Rsk based equty cost calculaton n bankng * Mark Wahrenburg Unversty of Cologne, 50923 Cologne, Germany Rajeev De Mello McKnsey and Company, 8703 Erlenbach/Zürch, Swtzerland August 1993 The objectve of bank rsk management s to optmze the rsk/return poston of the bank. Snce a bank conssts of a portfolo of dfferent and often nterdependent rsks, ths objectve can only be acheved by ncorporatng rsk nterrelatons nto the analyss. In ths artcle, we propose a practcal method how the total rsk of a bank ncludng rsk nterrelatons can be determned and how the rsk/return stuaton of ndvdual busness unts can be evaluated from the total bank perspectve. It s shown that the evaluaton can be easly ncorporated nto the bank's accountng system by employng busness unt equty costs that are based on the unt's contrbuton to the total bank rsk. The proposed concept provdes senor management wth a smple tool to allocate the bank's captal such that the bank's rsk/return trade off s optmzed and to provde busness unt managers wth ncentves to optmze the total bank rsk exposure. JEL Classfcaton code: G11, G21, M41 Keywords: Rsk management, Portfolo analyss, Rsk adjusted performance measurement C orrespondence to: Mark Wahrenburg, Staatswssenschaftlches Semnar, Unverstät zu Köln, 50923 Köln, Federal Republc of Germany, Tel +49 221 4704354, Fax +49 221 4705151. * We would lke to thank Tom Copeland, Carl Chrstan von Wezsäcker and Tom Wlson for helpful comments and suggestons. The usual dsclamer apples.
2 1. Introducton Rsk management s recevng ncreased attenton from senor bank management. The growng volatlty of many fnancal varables such as exchange rates, nterest rates or real estate prces has drven up the rsk exposure of most banks. The resultng deteroraton of bank ratngs demonstrate the need to control rsk n order to reduce the cost of captal. At the same tme, the ongong process of regulatory reform ncreases equty requrements for many rsky products and further contrbute to the growng rsk conscousness of bank management. 1 For a long tme, fnance theory offered lttle help to explan bank managers desre to lmt bank rsk. Although t has been acknowledged that banks perform a portfolo management functon, ths functon n prncple falls under the Modglan-Mller theorem on the rrelevance of pure fnancng decsons. 2 Thus n a world of perfect captal markets, bank rsk management does not add to shareholder value. Followng ths vew, Saunders, Strock and Travlos (1990) explan the rsk management actvtes of banks as an outflow of managers self nterest n preservng ther frm specfc captal. However, new theoretcal fndngs cast deep doubts on the approprateness of ths vew and suggest that there s a ratonale for controllng bank rsk. Bankruptcy costs and the tax system cause an nverse relaton between a frm's total rsk and ts expected cash flow, whch make reductons of total rsk valuable even for perfectly dversfed shareholders. 3 On the other hand, Froot, Scharfsten and Sten (1992) show that cash constrants due to unfavourable outcomes of nvestment projects can ncrease the cost of captal and may prevent a frm from explotng valuable nvest- 1 The desre of bank management to lmt total rsk exposure has been proven emprcally by Shrves/Dahl (1992) and Heggestad/Houston (1992). 2 Fama (1980). 3 Orgler/Taggart (1983), Smth/Stulz (1985), Shapro/Ttman (1986).
3 ment opportuntes. 4 Agency theory furthermore suggests that hgh dosyncratc rsk s assocated wth perverse ncentve effects. A hgh probablty of bankruptcy deterorates ncentves from optmal compensaton contracts [Campbell/Kracaw (1987)] and gves management ncentves to nvest n overly rsky projects [Campbell/Kracaw (1990)]. 5 In addton, hgh rsk presses management towards short-sghted decsons [Narayanan (1985)] and produces ndrect bankruptcy costs when tradng partners or prospectve employees are reluctant to deal wth a frm for fear of ts future default rsk [Ttman/Wessels (1988), Shapro/Ttman (1986)]. The agency costs of hgh dosyncratc rsk are presumably of partcular mportance n the bankng sector because banks can easly change the rskness of ther nvestments and because a long tme horzon s an ndspensable requrement for good credt decsons. Slovn, Sushka and Polonchek (1993) also show that the Contnental Illnos Bank falure caused sgnfcant negatve share prce effects for Contnental's clents suggestng that bank durablty has a postve value for bank clents and thus ultmately also for bank shareholders. We therefore conclude that banks have good reasons to control ther total rsk exposure. Havng establshed the need to manage the total bank rsk, the natural second queston s how to mplement such a system n a bank. Snce the bank conssts of a portfolo of dverse rsks, the consderaton of dversfcaton effects obvously s crucal for the evaluaton of the total bank rsk. For example, there may be two busness unts wth offsettng rsks. Whle both unts seem rsky when vewed n solaton, they are n effect rskless when vewed from the total bank perspectve. A central decson maker who optmzes the bank's overall rsk poston wll of course recognze these nterrelatons n hs rsk takng decsons. For example, senor management mght fnd t advantageous to ncrease foregn currency deposts because the bank already has 4 In a related model, Stulz (1990) consders also the possblty of overnvestment when cash flow s hgh. 5 Lambert (1986) shows that the reverse may also be true: rsk may prevent a manager from gatherng nformaton about rsky projects and nduces hm to nvest n overly safe projects.
4 offsettng credt clams n that currency. However, rsk takng decsons are more and more decentralzed n order to make use of the superor nformaton of decentralzed busness unt managers and to mprove the nternal ncentve structure of the bank. Decentralzed decson makng of course prevents a central optmzaton of rsk takng decsons. In the above example, a foregn branch manager who s not responsble for nternatonal credts may fnd depost rasng too rsky because he does not take nto account the dversfyng effect for the bank as a whole. Decentralzed decson makng under these crcumstances does not lead to an optmzaton of the total bank rsk because busness unt managers care only about the solated rsk of ther unts. In ths artcle we demonstrate how banks wth decentralzed decson makng nevertheless can optmze ther overall rsk/return trade off. Senor management n prncpal has two optons how to enforce the optmzaton of the total bank rsk: t can control the allocaton of captal among the bank's busness unts and t can provde busness unt managers wth ncentves to optmze the overall bank performance n respect to rsk and return. We show that both objectves can be acheved by employng busness unt equty costs that are based on the rsk contrbuton of a busness unt to the total bank rsk. In the example of the foregn branch manager, depost rasng wll be assocated wth low equty costs due to ts dversfyng effect for the bank as a whole and wll make depost rasng attractve. In the followng sectons, we develop a practcal approach to mplement calculatory equty costs based on the rsk contrbuton of a busness unt. However, tradtonal portfolo theory n the sprt of Markowtz (1959) cannot be appled for the evaluaton of bank rsk snce the rsk of nterest rate movements and especally the rsk from dervatve nstruments such as optons and futures cannot be ncorporated nto the standard portfolo model. For our analyss we choose a contnuous tme model as developed n Merton (1990), n whch a set of random rsk factors drves the bank proft. Secton 2
5 frst dscusses exstng approaches to calculate busness unt equty costs and ther shortcomngs. In sectons 3 and 4, we derve the total bank rsk n terms of the standard devaton of the bank's proft and show, how ths rsk can be allocated to the dfferent busness unts. Secton 5 derve a measure for busness unt equty costs based on the contrbuton to the total bank rsk and demonstrate ts scope for bank rsk management. In secton 6, we outlne other promsng applcatons of our concept and especally dscuss ts applcablty for rsk adjusted performance evaluaton of busness unt managers resp. traders. Fnally, the man results are summarzed n secton 7. 2. Exstng approaches to calculate busness unt equty costs Among the varous methods to calculate busness unt equty costs, we wll dscuss only the four most promnent: the allocaton of equty costs accordng to poston sze, accordng to regulatory captal requrements, based on beta factors derved from the captal asset prcng model and fnally based on poston rsk as measured by the captal at rsk concept. The allocaton of the bank's total equty costs to assets and labltes accordng to ther sze clearly s unacceptable from a rsk management perspectve because every poston s charged the same requred rsk premum regardless of ts actual rsk. Furthermore, ths approach does not allocate any equty costs to off-balance sheet products, treatng them mplctly as rskless. The concept most often used n practce s to allocate captal accordng to regulatory captal requrements and to apply the total equty cost rate of the bank to every unt. The major weakness of ths method s the crude relaton between regulatory captal and rsk. For example, dfferent loans may have very dfferent degrees of rsk but nonetheless requre the same amount of regu-
6 latory captal. In addton, some major rsks lke the nterest rate rsk dervng from an asset lablty msmatch do not requre any regulatory captal at all. A large part of the rsk borne by equty holders s therefore completely gnored n the cost calculaton. Although regulatory changes wll brng regulatory captal n a closer lne wth rsk, ths method wll necessarly reman crude because the bank's nsde knowledge about ts rsks s neglected. Besdes these weaknesses, both methods mentoned share the total neglect of rsk nterrelatons between dfferent rsks. A busness unt receves the same amount of equty costs regardless of whether t actually ncreases or decreases the total bank rsk. Under the assumptons of the captal asset prcng model, the systematc rsk measured by the beta factor s the relevant measure for the rsk of a busness unt and equty costs should thus be a functon of beta [Leemputte/Kearney (1990), Harrs/O Bren/Wakeman (1989)]. Besdes the practcal problem of estmatng beta when no stock wth comparable rsk characterstcs trades n the market 6, t s hghly questonable whether the captal asset prcng model s an approprate model for gudng rsk management at all. Accordng to the captal asset prcng model, there s no need to manage the dosyncratc rsk of the bank, because only ts systematc rsk n the stock market matters. Emprcal studes nvestgatng the determnants of stock market returns however queston the valdty of the captal asset prcng model to explan stock market returns. Fama and French (1992) found that the beta factor has only a weak relaton to stock returns and that nstead sze and book-to-market equty powerful explan returns. Bhandar (1988) found that leverage helps to explan stock returns even after controllng for beta suggestng a postve relaton between dosyncratc rsk and captal costs. Chan and Chen (1991) conclude from ther nvestgaton of the small frm effect that the probablty of fnancal dstress mght be a good predctor of the cost of captal. Gven these emprcal fndngs and the theoretcal argu- 6 Harrs/O Bren/Wakeman (1989) dscuss ways how to crcumvent ths problem.
7 ments for controllng total bank rsk outlned n the ntroducton, we reject the CAPM approach for bank rsk management. In an attempt to base equty costs on the true rsk of a busness unt, Bankers Trust developed RAROC or Rsk Adjusted Return On Captal. RAROC s based on the dea that the allocated equty captal of a busness unt should equal ts captal at rsk, whch s defned as the worst possble loss wthn a gven confdence nterval over a gven tme perod. Whle RAROC s superor to the regulatory approach n terms of recognzng all types of rsk and dfferentatng exactly between dfferent degrees of rskness, t has the severe shortcomng that captal costs are ndependent of the actually nvested equty captal. Consder for example two bond portfolos wth equal rsk. The frst one s self-fnancng and consequently has a lqudaton value of zero whle the second one has a net worth of $1 mllon. Snce both portfolos have equal rsk, RAROC allocates the same equty costs to both portfolos. An nvestor should of course expect a hgher return on the second portfolo snce he expects an extra return on the nvested captal. The shortcomng of RAROC s the neglect of the tme component of equty costs. Whle basc fnancal theory states that captal costs nclude both a tme component and a rsk component, RAROC consders only the latter. Besdes ths weakness, RAROC also shares the neglect of dversfcaton effects between dfferent busness unts because t regards only the solated rsk of one unt. In the followng sectons we develop a method that s free of the shortcomngs of the approaches dscussed above and that ncorporates nterrelatons between dfferent rsks by measurng the rsk of a poston n terms of ts contrbuton to the total bank rsk. The analyss proceeds n three steps: Frst we develop a method to calculate the total rsk of a bank. Second, the contrbuton of a busness unt to the total bank rsk s determned. Fnally, we show how busness unt equty costs can be calculated usng the rsk contrbuton concept.
8 3. Measurng the total bank rsk In order to derve a measure of the total bank rsk, we frst have to defne the relevant performance ndcator. Tradtonally, bank management focused on accountng proft as the approprate performance ndcator. Rsk management consequently had the task of stablzng the bank's accountng proft. However, t s well known that accountng fgures provde only an mperfect pcture of actual bank performance due to the scope to manpulate the stated proft. It s now wdely accepted that market value changes are a superor ndcator of performance snce market values leave no scope for manpulaton. We therefore choose the change of the market value of the bank's assets and labltes as the relevant measure of performance. For tractablty reasons, we measure rsk n terms of the standard devaton of outcomes. We consequently defne bank rsk as the standard devaton of future market value changes of the bank's assets and labltes. 7 The market value of assets and labltes depends on the constellaton of many dfferent rsk factors such as nterest rates, stock prces, exchange rates etc. In order to derve the standard devaton of market value changes, we frst need a model of the stochastc behavor of the underlyng rsk factors. We assume a geometrc brownan moton process for rsk factor movements, whch states that any rsk factor S follows the process µ t+ σ tω S = S e, (1), t+ t, t % 7 See also Elton/Gruber (1992) for a justfcaton of ths assumpton n a smlar settng.
9 where %ω s a standard normal varate wth zero mean and unt varance. The brownan moton assumpton has the advantage that the process depends only on two varables, the drft factor µ and the standard devaton σ, whch can easly be estmated from tme seres data. The dstrbuton functon of S t+ t s lognormal whch has the favourable property that negatve future rsk factor values have zero probablty. A further advantage of the brownan moton assumpton s the fact that t s used n most opton prcng models. It follows that mpled volatltes derved from opton market prces can be used for the calculaton of the total bank rsk. Snce mpled volatlty s the market's assessment of future volatlty, t represents a better estmate of future rsk than estmates from hstorcal data and has the addtonal advantage that t s objectve and non manpulable. Usng Ito s lemma, we can derve the change of the rsk factor over small perods of tme as S = S ( µ + 1 2 σ ) t + S σ t ω%. (2) Havng defned a stochastc process for rsk factors, we need to determne the mpact of movements by the varous rsk factors on the market value of the bank's assets and labltes. One obvous way to do ths s to set up a prcng functon that relates the market value of a poston to the value ts underlyng rsk factors. The mpact of rsk factor changes could then be calculated by smply evaluatng the prce functon at dfferent rsk factor levels. However, workng wth complete prce functons s cumbersome and often mpossble because closed form solutons do not exst for many (especally dervatve) products of banks. In order to smplfy the analyss, we calculate the mpact of rsk factor moves usng a frst order Taylor seres approxmaton. If P k (S,t) denotes the market value of all postons of busness unt k as a func-
10 ton of the underlyng rsk factors and tme, ts proft or loss due to rsk factor movements and the passage of tme can be expressed as Pk = δ, k S + δ t, k t + o( 2 ). (3) In ths expresson δ,k denotes the senstvty of the k's busness unt postons to changes of rsk factor and δ t,k s the senstvty to the passage of tme. The term o(2) summarzes effects of second and hgher order and can be neglected for small changes of S and t. Snce the senstvty of a portfolo of postons s just the sum of the ndvdual poston senstvtes, δ,k can be determned by smply addng the δ-factors of the sngle postons of the busness unt. For many postons, the neglect of hgher order effects has no mpact because many market values, e.g. the value of a stock portfolo or foregn exchange postons, are lnear functons of the underlyng stock prces resp. exchange rates. For nterest rate senstve postons however, the Taylor seres approxmaton neglects the convexty rsk and thus s an approprate estmate of profts only for small changes of nterest rates, when convexty has a neglectble mpact. Another problematc area are opton values whch are generally non-lnear functons of the underlyng rsk factors. For these reasons, the proposed concept should not be used for rsk estmatons over long tme perods, when the error ntroduced by a lnear approxmaton becomes mportant. Because δ-factors are addtve, we can sum up the senstvtes of dfferent busness unts to obtan the total bank senstvty to a shft of rsk factor : δ =Σ δ,k. The total bank proft P thus can be expressed analogously as
11 P = Pk = δ S + δ t t. (4) k After nsertng (2) nto (4) we can compute the total bank rsk n terms or the standard devaton of the bank proft P: σ P = δδ j S S j t σ, j (5) j In ths expresson σ,j denotes the covarance between two rsk factors and j ~ ~ ( ) σ = σ σ E ω ω, (6), j j j and E(% ω ω % j ) s the correlaton coeffcent between rsk factor and rsk factor j. From (5) we can assess the data requrements to calculate the total bank rsk: the necessary data are frst the senstvtes of every poston to every rsk factor, second the actual rsk factor values and ther standard devatons and fnally the correlaton matrx of all rsk factors. It s worthwhle to consder whch set of rsk factors should be ncluded n a practcal real world applcaton of the model. In order to keep data requrements and computatonal effort at a reasonable level, the huge amount of potental rsk factors has to be reduced to a tractable set of the most mportant drvers of the bank proft. For stock market rsk for example, t would not be realstc to treat every stock prce as a dfferent rsk factor. Instead, a practcal applcaton would use stock prce ndces to
12 model stock prce rsk. For foregn exchange rsks, the exchange rates are the natural rsk factors. Countres wth hghly correlated exchange rates could be treated as one currency block n order to smplfy the analyss. For real estate holdngs, the approprate rsk factor would be a sutable real estate prce ndex. In the area of nterest rate rsks, at least a short and a long term rate should be ncluded for every currency n order to nclude the yeld curve rsk n the calculaton. The behavor of nterm nterest rates can be modelled usng a two factor yeld curve model 8, n whch the change of nterm rates s modelled as a lnear combnaton of both the change of the long and the short term rate. A dffcult task s the choce of rsk factors for credt rsk. A frst step for a practcal estmaton of credt rsk would be the groupng of credts wth smlar rsk characterstcs nto classes and to estmate the credt rsk for every class as a whole nstead of nvestgatng every sngle credt. A bank that already ntroduced a sophstcated ratng system could use ths system for classfcaton. The second problem les n the determnaton of the relevant rsk factors. The obvous frst rsk factor to consder s the default rate. However, changes of the market value of a credt portfolo can also come through a change of credt spreads demanded n the market due to up- or downgradngs or due to changes of the market spread of a gven rsk class. A sophstcated system would therefore use the default rate and the spread of every rsk class as the relevant rsk factors. If a bank decdes to proceed ths way, the major problem arsng s the estmaton of the varance and covarances of default rate and spread movements for every rsk class. These parameters n prncple could be derved from bond prces n the US bond market. However, a bank may worry whether the rsk characterstcs of the bond market are comparable wth ts own credt portfolo. A better soluton mght be to develop a data base from nternal and other bank's data on credt experence. Fnally, a last set of potental rsk factors are mpled volatltes. Banks wth large dervatve departments are heavly exposed to changes of mpled volatltes and therefore should also 8 See for example Ingersoll (1983), Elton/Gruber (1988).
13 nclude mpled volatltes as dstnct rsk factors for rsk measurement. Fortunately, tme seres of mpled volatltes are publcly avalable so that ther ncluson nto the model ntroduces few addtonal data problems. 4. Measurng the rsk contrbuton of busness unts Untl now, we regarded the rsk of the bank as a whole. Now we turn to our orgnal queston and analyse how ths total rsk can be allocated between the dfferent busness unts of a bank. We search for a busness unt rsk measure that meets two requrements: frst the sum of the ndvdual rsks should equal the total bank rsk and second the busness unt rsk should represent ts contrbuton to the total bank rsk. We proceed by nvestgatng the mpact on total bank rsk of a small change of the total bank senstvty δ by takng the frst dervatve of the total bank rsk n respect to δ : σ P δ = δ S S t σ, j j j j σ P (7) We can nterpret ths expresson as the margnal rsk contrbuton of one unt of senstvty δ to the total bank rsk. The margnal rsk contrbuton of a busness unt wth senstvtes δ k, can be derved by multplyng equaton (7) wth the actual senstvtes of the busness unt:
14 MRC k = δ k, j δ S S t σ j j, j σ P (8) It turns out that the sum of all margnal rsk contrbutons from every busness unt equals the total bank rsk: k MRC k j δδ S S tσ j j, j = = σ P σ P (9) The margnal rsk contrbuton thus fulflls the requrements for an allocaton rule of the total rsk to the sngle busness unts: the measure of busness unt rsk s based on ts contrbuton to the total bank rsk and the sum of the ndvdual busness unt rsks equals the total bank rsk. In the jargon of portfolo theory, we can nterpret the margnal rsk contrbuton of a busness unt as the systematc rsk vewed from the total bank portfolo perspectve. The margnal rsk contrbuton s the part of a busness unt's rsk that s not dversfed away by offsettng rsks n other unts of the bank. 5. The calculaton of busness unt equty costs Havng derved a rule how to allocate the total bank rsk between ts busness unts, we now want to show how busness unt equty costs based on the contrbuton to the total bank rsk can be derved. We assume that the bank knows ts equty cost rate and the only problem s to allocate the total equty costs to ts busness unts.
15 Approaches to calculate equty costs of corporatons have been wdely dscussed and wll not be outlned here. Standard methods nclude the captal asset prcng model, mult-factor arbtrage prcng models or a smple target return on equty that s strategcally chosen by senor bank management. From the dscusson of RAROC t became clear that equty costs have two components: a tme component and a rsk component. The tme component s the remuneraton of the nvested equty captal n the absence of rsk. Snce equty holders cannot expect to earn more than the rsk free rate of nterest on a rskless nvestment, ths component s the rsk free nterest on the nvested captal. It should be noted that the tme component of many busness unts wll be negatve because busness unts operatng on the lablty sde of the balance sheet wll typcally have negatve lqudaton values. The second component s the rsk component of equty costs. If we denote the total market value of all postons of the bank as γ, the rskless rate of nterest as R f and the requred rsk premum as RP, the total equty costs of the bank C E can be expressed as ( f ) C = γ R + RP. (10) E The margnal rsk contrbuton concept can now be appled to allocate the total rsk costs γrp among the bank's busness unts. Snce the margnal rsk contrbutons add up to σ P, the share of the total rsk costs allocated to a busness unt must be 1/σ P of ts margnal rsk contrbuton. Let γ k denote the market value of the postons of busness unt k. The resultng equty costs of a busness unt C E;k are
16 C = γ R + δ E, k k f k, δ jss j t σ, j γ RP. (11) 2 σ j P As can be easly verfed, the equty costs of all busness unts from (11) sum up to the bank's total equty costs n (10). Equaton (11) thus defnes a rule how the bank's total equty costs can be allocated to busness unts based on a unt's rsk contrbuton. What are the advantages of ths equty cost calculaton over exstng approaches? We see the man advantage of our approach s ts superor ablty to gude captal allocaton and to provde ncentves to busness unt managers such that the bank's rsk/return trade off s optmzed. Consder the example of a busness unt that actually reduces the total bank rsk and earns a return on captal equal to the rsk free rate. Ths unt should be sustaned even f ts return on nvested equty captal s slghtly below the rsk free rate of nterest due to ts dversfyng mpact on the bank. Equty cost rules as appled today however allocate postve rsk costs to such unts, producng calculatory losses although performance good from the total bank perspectve. When captal allocaton decsons are based on tradtonal equty costs, ths busness unts wll be scaled down snce t produces calculatory losses, although t should n fact be expanded. On the contrary, captal allocaton decsons based on our proposal automatcally optmze the bank's rsk/return trade off. We demonstrate ths by nvestgatng the optmal rsk takng polcy of a rsk averse bank that wll maxmze the rato of profts from rsk takng and the resultng rsk: δ E( S ) σ P max! (12) δ
17 The frst order condton of a maxmum s j δ S S tσ j j, j E( S ) = δ E( S ). (13) 2 σ P (13) states that for the effcent portfolo, the expected move of a rsk factor equals the (relatve) margnal rsk contrbuton of the rsk factor multpled by the expected total bank proft from rsk takng δ E( S ). From the concavty of the objectve functon n respect to δ t follows that the bank should ncrease (decrease) ts exposure δ f the expected move of rsk factor s smaller (larger) than the expected total bank proft from rsk takng multpled by the (relatve) margnal rsk contrbuton of the rsk factor. We can translate expected rsk factor change nto expected busness unt profts from rsk takng by multplyng both sdes of the equaton wth the busness unts senstvty and summng up for every rsk factor. The resultng modfed frst order condton s δ SS tσ j j, j E j δ ( S ) = δ δ E S k, ( ) k,. (14) 2 σ P Note that the rght hand sde of (14) s equal to the rsk component of equty costs that was derved n (11) f δ E ( S ) = γ RP holds. We can therefore conclude that captal allocaton decsons based on our proposed equty cost calculaton method optmze the bank's rsk/return trade off f the bank uses a total bank equty rsk premum that s far n the sense that t s the premum that the bank actually expects
18 to earn. When ths condton s met, the proposed equty cost concept provdes senor bank management wth an extremely smple tool to optmze the total rsk poston of the bank: The only task of senor management s to observe busness unt profts and to expand the allocated captal where profts are hgh and to reduce captal where profts are low. 9 Besdes usng the equty cost concept for captal allocaton decsons, the concept can also be used to provde busness unt managers wth drect ncentves to optmze the bank's total rsk poston. If fnancal benefts and/or promotons are based on profts from the above equty cost calculaton, busness unt managers wll realze that they can gan from takng rsks that have a dversfyng effect for the bank. Dversfyng postons become attractve because they lower the equty costs allocated to a busness unt. However, busness unt profts derved from our proposal should not consttute the sole ncentve devce for managers. The reason s that the proposed equty costs create a far benchmark for rsk adjusted performance evaluaton only n the specal case of a mean varance effcent total bank portfolo. The approprateness of our concept for rsk adjusted performance evaluaton wll be dscussed n depth n the followng secton. 6. Other applcatons Besdes the optmzaton of the bank's rsk/return trade off, the outlned framework has further promsng applcatons n the feld of rsk measurement and rsk management. We dscuss how the concept can be used to vsualze rsk through scenaro analyss, to montor the bank's rsk exposure, to evaluate the adequacy of the bank's 9 Ths argument of course assumes that past profts or losses ndcate future expected profts or losses.
19 equty captalzaton, to evaluate the rsk adjusted performance of busness unt managers or traders and fnally to set prces based on the bank's current rsk stuaton. Scenaro analyss s currently the most often appled tool for measurng the rsk exposure of a bank. Whle we argued that rsk should be measured by the standard devaton of the bank's profts, scenaro analyss stll may be a useful tool for demonstratng bank management potental rsk factor constellatons that represent dangerous threats to the bank. A bank that uses the proposed equty cost calculaton can mmedately evaluate the proft mpact of dfferent scenaros: t smply has to multply the consdered movements of every rsk factor by the assocated total bank senstvty of that rsk factor to derve ts mpact on bank proft. The second applcaton concerns the control of the bank's total rsk exposure. Whle most banks would lke to lmt ther total rsk exposure, ths s currently often mpossble because no sngle measure of the total bank rsk ncludng rsk nterrelatons s avalable. As a consequence, banks set smple poston lmts on ther aggregate postons n order to lmt rsk. The choce of these lmts however ntroduces arbtrarness, snce the bank does not know how much a specfc lmt contrbutes to the total bank rsk. If the bank for example wants to ncrease ts currency rsk exposure wthout changng ts total rsk level, t does not know how far other lmts have to be reduced n order to acheve ths goal. The approprate way to lmt the total bank rsk that preserves the flexblty to relocate rsks wthout changng the total bank rsk consst n the settng of lmt on the total bank rsk as derved n secton 3. 10 Equpped wth the total bank rsk, senor management has also a valuable tool to evaluate the adequacy of ts equty captalzaton. A reasonable rule for rsk management would be to set an upper lmt on the relaton between total rsk and equty captalzaton. 10 Equvalently, n order to express the lmt n terms of losses, the bank could lmt ts total captal at rsk,.e. the worst loss that can occur wthn a gven confdence nterval over a gven tme perod. Ths loss can easly be calculated from (4) and (5) usng the propertes of the normal dstrbuton.
20 Ths rule automatcally trggers new equty ssues or rsk reductons once equty captal becomes too small to cover the bank rsk. A further possble applcaton of the proposed equty cost calculaton mght le n rsk adjusted performance evaluaton of busness unt managers or traders. It s well known that the current use of accountng profts s msleadng for the evaluaton of performance because expected proft s generally an ncreasng functon of rsk whle the presently allocated equty costs reflect dfferent degrees of rsk only to a very lmted extent. Because equty costs do not adjust approprately for rsk, profts (losses) may reflect hgh (low) market rsk premums nstead of superor (nferor) performance of managers. The proposed equty costs properly reflect the rsk contrbuton of every unt. However, does t produce a far benchmark for evaluatng busness unt performance? For ths to be the case an unsophstcated nvestor who earns the expected market return should make zero proft. From equaton (14) we can see that two condtons must be met for ths to be the case: the total bank portfolo must be mean varance effcent and the equty rsk premum used n the calculaton must be the actually expected market rsk premum on the bank's portfolo. We can therefore conclude that the proposed method does not consttute a far benchmark for performance evaluaton unless the bank holds a mean varance effcent portfolo whch we thnk s unlkely to hold n practce. It s of course stll possble that performance evaluaton based on our proposed equty costs may well be superor to currently used approaches. As a fnal potental applcaton of the proposed framework we consder ts use as a prcng tool that helps a bank to set prces on rsky products on the bass of ts current portfolo poston. The need to adjust prces to the current portfolo stuaton s wdely acknowledged but rarely appled n banks. To understand why prces should depend on current portfolo postons, consder the prcng behavor of a stock market
21 maker: when a market maker observes a sharp ncrease n hs nventory, he wll ncrease hs bd and ask prce n order to reduce hs holdngs. By settng prces accordng to the current nventory, he s able to mantan a postve but lmted nventory of stocks and thus lmts hs rsk from stock prce movements. For banks that trade a stock smultaneously n dfferent markets or also trade stock prce dependent dervatve products, prces should be set accordng to the current total bank exposure from all stock and dervatve holdngs n order to lmt the bank's total stock prce rsk. We can take the argument one step further and clam that t s not the solated rsk of the stock but rather ts contrbuton to the total bank rsk that the bank really wants to control. Ths mples that prces should not depend on solated poston rsk but rather on the poston's contrbuton to the total bank rsk. Portfolo based prcng can be ntroduced n a bank by applyng equaton (14), whch states a benchmark proft whch makes rsk takng attractve n the bank's current portfolo stuaton. Applyng the δ-factors of the consdered transacton, equaton (14) delvers a benchmark expected proft that the bank should demand n return for assumng the rsk. If market condtons are such that the expected return s hgher than (14) ndcates, the bank should set ts prces slghtly above market condtons n order to attract deals. If the reverse s true, a bank should decrease ts rsks by slghtly underprcng the market. 7. Concluson The proposed method of calculatng busness unt equty costs based on ts contrbuton to the total bank rsk s an effectve tool for combnng decentralzed decson makng wth the optmzaton of the total bank rsk exposure. By basng the equty costs of busness unts on ther contrbuton to the total bank rsk, the bank's captal can be allocated such that ts rsk/return stuaton s optmzed. Busness unt managers can be provded wth ncentves to optmze the total bank rsk/return trade off,
22 whch makes drect senor management nterventon nto busness unt decsons superflouos. As a sde product, the proposed method delvers valuable tools for controllng and managng the total bank rsk. A bank that wants to mplement the proposed concept should be aware of potental resstance from busness unts. Whle the potental ncrease of authorty makes the rsk contrbuton concept attractve to many busness unt managers, those busness unts that are currently responsble for large contrbutons to the total bank rsk wll expect rsng equty costs and presumably oppose the concept. A way to gan acceptance even from these unts s a clear separaton of performance evaluaton and equty cost calculaton. It has been demonstrated that equty costs based on the rsk contrbuton of busness unts can serve as a far performance benchmark only n the specal case that the bank portfolo s mean varance effcent. Ths mples that f the bank for strategc reasons chooses an neffcent portfolo, calculatory profts or losses do not automatcally ndcate good or poor performance of busness unt managers. Whle currently accountng profts are used both to gude rsk takng decsons and to evaluate manager performance, future rsk management systems should use separate systems for the rsk takng functon and for rsk adjusted performance evaluaton. Snce bank management wants to reward busness unts for both beatng the market and for mprovng the total bank rsk poston, t needs two dstnct ncentve systems to pursue both goals.
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