t determnes the amount of eonom atal that s requred to suort frm-wde onsoldated rsks and t s the key ngredent of rsk-adjusted return (RAROC) measures. The exstng lterature, however, offers varous defntons of RAROC. In addton most aroahes assume a jont-elltal world. Eseally n the ontext of redt rsk, where loss dstrbutons are skewed, ths s not realst. Moreover ths leads to bases n estmatng the rsk ontrbutons of ortfolo omonents and n determnng the subsequent alloaton of eonom atal. In ths aer we study atal alloaton and rsk-adjusted erformane measurement (RAPM) n a oherent and non-arametr framework. Our results an readly be used n a smulaton ontext and serve as a benhmark to evaluate the orresondng CredtMetrs, CredtRsk+ and KMV aroahes. We frst dsuss the alloaton of eonom atal over busness unts or ortfolo omonents aordng to ther rsk ontrbutons. We then show that the relevant RAROC measure, based on relatve rsk-return ontrbutons, atually emerges from the soluton to a sutable CaR-onstraned ortfolo otmzaton roblem. Ths mled RAROC s mortant as a deson measure for shang ortfolo omoston ex ante fato; as a erformane measure t serves to evaluate and to attrbute ortfolo erformane ex ost fato. However, dfferent deson roblems mly dfferent RAROC measures. The relevant defnton of a RAROC measure deends on the sef deson ontext at hand and, onsequently, no generally vald rees an exst. Hene we roose a unfed aroah to ortfolo otmzaton, eonom atal alloaton and RAPM. In rate, there are restrtons on ortfolo revsons and flexblty n ortfolo omoston s lmted. Attenton thus shfts from fully-fledged ortfolo otmzaton to ortfolo enhanement. In ths ontext our results an be used () to estmate the rsk-return trade-off that s mled by a gven sub-otmal ortfolo, () to gauge the degree of ts sub-otmalty, and () to mrove the ortfolo n aordane wth the estmated rsk-return trade-off. Key words: Catal-at-Rsk, rsk-adjusted erformane evaluaton, RAROC, ortfolo otmzaton, non-arametr methods JEL lassfaton: C13, C14, C15, D81, G11, G20 2

3 1. Introduton Market rsks, redt rsk and oeratonal rsks are the man rsk ategores faed by fnanal nsttutons. To an nreasng degree the allowed exosures from these rsks are subjet to regulaton (notably by the Basle II roosals). Under the denomnator of Enterrse-Wde Rsk Management (EWRM) these rsks are analyzed n a oherent way. Ths s a hallengng task. 1 Fnanal nsttutons hold reserves and rovsons n order to over exeted losses nurred n the normal ourse of busness. In order to rovde a ushon aganst unexeted losses they must hold some amount of atal. The mnmum amount of atal requred by BIS regulatons s termed regulatory atal. Fnanal frms also sefy nternal atal requrements n order to ensure solveny. The mnmum amount of nternal atal s termed eonom atal. Defned as a one-sded onfdene nterval on otental ortfolo losses over a sef horzon, Value-at-Rsk (VaR) serves the role of settng the atal requrement for market rsks. Beause of the frequent ortfolo revsons the VaR horzon s hosen farly short, rangng from one to twenty tradng days. In the ontext of redt rsk VaR s often denoted as Credt-VaR; n the general ontext of enterrse-wde rsk the VaR measure s termed Catal-at-Rsk (CaR). These metrs serve to set the amount of eonom atal. Comared to VaR the fous s more on solveny than on lqudty so that the horzon s longer, tyally one year. 2 Also, sne the ontnuty of the frm s at stake the onfdene level s set farly hgh, tyally 99% or even 99.5%. 3 For smlty we heneforth gather the onets of VaR, Credt-VaR and CaR under the gener term of CaR. Probablst analyses of otental ortfolo losses date bak more than a entury. An analyss of CaR avant la lettre s rovded by Edgeworth [1888] who nvoked the entral lmt theorem and used quantles of the normal dstrbuton to analyze otental bank losses and to evaluate bank solveny. Motvated by the BIS rules and the EU s Catal Adequay Dretves, the release of RskMetrs by J.P.Morgan [1994] n Otober 1994 surred the develoment of the VaR onet. Nowadays a wde varety of analytal and smulatonbased estmaton methods s avalable for market rsk and redt rsk analyss. 4,5 Extendng the VaR onet from a tradng envronment to a redt rsk and more general, an EWRM ontext rases some nterestng roblems. EWRM entals several stes, vz. rsk analyss, ortfolo otmzaton, eonom atal alloaton, and rsk-adjusted erformane (RAP) evaluaton. In the frst ste the market and redt rsks must be analyzed n a onsstent way, reognzng the nterdeendeny of these rsks. The arametr assumton of symmetral (vz. elltal) dstrbutons may be analytally onvenent n CaR analyses 6, but s not arorate. The deomoston of ortfolo CaR wth reset to 1 See for examle Bookstaber [1997] and Cummng & Hrtle [2001]. 2 Horzon ssues are dsussed n Kue [1999] and Shen [2001]. 3 Suh hgh onfdene level renders model valdaton by baktestng vrtually mossble, eseally when ombned wth a long horzon. See also footnote 2. 4 The hoe for a sef estmaton method deends on both the degree of non-lnearty of the nstruments omrsed n the ortfolo and the wllngness to make restrtve assumtons on the underlyng statstal dstrbutons. See for examle Duffe & Pan [1997] and Joron [2000] for an overvew. 5 Kue [2001] dsusses estmatng Credt-VaR vs à vs dfferent alatons. 6 See for examle Sata [1999], Stoughton & Zehner [1999,2000] and Dowd [1998,1999,2000] who assume normalty throughout. In resonse, Tashe & Tblett [2001] relax ths assumton by nvestgatng 3

4 ndvdual atvtes s then troublesome, and may dstort the alloaton of eonom atal over ortfolo omonents, ortfolo otmzaton, and the RAP analyss. In CredtSusse s CredtRsk+ [1997], for examle, the alloaton s erformed roortonally to standard devaton and ths aroah breaks down when the mled elltty assumton s volated. 7 In KMV s [1997] ortfolo otmzaton roedure a onventonal Share [1966,1994] rato s used, whh s also based when returns are not elltal. Also the erformane measurement on the bass of onventonal RAROC measures s troublesome. Frstly beause these measures may be based on a unsutably arametrally estmated CaR. Seondly beause throughout the lterature RAROC measures are defned and not derved. 8 Hene there exsts substantal ambguty n ex athedra roosed RAP measures. We here argue that the relevant rsk-adjusted erformane measure s mled by the underlyng otmzaton roblem. Hene the relevant defnton of a RAROC measure deends on the sef deson ontext at hand. Consequently no generally vald rees an exst. In ths aer we study ortfolo otmzaton, atal alloaton and rsk-adjusted erformane measurement (RAPM) n a oherent and non-arametr framework. Our results an readly be used n a smulaton ontext and serve as a benhmark to evaluate the orresondng CredtMetrs, CredtRsk+ and KMV aroahes. 9 The outlne of the aer s as follows. In seton 2 we brefly revew the onets of eonom atal, CaR, and RAROC, and we dsuss the alloaton of eonom atal over busness unts or ortfolo omonents aordng to ther rsk ontrbutons. In seton 3 we analyze ortfolo otmalty and RAPM n some smlfed deson ontexts. We show that the mled ortfolo otmalty ondtons gude the hoe of the arorate RAROC metr. More sefally we show that the relevant RAROC measure, based on relatve rskreturn ontrbutons, atually emerges from the soluton to the sutable CaR-onstraned ortfolo otmzaton roblem. Ths mled RAROC s mortant as a deson measure for shang ortfolo omoston ex ante fato; as a erformane measure t serves to evaluate and to attrbute ortfolo erformane ex ost fato. Hene we roose a unfed aroah to ortfolo otmzaton, eonom atal alloaton and RAPM. In rate, there are restrtons on ortfolo revsons and flexblty n ortfolo omoston s lmted. Attenton thus shfts from fully-fledged ortfolo otmzaton to ortfolo enhanement. In ths ontext our results an be used () to estmate the rsk-return trade-off that s mled by a gven subotmal ortfolo, () to gauge the degree of ts sub-otmalty, and () to mrove the ortfolo n aordane wth the estmated rsk-return trade-off. Seton 4 onludes the aer and resents lnes for further researh. sutable aroxmatons. In Hallerbah [1999] we also resent a non-arametr aroah. 7 See Hallerbah [1999]. 8 See Besss [1998] and Matten [2000], e.g. 9 For a omarson of these Credt-VaR models we refer to Crouhy, Gala & Mark [2000]. 4

5 2. Prelmnares In ths seton we ntrodue notaton and resent some useful results. We frst dsuss the theoretal onets of overall CaR, margnal CaR and omonent CaR. 10 Sne we want to dsuss CaR and RAROC n the most general ontext, the only (and very weak) assumton we make s that all relevant return dstrbutons have fnte frst moments. We then summarze the onet of RAROC and outlne ts role n RAPM. defnng overall CaR Consder a ortfolo wth urrent value V, onsstng of N omonents. In the broad ontext of EWRM the ortfolo s the overall frm and the omonents reresent the arttonng of the frm s busness atvtes aordng to searate atvtes, organzed atvtes or busness unts, e.g. In the ontext of redt rsk the omonents are the ndvdual redts or loans omrsed n the ortfolo. Gven the urrent values { V } of the dollar ostons n eah of the omonent atvtes, the hange n ortfolo value over a holdng erod t equals: = wth : = V (2.1) V Vr Vr V where r and r denote the t return on the ortfolo and atvty, resetvely. A tlde marks a stohast varable. 11 All returns denote total returns and reflet both hanges n market value ( re return ) and ash flows ( ash return ) durng the erod. 12 The ortfolo omoston s assumed onstant over the erod t. Gven the ortfolo, ts exeted dollar return s: (2.2) V Vr Vr = where r s the exeted erentage return on atvty. Gven a onfdene level and an evaluaton horzon of erentage return t, we defne the quantle dollar return V and the quantle r (gven V ) that satsfy the sefed onfdene level: 10 The followng results (and notably the orresondene between CaR ontrbutons and ondtonal exetatons) have been frst derved n Hallerbah [1999]. See also Tashe [1999] for a detaled dsusson. In Hallerbah [1999] we also show how these metrs an be estmated n a non-arametr ontext. V and 11 We defne the varates { } V on the robablty sae (Ω,F, Pr( )), where the σ-feld F ontans subsets of the samle sae Ω. 12 When a market value s not avalable, for examle for non-traded redts, the mark-to-market valuaton s relaed wth mark-to-model valuaton. Ths ntrodues model rsk. 5

6 (2.3) { } { } = Pr V = r V = 1 r Pr r r 1 Overall ortfolo CaR s now gven by of losses. 13 When V s ntally gven, fous an be on CaR V = rv sne CaR s defned n terms r nstead of on V. defnng margnal exeted return and margnal CaR When studyng ortfolo otmalty n seton 3 we need nformaton about the margnal exeted returns and the margnal CaRs of the ndvdual seurtes omrsed n the ortfolo. From (2.2) t follows that margnal exeted return of seurty s: (2.4) ( rv ) V = r The margnal CaR MCaR s the hange n ortfolo CaR resultng from a margnal hange n the dollar oston n omonent atvty : (2.5) MCaR CaR V Note that eq.(2.5) also ales to an atvty N not yet nluded n the frm s ortfolo. The ntal ortfolo then omrses N 1 omonent atvtes and we onsder ths (N 1)- element ortfolo as an N-element ortfolo where V N = 0 ntally. To evaluate margnal CaR we start from eq.(2.1) whh dentfes the ortfolo dollar return as a onvex ombnaton of the dollar returns on the ndvdual omonents. Beause of the ortfolo arttonng and by the very defnton of ondtonal exetatons we have: (2.6) V = E{ rv V } = VE { r } Note that the ondtonal exetaton E { r } s a random varable. 14 Also note that the ortfolo dollar return s a lnearly homogeneous funton of the ostons { V }. Sne ths funton s ontnuous and analyt we an aly Euler s theorem: 13 We obtan hanges n dollar values by ombnng returns wth mark-to-market values. Whether fousng on returns or on dollar ostons, transatons wth zero ntal value (suh as redt swas) have to be deomosed nto non-zero long and short ostons (mang). 14 Hene Er { } s to be nterreted as the exetaton of r ~ ondtonal to the σ-feld F relatve to whh s defned. 6

7 (2.7) V = V = Vr V Substtutng eq.(2.7) n (2.6) and ondtonng on V = yelds: (2.8) CaR = V = VE V = V = VEr { } V where we have added a mnus sgn sne CaR s defned n terms of losses. Sne the ortfolo return now takes the artular value V the ondtonal exetatons beome determnst. Note that the ondtonal exetaton n the frst equalty of eq.(2.8) ndates MCaR. Hene: (2.9) MCaR = E{ r } The ntuton behnd eq.(2.9) s lear. When there s a ostve (negatve) nterdeendene between and then large negatve ortfolo returns wll on average be assoated wth large negatve (ostve) omonent returns. Inreasng (dereasng) the sze of the atvty oston V wll then lower the ortfolo value even more, thus nreasng the ortfolo s CaR. defnng and relatng omonent CaR Sne: (2.10) { } CaR = VEr V = VMCaR eah term measures the total ontrbuton of asset to the overall ortfolo VaR. Hene V MCaR = CCaR s the omonent CaR of atvty. These omonent CaRs an unquely be attrbuted to eah of the ndvdual omonents of that ortfolo and aggregate lnearly nto the total dversfed ortfolo CaR: (2.11) CaR CCaR = VMCaR Note that beause of return nterdeendenes and dversfaton effets the omonents 15, 16 stand-alone CaRs do not add u to the dversfed ortfolo CaR. 15 The break-down of VaR aordng to ortfolo omonents or market rsk fators as suggested by Fong & Vasek [1997], for examle, suffers from ths shortomng and s hene not useful. 16 The sum of the stand-alone VaRs an be larger than the ortfolo VaR but also smaller. The latter henomenon ndates that VaR s not sub-addtve ; see Artzner et al. [1999]. 7

8 Eq.(2.11) s a owerful result. It does not deend on any dstrbutonal assumtons but revals sne the ortfolo oerator s lnear. Wthout loss of generalzaton the omonent atvtes may be maed n a non-lnear fashon onto standardzed ostons or underlyng state varables (lke default roesses or reovery rates, as n JPMorgan s [1997] CredtMetrs ). eonom atal, RAPM and RAROC Sne eonom atal s neessary to over otental losses from the frm s atvtes ostons { V }, the RAP s measured by relatng generated nome to eonom atal. The resultng RAP metr, termed RAROC, was frst roosed by Bank of Amera. 17 It takes the form: (2.12) adjusted nome RAROC = CaR The denomnator s rsk-adjusted or eonom atal. In the numerator nome s revenues mnus osts mnus exeted losses. The adjustment for exeted loss s generally onsdered as a rsk orreton (although t s a rovson for exeted losses, whh by defnton does not reresent rsk). For ths reason, (2.12) s sometmes alled a RARORAC (rsk-adjusted return on rsk-adjusted atal) measure. Varous sef defntons exst (suh as RAROC, RARORAC and RORAC), but most varatons are due to the sefaton of the numerator. The numerator ndeed rases some questons. Should fnanng (oortunty) osts be taken nto aount? The numerator then reresents the eonom roft. More mortantly we ask ourselves why the fous s on ash nome? Return an also be generated from atal gans or losses? Defnng nome on a mark-to-market bass an orret for ths. In ex ost alatons, how s the numerator measured? Sne rsk s nvolved we would lke to sefy the numerator as an exeted return. In ex ost alatons the far game assumton ould be nvoked to estmate the exeted value by means of an hstoral average. The denomnator rases the ssue of how to alloate the amount of eonom atal that relates to the overall dversfed frm ortfolo over the dfferent sub-levels wthn the frm (rangng from busness unts, va geograhal loatons to tradng desks, ndvdual traders and ultmately to ndvdual busness transatons). These ambgutes all for a fool roof defnton of RAROC. However, n the next seton we argue that the relevant defnton of the RAP measure deends on the deson ontext, omrsng the ursued objetve(s) and the mosed onstrants. 3. Portfolo otmzaton, RAROC and RAPM In ths seton we show how the relevant defnton of rsk ontrbutons and varous RAROC measures are mled by a ortfolo otmzaton model. Eah RAROC measure s relevant wthn the sef underlyng otmzaton ontext. We assume that the frm strves to 17 See Zak, Walter & Kellng [1996]. For general exoses we refer to Besss [1998], Matten [2000] or Smthson & Hayt [2001]. 8

9 mze the exeted return on ts atvtes ortfolo subjet to a onstrant on the requred eonom atal. Eonom atal s measured by the frm s CaR over the horzon. We lassfy the ossble models usng two dmensons. The frst dmenson s defned by the sale of oeratons. The atal nvested n the frm atvtes (or the budget avalable for the frm s busness ventures) V may ether be fxed or free. In the former ase the avalable atal s restrted (lke n a standard ortfolo nvestment roblem); n the latter ase the frm may nrease (derease) the sale of ts atvtes by rasng more (less) atal. The seond dmenson s defned by the tye of CaR onstrant. 18 Ths onstrant on eonom atal may be formulated n ether absolute or relatve terms. The absolute CaR onstrant s gven by the ortfolo dollar CaR level CaR that should not be exeeded. The relatve onstrant s defned n erentage terms ombnatons are summarzed n Exhbt 1., r of V (see eq.(2.3)). The ossble Exhbt 1: A smle tyology of CaR-onstraned otmzaton roblems CaR restrton frm s atal absolute ($) relatve (%) V free I: V = s.t. CaR IV: V = s.t., V r V fxed II: V = rv s.t. CaR V V III: V = rv s.t., V r V V In all ases we may wsh to restrt short ostons, V 0,. In that ase Kuhn- Tuker ondtons wll aly and only ostve ostons are onsdered. 19 When, V s fxed, CaR r V so the otmzaton roblems II and III are equvalent. When the CaR restrton s formulated n relatve terms and V s not fxed, as n IV, the roblem beomes ndetermnate and an only be solved for { V } gven some level of V. Atually, IV s not realst under our smle assumtons sne n rate there wll be some lmt to the frm s atvtes anyhow. Should we allow for a trade-off between exeted return and CaR, or when we would add other restrtons, IV would beome a relevant startng ont. 18 There s some debate whether CaR should be dsounted over the horzon or not. When CaR should over otental losses at the end of the horzon the dsountng argument s lear. When also ntermedate losses should be overed the ase s not lear. In the followng we refran from dsountng CaR, but the neessary adjustment s obvous. 19 Throughout the aer we assume that seond order ondtons are satsfed. Hene we assume that the feasble CaR regon s onvex. 9

10 But for now we are left wth two dfferent ases: on the one hand we have roblem I, and on the other roblems II and III. Another aset that roves to be mortant s whether rskfree atvtes (rskfree borrowng and/or lendng) are avalable to the frm. In seton 3.1 we assume that all ortfolo omonents are rsky, so there does not exst a rskfree rate. In seton 3.2 we dro ths assumton and allow for rskfree nvestment oortuntes. 3.1 Portfolo otmzaton wthout rskfree rate roblem I In stuaton I the frm strves to mze the exeted dollar return over the hosen tme horzon subjet to an absolute CaR onstrant. The rsk and hene the eonom atal of the frm s atvtes s restrted by the mum admssble CaR level otmzaton roblem beomes hoosng { V } suh that: CaR. The (3.1) { V } st.. CaR We an safely assume that there exst suffent roftable busness ventures so that the CaR onstrant s bndng. Hene the mum allowed amount of eonom atal wll be emloyed. Formng the Lagrangan and takng the artal dervatves to V leads to the followng frst order ondtons (FOCs heneforth): (3.2) r λ MCaR = 0 * together wth the orgnal CaR onstrant. λ s the Lagrange multler and and an astersk refers to the otmum. Multlyng wth V and summng over * yelds, n ombnaton wth (3.2): (3.3) * j * * = =, j * CCaR CCaR CaR * j* * rovded that MCaR * 0, *. Eq.(3.3) s the ortfolo otmalty ondton. When the ortfolo s otmal the rato of margnal (total) return ontrbuton and margnal (total) CaR ontrbuton s onstant over all atvtes n *. To allow for zero margnal CaRs we rewrte (3.3) as: (3.4) * CCaR* V = * * CaR * 10

11 Otmal alloaton of atal s aheved when (3.3) (or (3.4)) s satsfed. Note that for eah atvty ts exeted dollar return should be related to ts total ontrbuton to the dversfed ortfolo CaR. In the last term of (3.3) we reognze the famlar frm-wde RARORAC (or RAROC). The numerator s the exeted dollar return on the atvty ortfolo. By defnton ths return () s net of the exeted loss and () nludes re returns (atal gans/losses). In the onventonal defnton, only the ash return (.e. nome) s onsdered (ontrastng ()) and subtratng the exeted loss s meant to yeld the rsk-adjusted return. Sne the exeted loss s exeted by defnton, ths s not a rsk orreton at all. The frst term of (3.3) ndates how to arase the ex ante erformane of atvty : by relatng ts exeted dollar return to ts ontrbuton to overall eonom atal. Now suose that gven some ortfolo we fnd that: j (3.5) > >, j CCaR CCaR CCaR j Obvously s not otmal. Ths mles that an be enhaned by nreasng the oston n atvty and dereasng the oston n j. Ex ante erformane analyss s thus relevant for evaluatng the otmalty of some (ntal) ortfolo and dervng ortfolo revson rees. Under a far game assumton atvty s ex ost erformane an be gauged by relatng ts average realzed dollar return to ts ontrbuton to overall eonom atal. In rate the resrton s to mze the frm s RAROC. As the Aendx shows, mzng RAROC yelds the same FOCs eqs.(3.3) and (3.4). Obvously the unonstraned mzaton of onventonal RAROC assumes the underlyng otmzaton roblem (3.1). Conversely, mzng RAROC an be justfed on the bass of (3.1). But suose now that the frm s total atal V s fxed, or that rskfree ventures exst. Obvously the ortfolo otmalty ondtons wll hange and hene the mled rskadjusted erformane measure. Ths s nvestgated below. roblems II and III In stuatons II and III the total avalable atal V s fxed and the frm strves to mze the exeted dollar return over the hosen tme horzon subjet to an absolute or relatve CaR onstrant. The otmzaton roblem now beomes hoosng { } V suh that: (3.6) { V } V = rv, st.. CaR = V r V V From the Lagrangan the FOCs are: (3.7) r λ MCaR θ = 0 * 11

12 together wth the orgnal onstrants. λ and θ are the Lagrange multlers of the CaR and the atal onstrants, resetvely. Multlyng wth V and summng over * yelds, n ombnaton wth (3.7): (3.8) * * j θ j * θ * * * θv V V = =, j * CCaR CCaR CaR * j* * wth CaR * = CaR, rovded that MCaR * 0, *. Ths mled ortfolo otmalty ondton s equvalent to: (3.9) r θ r θ r θ = =, j * MCaR MCaR r j * * j* * When the atvty ortfolo s otmal the rato of margnal (total) adjusted return ontrbuton and margnal (total) CaR ontrbuton s onstant over all atvtes n *. Exeted returns are adjusted wth a fator θ (the shadow re of relaxng the atal onstrant) ndatng the erentage oortunty ost of obtanng addtonal funds. To allow for zero margnal CaRs we rewrte (3.8) as: (3.10) CCaR V = V * * * * θ * * CaR θ * Otmal alloaton of the avalable restrted atal s now aheved when (3.8) (or (3.9) or (3.10)) s satsfed. The last term of (3.8) (or (3.9) n erentage terms) s the relevant mled rskadjusted erformane metr. The numerator s the exeted adjusted return on the atvty ortfolo. Agan, ths return () s net of the exeted loss and () nludes re returns (atal gans/losses). Moreover t s () adjusted for the mled shadow ost θ of obtanng addtonal funds. The frst term of (3.8) ndates how to arase the erformane of atvty : by relatng ts exeted or average adjusted dollar return to ts ontrbuton to overall eonom atal. From an ex ante ersetve, devatons for the FOCs an be used to gude ortfolo revsons n order to enhane the sub-otmal ortfolo. When the adjustment sub () s gnored, otmal alloaton s not guaranteed. Lkewse, ex ost erformane analyss s dstorted. Conventonal RAROC analyss on the bass of (3.3) for the ortfolo, or the ndvdual atvtes omrsed theren, wll fal n ths ase. As shown n the Aendx, the unonstraned mzaton of onventonal RAROC s at odds wth the underlyng otmzaton roblem (3.6). Let θ now be the exltly sefed erentange ost of nreasng the atal base. The otmzaton roblem beomes rv θ V V subjet to CaR (wth V no longer restrted). Ths alternatve roblem resembles stuaton I where V s not fxed, but has the same FOCs (3.8) and (3.10) as above. Alternatvely, the total fnanng 12

13 osts over r θ V subjet to the CaR ontstrant. Agan ths results n the same FOCs (3.8) and (3.10). V (not restrted) an be taken nto aount, leadng to ( ) 3.2 Portfolo otmzaton allowng for rskfree atvtes We now assume that rskfree borrowng and lendng oortuntes exst for the frm. Denotng atvty 1 as rskfree, ts return s the rskfree rate r f. In general, the dollar return on the frm s total atvty ortfolo s: N (3.11) V = Vr 1 f + Vr = Vr 1 f + q wth V q = Vr q q = 2 where V q s the rsky art of ortfolo, satsfyng V q N = 2 Vr. When V s fxed, we have V 1 V Vr 2 = as a funton of the rsky ventures. Defnng the weght w of the rsky atvtes n the total ortfolo, the exess total ortfolo return s: (3.12) r rf = w ( rq rf ) = wth w V / V Portfolo s exess dollar return CaR s: (3.13) ( ) CaR V r r = CaR + Vr f f f q N It fnally readly follows that the exess dollar return CaRs of and q are equal: 1 (3.14) ( ) ( ) CaR = V r r = V wr r CaR w f f q q f qf We now revst the four roblems n Exhbt 1. roblem I In ths stuaton the frm strves to mze the exeted dollar return over the unrestrted atal V subjet to the absolute CaR onstrant: { } (3.15) { } q ( 1 f ) Pr CaR = Pr CaR + Vr = 1 where we have used (3.11). Hene: (3.16) q = q = + 1 f CaR V CaR Vr 13

14 Eq.(3.16) shows that ths otmzaton ase s not nterestng. The absolute CaR restrton on an be satsfed wth any rsky ortfolo q smly by addng suffent rskfree nvestment V 1 > 0. roblems II and III When onsderng V fxed, the otmzaton roblem s gven by eq.(3.6) wth FOCs (3.7). For the rskfree atvty =1 we have MCaR1 = rf, so the FOC beomes: (3.17) θ = ( + λ) 1 rf Substtutng n (3.7) yelds: (3.18) f λ ( f ) r r = MCaR + r * * Multlyng wth V and summng over * gves: (3.19) V( r* rf ) = λ ( CaR* + Vr f) The LHS of (3.19) s the exeted exess dollar return (.e. dollar rsk remum) on ortfolo, and the term n arentheses on the RHS s s exess dollar return CaR: (3.20) CaRf CaR + Vr f = V( r rf ) Eqs.(3.18) and (3.19) translate nto: (3.21) * * * * * f j f j * f rv rv rv = = q* CCaR CCaR CaR f* jf* f * wth CaR f* f = CaR n exess return form, rovded that MCaR * 0, *. Ths s the mled ortfolo otmalty ondton, equvalent to: f (3.22) r r r r r r r r = = = q* MCaR r MCaR r r r r r f j f * f q* f * + f j* + f * + f q* + f The last equalty follows from (3.12) and (3.14).The notable dfferene wth (3.9) s that the denomnators are the exess return CaR ontrbutons. Dowd [2000,.221] dsqualfes onventonal RAROC sne t an beome nfntely large by nvestng all atal n rskfree ventures. But we see that the relevant RAROC measure n ths lmt ase beomes ndetermnate. To allow for zero margnal CaRs we rewrte (3.22) as: 14

15 (3.23) MCaR + r r r = r r * * f f * f CaR* + V* r f Multlyng both sdes of (3.23) wth V translates the exresson nto dollar terms as n (3.21).Note that (3.23) orresonds to (3.4) ast n exess return form. The FOC eq.(3.22) s dental to the FOC for the mean-var ortfolo seleton roblem as derved n Grootveld & Hallerbah [2000]. It reveals lnear two-fund searaton,.e. the otmal alloaton wthn the rsky ortfolo q s ndeendent of the total ortfolo. For any value of the mum admssble CaR the orresondng otmal ortfolo alloaton onssts of a lnear ombnaton of the rsk free nvestment and only one sngle rsky ortfolo q. 20 Note that ths searaton roerty s fundamentally dfferent from that mled by the mzaton of exeted utlty; for the latter otmzaton ase the results of Cass & Stgltz [1970] are defntve. The otmal alloaton of the avalable atal s aheved when (3.21) or (3.22) s satsfed, and these ondtons aly for ortfolo q. Gven ortfolo q and the CaR onstrant, ortfolo readly follows. The last term of (3.21) (or (3.22) n erentage terms) s the relevant mled rskadjusted erformane metr for ths ase. The numerator s the (dollar) rsk remum on the atvty ortfolo. 21 The frst term of (3.21) or (3.22) shows how to arase the erformane of atvty : by relatng ts rsk remum (or average exess return) to ts ontrbuton to overall exess eonom atal CaRf = CaRqf. Agan, usng a onventonal RAROC measure wll loud the erformane analyss. An nterestng aset of ths artular roblem s that t smly otmzaton roblem I as studed n seton 3.1, but now fully ast n exess returns. Hene the result from the Aendx ales: unonstraned mzaton of the mled adjusted RAROC measure (the last term of (3.21) or (3.22)): (3.24) rv f r rf = CaR r r ( ) f f yelds the same FOCs as n the two LHS terms of (3.21) or (3.22): (3.25) * * * * f j f j rv rv = * CCaR CCaR f * jf * Note that the mand n (3.24) s n srt smlar to the Share rato 22 : t measures the exeted exess return er unt of rsk where rsk s here defned n terms of the CaR of the 20 See also Arza & Bawa [1977] and Tashe [1999] on ths ont. 21 Crouhy, Turnbull & Wakeman [1999] defne an adjusted RAROC measure n the form of RAROC mnus the rskfree rate, dvded by CAPM beta. Ths s to aount for systemat (red) rsk nstead of total frm rsk. However, they do not derve the metr nor ndate how t would be. 22 See Share [1966,1994]. 15

16 exess returns. Wthout restrtng referenes, mean-varane analyss s vald when ortfolo returns are elltally dstrbuted wth fnte varane. 23 Suh elltal dstrbutons are fully defned by the loaton and sale arameters and do not exhbt skewness. 24 In an elltal world the exess CaR of ortfolo s smly defned by the frst two statstal moments of the ortfolo return dstrbuton: f f f f (3.26) CaR = r = r + k () σ (wth σ f < ) whereσ s the standard devaton of the exess return on ortfolo and k () s the f roortonalty fator belongng to the CaR onfdene level. For examle when ortfolo 1 returns are normally dstrbuted we have k () = N () where N () s the standard normal dstrbuton funton. Inororatng (3.26) n (3.24) and some rearrangng yelds: (3.27) 1 f σ f f r r r r = 1 + k () rf + k () σ f r rf σ f gven. So n an elltal world, the otmzaton roblem (3.24) entals fndng the ortfolo that mzes ts Share rato (the last term n (3.27)). Hene n an elltal world the two ortfolo otmzaton roblems are omletely equvalent. Ths ontradts Cambell, Husman & Koedjk [2001]. However, as we exet that the underlyng dstrbutons n a RAPM ontext are asymmetr, ths dsqualfes the onvenent elltal arametr assumton. For onvenene, the relevant RAROC measures mled n eah of the deson stuatons s summarzed n Exhbt 2. (Reall that IV s ndetermnate.) 23 See Owen & Rabnovth [1983]. 24 The general lass of (both fnte and nfnte varane) elltal dstrbutons nludes the Student t dstrbuton, the exonental dstrbuton, symmetr stable (Pareto-Lévy) dstrbutons wth haraterst exonent smaller than two and the normal dstrbuton. Also non-normal varane mxtures of multvarate normal dstrbutons belong to the elltal lass, see for nstane Chmelewsk [1981] and Fang, Kotz & Ng [1990]. 16

17 Exhbt 2: Adjusted RAROC measures mled n eah of the deson stuatons (see n Exhbt 1) of CaR-onstraned otmzaton. rskfree ventures frm s atal no yes ( r f ) V free I: CaR * * I: NA V fxed II, III: θv * * CaR * II, III: * * rv f CaR f * 4. Conlusons We argue that wthout exltzng the relevant EWRM deson ontext (n the form of learly and unambguously stulatng the objetves and onstrants), t s not feasble to: ex ante otmze the frm s atvtes ortfolo; ex ante alloate eonom atal over atvtes omrsed n the frm-wde ortfolo aordng to ther rsk ontrbutons; ex ante evaluate the qualty of the atvtes ortfolo n the lght of ursued objetves and mosed onstrants; ex ost evaluate overall ortfolo erformane; ex ost attrbute erformane to ndvdual atvtes. We llustrated our argument wth varous smle otmzaton examles. Even though the resented deson ontexts are smlfed, our results learly show that alyng RAPM on the bass of onventonal RAROC measures (as resented n the lterature) more often than not may lead to erroneous onlusons and atons. For dervng relevant RAP measures we suggest the followng ree : dentfy objetve(s) and onstrants; derve the mled ortfolo otmalty ondtons; aly the mled relevant RAP measure for ex ante ortfolo enhanement and ex ost reformane evaluaton and attrbuton. A hallengng route for further researh s to unover more omlex tyologal deson ontexts as we may enounter them n rate and to reveal the mled and hene adequate RAP metrs. 17

18 Aendx Let A(x) and B(x) be analyt funtons of x=[x ], homogeneous of degrees g and h, resetvely. Consder the followng onstraned otmzaton roblem: (A.1) A( x ) st.. B( x) B { x} where B s a (negatve) number. The FOCs are: (A.2) A( x) + λ B( x) = 0 together wth the orgnal onstrant, where λ s the Lagrange multler and s the gradent oerator. We assume seond order ondtons are satsfed. Premultlyng (A.2) wth x, solvng for λ and substtutng bak yelds the ortfolo otmalty ondton: (A.3) B( x* ) A( x* ) = A( x* ) h B( x* ) g where astersks denote the otmum. Now onsder the frst unonstraned roblem, stulatng a fxed trade-off between A(x) and B(x) governed by the arameter γ > 0: (A.4) A( x) +γ B( x) { x} Ths yelds the same FOCs (A.2) and ortfolo otmalty ondton (A.3) Next onsder the seond unonstraned roblem: (A.4) A( x) B( x) 0 { x} B( x) The FOC s: (A.5) A( x* ) A( x* ) B( x* ) = 0 B( x* ) whh translates nto (A.3) when g = h = 1 (.e. lnear homogenety). The thrd unonstraned roblem: (A.6) A( x) θ B( x) 0, θ 0onstant { x} B( x) has FOCs: 18

19 (A.7) A( x* ) θ A( x* ) B( x* ) =0 B( x* ) whh s nomatble wth (A.3), even when g = h = 1 ortfolo CaR Now let the exeted ortfolo return orresond to A(x) above, and the CaR (defned n terms of losses) orresond to -B(x). Sne CaR are lnearly homogeneous n the atvtes { } onstrant on CaR s equvalent to mzng V, mzng RAROC V and V subjet to only a unonstraned. CaR Moreover, sne RAROC s homogeneous of degree zero, ths measure an be mzed wthout takng nto aount any restrton on V. The soluton an smly be saled to satsfy ths restrton. However, from the thrd roblem above we have: (A.8) CaR θv CaR 19

20 Referenes Artzner, P., F. Delbaen, J.-M. Eber & D. Heath, 1999, Coherent Measures of Rsk, Mathematal Fnane 9/3, Arza, E.R. & V.S. Bawa, 1977, Portfolo Choe and Equlbrum n Catal Markets wth Safety-Frst Investors, Journal of Fnanal Eonoms 4, Besss, J., 1998, Rsk Management n Bankng, John Wley & Sons, Chhester NY Bookstaber, R., 1997, Global Rsk Management: Are We Mssng the Pont?, The Journal of Portfolo Management Srng, Cambell, R., R. Husman & K. Koedjk, 2001, Otmal Portfolo Seleton n a Value-at- Rsk Framework, Journal of Bankng & Fnane 25, Cass, D. & J.E. Stgltz, 1970, The Struture of Investor Preferenes and Asset Returns, and Searablty n Portfolo Alloaton, Journal of Eonom Theory 2/1, Chmelewsk, M.A., 1981, Elltal Symmetr Dstrbutons: A Revew and Bblograhy, Internatonal Statstal Revew 49, Credt Susse Fnanal Produts, 1997, Credt Rsk+: A Credt Rsk Management Framework, London UK ( Crouhy, M., D. Gala & R. Mark, 2000, A Comaratve Analyss of Current Credt Rsk Models, Journal of Bankng & Fnane 24, Crouhy, M., S.M. Turnbull & L.M. Wakeman, 1999, Measurng Rsk-Adjusted Performane, The Journal of Rsk 2/1, Fall,.5-35 Cummng, C.M. & B.J. Hrtle, 2001, The Challenges of Rsk Management n Dversfed Fnanal Comanes, FRBNY Eonom Poly Revew,.1-17 Dowd, K., 1998, VaR by Inrements, Enterrse-Wde Rsk Management Seal Reort, RISK, November, Dowd, K., 1999, A Value-at-Rsk Aroah to Rsk-Return Analyss, The Journal of Portfolo Management Srng, Dowd, K., 2000, Adjustng for Rsk: An Imroved Share Rato, Internatonal Revew of Eonoms and Fnane 9, Duffe, D. & J. Pan, 1997, An Overvew of Value at Rsk, The Journal of Dervatves 4/3, Srng,.7-49 Edgeworth, F.Y., 1888, The Mathematal Theory of Bankng, Journal of the Royal Statstal Soety 51/1, Marh, Fang, K.T., S. Kotz & K.W. Ng, 1990, Symmetr Multvarate and Related Dstrbutons, Chaman and Hall, London UK Fong, G. & O.A. Vasek, 1997, A Multdmensonal Famework for Rsk Analyss, Fnanal Analysts Journal July/August, Grootveld, H. & W.G. Hallerbah, 2000, Ugradng Value-at-Rsk from Dagnost Metr to Deson Varable: A Wse Thng to Do?, reort 2003, Erasmus Center for Fnanal Researh, June Hallerbah, W.G., 1999, Deomosng Portfolo Value-at-Rsk: A General Analyss, dsusson aer TI /2, Tnbergen Insttute Joron, Ph., 2000, Value at Rsk: The Benhmark for Controllng Market Rsk, MGraw-Hll, Chago Ill. 20

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