Risk Capital Aggregation: the Risk Manager s Perspective

Transcription

1 Rsk Captal Aggregaton: the Rsk Manager s Perspectve Francesco Sata Assocate Professor Newfn Research Center and IEMIF Unverstà Boccon Vale Isonzo Mlano MI Italy Tel (dr.) Tel (secr.) Fax E-mal: francesco.sata@un-boccon.t Frst verson: November 2003 Ths verson: September 2004

2 Rsk Captal Aggregaton: the Rsk Manager s Perspectve Abstract Rsk aggregaton, defned as the development of quanttatve rsk measures that ncorporate multple types or sources of rsk amed at measurng the overall captal at rsk for a fnancal nsttuton, s a crtcal topc both for banks and for ther regulators. Ths paper ponts out the crtcal role that the choce of the noton of captal that s consdered at rsk may have, therefore dscussng the ssue of busness rsk, that has so far receved very lttle attenton n the lterature. The paper then dscusses alternatve rsk aggregaton technques and some of the problems that arse n the estmaton of ther parameters. Parameter estmaton appears to be a major concern when decdng whch aggregaton technque to adopt, especally consderng the mplcatons for rsk-adjusted performance measurement and therefore for decsonal processes that may derve from the rsk aggregaton exercse.

3 Rsk Captal Aggregaton: the Rsk Manager s Perspectve *. Introducton Defnng the optmal level of captal and ts best possble allocaton across busnesses s mportant for any knd of frm. In the fnancal sector, ths problem has ganed ncreasng attenton and has attracted a substantal amount of nvestments by banks and other fnancal nsttutons due to the parallel and ntertwned development of regulatory captal requrements and of rsk measurement technques (typcally based on the concept of Value at Rsk). At present, most f not all large bankng groups have specfc models for evaluatng market and credt rsk for at least a large part of the legal enttes they comprse, and are nvestng n models of ncreasng sophstcaton to assess and quantfy operatonal rsk. In the past few years, ther efforts centered on credt rsk models and nternal ratng systems, partly as a consequence of the crucal phase of the New Captal Accord. However, the ssue of rsk aggregaton,.e. the development of quanttatve rsk measures that ncorporate multple types or sources of rsk, s now recognzed as crtcal both for banks and for regulators. In fact, f a rsk manager were able to measure ndvdual rsks on a stand-alone bass only, t would be very hard for hm to address the two key questons about () whether the bank has the rght amount of captal and () how to allocate captal across dfferent busness unts and dfferent rsks. Both ssues requre to be able to determne an ntegrated measure of the economc captal requred by the bank n order to cover potental losses. Anyway, t s clear * I am grateful for very useful comments to Luca Erzegoves, Maro Masn, Davde Maspero, Paolo Mottura, Ramon Rabnovtch, Andrea Sron, and to partcpants at the 2004 EFMA Conference n Basle, and n partcular to Amrt Judge. The usual dsclamer apples. See Jont Forum Workng Group (2003). The Workng Group makes a dstncton between rsk ntegraton (.e., developng a common rsk measurement framework n terms of polces and procedures, tasks and responsbltes, systems, and so on) and rsk aggregaton. 2

4 that rsk aggregaton and economc captal methods are stll n early stages of evoluton, as clearly stated n the recent Jont Forum Workng Group (2003) survey on Trends on rsk ntegraton and aggregaton. The topc of rsk aggregaton s therefore attractng an ncreasng nterest from researchers. Matten (996) has been the frst one to address the ssue of dervng an overall measure of captal at rsk combnng dfferent knds of rsks, adoptng a pragmatc perspectve from the bank s nternal vewpont 2. More recent papers have nstead ether revewed banks best practces concernng rsk aggregaton (Jont Forum Workng Group 2003) or proposed aggregaton technques to be adopted (see Alexander and Pezer 2003, Dmakos and Aas 2003, Pezer 2003). Of course, consderng the ptfalls of common dependence measures such as lnear correlaton when aggregatng return dstrbutons see n partcular Embrechts, McNel and Straumann (999) there s a lot of nterest about the role of copulas n rsk aggregaton. Recently, Rosenberg and Schuermann (2004) have developed an emprcal test on the aggregaton of market, credt and operatonal rsk wth a detaled senstvty analyss of the mpact of dfferent copula functons and correlaton parameters on aggregate VaR estmates. The same knd of methodology has been dscussed and used by actuares wth reference to rsk aggregaton n the nsurance busness or among conglomerates: see n partcular Wang (2002), Ward and Lee (2002), Venter (2003). Some contrbutons have also drectly addressed the problem from regulators pont of vew concernng whether and how dversfcaton benefts among busnesses or rsks may and should be taken nto account n settng mnmum captal requrements (see Kurtzkes, Schuermann and Wener 200 and 2002, and agan, Jont Forum Workng Group 2003). 2 The related ssue concernng dversfed or undversfed measures of Value at Rsk for ndvdual busness unts has been dscussed n the same perod also n Merton and Perold (993), James (996), Sata (999), Culp (2000). Froot and Sten (998) and Perold (999) have addressed the topc of captal allocaton wth a more 3

5 Consderng both the techncal hurdles that need to be overcome n order to obtan an aggregated rsk measure and the relevance of the topc from a regulatory vewpont, the focus adopted by most of the contrbutons about rsk aggregaton s completely understandable. Yet, the dscusson has left n the background some problems that may be extremely relevant from the rsk manager s vewpont. The am of ths paper s to gve a contrbuton precsely n ths drecton, tryng to gve an useful complement to the exstng lterature n the feld. In partcular, the man contrbutons of the paper are the followng. Frst, whle the ssue of rsk ntegraton and aggregaton s related typcally to market, credt and operatonal rsk, ths paper wll dscuss the role of busness rsk, defned here as the rsk assocated to earnngs volatlty not determned by event-drven operatonal rsks. We wll dscuss the key ssues n measurng earnngs volatlty and propose new methods to translate earnngs volatlty nto a measure of captal at rsk. Second, the paper wll show that the aggregaton of captal at rsk measure may be analyzed n the context of dfferent notons of captal. In partcular, there s a dfference between the book value of captal and the market captalzaton that may be relevant especally when addressng the role of busness rsk. Whle regulators may correctly be worred manly about the rsks that mght cause a bank to fal, the rsk manager should be concerned also about rsks that whle generatng moderate losses n the short term (and therefore wth lttle mpact on the bank s short-term lkelhood to face a dstress) may mpact substantally on ts forecasted earnngs and therefore on ts market captalzaton, destroyng value for the shareholders. Thrd, the paper wll dscuss the dfferent purposes for whch an ntegrated rsk measure may be necessary, and hence whch crtera the best measure should deally satsfy. Snce such aggregated rsk measures should mpact on real decson makng processes and may enter nto busness unts evaluaton, precson whle remanng crucal formal approach, and have clarfed the potental mpact of economc captal and rsk-adjusted performance measures on captal management and captal allocaton decsons. 4

6 may not be the only desrable attrbute. Alternatve aggregaton technques wll be analyzed n front of such crtera. Fourth, the paper wll explctly address calbraton ssues n the debate concernng the optmal rsk aggregaton technque, pontng out the rsks nherent n usng short term (e.g., monthly) earnngs data to calbrate the models. Two bref premses are needed. Frst, ths paper does not support the dea that a sngle number of aggregate rsk captal at rsk can be scentfcally derved, and magcally solve any problem for a CEO that has to take a decson on a bank s captal structure. Estmates of rsk captal may be dffcult even at sngle rsk level and become much more uncertan at bankwde level; and anyway, even f a perfect sngle number could be obtaned, captal management choces would never be mechancal. Therefore, whle t s mportant to try to ncrease the qualty of overall rsk estmates, overconfdence n the fnal number(s) a rsk aggregaton technque may produce remans extremely dangerous. Second, even f throughout the paper we wll manly consder value at rsk (or equvalently captal at rsk) as the typcal rsk metrc, we do not clam that t should be vewed as the best or only rsk measure. Startng from the semnal paper of Artzner, Eben, Delbaen and Heath (999), Value at Rsk, defned as the maxmum amount that may be lost over a certan tme horzon wthn a gven confdence nterval, has been ncreasngly crtczed snce t does not fulfll all desrable attrbutes a coherent rsk metrc should possess. In partcular, Value at Rsk s not subaddtve,.e. gven two portfolos X and Y t s not always true that VaR(X+Y) VaR (X) + VaR (Y). Consequently, alternatve coherent measures have been proposed, wth partcular reference to expected shortfall (ES) that can be defned as the average loss over a certan tme horzon wthn a gven porton of the left tal of a portfolo s return dstrbuton. Expected shortfall s subaddtve, and ths s clearly a hghly desrable qualty n the context of rsk aggregaton. In the paper we wll manly refer to captal 5

7 at rsk for the reason that, despte these crtques, captal at rsk remans at present the typcal rsk metrcs adopted n almost all banks. Yet, many of the comments that we wll make could also be appled to other alternatve rsk metrcs. The paper s organzed as follows. Secton 2 analyses the dfferent ams of an aggregated rsk measure, explans the problem of alternatve notons of captal at rsk that may be adopted, and dscusses the crtera that may be used n evaluatng aggregaton technques. Secton 3 s devoted to busness rsk, analyzng how a captal at rsk measure may be derved dependng on whether t s evaluated n terms of book value of captal or market captalzaton. Secton 4 descrbes alternatve rsk aggregaton technques and the ssues related to parameters estmaton. Secton 5 concludes. 2. Rsk aggregaton from the rsk manager s vewpont 2.. The possble ams of an aggregated rsk measure Why s an aggregated rsk measure useful n a bank? A frst clear objectve s to assess whether avalable captal s adequate to cover the bank s rsks. An aggregate measure of captal at rsk may therefore support captal management and captal structure decsons (even f one mght then queston whether a normal or an extreme aggregate captal at rsk measure s warranted) 3. Second, t may be used n order to measure the overall rsk of the bank so to then decompose t nto each busness unt s contrbuton, defnng a dversfed rsk measure as a bass for (a) evaluatng busness areas so to support top managers decsons and (b) evaluatng busness unts and maybe defnng the bonuses for those who run them. Thrd, t may be used to assess margnal contrbutons of busness unts or even ndvdual deals to the overall rskness of the bank, so to defne rsk prcng rules whch may be consstent wth the margnal rsk assocated to any transacton. Keepng n mnd these 6

8 dfferent purposes s relevant snce many authors note that the choce of the rsk metrc and of the aggregaton technque should be purpose-specfc (see Venter 2003, Pezer 2003). The complexty of the dfferent ams that may be assocated to rsk aggregaton s relevant for two mplcatons, that are typcally not dscussed n exstng lterature on rsk aggregaton. The frst s that one should queston whch captal has to be aggregated, snce dfferent notons of captal may be used and are relevant dependng on the purpose of the rsk manager. The second one s that rsk aggregaton n the real world s not a purely techncal exercse, snce t may mpact outsders perceptons of the bank s rsk, nternal allocaton of captal and other resources, ndvdual busness unt leaders evaluaton and compensaton. Worres about these topcs may (and should) be consdered by the rsk manager when choosng an aggregaton technque Defnng the crtera to evaluate rsk aggregaton technques Consderng the dfferent possble objectves of an aggregate rsk estmate, there are dfferent crtera that may be consdered when evaluatng alternatve solutons. Some of them are objectve and structural (.e, the theoretcal soundness of the model, the ntrnsc dffculty n parameters calbraton), others are stll objectve but may be temporary and transent (e.g. the ablty to reduce samplng error or model rsk when parameters are estmated wthout adequate tme seres of returns), whle others are subjectve snce they depend on the purposes for whch the rsk manager wll use the aggregaton technque. For nstance, attrbutes such as transparency, perceved farness, or the ablty to produce dversfed CaR measures n an easy and consstent manner wll matter more or less dependng on whether the rsk aggregaton exercse s ntended to support only top management s decsons on captal structure, or nstead also to gude the nternal captal 3 See Pezer (2003). 7

9 allocaton process, or even to mpact (through measures of dversfed captal at rsk) busness unts leaders evaluaton and compensaton. Ths also means that choces of dfferent technques made by dfferent banks mght fnd a ratonale n the dfferent breadth of purposes to be pursued. Some of the crtera may need to be analyzed more closely. Whle the theoretcal attrbutes and the ease of calbraton may be relatvely clearer, and wll be dscussed n Secton 4 n more detal, the senstvty to samplng error s an ssue snce n most cases the tme seres over whch to try to estmate the parameters (whether they are smple lnear correlatons or sophstcated copula parameters) may be short or extremely short 4. Addng new data on a short seres could lead the rsk manager to change correlaton parameters estmates, therefore makng the aggregated rsk measure unstable n early perods even n absence of real market condton changes. Potental jumps of the aggregate rsk measure would be dffcult to explan for the rsk manager to the board and to top managers and f the rsk aggregaton technque were used also to produce measures of dversfed captal at rsk for ndvdual busness unts may cause dscontent and face greater opposton by busness unts managers, especally by those who feel to be negatvely affected by the new technque to allocate dversfcaton benefts 5. A rsk manager may then favour at least for the frst few years a technque whch s less sophstcated but appears to produce results that are more stable as the sample for correlaton estmates changes, rather than resortng to a theoretcally superor approach that may cause hgher nstablty of estmates. 4 Rosenberg and Schuermann (2004) adopt for nstance n ther nterestng smulaton on rsk aggregaton a method n order to reproduce a longer tme seres of quarterly returns from market and credt rsk than they orgnally have by analyzng the relatonshp between market and credt rsk returns and certan macro factors, for whch they have a longer tme seres. Whle suggestng an nterestng soluton, ther choce also pnponts the problems concernng the length of avalable data seres. 5 On ths topc see also Hall (2002). Of course ths poltcal problem would not exst f dvson managers were evaluated on a stand-alone, undversfed measure of the captal at rsk of ther unt, wthout consderng dversfcaton benefts. 8

10 Transparency and perceved farness agan are relevant f the measures that are produced also have as a byproduct rsk measures for ndvdual busness unts. It s dffcult to motvate people to pursue challengng RAROC or EVA targets f they feel that a large part of the result depends on how dversfcaton benefts are allocated to busness unts based on some mysterous formula whose parameters are secretly handled by the rsk management unt. Agan, ths may lead to favour smple technques untl the more sophstcated ones have been clearly accepted by most managers The noton of captal at rsk and the role of busness rsk The frst key decson that an aggregate rsk measure should support concerns the optmal amount of captal that a bank should hold so to be adequately protected aganst the rsks dervng from ts actvty. One clear condton the bank should satsfy s that hs regulatory captal should be greater than ts mnmum regulatory requrement. At the same tme, a bank could produce an nternal estmate of captal at rsk and compare t wth ts avalable captal. The nternal estmate mght consder also on one hand the rsks that are not lnked to a pllar one captal requrement drectly, and on the other hand the dversfcaton benefts that are not consdered at present n defnng mnmum regulatory captal charges, due to the adopton of a buldng block approach 6. The noton of avalable captal may vary, snce n order to run an nternal evaluaton a bank may decde to adopt a noton of captal that does not contan, for nstance, all Ter 2 and Ter 3 components of regulatory captal. The typcal problem n ths case s to check the ablty of the bank to wthstand severe adverse condtons wthout falng; the choce of a book value defnton of avalable captal and of a one-year horzon as the common denomnator to estmate aggregate losses seems reasonable gven the objectve. 9

11 At the same tme, yet, n order to defend shareholders nterests, the rsk manager may be wllng to measure captal at rsk also n terms of market captalzaton of the bank. Adoptng ths vew would mply a completely dfferent judgement about the rsk of certan busnesses, snce there are some that may generate small short term losses but have a much greater mpact on long term earnngs expectatons by analysts and nvestors, and may therefore nfluence market cap sgnfcantly. Consder for nstance the case of an asset management frm nsde a bankng group. If we consder the mpact the busness may have n terms of potental losses of book captal over a one-year horzon, ts rsk s low, snce drect losses would be ncurred only f the total fees pad by the customers were unable to cover ts varable and fxed costs. At the same tme, yet, f the asset management company faced a sharp reducton of management and other fees analysts may revse downwards earnngs expectatons for the bankng group n the future, and ths may have a much more vsble effect on target prces and eventually on stock market captalzaton of the bank holdng company. Ths may be true for many other fee-generatng busnesses. Of course, we are not sayng that regulators should be concerned wth market captalzaton at rsk ; systemc rsk s lnked to the possblty that a bank may fal, and from ths vewpont a busness that may gve a very small contrbuton to a bank s falure should be consdered as a low rsk busness. It s therefore perfectly correct that fee-generatng busness be assocated at present only to an operatonal rsk captal requrement, wth no specfc requste for ntrnsc busness rsk. Instead, from the shareholder s perspectve and hence deally also from bank top management s vewpont a sharp reducton n market cap caused by a fee-based busness s clearly bad news even f t s not assocated to any change n the bank s default probablty. Top managers should at the same tme try to prevent the bank from falng and try 6 Dscussng the relatonshp that should exst between dversfcaton benefts and mnmum captal requrements s beyond the scope of ths paper. The reader may refer to Kurtzkes, Schuermann and Wener 0

12 to ncrease the value of the frm n the market whle controllng the volatlty of ths value. Therefore havng an dea of how great a porton of market captalzaton may be at rsk n case of adverse events (dfferent from operatonal rsk ones) nsde a sngle busness may be useful nformaton for them. As a consequence, an nterestng queston s whch noton of captal should be used when calculatng rsk-adjusted performance measures, such as RAROC and EVA, for the bank s busness unts 7. Of course, the contrbuton of fee-based busness unts wll be remarkably dfferent dependng on whch knd of captal s used. Even f both a measure based on book value captal at rsk and measure base on market captalzaton at rsk may be useful, probably the latter s more adequate n provdng a support to top managers decsons for the allocaton of captal and other crtcal resources f they want to make the value of the frm greater and at the same tme less volatle. In a shareholder s perspectve, then, measurng the contrbuton of busness rsk may be much more mportant than t s n the lght of guaranteeng systemc stablty: ths s why n secton 3 we wll dscuss how busness rsk can be measured and how t can be translated nto a measure of captal at rsk, dependng on whch noton of captal s adopted. 3. Measurng busness rsk The startng pont n any effort to defne an ntegrated measure of captal at rsk s the dentfcaton of all relevant rsks and the defnton of a model to quantfy ther mpact under a common metrcs. In practce, ths metrcs s largely dentfed by captal at rsk or Value at (200, 2002), and Jont Forum Workng Group (2003). 7 RAROC (rsk-adjusted return on captal) s ntended here as a generc rsk-adjusted performance measure equal to the rato between the proft, or margn, produced by a busness unt and the captal at rsk assocated to the same unt. EVA (economc value added), n the form n whch t s typcally appled to banks, s ntended as the dfference between the proft or margn of the unt and the cost of captal, calaulated as the product of ts

13 Rsk. In the past years, methodologes for market and credt VaR have been developed and mproved so to become extremely wdespread, whle loss databases and measurement methodologes for operatonal rsk are currently under development. Busness rsk, nstead, has receved so far less attenton, also due to the fact that there s no mnmum captal requrement lnked to t. Yet, the volatlty of revenues dervng from fee-generatng busness such as asset management and brokerage may have a substantal mpact on a bank s P&L, and may nfluence equty analysts valuatons and the bank s market captalzaton. From the rsk manager s pont of vew, ndependently from the absence of any regulatory constrant, quantfyng ths knd of rsk s therefore an mportant task. In order to measure busness rsk, a bank can n theory adopt one of the followng solutons 8. (a) Analyze the average captal of a sample of mono-busness compettors and set captal at rsk for the fee-generatng busness unt equal to some smple or weghted average of that sample, usng, therefore, a measure of benchmark captal. (b) Identfy rsk wth a measure of earnngs at rsk, e.g. a certan multple of the volatlty of earnngs of the fee-generatng busness unt 9. (c) Calculate earnngs at rsk as n case (b) and then translate ths measure n some way nto a measure of captal at rsk. The frst soluton s conceptually smple but faces several problems. Frst, t may not be possble to fnd pure mono-busness compettors. Second, even f they exst, t may be necessary to scale up or down ther captal - that may cover not only busness but also other captal at rsk multpled by a target return on captal. For a more detaled analyss of rsk-adjusted performance measurement see James (996), Matten (996), Sata (999), Culp (2000). 8 See Matten (996). The Jont Forum Workng Group (2003) survey reports that only some among the banks that have been contacted are really measurng n some way busness rsk ntegratng t n the overall rsk management framework. 9 See agan Matten (996) for a more detaled descrpton. Dependng on data avalablty, n theory one could also decde to derve analytcally the desred percentle on the hstorcal dstrbuton of earnngs, but typcally data are nsuffcent. In other cases, due to the lack of proper data wth a reasonable frequency, a measure of 2

14 rsks, such as operatonal rsk - accordng to dfferences n sze between the nternal busness unt and the set of peers that has been selected. Thrd, ther level of captalzaton may be nfluenced by mnmum regulatory requrements that may be bndng for a mono-busness compettor whle ths would not happen f the same actvty were conducted wthn a dversfed group. Fnally, snce the measure depends on external data only, there s no lnk between the behavour of the busness unt and the benchmark captal at rsk t s assgned. These problems can be overcome adoptng approaches (b) or (c). Snce determnng an Earnngs at Rsk measure s a prerequste for both, we wll frst dscuss how such a measure can be derved and then ts possble translaton nto a captal at rsk mesaure. 3.. Earnngs at Rsk estmaton In theory, earnngs at rsk (EaR) can be dentfed wth the losses generated by a gven busness unt under the maxmum adverse varaton n ts earnngs wthn a gven tme horzon and confdence nterval. For nstance, for the asset management busness, one could try to buld a dstrbuton of the net margn produced by the fund management company and dentfy EaR at a 99% confdence level as the 0.0 quantle of the margn dstrbuton. In practce, yet, defnng busness rsk properly s the frst task for the rsk manager. In partcular, the rsk manager should decde whether to dentfy Earnngs at rsk wth the devaton from expected earnngs or only wth actual losses. Imagne for nstance that for the busness unt Alpha expected earnngs were equal to 50 and the maxmum adverse reducton n earnngs at a 99% confdence level s equal to 20: Earnngs at Rsk should then be 20 (the maxmum potental reducton n earnngs) or just 70 (the actual loss n the worst case scenaro)? Usng actual losses has the dsadvantage that the measure may become very volatlty of earnngs can be derved by measurng the volatlty of revenues and then ntroducng careful assumptons about the structure of varable versus fxed costs for the busness unt. 3

15 senstve to changes n the nternal cost allocaton: a bank that allocates to fnal busness unts a hgher share of ts ndrect costs would measure everythng else beng equal hgher potental actual losses for each busness unt than an dentcal bank that allocates only drect costs. Ths may both mpact evaluaton of dfferent busness unts and overall busness rsk estmates (n ths case, only provded that there s no correcton at aggregate level for the share of unallocated costs). After defnng whether or not to consder expected earnngs, the rsk manager has to derve that measure resortng typcally to a tme seres of a busness unt s reported earnngs. As n the case of operatonal rsk, data avalablty may be a major ssue. Frst, the method cannot be appled at all when there s no tme seres of earnngs avalable. Ths happens for nstance when busness unts have been restructured, or merged, and the structure of the nformaton system contanng accountng data s not flexble enough to reconstruct a theoretcal tme seres for the new busness unts 0. Second, the tme seres of earnngs s lkely to nclude the effect of some adverse events that may be attrbuted to operatonal rsk. It would therefore be possble to consder the same source of rsk twce, once by ncludng the event n the operatonal rsk database and the second tme by consderng ts adverse effect on earnngs volatlty that leads to busness rsk estmaton. Cauton s therefore necessary to derve a tme seres that flters out the effects that are lnked to rsks that are separately measured. Thrd, the rsk manager has to buld a seres of returns from earnngs data, whch mples the defnton of a proper measure of exposure to busness rsk. There are two reasons why returns nstead of pure earnngs s needed. If for nstance the sze of the busness unt had substantally changed through tme, then hgher or lower earnngs n certan perods may reflect a sze change rather than earnngs volatlty. At the same tme, when estmatng 4

16 correlaton across busness unts busness rsk, the dependence across busness rsk n dfferent unts should be measured through returns, that do not depend dfferently from earnngs on the relatve sze of dfferent busness unts. Transformng earnngs nto returns has yet the shortcomng that t s necessary to dentfy a sze, or exposure, measure, that should be proportonally lnked wth returns. Ths measure should also be used at the end of the process n order to translate a percentage EaR measure nto a dollar EaR value. Yet, the choce of such scalng factor may be often judgemental and arbtrary. Ths explans why rsk managers may be tempted to use dollar earnngs dstrbutons n order to estmate earnngs at rsk drectly from these seres. Fourth, t s dffcult to select the rght tme perod from whch to extract data to estmate earnngs volatlty. If the perod s too long, then the measure may depend on old data that may no longer be representatve of the volatlty of the busness unt. If, on the contrary, one decdes to consder a relatvely short tme nterval, than t s almost a must to use monthly earnngs data n order to have a sample of a mnmum sze. Ths n turn has two shortcomngs. On one hand, wth such a small number of data, t s very dffcult to estmate the exact shape of the dstrbuton of earnngs emprcally; hence, the dentfcaton of the desred dstrbuton percentle used to derve an EaR measure cannot be based on a sound ground (and many may end up assumng that the earnngs dstrbuton of any busness unt s magcally normally dstrbuted). On the other hand, usng monthly data can lead to an ncorrect estmate of earnngs volatlty f as t may easly happen there s some seral postve autocorrelaton n earnngs changes, or f the way n whch accountng earnngs are calculated tends to alter n some way the volatlty of the real results of the busness unts. Yet, snce t s not always 0 Ths stuaton s typcal n the case of banks whch have recently experenced a merger. Ths phenomenon may not depend on a precse wll to nfluence fnancal results, but t may smply derve from the way n whch costs are attrbuted (e.g. beng equally splt n each month even f they are partally varable). 5

17 possble to scale monthly return volatlty nto annual volatlty n a smple manner, the annual earnngs at rsk fgure may be derved by numercally smulatng a seres of monthly shocks, and autocorrelaton n case t exsts n monthly returns could be smply taken nto account through the same smulaton procedure From Earnngs to Captal at Rsk After estmatng an EaR measure, the other problem s whether such a measure can be properly compared wth measures of captal at rsk. Matten (996) observes that the EaR measure s typcally dsproportonate to captal at rsk measures for rsks such as market or credt rsk, and ths would mply the attrbuton of an equally dsproportonate RAROC 2 to fee-generatng busness unts. He therefore suggests to translate earnngs at rsk nto captal at rsk by dvdng EaR by the rsk-free rate: captal at rsk would therefore be conceved as the amount of captal that should be nvested at the rsk-free rate n order to cover the loss comng from the fee-generatng busness unt. Therefore captal at rsk (CaR) assocated to the busness unt s defned so that CaR r EaR () and hence CaR EaR r (2) where r s the rsk-free rate. Ths method s smple and requres only one nput: the rsk-free rate. However, the amount of captal that s obtaned n ths manner s not captal that s lost forever (as t would happen, for nstance, for a market rsk CaR n case of an adverse move of the markets) but only an amount of captal for whch the nvestor faces an opportunty cost. 2 We defne here RAROC (rsk-adjusted return on captal) n general terms as a rato between a measure of earnngs of a gven busness unt (or busness area, or even transacton) and the captal at rsk assocated to t; we consder therefore RAROC as the classc rsk-adjusted performance measure. 6

18 As an alternatve to ths smplfed method, we therefore propose here some new methods to translate the EaR measure nto a CaR measure, and that can be derved by analyzng the mpact on the economc value of the bank of a loss equal to EaR. The key dea s to adopt an equty analyst s pont of vew as f we wanted to evaluate how the reducton n earnngs would affect the target prce and therefore the theoretcal economc value of the bank. In ths case, there are dfferent possble solutons dependng on the crtera that the analyst s assumed to follow. For nstance, let us assume that the analyst smply evaluates the value of the equty stake n the bank through a multple, such as the prce/earnngs rato (P/E). Then, the adverse change n the market captalzaton of the bank (.e., the change of economc captal n the worst case scenaro, EC wcs ) that would derve from an unpredcted reducton n earnngs would be equal to CaR x EC wcs,x * P E x Earnngs wcs * P E x EaR (3) where the term * P E x denotes the far prce/earnngs rato that the analyst attrbutes to busness unt x. Ths term could therefore vary across busness unts, snce they can be valued by analysts at dfferent multples. A smlar logc could be followed f analysts were assumed to adopt a dvdend dscount model. In the smple case of a constant growth dvdend dscount model, defnng k e the dscount rate, g the estmated dvdends perpetual growth rate, and D the next dvdend per share, the prce P of the share should be equal to 7

19 D P k g e (4) so that expressng the dvdend D as the earnngs per share multpled by the payout rato, p, and multplyng both terms by the number of outstandng shares, the market captalzaton EC s equal to EC p Earnngs k e g (5) If we assume that the total economc value of the bank s evaluated n ths way by the market, then the reducton n the economc captal of the bank that would follow a reducton n the value of earnngs equal to EaR would be CaR x EC wcs,x p Earnngs k g e wcs,x k e p EaR g x (6) In ths case 3, the Captal at Rsk estmate could even be too conservatve, snce t mplctly assumes that the unexpected reducton n earnngs represented by EaR would negatvely affect the level of future earnngs of each year n the future (snce they wll be calculated captalzng earnngs n year at the rate g). However, reducton n earnngs are sometmes combned wth a downward revson of the growth expectatons for certan areas of busness. Thus, at least for certan busness unts a cautous rsk manager could queston whether the potental adverse varablty of g should also be taken nto consderaton. Translatng the mpact of earnngs volatlty nto captal at rsk adoptng the analyst s vew can also lead, under certan smplfyng assumptons, to a soluton that s reasonably close to Matten s (996) proposal. To see ths, let us assume the analyst evaluates the total value of the bank as the sum of dscounted cash flows, so that the value of the bank V s 3 If one accepts the assumptons of the constant growth dvdend dscount model, then the term p/(k e -g) n equaton (6) s the theoretcal value of the correct P/E, so that equaton (6) s smply a dfferent formulaton of equaton (3), suggestng an alternatve way to estmate the correct multple when a drect estmate s not feasble. 8

20 V Ft ( + t t ) (7) where s the proper dscount rate. Snce a change n V would affect drectly the shareholders wealth, CaR s equal to V. If we assume also that (a) a reducton n earnngs equal to EaR x n busness unt x would mply an equal reducton n cash flows and (b) ths reducton s perpetual, then labelng V 0 the value before the shock n earnngs and V wcs the post-shock value, captal at rsk s equal to CaR x V 0 t V wcs EaRx ( + ) t t Ft t ( + ) EaR x t Ft EaRx t t ( + ) EaRx t ( + ) (8) Ths soluton s close to the one suggested by Matten (996) but (a) t has a completely dfferent dervaton and (b) t uses the dscount rate rather than the rsk-free rate (and therefore produces a lower equvalent captal at rsk, snce >r). Ths also clarfes why Matten s proposal, although clearly gong n the rght drecton, could be consdered too pessmstc: n fact, t assumes that captal at rsk s even hgher than the loss n economc value that the bank would face f the loss of cash flows were perpetual. Of course, by assumng equal changes n cash flows and earnngs 4 and that a loss equal to EaR would persst for n years (nstead of beng perpetual) the estmated captal at rsk would be reduced to CaR V 0 n t V wcs Ft (+ ) t t n Ft ( + ) t Ft EaR t ( + ) t t n t Ft EaR + t ( + ) n EaR (+ ) t t n+ Ft (+ ) EaR t ( + ) n (9) 4 Of course, f a bank wanted to adopt ths method, a drect estmate of cash flows varablty would be advsable. 9

21 The assumpton whether a loss equal to EaR today would be consdered by analysts as transent and recoverable through tme or, nstead, as a sgn of permanently lower earnngs n the future becomes therefore crtcal 5. In general, the alternatve solutons proposed here requre more nputs than Matten s method, and may lead to dfferent CaR values startng from smlar EaR measures, because dfferent busness unts may have dfferent prce/earnngs multples or dfferent customer relatonshps average duraton. Therefore, they may be more easly questoned by the busness unts under scrutny, whch mght also try to negotate the parameters wth the rsk management functon. In any case, they have a sound ratonale and can be made compatble wth any possble evaluaton crteron that banks equty analysts favour n a gven context. As a fnal but very mportant remark, the translaton of EaR nto CaR mplctly assumes that the bank wants to measure ts captal at rsk n terms of the reducton n ts market captalzaton due to an adverse event: n fact, we have dentfed CaR through the mpact of potental losses on the target prce of the bank s stock. Ths pont wll be dscussed n more detal n the next paragraph Harmonzng captal at rsk measures We showed how a measure of captal at rsk can be derved for all dfferent rsks, but then the queston stll remans of how homogeneous those measures can be. As t s well known see also Jont Forum Workng Group (2003) there are at least two choces that requre harmonzaton: the tme horzon used for the CaR estmate, whch s usually daly for market rsk and yearly for credt rsk, and the confdence nterval, where 99% s the standard for market rsk, whereas n the case of credt rsk and at a corporate level a hgher rate such as 5 Ths could be based on emprcal data concernng the average duraton of a customer relatonshp, f adequate data are avalable. 20

22 99.97%, consstent wth the target ratng of the bank, s generally preferred. Whle these effects are qute well-known, the thrd ssue related to the dfferent concept of captal underlyng the measures of captal at rsk s typcally neglected. The harmonzaton of tme horzon s normally acheved by adoptng a one-year horzon for all rsks and by scalng up market captal at rsk by multplyng t by the number of tradng days n a year 6. Alternatvely, when a rsk manager does not need to calculate the annual equvalent of a sngle value (e.g., market CaR calculated on December 3 st ) but rather the annual equvalent of the market rsk the bank has faced over a certan perod of tme (based on a strng of daly CaR values), one could calculate annual CaR as the square root of the sum of n daly CaR values n each day throughout the year accordng to the formula 7 2 CaR annualzed CaR daly, n (0) The harmonzaton of confdence nterval s sometmes obtaned by rescalng the 99% VaR measured for market rsk to the desred and typcally hgher percentle usng a fxed scalng factor as f the dstrbuton were Gaussan. Needless to say, ths can be false n practce, but a drect estmate at the 99.97% confdence level may be very unstable and questonable, e.g. f the bank uses an hstorcal smulaton method whch would requre nterpolaton to determne 6 See for nstance Jont Forum (2003). 7 See Sata (999). The formula s derved consderng annual CaR as a multple of the volatlty of yearly proft and losses, conceved as a random functon equal to the sum of n ndependent random functons represented by daly proft and losses. The formula therefore replcates the relaton between the standard devaton of a varable Y whch s the sum of n ndependent random varables X, X 2, X n. It can be noted that ths soluton s dfferent from annualzng daly CaR values and then takng ther average as the annualzed CaR for, say, the whole year. In fact, ths latter and apparently smpler soluton would not properly take nto account the rsk related to peak exposures. Consder a perod of 25 tradng days and two traders. Let us assume that trader A had a daly CaR equal to 0 durng each tradng day, whle trader B had no rsk (and zero CaR) for 24 tradng days and a CaR equal to 250 n one tradng day only. Even f both have an average daly CaR equal to 0, t s evdent that the second generates a hgher rsk of falure for the bank: n fact, hs trades are not dversfed through tme, and there would be no chance for the bank to ntervene and stop tradng after a certan amount of losses as t would happen for the frst trader. Whle by annualzng daly CaR values and then averagng them one would obtan the same equvalent CaR estmate for both traders, by usng equaton (0) trader B would properly be assgned a much hgher perod-equvalent CaR than hs fellow. In fact, 2

23 that exact percentle. Moreover, snce dfferent busness unts would have dfferent mxtures of dfferent rsks that have n turn dfferent shapes of return dstrbutons, the choce of the confdence level s not neutral wth respect to the relatve captal at rsk absorpton of dfferent busness unts (see Hall 2002), and therefore t may nfluence not only the absolute level of reported RAROC for a unt, but also ts relatve rankng nsde the organzaton. Ths s why defnng the confdence nterval n an objectve manner (e.g. lnkng t to the target ratng the bank s wllng to mantan) s very mportant. The thrd relevant ssue n harmonzng CaR measures s related to the noton of captal that underles captal at rsk measures for dfferent knd of rsks. There are at least three dfferent versons of captal that can be and are used:. Market captalzaton value of captal (.e. the value of the equty of the bank on the stock market). 2. The market value of captal (.e. the dfference between the value of marked-to-market assets and the value of marked-to-market labltes, wthout ncludng any form of goodwll or badwll). 3. The book value of captal (.e. the dfference between the value of assets and of labltes at book values). Tryng to smplfy and reduce the possble behavour of a bank to a few most common alternatves, we can say that: market CaR s measured n mark-to-market terms; credt CaR may be measured ether at book values (especally as far as loans are concerned) or at mark-to-market values, or both; the overall CaR measure for the whole perod accordng to equaton (0) would be equal to 250 for trader B and to 0 25 only for trader A. 22

24 busness rsk CaR can be measured ether at book values (e.g. as a measure of oneyear Earnngs at Rsk) or as the market cap value of captal (by applyng some of the methods descrbed n the prevous paragraph to convert an EaR nto a CaR measure). Ths lack of consstency between captal at rsk measures could be solved dfferently dependng on the objectve. If the goal s to have a consstent measure n order to calculate rsk-adjusted performance (RAP) measures to support captal allocaton, then market cap value of captal s lkely to be the correct soluton snce t s the most consstent wth the shareholder s vew of the bank. If one wants to check whether the bank s suffcently captalzed or not, n order to support decsons about the bank s optmal captal structure polcy, then t can be argued that n theory one should lke CaR measures not only to be ntrnscally homogeneous, but also to be consstent wth the measure of avalable captal wth whch they are compared. It s therefore possble to compare for nstance a book value measure of captal at rsk and compare t to avalable captal at book values, and an economc measure of captal at rsk and compare t to captal measured as the current economc captal of the bank at stock market prces. The same may happen at mark-to-market prces, so to derve a set of co-exstng constrants for the bank s survval, (havng of course the fourth key boundary of mantanng regulatory captal hgher than mnmum captal requrements). Ths approach dffers from the common practce to assume that all albet dfferent measures of captal at rsk have to be compared aganst a sngle measure of avalable captal. In practce, t may clearly be hard and costly to mantan dfferent measures of captal at rsk accordng to dfferent crtera, and try to correctly explan the dfferences to the board members that have to take key decsons. Yet, t s mportant to try to attan consstency of the CaR measures that are used; moreover, acceptng ths vew some problems become easer to be managed. For nstance, whle for busness rsk the rsk manager apparently has to choose between a pure 23

25 EaR measure or a much hgher CaR measure 8, f dfferent notons of avalable captal are consdered then the two measures become an accountng captal at rsk measure and a market cap captal at rsk measure that must be compared wth dfferent amounts of avalable captal (the larger CaR measure beng compared wth the hgher amount of captal). 4. Frm-wde ntegraton and the ssue of correlaton across rsks and busness unts. Aggregatng dfferent rsk measures nto a sngle number for the whole nsttuton requres to tackle at least three dfferent ssues: (a) dentfyng the components that have to be aggregated, (b) dentfyng the aggregaton technque or algorthm to be used, and (c) calbratng the parameters (e.g. correlaton coeffcents) needed to derve the sngle rsk measure. We wll approach these ssues precsely n ths sequence. 4.. The choce of the components to be aggregated: busness unts versus rsk types Gven a homogeneous measure for all rsks, the problem s how to aggregate ndvdual rsks to obtan a sngle captal at rsk measure. The frst ssue s the choce between busness unts and rsk types as the startng pont for rsk aggregaton 9. Aggregatng across busness unts has the advantage of a clear lnk between the ndvdual dvsons CaR measures and the overall captal requrement of the bank. Moreover, correlaton coeffcents can be estmated through the seres of the P&Ls of the busness unts. At the same tme, there are two clear dsadvantages. Frst, snce a sngle busness unt s typcally exposed to multple rsks, the need to defne or assume a correlaton coeffcent across rsks n not avoded at least when calculatng stand-alone CaR for the ndvdual busness unt. Secondly, f the same knd of rsk s common to two or more busness unts, ths 8 Consder for nstance that f one adopted Matten s suggeston, f the rsk free rate s equal to 4% then CaR would be equal to EaR multpled by

26 method may fal to capture the compensatons between exposure n dfferent busness unts (magne the case n whch busness unt A s exposed to the rsk of ncreasng n nterest rates whle B has an opposte rsk profle). Opposte exposures may n fact be netted correctly only f there s frst a groupwde rsk mappng and then an aggregaton among rsk types 20. In theory, therefore, an aggregaton across rsk types should be preferred at least when the man purpose s to derve the best possble measure of the total captal at rsk at groupwde level. Yet, the rsk manager often has to handle also rsk aggregaton across busness unts, snce he s frequently asked to evaluate busness unts not only accordng to ther undversfed, stand-alone CaR, but also to some measure of dversfed captal at rsk. Therefore, she or he cannot avod the problem of estmatng correlaton coeffcents across busness unts (ether drectly or ndrectly, through the correlaton among rsk factors and rsk factor exposures across busness unts) The aggregaton algorthm Apart from the choce between rsk types and busness unts aggregaton, the rsk manager has to choose an aggregaton technque and to dentfy ts parameters. As Rosenberg and Schuermann (2004) clearly descrbe, the portfolo CaR of the whole bank n percentage terms can be wrtten as CaR ( α) µ + σ F ( ) () p,% p p p α 9 See Jont Forum Workng Group (2003). 20 Not surprsngly, yet, aggregatng group-wde rsk factor exposures becomes more dffcult n the case of a fnancal conglomerate, so that accordng to the Jont Forum survey they seem to aggregate exposures wthn all the bankng group but excludng the nsurance dvsons. If we consder for nstance a lfe nsurance company and the tradng dvson of a bankng group, they are both exposed to market rsks, but n a dfferent manner. Frst, the lfe nsurance company may be senstve to market rsk especally over long tme horzons rather than over the short term; secondly, n a lfe nsurance company the market rsk may be so ntertwned wth other typcal rsks (e.g. dervng from actuaral assumptons made n defnng the nvestment strategy, or from customers behavour regardng the surrender or extenson optons often embedded n lfe nsurance contracts) that t may be necessary to estmate ts mpact through a jont smulaton of all relevant varables. 25

27 26 where ) ( p α F s the α-quantle of the standardzed return dstrbuton. The varance of portfolo return can be wrtten as + n J j j j n p w w w ρ σ σ σ σ (2) where w, w j represent the weghts of the portons of the portfolos that are beng aggregated, σ, σ j are the standard devaton of ther returns and ρ s the correlaton coeffcent between returns on assets and j. Therefore, by mergng () and (2) we get [ ] [ ] n J p j j j n p p n J j j j n p p p F w w F w w w w F CaR ,% ) ( ) ( ) ( ) ( α ρ σ σ α σ µ ρ σ σ σ α µ α (3) If one assumes that the quantles of standardzed returns of each porton of the portfolo ) ( ), ( ), ( α α α z y x F F F are dentcal to the quantles of the portfolo returns, then one could substtute ) ( p α F wth the approprate quantle and express (3) n terms of ndvdual CaR values n percentage terms (CaR,%, CaR j,% ): [ ] [ ] [ ] [ ] [ ][ ] n J j j j j n p n J j j j j n p n J p j j j n p p p CaR CaR w w CaR w F F w w F w F w w F w CaR,%,% 2,% ,% ) ( ) ( ) ( ) ( ) ( ) ( ρ µ µ µ µ ρ α α σ σ α σ µ α ρ σ σ α σ µ α (4) A further smplfcaton of the formula could be represented by neglectng the expected return terms (both for prudental purposes and consderng ther dffcult estmaton) and by expressng CaR n absolute rather than percentage terms, multplyng both terms for the value of the whole portfolo V. Ths smply leads to the well-known formula

28 CaR p n CaR 2 + n j CaR CaR ρ j j (5) Equaton (5) depcts the smplest possble technque for rsk aggregaton, and corresponds to ether Normal VaR or Hybrd VaR, accordng to the defnton of Rosenberg and Schuermann (2004), dependng on how ndvdual CaR s calculated. The formula reles on a crtcal assumpton concernng the relaton between ndvdual dstrbuton quantles and portfolo quantles, that s satsfed n the case of ellptc dstrbutons (a partcular case of whch s a jont normal dstrbuton). If ths assumpton s not satsfed, then more sophstcated technques, such as copulas, would be necessary. Copulas, n fact, can be much more flexble. Unfortunately, snce there are many dfferent possble copula functons, the problem of defnng whch s better remans. Unfortunately, the rsk manager lacks the amount of data that would be needed to support the choce wth clear emprcal evdence. The choce of the copula functon that s best suted to represent the jont dstrbuton of two or more stock ndexes returns s a problem that can be solved by analyzng many years of daly data. When, on the contrary, the problem s the aggregaton among busness unts or rsk types, t s vrtually mpossble to have long enough tme seres of returns/earnngs so as to conduct a smlar analyss. Consequently, the choce of a partcular copula functon would at the end be partally judgmental and arbtrary. Ths could be a major problem f the aggregated captal at rsk measure s expected to enter nto the bank s decsonal processes. For nstance, f busness unts were evaluated on a rsk-adjusted performance measure based on dversfed captal at rsk, then the aggregaton technque may nfluence the allocaton of dversfcaton benefts among dfferent busness unts, and t could be very dffcult for the rsk manager to defend a subjectve choce aganst the crtques of 27