Modelling and Measuring Business Risk

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 79 Modellng and Measurng Busness Rsk 7 Klaus Böcker UnCred Group As can be seen for exaple by he cung edge secon of Rsk agazne, research papers anly focus on arke rsk, cred rsk, and wh a lle less aenon operaonal rsk. Alhough hese rsk ypes are very poran for fnancal nsuons, he rue landscape of rsk s uch ore coplex and far fro beng well explored and undersood. There s a varey of oher rsks loong on he horzon, whch serously hreaen a banks profably or whch can dsrup or even desroy s busness copleely. Moreover, such rsks ofen reflec an under-researched area of fnancal rsk anageen, and esablshed and ready-o-use easureen echnques are rarely avalable. Also bankng supervsors deand ha ore aenon s beng pad o such hard-o-easure rsks as he followng Pllar II passages of he new nernaonal regulaory fraework of Basel II (Basel Coee on Bankng Supervson 4) show: 73: Sound capal assessen nclude polces and procedures desgned o ensure ha he bank denes, easures, and repors all aeral rsks. 74: Alhough he Coee recognses ha oher rsks [ ] are no easly easurable, expecs ndusry o furher develop echnques for anagng all aspecs of hese rsks. Ths vew has also been confred by dfferen European supervsors, confer for exaple, The Coee of European Bankng Supervsors (4). Capurng all aeral rsks of a fnancal nsuon requres a broad rsk self assessen o fnd ou whch are he os relevan rsk drvers for he 79

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 8 PILLAR II IN THE NEW BASEL ACCORD bank. One of he os obvous varables o be onored n hs conex are earnng heselves. However, none of he Pllar I rsks ake earnngs volaly drecly as a prary drver no accoun, nsead, hey usually focus on aspecs of he busness envronen ha only ndrecly affec he nsuon earnngs by vrue of for exaple, faled processes, cred defauls, drop n share prces, or neres rae changes. For an all-encopassng rsk assessen s herefore necessary o nroduce an addonal knd of rsk ha s drecly lnked o he uncerany of specfc earnngs coponens no ye assocaed o oher rsk ypes. Usually, such an earnngs-relaed poenal loss, whch can also hreaen a banks arke capalsaon, s referred o as busness rsk. Evdence for he growng porance of busness rsk was recenly also gven n a survey underaken by he IFRI/CRO Foru abou econoc capal pracces n leadng fnancal nsuons (The Insue of he Chef Rsk Offcers (CROs) and Chef Rsk Offcers Foru (CRO Foru 7)) where 85 % of he parcpans saed o nclude busness rsk n her aggregaed econoc capal assessen. Ye surprsngly, here s no coon agreeen on a precse defnon, specfc rsk drvers, and easureen ehodology for busness rsk, even hough s absolue sze n er of econoc capal s coparable o ha of operaonal rsk, see agan The Insue of he Chef Rsk Offcers (CROs) and Chef Rsk Offcers Foru (CRO Foru 7). Wh hs regard, we also perfored a benchark exercse on a saple of 5 nernaonal banks by analysng her rsk anageen pracse as dsclosed n her offcal fnancal annual repors fro 4 o 6. Agan, we found ha an ncreasng nuber of nsuons are ryng o quanfy busness rsk n soe way, even f dfferen defnons and assupons are adoped. Broadly speakng, approaches for busness rsk quanfcaon can be dvded no wo an caegores; opdown and boo-up. Top-down echnques are lnked o he general rend of he busness envronen and benchark analyss based on exernal daa s used for approxang busness rsk. In conras o ha, boo-up approaches ry o explcly deerne he volaly of parcular, banknernal econoc e seres (such as volues, earnngs, revenues, and expenses) a a ore granular level, whch s hen ransfored no a easure of busness rsk. Here we we propose a boo-up approach for odellng and easurng busness rsk where he dynac of he underlyng earnngs s descrbed n a connuous-e odel. The reander of hs chaper s srucured as follows. Afer soe prelnares such as forulang he dscouned-cash- 8

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 8 MODELLING AND MEASURING BUSINESS RISK flow ehod n connuous e, we nroduce a frs sochasc odel for quanfyng busness rsk laer n he chaper where we also brefly pon ou how such odel could be pleened n pracce. The resuls obaned are hen used o nvesgae he relaon beween he earnngs-a-rsk (EaR) easure and he so-called capal-a-rsk (CaR) easure n greaer deal. Fnally, we propose a possble exenson of he sple busness rsk odel. MODELLING BUSINESS CAR: A SIMPLE APPROACH Seng he Scene Overlap wh oher rsk ypes Of course, he concep of revenues and expenses as we used so far s oo general for easurng busness rsk. In parcular, n order o avod double counng and rsk overlap, revenue and cos coponens ha ener he busness rsk odel us no drecly or ndrecly be used for he quanfcaon of oher rsk ypes. To gve an exaple, as poenally relevan revenues one ay consder cusoer relaed provsons and ne neres rae ncoe, whle on he cos sde adnsrave expenses and deprecaons ay be ncluded no busness rsk quanfcaon. On he oher hand, earnngs relaed o radng acves would clearly cause an overlap wh arke rsk, and should herefore no be ncluded. Soehng slar holds for loan loss provsons, when hey are capured whn he banks cred porfolo odel. However, he queson whch revenue and cos coponens are really relevan for odellng a parcular frs busness rsk, and whch pars have o be excluded, s no an easy one. The answer crucally depends on he frs defnon of ohers rsk ypes and s econoc capal fraework n general, and herefore seng up a busness rsk odel should always be an negral par of he banks overall rsk-defnng and assessen process. Moreover, s necessary o be aware ha he avalably and granulary of revenue and cos daa ay also depend on he frs accounng rules, conrollng sandards, and IT nfrasrucure. As a consequence, he qualy of daa ay dffer fro one legal eny o he oher, and n order o acheve relable resuls a aggregaed level, grea aenon should be pad wh regard o daa selecon and preparaon. Hereafer, when we alk abou earnngs, we acually always ean non-cred and non-arke earnngs so ha here s no double counng wh oher rsk ypes ha are already easured whn a banks econoc capal odel. 8

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 8 PILLAR II IN THE NEW BASEL ACCORD EaR versus CaR Busness rsk can be defned as he poenal loss n he copany s earnngs due o adverse, unexpeced changes n busness volue, argns, or boh. Such losses can resul above all fro a serous deeroraon of he arke envronen, cusoer shf, changes n he copeve suaon, or nernal resrucurng. On one hand, hese effecs can lead o a drop n earnngs n he shor-er, for exaple whn he nex budge year, and are ofen easured n ers of earnngs volaly, ore general by EaR. On he oher hand, volue or argn shrnkng probably leads o a longer-lasng weakenng of he earnngs suaon, hereby serously dnshng he copany s arke capalsaon, and hs rsk s ofen referred o as CaR. As poned ou by Saa (4, 7), he recognon of such negave long-er effecs on earnngs and he resulng pac on he arke capalsaon s parcular poran for he shareholder perspecve on capal and should also be used n he conex of rsk-adjused perforance easureen, for exaple by eans of RAROC, EVA, or relaed conceps. A convncng analyss provng hs lnk beween earnngs relaed rsk and a copany s loss n arke value s gven n Morrson, Quella and Slywozky (999). They found ou ha durng a perod of fve years, % of Forune, copanes los (a leas once) 5% of her shareholder value whn a one-onh perod, and ha nearly all of hese sock drops were a resul of reduced quarerly earnngs or reduced expeced fuure earnngs. Moreover, he ajory of hese earnngs-shorfalls (abou 58%) were no owng o classcal fnancal rsks or operaonal losses bu raher o wha Quella e al refer o as sraegc rsk facors, such as rasng coss and argn squeeze, eergng global copeors, and cusoer prory shf ec. If busness rsk s consdered as a aer of arke capalsaon and herefore s easured by CaR, he uncerany of (all) fuure earnngs have o be aken no accoun. As enoned above, such earnngs flucuaons, e, he devaons of he realsed earnngs fro he planned earnngs rajecory, ay be he resul of any dfferen facors. However, for he odel we sugges here, s no necessary o explcly lnk all hese rsk facors o fuure earnngs. Insead we suppose ha all rsk facors ogeher consue soe rando nose effec, xng wh he expeced earnngs pah; e, for fuure cuulaed earnngs E() can be wren as E() = f() + nose, where f s a nonrando funcon descrbng he planned earnngs rajecory. Consequenly, furher n he chaper we odel fuure earnngs as a sochasc process (E()). 8

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 83 MODELLING AND MEASURING BUSINESS RISK Dscouned-cashflow ehod Before we go on, however, s worhwhle o recall soe basc facs abou copany valuaon, especally abou he dscouned-cashflow ehod where expeced fuure earnngs are dscouned o oban he copany s arke value (see for exaple Goedhar, Koller and Wessels (5) or Pra, Relly and Schwehs () for a ore dealed descrpon of hs subjec). Denoe by E R he copany s earnngs ha are planned o be realsed n =, e, beween he fuure e perods and, defned for all = < < < T. The presen value or arke value of he copany s usually defned as ( ) = VT T E = ( + r ) (7.) where r R + s a rsk adjused dscoun rae. Expresson 7. sply says ha a copany s arke value s jus he su of s dscouned expeced fuure earnngs. For our purposes, however, a connuous-e seng of he presen value 7. s ore feasble. Defnon 7. (Presen value n connuous e) The presen value of all earnngs cuulaed unl he e horzon s gven by wh dscoun rae r() = r (τ)dτ for, where r ( ) s a nonrando posve funcon, represenng he shor-er dscoun rae and are he cuulaed fuure earnngs as hey are expeced o be realsed up o a e horzon. Furherore, we defne P( ): = l P( ) provded ha he l exss. ( ) = ( ) + ( ) rs ( ) P P e des, ( ) = ( ) + ( ) P P des, Model defnon and frs resuls We begn our analyss of busness CaR wh a sple odel based on Brownan oon. Such a odel allows for closed-for soluons for busness CaR and s herefore parcularly useful o undersand he naure and general properes f hs poran rsk ype. 83

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 84 PILLAR II IN THE NEW BASEL ACCORD Sochasc odellng of fuure cashflows We defne our odel n a ulvarae fashon n ha akes he dependence beween dfferen cells no accoun. Each cell could sply reflec a legal eny, busness dvson, geographcal regon, or a cobnaon of he. If earnngs are explcly spl up no revenues and coss, each of he are represened by dfferen cells. Hence, for each cell, we can defne a cashflow process descrbng he sochasc evoluon of revenues, coss, or earnngs. In he followng we rea revenues as posve and coss as negave varables. Defnon 7. (Brownan oon cashflow (BMC) odel) Consder a d-densonal sandard Brownan oon (W (),, W d ()) on a probably space (Ω, F, P). Then, he BMC odel consss of: () Cashflow processes For each busness rsk cell, ndexed by =,,, cuulaed fuure cashflows X () for are descrbed by a cashflow process, whch s he srong connuous soluon o he Iô-sochasc-dferenal equaon d ( ) = ( ) + j ( ) j ( ) j= dx α d σ dw, (7.) The bank s oal aggregaed cashflow s gven by he aggregae cashflow process ( ) = ( ) = ( ) + ( ) = = = X X X dx s Here, α ( ) >, =,,, and σ j ( ), =,, ; j =,, d, are non-rando funcons of e, sasfyng he negrably condons α (s) ds < and σ j (s) ds <. () Value process Le r( ) > be a non-rando dscoun rae so ha ( α (s) e r(s) + σ j (s)e r(s) )ds <. Then, he aggregae value process (P()) s defned by (seng P() = ) ( ) = ( ) = rs ( ) P e dx s, = ( ) = d + j ( ) j ( ) rs ( ) rs ( ) α se ds σ s e dw s, j= = (7.3) 84

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 85 MODELLING AND MEASURING BUSINESS RISK Reark 7.3 (a) For every, he BMC odel descrbes ulvarae norally dsrbued cuulaed cashflows X ( ), =,,, wh expecaon ( ) = ( ) + ( ) and cross-covarance funcon of X ( ) and X k ( ),, k =,,, gven by d ( ( ) ( )) = ( ) ( ) = j= covar X, X σ s σ s ds : s k j kj ds, k EX X α s ds, ( ) (7.4) Here, Σ := (Σ k ()) k s called nsananeous cashflow covarance arx, whch s assued o be posve defne for all. (b) The varance of fuure cashflows X ( ), =,, d, can be wren as ( ( )) = ( ) = ( ) var X sds : σ sds, (7.5) where σ ( ); =,,, are referred o as he nsananeous cashflow volales. (c) For non-zero σ ( ), he cross-correlaon funcon beween X ( ) and X k ( ) s ( ( ) k ( )) = corr X, X Inforally, we denoe he nsananeous correlaon beween dx ( ) and dx k ( ) as ( ) = ( ( ) ( )) ( k ( )) covar X, X k var ( X ( )) var X k corr dx dx ( ), k ( ) k : ρ σ σ, ( ) ( ) k ( ) = ( ) (d) The value of he aggregae cashflow process X() a e gves he oal earnngs of he bank ha have been realsed beween and (cuulaed earnng). Is varance s gven by, (7.6) ( ) = ( ) d var X ( ) s ds j σ j= = d = σ s j ( ) σ j= = k= = k ( ) = k = sds kj ( ) sds = : σ ( s) ds, (7.7) where we call σ ( ) he nsananeous aggregae cashflow volaly. 85

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 86 PILLAR II IN THE NEW BASEL ACCORD (e) Noe ha he nuber d of ndependen Brownan oons W ( ) can be dfferen fro he nuber of busness rsk cells. Therefore, our odel also allows for such realsc scenaros where he nuber of rsk facors (represened by dfferen Brownan oons) s greaer han he nuber of clusers; hnk for exaple of a bank wh wo legal enes ha are exposed o hree rsk facors, whch affec her non-cred and non-arke earnngs and hus busness rsk. Exaple 7.4 [Bvarae BMC odel wh consan paraeers] Consder a sple bvarae BMC odel wh consan drf and dffuson paraeers where he cashflow processes are gven by ( ) = + ( ) ( ) = + ( ( ) + ( )) dx α d σ dw dx α d σ ρdw ρ dw, Fro Reark 7.3 follows ha (7.8) = σ ρσ σ ρσ σ σ plyng for he varance of he -h cuulaed fuure cashflow (7.9) Moreover, he correlaon beween X ( ) and X ( ) and he nsananeous correlaon are gven for all by corr X, X corr dx, dx ρ Snce all paraeers are e-ndependen, hs odel can be calbraed que easly. Afer dscresaon of 7.8 by usng he Euler ehod, σ and σ can be calculaed drecly fro he sandard devaons of he dscree ncreens X ( ) and X ( ). Then, accordng o 7.9, volales a a larger e-scale can be derved by usng he -scalng law. Fnally, α and f α can be esaed hrough he saple eans of he dscressed ncreenal cashflows 7.8, or, alernavely, hey can be obaned fro he cuulaed cashflows for =, by regresson analyss. ( ( )) = = var X σ,,, ( ( ) ( )) = ( ) ( ) EX α cons., ( ) = + ( ) = 86

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 87 MODELLING AND MEASURING BUSINESS RISK Exaple 7.5 [Bvarae BMC odel wh e-dependen paraeers] We use a slar se up as n 7.8 bu wh e-dependen paraeers for of α () = α α and σ () = σ b for α, b ; =, : (7.) The expecaons and varances of fuure cashflows can be calculaed fro 7.4 and 7.5; for =, we oban and ( ) = + ( ) ( ) = + ( ) + ( ) α b dx α d σ dw α b dx α d σ ρdw ρ dw, α + α EX, α ( ( )) = var X The nsananeous correlaon n hs odel s sll ρ, however, usng 7.6 we derve b + b + corr ( X X ( ), ( )) = ρ, + b + b We now defne he sgnal-o-nose rao, also referred o as Sharpe rao, for each cashflow process as he rao of s expeced growhs o he flucuaons, e, for =, ( ) = ( ) ( ) = + σ b + b + ( ) ( ) EX X, η var ( X ( )) (7.) Insead of consan volales σ ( ) = σ as n he Exaple 7.4, we are now askng for consan Sharpe raos η ( ) = η. Obvously, he Sharpe raos of X ( ) and X ( ) are here consan for b = a +. A ypcal suaon as ay occour n pracce s depced n Fgure 7., whch shows he onhly earnngs over fve years for wo hypohecal legal enes., Calculang busness CaR For he purpose of CaR calculaons we have o learn ore abou he value process (P()). The followng resul s well-known and descrbes s dsrbuonal properes. 87

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 88 PILLAR II IN THE NEW BASEL ACCORD Proposon 7.6 Consder he BMC odel of Defnon 7. wh value process (P()). Then, for every >, he value P() has noral dsrbuon funcon Φ P wh expeced value and varance ( ) = ( ) = r( s) EP α se ds, ( ( )) = ( ) var P rs ( ) σ s e ds, (7.) where σ ( ) s he nsananeous aggregae cashflow volaly defned n 7.7. Proof Noe ha on he rgh-hand sde of Equaon 7.3 he negrals n he frs er are sandard Reann whereas hose of he second er gven by I() = σ j (s)e r(s) dw j (s) are Iô negrals wh deernsc negrands. Then I() s norally dsrbued wh E() = and var(i()) = E(I() ), see for exaple Shreve (4), Theore 4.4.9. Usng Iô s soery we furher oban ( ) = ( ) ( ) d rs ( ) var P ( ) var se dw s j j σ j= = d rs ( ) = E se dw s j ( ) j ( ) j σ = = = d j= σ j = rs ( ) ( se ) ds rs σ se ds, = ( ) ( ) Snce we now know ha n he BMC odel he banks oal arke value P( ) s norally dsrbued wh dsrbuon funcon Φ P, s sraghforward o calculae busness CaR. Before, however, we wan o precsely defne busness CaR for general dsrbuons of P( ). Defnon 7.7 (Busness CaR) Consder dfferen busness rsk cells wh cashflow processes X ( ), =,,, ha are no arbuable o oher rsk ypes, and defne her correspondng arke value process P( ) accordng o 7.3. For >, suppose ha F s he dsrbuon funcon of he value P() wh ean value EP() <. Then, busness CaR a e horzon and confdence level κ (, ) s gven by 88

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 89 MODELLING AND MEASURING BUSINESS RISK Fgure 7. Illusraon of onhly earnngs of wo dfferen legal enes as descrbed n Exaple 7.5. In conras o legal eny, he onhly earnngs of legal eny sees o have a posve lnear drf whch eg, could be deerned by lnear regresson (showed as dashed lnes). Hence, α = and α = could be se n 7.. Regardng he volales eher b = b = (consan absolue volales) or b =.5 and b =.5 (consan Sharpe raos) can be used. Fgure 7. Busness CaR s defned as he dfference beween he expeced arke value EP() of he bank s aggregae value process and a very low quanle of s arke value dsrbuon F. CaR CAR EP F κ, κ ( ) = ( ) ( ) (7.3) where F (κ) = nf{x R : F (x) κ}, < κ <, s he generalsed nverse of F. If F s srcly ncreasng and connuous, we ay wre F ( ) = F ( ). In he conex of econoc capal calculaons, he confdence level κ s a nuber close o, eg, κ =.999. In he case ha he probably densy func- 89

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 9 PILLAR II IN THE NEW BASEL ACCORD on (pdf) of F exss, he defnon of busness CaR s llusraed n Fgure 7.. In general, F and hus busness CaR canno be calculaed analycally. For he BMC odel, however, a closed-for expresson for busness CaR s avalable. Theore 7.8 (Busness CaR for he BMC odel) Assue ha fuure cashflows are descrbed by a BMC odel wh nsananeous aggregae cashflow volaly (see Equaon 7.7) d ( ) = j ( ) kj ( ) = j= k = σ σ σ, (7.4) and non-rando dscoun rae r( ). Then, busness CaR a e-horzon and confdence level κ (, ) s gven by ( ) = ( ) ( ) r s CAR Φ κ σ s e ( ) ds, κ (7.5) where Φ s he sandard noral dsrbuon funcon. Proof The asseron follows drecly fro Proposon 7.6 ogeher wh he defnon of busness CaR 7.3. Ths analycal expresson for busness CaR s anly a consequence of usng non-rando dscoun raes. If nsead r( ) s allowed o be soe connuous adaped neres rae process, he dsrbuon funcon F,, of he aggregae value process (P()) s n general no noral anyore, and he resul for busness CaR wll rarely be avalable n closed-for. Noe ha 7.5 only depends on he cashflows covarance arx and no on oher odel paraeers such as drf paraeers α ( ) or he nal values X (). As a consequence hereof, he BMC odel can be calbraed easly, confer also Exaples 7.4 and 7.5. The Relaonshp Beween EaR and CaR Earnngs-a-Rsk In he lgh of Defnon 7. we now can qualfy our noon of EaR, whch we have already nroduced. Fro Reark 7.3 (d) we know ha var(x()) s he volaly of he banks oal aggregae cashflows accuulaed beween e and. I s also well-known ha X(),, s norally dsrbued, see for exaple Shreve (4), Theore 4.4.9, so ha EaR of he cuulaed earnngs X() a and confdence level κ (, ) s sply gven by 9

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 9 MODELLING AND MEASURING BUSINESS RISK EAR Φ κ var X κ ( ) = ( ) ( ( )) = Φ ( κ ) σ ( sds ), (7.6) where Φ s he sandard noral dsrbuon funcon. In conras o 7.5, we see ha n 7.6 he volaly s no dscouned. Moreover, should be enoned ha accordng o wha we have sad earler n he chaper, he e paraeer n 7.6 should be chosen n a way ha reflecs a shorer horzon so ha dscounng effecs can be negleced. Fnally, we defne he nsananeous EaR as ( ) = ( ) ( ) ear κ Φ κ σ (7.7) EaR-CaR-ransforaon facors An neresng queson concerns he relaon beween EaR and CaR. Inuvely, CaR should be hgher han EaR snce CaR akes n conras o EaR he long-er uncerany of fuure cashflows no accoun. I has been suggesed for exaple by Maen (996) or Saa (4, 7) ha CaR s a consan ulpler of EaR, and ha he ulplcaon facor depends only on a (rsk-adjused) neres rae (dscoun facor) and he e horzon. Saa based hs analyss of he EaR-CaR relaonshp on a dscree-e cashflow odel slar o 7. where EaR reflecs he uncerany of he E, and Saa (7), secon 5.8, gves a very readable overvew abou hs opc. In he case of he BMC odel, we see by coparng 7.5 wh 7.7 ha such a proporonaly beween EaR() and CaR() does no hold for all because of he e dependence of he nsananeous aggregae-cashflow volaly σ ( ). However, he ean value heore ensures ha ξ (, ) can always be chosen so ha 7.5 can be wren as CAR κ ( ) = ( ) ( ) rs ( ) Φ κ σ ξ e ds = ear ( ξ ) κ rs ( ) e ds, and we can hnk of ear κ (ξ) as an average EaR of he e nerval [, ]. The followng wo exaples llusrae he relaonshp beween EaR and CaR, n parcular showng ha even n he que sple fraework of BMC odels EaR-CaR-ransforaon crucally depends on he specfcaons of he cashflow processes X (), =,, d. 9

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 9 PILLAR II IN THE NEW BASEL ACCORD Exaple 7.9 [BMC odel wh consan dffuson paraeers] Consder a BMC odel wh σ j ( ) = σ j = cons. for =,, ; j =,, d. Then also he aggregae-cashflow volaly 7.4 s consan and we oban CAR wh EaR-CaR ransforaon facor where we ndcaed he explc dependence of k on he dscoun rae r() by r. Hence, n hs specal case busness CaR s proporonal o he (consan) nsananeous EaR. If furherore he shor-er dscoun rae s consan, r ( ) = r we have ha r() = r ds = r and we arrve a exp r ˆ k r (7.8) ( ˆ, ) = rˆ whch s a sple funcon of he e horzon and he dscoun rae r. The hgher r s, he saller k wll be because fuure cashflows (and so her flucuaons) end o have lower pac on he arke value (and so on s uncerany). Slarly, longer e horzons lead o a growng k because ore unceran fuure cashflows are aken no accoun. In he l expresson 7.8 splfes o l k rˆ, ( ) = (7.9) Exaple 7. [BMC odel wh consan Sharpe rao] Consder a BMC odel wh cuulaed cashflows X ( ) for =,,. As n Exaple 7.5 we consder he Sharpe raos η ( ) gven by 7.. Adopng a lnear-growh odel wh consan drf paraeers α ( ) = α >, =,,, we know fro Exaple 7.5 ha consan Sharpe raos requre squareroo-of-e scalngs of he nsananeous cashflow volales, e, σ () = c for soe consans c, =,,. Ths ples ha he aggregae-cashflow volaly 7.4 for can be wren as σ () = σ, resulng n saple pahs of X ( ) ha are n general ore nosy han he one obaned n Exaple 7.9, a fac ha s llusraed n he op char of Fgure 7.3. Fnally, usng 7.5 we can calculae busness CaR n he case of a consan shor-er dscounr rae r o wh EaR-CaR ransforaon facor κ ( ) = k ( r ), Φ ( κ ) σ = k ( r, ) ear, κ r( s) k r, e ds ( ) = ( ) rˆ ( ) = ( ) ( ) CAR κ k r ˆ, Φ κ σ, 9

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 93 MODELLING AND MEASURING BUSINESS RISK and rs ˆ k rˆ, se ds rˆ ( ) = = + l k rˆ, ( ) = ( ) ( ) r ˆ exp r ˆ rˆ Coparng he dfferen BMC-odel specfcaons of Exaple 7.9 and 7., could be expeced ha k s greaer han k because n he prevous exaple he Sharpe rao was acually ncreasng wh (plyng a decrease of fuure cashflow flucuaon) whereas here s consan by consrucon. Hence, k accuulaes ore fuure uncerany han k. As we see n he boo char of Fgure 7.3, hs s ndeed he case f exceeds a ceran hreshold, e, > (r ), whch for r =. s approxaely gven by (.) =.. Moreover, by coparng 7.9 and 7. follows ha for all (r ) we have ha k rˆ, k rˆ, rˆ ( ) ( ) A MODEL WITH LEVEL-ADJUSTED VOLATILITY In he BMC odel he absolue changes of fuure cashflows X ( ) are drecly odelled by a Brownan oon, see Equaon 7., whch n parcular eans ha he uncerany of a busness cells cashflow s ndependen fro s absolue level. There s, however, no raonal for hs behavour and, nuvely, hgher earnngs should acually be ore volale han low earnngs. As a possble reedy, fuure cashflows could be descrbed by a geoerc Brownan oon as s used for exaple, for sock prces n he faous faous Black-Scholes-Meron seng. Then, for cuulaed cashflows X ( ), =,,, would be gven by d d X X s s ds s ( ) = ( ) exp α ( ) σ j ( ) + σ j ( ) dw ( s) j j= j = plyng an expeced exponenal cashflow growh of ( ) = ( ) ( ) EX X exp s ds α whch, however, gh be consdered as oo exree and opsc for os busnesses. Alernavely, we sugges a odel wh sll oderae growh bu a knd of cashflow level adjused volaly. More precsely, we have he followng defnon. 93

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 94 PILLAR II IN THE NEW BASEL ACCORD Fgure 7.3 Top panel: Typcal cashflow pahs X () wh consan absolue cashflow volaly (sold lne) and consan Sharpe rao (dashed lne) referrng o Exaples 7.9 and 7., respecvely. The paraeers are X () =, α =. and σ =.4 wh a e horzon of years and onhly ncreens. Moreover, he doed pah s obaned by he advanced odel dscussed n secon 3, especally Exaple 7., and we se σ = σ / X () =.4. We used he sae seeds of noral rando varables for all hree pahs. Boo panel: EaR-CaR-ransforaon facors k (rˆ,) and k (rˆ,) accordng o (7.8) and (7.), respecvely, as a funcon of e for a shor-er dscoun rae rˆ =.. The doed-dashed lne ndcaes he asypoe of k (rˆ,) when. 94

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 95 MODELLING AND MEASURING BUSINESS RISK Defnon 7. (Level-adjused BMC odel) Consder a d-densonal sandard Brownan oon (W (),, W d ()) on a probably space (Ω, F, P). Then, he level-adjused BMC odel consss of: () Cashflow processes. For each busness rsk cell, ndexed by =,,, cuulaed fuure cashflows X () for are descrbed by a cashflow process, whch s he srong connuous soluon o he Iô-sochasc-dferenal equaon d ( ) = ( ) + ( ) j ( ) j ( ) j= dx α d X σ dw, (7.) wh aggregae cashflow process ( ) = ( ) = ( ) + ( ) = = = X X X dx s Here, α ( ) >, =,,, and σ j ( ), =,, ; j =,, d, are nonrando funcons of e, sasfyng he negrably condons α (s) ds < and σ j (s) ds <. The arx (σ j()) j s assued o be posve defne for all. () Value process Le r( ) > be a nonrando funcon so ha ( α (s) e r(s) +σ j (s)e r(s) ) ds <. Then, he aggregae value process (P()) s defnedby (seng P() = ) ( ) = ( ) = rs ( ) P e dx s, Le us frs consder he cashflow process and copare 7. wh 7. of he BMC odel. For he laer, he dffuson paraeers σ j play he role of an absolue easure of uncerany for he ncreens of X ( ), whereas n 7. he σ j descrbe he ncreen s flucuaons relave o he level of X ( ). Furherore, nsead of 7. we ay wre d ( ) = ( ) + ( ) + ( ) j ( ) j= X X α s ds X s σ s dw ( s), j (7.) and fro he arngale propery of he Iô negral edaely follows ha he expecaon of X ( ) s gven by ( ) = ( ) + ( ) EX X α s ds, 95

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 96 PILLAR II IN THE NEW BASEL ACCORD e, s he sae as for he BMC odel, and, n parcular, he odel does no exhb exponenal growh as would be he case when geoerc Brownan oon s used. We close hs secon wh an exended exaple ha llusraes soe properes of he level-adjused volaly odel. Exaple 7. [Consan drf and dffuson paraeers] For he sake of splcy we focus on he case of consan paraeers α ( ) = α and σ j ( ) = σ j. Noe however, ha he dffuson paraeer of he process 7. s rando and gven by X ( )σ j. In order o fnd a soluon for 7. we defne he funcon d d FW (, ( )): = exp σ W j j ( ) + σ j j = j=, Then, by usng Iô s forula he dfferenal of he produc FX can be calculaed as ( ( ( )) ( )) = ( ) ( ) dfw, X α FW, d whch afer negraon fnally yelds (seng W j () = for j =,, d), ( ) = ( ( )) ( ) + ( ( )) X F, W X α F s W s ds,, (7.) Accordng o 7., he cuulaed cashflows X () a e are no norally dsrbued as hey are n he BMC odel. A one-densonal exaple for a ypcal pah of X ( ) accordng o 7. s ploed as a doed lne n he op panel of Fgure 7.3. The Iô represenaon of he value process (P()) s gven by d rs ( ) ( ) = + j ( ) dw ( s) j = j= = rs ( ) P α e ds σ X se,, (7.3) whch canno be calculaed n closed for. Noe, however, ha he expecaon of P( ) s agan gven by 7. and herefore s he sae as for he BMC odel. Ths can also be seen n Fgure 7.4. In he op panel, we copare he dsrbuon of he presen value as obaned by 7.3 frsly o ha of a noral dsrbuon wh he sae ean and varance (dashed curve), and secondly o he norally dsrbued presen value calculaed fro he BMC odel of Exaple 7.9 (sold curve). I can be seen ha 7.3 leads o a dsrbuon ha s ore skewed o he rgh (posve skewness) and s ore peaked and heaver-aled han a noral dsrbuon wh he sae varance (kuross larger han 3). 96

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 97 MODELLING AND MEASURING BUSINESS RISK Fgure 7.4 Top panel: The hsogra shows he sulaed presen-value dsrbuon for = 5 years of he level-adjused volaly odel as dscussed n Exaple 7. (wh dscoun rae r = rˆ) as well as he ean, sandard devaon, skewness, and kuross paraeers of he sulaed daa. Ths s copared o a noral dsrbuon wh he sae ean and sandard devaon as he sulaed daa, ploed as a dashed lne. The sold curve represens he norally dsrbued presen value as obaned fro he BMC odel of Exaple 7.9. We se X () = and use he yearly paraeers α =., σ =.4, σ = σ / X () =.4, and a yearly neres rae of r =.8. Boo panel: EaR-CaR-ransforaon facor for he level-adjused growh odel of Exaple 7. as a funcon of he confdence level κ and dfferen values α =. (sold lne), α =. (dashed lne), and α 3 =.4 (doed lne) of he growh paraeer. The oher paraeers are he sae as used above. 97

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 98 PILLAR II IN THE NEW BASEL ACCORD EaR-CaR-ransforaon revsed We have seen ha for he BMC odel boh he cuulaed cashflows X ( ) and he presen value P( ) are norally dsrbued. Ths has wo poran consequences. Frs, busness CaR (and hus he EaR-CaR-ransforaon facor) of he BMC odel s ndependen of he expeced growh and hus of α ( ); =,,. Second, he ransforaon facor s nvaran under changes of he confdence level and equals he rao of he presen value volaly and he earnngs volaly (see Exaples 7.9 and 7.). Ths s a consequence of he well-known propery of ellpcally dsrbued rando varables, whch says ha her quanles can always be expressed n ers of her sandard devaon. However, such a behavour canno be expeced for he level-adjused BMC odel because P( ) s no ellpcally dsrbued. Fgure 7.4 serves o llusrae hs for he odel of Exaple 7.. I shows he resuls of a sulaon sudy where he ransforaon facor beween CaR and EaR (calculaed a = where EAR s norally dsrbued wh sandard devaon σ = σ X ()) s ploed as a funcon of her confdence level. The growh paraeer s se o be α =.,., and.4. Noe ha he hgher he growh rae s, he lower he ransforaon facor and herefore he rao beween CaR and EaR wll be. In conras, f we copare he rao of he volales of he presen value and EaR, e, he volaly of he sulaed hsogra daa o he nal absolue cashflow volaly σ = σ X () =.4, we oban.9,.38, and.46 for α =.,., and.4, respecvely. Moreover, we see ha an ncreasng confdence level leads o a decreasng ransforaon facor. Sung up we can conclude ha n general he queson of how a EaR can be convered no a CaR s no sraghforward o answer. Whle for he BMC odel hs sees o be easer and nuvely easer o grasp (snce ndependen of he confdence level and he expeced growh rae) becoes raher nvolved for ore general odels lke he one dscussed n hs secon. CONCLUSION AND OUTLOOK In hs chaper we suggesed a ulvarae connuous-e seng for assessng busness rsk usng sochasc odels for he fuure cashflows of (non-cred, nonarke, ec.) earnngs. In conras o scenaro-based ehods for esang busness rsk, our odel has he advanage ha resuls n a e-dependen probably dsrbuon of fuure earnngs, 98

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page 99 MODELLING AND MEASURING BUSINESS RISK whch allows for an accurae defnon of VaR-lke rsk easures a any confdence level and e horzon. We also nvesgaed he relaonshp beween EaR and CaR, whch for general cashflow processes urns ou o be no sraghforward, and, n parcular, a consan ulpler converng EaR no CaR s usually no avalable. However, a sple EaR-CaR-ransforaon facor only dependng on he e horzon and he dscoun rae can analycally be derved for he sple BMC odel. Such a resul ay be useful when a fas and approxave VaR esaon based on soe EaR fgure s needed. Snce n our odel he dependence srucure beween dfferen legal enes or busness uns s refleced by her correlaon arx, rsk-reducng sraeges such as known fro sock porfolo analyss can be sraghforwardly appled. Several enhanceens of he BMC odel can be hough of. A parcular neresng possbly would be o nroduce jups n he earnngs processes represenng sudden and sharp falls n a copany s earnngs caused by, for exaple, a change n he copeor envronen or a cusoer shf. A parcular poran class of jup process are he Lévy processes, whch have becoe que popular also n fnancal odellng, see for exaple Con and Tankov (4). Then, n addon o he covarance srucure of he ulvarae Brownan oon we already dscussed, jup dependence beween fuure earnngs could be odelled by so-called Levy copulas, see for exaple Böcker and Klüppelberg (8) for an applcaon of hs echnque o operaonal rsk. However, snce every odel should be seen also n he lgh of he daa needed for s calbraon, ay be wse o sar wh a well-known and esablshed Brownan approach. In our opnon, he developen of advanced busness rsk odels wll be an poran ask n quanave rsk anageen, despe he dffcules and coplexy dscussed above. Equally poran s o work on a haronsed defnon of hs aeral rsk ype, whch clearly requres a closer collaboraon beween praconers and acadecs. Dsclaer The opnons expressed n hs chaper are hose of he auhor and do no reflec he vews of UnCred Group. Moreover, presened rsk easureen conceps and rsk conrol echnques are no necessarly used by UnCred Group or any afflaes. 99

7 Bo?cker:Pllar II n he New Basel Accord 7//8 5:58 Page PILLAR II IN THE NEW BASEL ACCORD Acknowledgeen The auhor would lke o hank Alessandra Cr fro UnCred Group for helpful dscussons abou busness rsk. REFERENCES Basel Coee on Bankng Supervson, 4, Inernaonal Convergence of Capal Measureen and Capal Sandards, Basel. Böcker, K. and C. Klüppelberg, 8, Modellng and Measurng Mulvarae Operaonal Rsk wh Levy Copulas, The Journal of Operaonal Rsk, 3(). Coee Of European Bankng Supervsors (CEBS), 4, The Applcaon Of The Supervsory Revew Process Under Pllar. London. Con, R. and P. Tankov, 4, Fnancal Modellng Wh Jup Processes (Boca Raon: Chapan & Hall). Goedhar, M., T. Koller and D. Wessels, 5, Valuaon: Measurng and Managng he Value of Copanes (New York: John Wley & Sons). The Insue of he Chef Rsk Offcers (CROs) and Chef Rsk Offcers Foru (CRO Foru), 7, Insghs fro he jon IFRI/CRO Foru survey on Econoc Capal Pracce and applcaon suppored by Olver Wyan. Maen, C., 996, Managng Bank Capal, Capal Allocaon and Perforance Measureen (Chcheser: John Wley & Sons). Morrson, D.J., J.A. Quella and A.J. Slywozky, 999, Counerng sraegc rsk wh paern hnkng, Mercer Manageen Journal. Pra, S.P., R.F. Relly and R.P. Schwehs,, Valujng a Busness: he Analyss and Apprasal of Closely Held Copanes (New York: McGraw-Hll). Saa, F., 4, Rsk Capal Aggregaon: he Rsk Manager s Perspecve, Workng paper. Saa, F., 7, Value a Rsk and Bank Capal Manageen (San Dego: Elsever). Shreve, E.S., 4, Sochasc Calculus for Fnance II: Connuous-Te Models (New York: Sprnger).