Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C.
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1 Fnance and Economcs Dscusson Seres Dvsons of Research & Sascs and Moneary Affars Federal Reserve Board, Washngon, D.C. Prcng Counerpary Rs a he Trade Level and CVA Allocaons Mchael Pyhn and Dan Rosen OTE: Saff worng papers n he Fnance and Economcs Dscusson Seres (FEDS) are prelmnary maerals crculaed o smulae dscusson and crcal commen. The analyss and conclusons se forh are hose of he auhors and do no ndcae concurrence by oher members of he research saff or he Board of Governors. References n publcaons o he Fnance and Economcs Dscusson Seres (oher han acnowledgemen) should be cleared wh he auhor(s) o proec he enave characer of hese papers.
2 Prcng Counerpary Rs a he Trade Level and CVA Allocaons Mchael Pyhn 2 and Dan Rosen 3 ovember 2009 Absrac We address he problem of allocang he counerpary-level cred valuaon adjusmen (CVA) o he ndvdual rades composng he porfolo. We show ha hs problem can be reduced o calculang conrbuons of he rades o he counerpary-level expeced exposure (EE) condonal on he counerpary s defaul. We propose a mehodology for calculang condonal EE conrbuons for boh collaeralzed and non-collaeralzed counerpares. Calculaon of EE conrbuons can be easly ncorporaed no exposure smulaon processes ha already exs n a fnancal nsuon. We also derve closed-form expressons for EE conrbuons under he assumpon ha rade values are normally dsrbued. Analycal resuls are obaned for he case when he rade values and he counerpary s cred qualy are ndependen as well as when here s a dependence beween hem (wrong-way rs). The opnons expressed here are hose of he auhors and do no necessarly reflec he vews or polces of her employers. 2 Federal Reserve Board, Washngon, DC, USA. mchael.v.pyhn@frb.gov. 3 R 2 Fnancal Technologes and The Felds Insue for Research n Mahemacal Scences, Torono, Canada. dan.rosen@r2-fnancal.com and drosen@felds.uorono.ca
3 . Inroducon For years, he sandard pracce n he ndusry was o mar dervaves porfolos o mare whou ang he counerpary cred qualy no accoun. In hs case, all cash flows are dscouned usng he LIBOR curve, and he resulng values are ofen referred o as rs-free values. 4 However, he rue value of he porfolo mus ncorporae he possbly of losses due o counerpary defaul. The cred valuaon adjusmen (CVA) s, by defnon, he dfference beween he rs-free porfolo value and he rue porfolo value ha aes no accoun he counerpary s defaul. In oher words, CVA s he mare value of counerpary cred rs. 5 There are wo approaches o measurng CVA: unlaeral and blaeral (see Pcoul, 2005 or Gregory, 2009). Under he unlaeral approach, s assumed ha he counerpary ha does he CVA analyss (we call hs counerpary a ban hroughou he paper) s defaul-free. CVA measured hs way s he curren mare value of fuure losses due o he counerpary s poenal defaul. The problem wh unlaeral CVA s ha boh he ban and he counerpary requre a premum for he cred rs hey are bearng and can never agree on he far value of he rades n he porfolo. Blaeral CVA aes no accoun he possbly of boh he counerpary and he ban defaulng. I s hus symmerc beween he ban and he counerpary, and resuls n an objecve far value calculaon. Under boh, he unlaeral and blaeral approaches, CVA s measured a he counerpary level. However, s somemes desrable o deermne conrbuons of ndvdual rades o he counerpary-level CVA. The problem of calculang CVA conrbuons bears many smlares o he calculaon of rs conrbuons and capal allocaon (see Azz and Rosen 2004, Mausser and Rosen 2007). There are several possble measures of CVA conrbuons. We refer o he CVA of each ransacon on a sand-alone bass as he ransacon s sand-alone CVA. Clearly, when he gven porfolo does no allow for neng beween rades, he porfolo-level CVA s gven by he sum of he ndvdual rades sand-alone CVA. However, hs s no he case when neng and margn agreemens are n place. We refer o he ncremenal CVA conrbuon of a rade as he dfference beween he porfolo CVA wh and whou he rade. 6 Ths measure s commonly seen as approprae for prcng counerpary rs for new rades wh he counerpary (see Chaper 6 n Arvans and Gregory, 200 for deals). One problem wh ncremenal CVA conrbuons s ha hey are non-addve he sum of he ndvdual rade s CVA conrbuons does no add up o he porfolo s CVA. Hence neher sand-alone nor ncremenal conrbuons can be used effecve conrbuons of exsng rades n he porfolo o he counerpary-level CVA, n he presence of neng and/or margn agreemens. For hs purpose we requre addve CVA conrbuons. In hs case, we draw he analogy wh he capal allocaon leraure and refer o hese as (connuous) margnal rs conrbuons. 4 More precsely, LIBOR raes roughly correspond o AA rs rang and ncorporae he ypcal cred rs of large bans. 5 See Canabarro and Duffe (2003) or Pyhn and Zhu (2007) for an nroducon o counerpary cred rs and CVA. 6 Somemes hese are referred o as dscree margnal conrbuons. 2
4 The margnal CVA conrbuons wh a gven counerpary gve he ban a clear pcure how much each rade conrbues o he counerpary-level CVA. However, he use of CVA conrbuons s no lmed o an analyss a a sngle counerpary level. Once he CVA conrbuons have been calculaed for each counerpary, he ban can calculae he prce of counerpary cred rs n any collecon of rades whou any reference o he counerpares. For example, by selecng all rades booed by a ceran busness un or produc ype (e.g., all CDSs or all USD neres rae swaps), he ban can deermne he conrbuon of ha busness un or produc o he ban s oal CVA. We show how o defne and calculae margnal CVA conrbuons n he presence of neng and margn agreemens, and under a wde range of assumpons, ncludng he dependence of exposure on he counerpary s cred qualy. The heory of margnal rs conrbuons, somemes refer o as Euler Allocaons (see Tasche 2008), s now well developed and largely reles on he rs funcon beng homogeneous (of degree one). We show ha hs prncple can be appled readly for CVA when he counerpary porfolo allows for neng (bu does no nclude collaeral and margns). We furher exend hs allocaon prncple for he more general case of collaeralzed/margned counerpares For he sae of smplcy, we assume he unlaeral framewor hroughou he paper. However, an exenson of all he resuls o he blaeral framewor s sraghforward. The paper s organzed as follows. In Secon 2, we defne counerpary cred exposure for boh collaeralzed and non-collaeralzed cases. We show how counerpary-level CVA can be calculaed from he profle of he dscouned rs neural expeced exposure (EE) condonal on he counerpary s defaul. In Secon 3, we nroduce CVA conrbuons of ndvdual rades and relae hem o he profles of condonal EE conrbuons. In Secon 4, we adap he connuous margnal conrbuon (CMC) mehod ofen used for allocang economc capal o calculang EE conrbuons for he case when he counerpary-level exposure s a homogeneous funcon of he rades weghs n he porfolo. Ths s he case when here are no exposurelmng agreemens, such as margn agreemens, wh he counerpary. When such agreemens are presen, he CMC mehod fals because he counerpary-level exposure s no homogeneous anymore. In Secon 5, we propose an EE allocaon scheme ha s based on he CMC mehod, bu can be used for collaeralzed counerpares. In Secon 6, we show how o ncorporae EE and CVA conrbuon calculaons no exposure smulaon process. In Secon 7, we derve closed form expressons for EE conrbuons under he assumpon ha all rade values are normally dsrbued. We sar wh he case of ndependence beween exposure and he counerpary s cred qualy, and exend he resuls o ncorporae dependence beween hem (wrong-way rs). We also provde an nuve explanaon o our closed-form resuls. In Secon 8, we show several numercal examples ha llusrae he behavor of exposure (and hence CVA) conrbuons for boh, he collaeralzed and non-collaeralzed cases. 3
5 2. Counerpary cred rs and CVA In hs secon, we revew he basc conceps and noaon for counerpary cred rs, cred exposures and CVA. Counerpary cred rs (CCR) s he rs ha he counerpary defauls before he fnal selemen of a ransacon's cash flows. An economc loss occurs f he counerpary porfolo has a posve economc value for he ban a he me of defaul. Unle a loan, where only he lendng ban faces he rs of loss, CCR creaes a blaeral rs: he mare value can be posve or negave o eher counerpary and can vary over me wh he underlyng mare facors. We defne he counerpary exposure E( ) of he ban o a counerpary a me as he economc loss, ncurred on all ousandng ransacons wh he counerpary f he counerpary defauls a, accounng for neng and collaeral bu unadjused by possble recoveres. 2. Counerpary exposures Consder a porfolo of dervave conracs of a ban wh a gven counerpary. The maury of he longes conrac n he porfolo s T. The counerpary defauls a a random me τ wh a nown rs-neural dsrbuon P( ) Pr[ τ ]. 7 We furher assume ha he dsrbuon of he rade values a all fuure daes s rs neural. 8 Denoe he value of he h nsrumen n he porfolo a me from he ban s perspecve by V( ). A each me, he counerpary-level exposure E( ) s deermned by he values of all rades wh he counerpary a me, { V ( )} =. The value of he counerpary porfolo a s gven by V ( ) = V ( ) () When neng s no allowed, he (gross) counerpary-level exposure E( ) s = ( ) max{ 0, V ( ) } E = (2) = For a counerpary porfolo wh a sngle neng agreemen, he (need) exposure s { V } E( ) = max ( ), 0 (3) 7 The erm srucure of rs neural probables of defaul can be obaned from cred defaul swaps spreads quoed for he counerpary on he mare for dfferen of dfferen maures. See, for example, Schönbucher (2003). 8 See, for example, Brgo and Mase (2005). 4
6 When he neng agreemen s furher suppored by a margn agreemen, he counerpary mus provde he ban wh collaeral whenever he porfolo value exceeds a hreshold. As he porfolo value drops below he hreshold, he ban reurns collaeral o he counerpary. Collaeral ransfer occurs only when he collaeral amoun ha needs o be ransferred exceeds a mnmum ransfer amoun. The counerpary-level (margned) exposure s gven by { } where C( ) s he collaeral avalable o he ban a me. E( ) = max V ( ) C( ), 0 (4) Counerpary porfolos wh a combnaon of mulple neng agreemens and rades ousde of hese agreemens can be modeled n a sraghforward way by a combnaon of Equaons (2)-(4). 2.2 Models of Collaeral We sar modelng collaeral wh a smplfyng assumpon: we ncorporae he mnmum ransfer amoun no he hreshold H and rea he margn agreemen as havng no mnmum ransfer amoun. Ths approxmaon s raher crude, bu s very popular amongs bans because grealy smplfes modelng. We consder wo models of collaeral. In he nsananeous collaeral model, we assume ha collaeral s delvered mmedaely and ha he rades can be lqudaed mmedaely as well. Under hese smplfyng assumpons, he collaeral avalable o he ban s { } C( ) = max V ( ) H,0 (5) The nsananeous collaeral model s aracve because of s smplcy, bu s rarely used n pracce because s assumpons maerally affec he exposure dsrbuon. 9 However, we use hs model o show he smple, nuve nerpreaon of our resuls for collaeralzed neng. A more realsc collaeral model mus accoun for he me lag beween he las margn call made before defaul and he selng of he rades wh he defaulng counerpary. Ths me lag, whch we denoe byδ, s nown as he margn perod of rs. Whle he margn perod of rs s no nown wh cerany, we follow he sandard pracce and assume ha s a deermnsc quany ha s defned a he margn agreemen level. 0 We assume ha he collaeral avalable o he ban a me s deermned by he porfolo value a me δ accordng o { δ } C( ) = max V ( ) H,0 (6) 9 When he hreshold s no oo small, he nsananeous collaeral model wors reasonably well for expeced exposure. See Pyhn (2009). 5
7 We refer o hs more realsc model as he lagged collaeral model. Whle more dffcul o mplemen, s ofen used by bans o oban resuls, whch have more praccal value. 2.3 Cred losses and CVA In he even ha he counerpary defauls a me τ, he ban recovers a fracon R of he exposure E( τ ). The ban s dscouned loss due o he counerpary s defaul s T L = { } ( R ) E ( τ ) D τ ( τ ) (7) where { A } s he ndcaor funcon ha aes value when logcal varable A s rue and value 0 oherwse, D( ) s he sochasc dscoun facor process a me, defned accordng o D( ) = B0 B, wh B he value of he money mare accoun a me. The unlaeral counerpary-level CVA s obaned by applyng he expecaon o Equaon (7). Ths resuls n T CVA = ( R) dp( ) eˆ ( ) (8) where eˆ ( ) s he rs-neural dscouned expeced exposure (EE) a me, condonal on he counerpary s defaul a me : [ ] 0 eˆ ( ) = E ˆ D( ) E( ) E D( ) E( ) τ = (9) Throughou hs paper we use sar o desgnae dscounng and ha o desgnae condonng on defaul a me. oe ha we have no made so far any assumpons on wheher he exposure depends on he counerpary s cred qualy. 3. CVA Conrbuons from EE Conrbuons We would le o develop a general approach o calculang addve conrbuons of ndvdual rades o he counerpary-level CVA. We denoe he conrbuon of rade by CVA. We say ha CVA conrbuons are addve when hey sum up o he counerpary-level CVA: CVA = CVA (0) = 0 The margn perod of rs depends on he conracual margn call frequency and he lqudy of he porfolo. For example, δ = 2 wees s usually assumed for porfolos of lqud conracs and daly margn call frequency. 6
8 oe ha he recovery rae R and he defaul probables P( ) are defned a he counerpary level n Equaon (8). Thus, he problem of calculang CVA conrbuons reduces o ha of calculang conrbuons of ndvdual rades o he porfolo condonal dscouned EE, eˆ ( ), a each fuure dae. To oban addve CVA conrbuons, hen he condonal dscouned EE conrbuons mus sum up o he porfolo condonal dscouned EE: eˆ ( ) = eˆ ( ) () and he CVA conrbuon of rade can be calculaed from s EE conrbuon accordng o = T CVA = ( R) dp( ) eˆ ( ) (2) Thus, from now on we focus on defnng and calculang EE conrbuons. oe frs ha, whou neng agreemens, he allocaon of he counerpary-level EE across he rades s rval because he counerpary-level exposure s he sum of he sand-alone exposures (Equaon (2)) and expecaon s a lnear operaor. Furhermore, when here s more han one neng se wh he counerpary (e.g., mulple neng agreemens, non-neable rades), we can focus on frs calculang he CVA conrbuon of a ransacon o s neng se. The allocaon of he counerpary-level EE across he neng ses s hen rval agan because he counerpary-level exposure s defned as he sum of he neng-se-level exposures. Thus, our goal s o allocae he neng-se-level exposure o he rades belongng o ha neng se. To eep he noaon smple, we assume from now on ha all rades wh he counerpary are covered by a sngle neng se. 4. Addve EE Conrbuons for on-collaeralzed eng Ses In hs secon, we develop he basc mehodology o compue EE conrbuons and allocae porfolo-level EE for non-collaeralzed neng ses. 4. Connuous Margnal Conrbuons and Euler Allocaon We derve EE conrbuons by adapng he connuous margnal conrbuons (CMC) mehod from he economc capal (EC) leraure. EC s calculaed a he porfolo level and hen s allocaed o ndvdual oblgors and ransacons. Under he CMC mehod, he rs conrbuon of a gven ransacon o he porfolo EC s deermned by he nfnesmal ncremen of he EC correspondng o he nfnesmal ncrease of he ransacon s wegh n he porfolo (see Chaper 4 n Arvans and Gregory (200) or Tasche (2008) for deals). Ths follows from he fac ha he rs funcon s homogeneous (of degree one) and he applcaon of Euler s heorem. 0 7
9 A real funcon f ( x ) of a vecor x = ( x,..., x ) s sad o be homogeneous of degree β f for all c > 0, f ( c x) = c β f ( x ). If he funcon f ( ) s pecewse dfferenable, hen Euler s heorem saes ha: f ( ) β f ( x ) = x x (3) x = The rs measures mos commonly used, such as sandard devaon, value-a-rs (VaR) and expeced shorfall, are homogeneous funcons of degree one ( β = ) n he porfolo posons. Thus, Euler s heorem s appled o allocae EC and compue rs conrbuons across porfolos. If x denoes he vecor of posons n a porfolo, and EC( x ) he correspondng economc capal, hen Euler s heorem mples addve capal conrbuons = EC( x) EC ( x ) (4) = where he erms EC( x) EC ( x ) = x (5) x are referred o as he margnal capal conrbuons of he porfolo. 4.2 Connuous Margnal EE Conrbuons for need exposures whou collaeral Consder now he calculaon of EE conrbuons. Assume ha we can adjus he sze of any rade n he porfolo by any amoun. Defne he wegh α for rade as a scale facor ha represens he relave sze of he rade n he porfolo, V ( α, ) = α V ( ). These weghs can assume any real value, wh α = correspondng o he acual sze of he rade and α = 0 beng he complee removal of he rade. We descrbe adjused porfolos va he vecor of weghs α = ( α, K, α ). For adjused porfolos, we use he noaons E( α, ), eˆ ( α, ), and CVA( α ) for he exposure and EE a me and CVA. Furhermore, for convenence, denoe by = (, K,) he vecor represenng he orgnal porfolo. When here s no margn agreemen beween he ban and he counerpary, he counerpary-level exposure s a homogeneous funcon of degree one n he rade weghs: E( cα, ) = ce( α, ) (6) 8
10 The nuon behnd Equaon (6) s smple: f he ban unformly doubles he sze of s porfolo wh he counerpary by enerng no exacly he same rade wh he counerpary for each exsng rade, he ban s exposure doubles. We defne he connuous margnal EE conrbuon of rade a me as he nfnesmal ncremen of he condonal dscouned EE of he acual porfolo a me resulng from an nfnesmal ncrease of rade s presence n he porfolo, scaled o he full rade amoun: eˆ (, + δ u ) ˆ ( ) ˆ e e (, α) eˆ ( ) = lm = δ 0 δ α α = (7) where u descrbes a porfolo whose only componen s one un of rade. Snce he porfolo exposure s homogeneous n he rades weghs, he EE conrbuons defned by Equaon (7) auomacally sum up o he counerpary-level condonal dscouned EE by Euler s heorem (Equaon (3)). We can derve an expresson for he margnal EE conrbuons as follows. Frs, subsue Equaon (9) no Equaon (7) and brng he dervave nsde he expecaon. Ths resuls n ˆ E( α, ) eˆ ( ) = E D( ) α α= (8) where exposure of he adjused porfolo (wh wegh vecor α = ( α, K, α ) ) s gven by E( α, ) = max αv ( ), 0 (9) = Calculang he frs dervave of he exposure wh respec o he wegh α and seng all weghs o one, we have: E( α, ) V ( α, ) = max { V ( α, ), 0 } = = V ( ) { (, ) 0 } V (20) α α α α > { V ( ) > 0} = = = α α α Subsung Equaon (20) no Equaon (8), we oban he EE conrbuon of rade : ˆ ˆ e ( ) E ( ) ( ) = D V { V ( ) > 0 } (2) The EE conrbuon of rade s he expecaon of a funcon whch consders he dscouned values of he rade on all scenaros where he oal counerpary exposure s posve, or zero oherwse. As expeced, he EE conrbuons sum up o he counerpary-level dscouned EE: 9
11 = ˆ ˆ eˆ ( ) E ( ) ( ) E ( )max{ ( ),0} ˆ = D V = ( ) { V ( ) 0} D V = e > 5. Addve EE Conrbuons for Collaeralzed eng Ses Consder now a counerpary ha has a sngle neng agreemen suppored by a margn agreemen, whch covers all he rades wh he counerpary. As dscussed n Secon 2, he counerpary-level sochasc exposure s gven by Equaon (4), where he collaeral avalable o he ban s gven eher by he nsananeous collaeral model (Equaon (5)) or by he lagged collaeral model (Equaon (6)). In wha follows, we specfy addve EE conrbuons for boh models, sarng wh he smpler nsananeous collaeral model. 5. Insananeous Collaeral Model Subsung Equaon (5) no Equaon (4), we oban E( ) = V ( ) + H (22) { 0 < V ( ) < H} { V ( ) > H} As can be seen from Equaon (22), he expeced exposure n no a homogeneous funcon of he rades weghs and, hence, he CMC approach canno be appled drecly. From he mahemacal pon of vew, he condons of Euler s heorem are no sasfed, and he CMCs, as gven earler, do no sum o he counerpary-level dscouned EE anymore. To undersand concepually how he CMC mehod fals, noce ha when he porfolo value s above he hreshold, he counerpary-level exposure equals he hreshold. An nfnesmal ncrease of any rade s wegh n he porfolo s no suffcen o brng he porfolo value below he hreshold. Thus, he counerpary-level exposure s no affeced by he nfnesmal wegh changes, and he exposure conrbuon s zero for all scenaros wh he porfolo value above he hreshold. However, we sll would le o allocae he non-zero collaeralzed counerpary-level exposure (equal o he hreshold) o he ndvdual rades, so ha hese allocaons canno be all equal o zero. We can derve addve conrbuons for hs non-homogeneous case, whch are conssen wh he connuous margnal conrbuons as follows. Frs, noce ha, whle he exposure funcon n Equaon (22) s no homogeneous n he vecor of weghs α = ( α, K, α ), he funcon ( ) ( ) ( α ) E α, = V α, { 0 V, H + α H H < < α } { V ( α, ) > α H } (23) H H s a homogeneous funcon n he exended vecor of weghs α ' = ( α,..., α, α H ). Tha s, we consder scalng he each of he rades as well as he hreshold H. Thus, we can hn of he conrbuon of he hreshold self o he counerpary-level exposure. The frs dervaves of he exposure wh respec o he rade weghs s gven by 0
12 E( α, ) = V ( ) { 0 V ( ) H} α < α = (24) Smlarly, he dervave wh respec o he hreshold wegh s E( α, ) = H { V ( ) H} α > H α = (25) oe ha hese sum up o he counerpary-level exposure gven by Equaon (22), as expeced. By applyng dscounng and ang condonal expecaon of he rgh-hand sde of Equaons (24) and (25), we oban he EE conrbuons of he rades eˆ ˆ, H ( ) = E D( ) V ( ) { 0 < V ( ) H} (26) and of he hreshold eˆ ˆ H ( ) = H E D( ) { V ( ) > H} (27) whch sasfy =, H + H = eˆ ( ) eˆ ( ) eˆ ( ) (28) The conrbuon of he hreshold can be nerpreed as he change of he condonal dscouned EE assocaed wh an nfnesmal shf of he hreshold upwards scaled up by he acual sze of he hreshold. oe ha, when he hreshold goes o nfny, he las erm vanshes and we recover he uncollaeralzed conrbuons. As he fnal sep, we allocae bac he conrbuon adjusmen of he collaeral hreshold gven by Equaon (27) o he ndvdual rades, so ha Equaon (28) can be wren n erms of EE conrbuons only of he rades (as n Equaon ()): = = eˆ ( ) eˆ ( ) There are several possbles for allocang he amoun eˆ H ( ) n meanngful proporons o each rade. Gven ha ha he adjusmen of he rade conrbuons occurs when he porfolo value exceeds he hreshold, a meanngful weghng scheme s gven by he rao of he ndvdual nsrumen s expeced dscouned value when he hreshold s crossed o he oal counerpary dscouned value when hs occurs:
13 Ê D( ) V ( ) { V ( ) > H ˆ ˆ } E D( ) { ( ) } = E D( ) V H { V ( ) H} > (29) > = Ê D( ) V ( ) { V ( ) > H} Thus, he ndvdual rade conrbuons o EE are gven by H Ê D( ) { ( ) ˆ V > H} eˆ ( ) E ( ) ( ) ˆ = D V { 0 ( ) } + E D( ) V ( ) < V < H { V ( ) > H} Ê D( ) V { V ( ) > H} (30) Boh erms of Equaon (30) have a sraghforward nerpreaon: he frs erm s he conrbuon of all scenaros where he ban holds no collaeral a me, whle he second erm s he conrbuon of all scenaros where he ban holds non-zero collaeral a me. We refer o he allocaon scheme above as ype A allocaon. An alernave allocaon scheme (ype B) s obaned by brngng he weghng scheme of he hreshold conrbuon now nsde he expecaon operaor, so ha nsead of Equaon (29) we now have: (3) V ( ) ˆ ˆ ( ) E = ˆ V D( ) = E ( ) E ( ) { ( ) } D = V H { V ( ) H} D > > { V ( ) > H} = V ( ) = V ( ) Ths leads o he connuous margnal conrbuons gven by * ˆ ˆ V ( ) eˆ ( ) = E D( ) V ( ) { 0 ( ) } + H E D( ) < V < H { V ( ) > H V ( ) } (32) Boh erms on he rgh hand of Equaon (32) have he same nerpreaon as before. 5.2 Lagged Collaeral Model We now apply he formalsm developed for he connuous collaeral model o he lagged collaeral model. In hs case, he counerpary cred exposure s obaned by subsung Equaon (6) no Equaon (4): E( ) = V ( ) + [ H + δv ( )] (33) { 0 < V ( ) H + δv ( ) } { 0 < H + δv ( ) < V ( ) } where δv ( ) = V ( ) V ( δ) s he change of he porfolo value from he loo-bac me pon δ o he me pon of neres. oe ha as he margn perod of rs, δ, vanshes, hs expresson reduces o Equaon (22), whch gves he exposure wh nsananeous collaeral. 2
14 Indeed, a zero margn perod of rs mples a zero change of porfolo value from δ o, snce boh me pons are he same now. As defned for he nsananeous model, we rewre he exposure gven by Equaon (33) as a homogeneous funcon n he exended vecor of weghs α ' = ( α,..., α, α ) : E( α, ) = V ( α, ) + [ α (, )] { 0 V (, ) H H V (, ) H H + δv α (34) < α α + δ α } { 0 < αh H + δv ( α, ) < V ( α, ) } H where = = δv ( α, ) α δv ( ) (35) and δv ( ) = V ( ) V ( δ). The frs dervaves of exposure wh respec o he rades weghs and wh respec o he hreshold wegh are gven, respecvely, by E( α, ) α = V ( ) + δv ( ) { 0 ( ) ( ) V H δv } { 0 H δv ( ) V ( ) } α < + < + < = E( α, ) α = H { 0 H δv ( ) V ( ) } α < + < = (36) (37) The sum hese frs dervaves across all he rades and he hreshold, gves he counerparylevel exposure, Equaon (33). By applyng dscounng and ang he condonal expecaon of he rgh-hand sde of Equaons (36) and (37), we oban he EE conrbuons of he rades eˆ ˆ ˆ, H ( ) = E D( ) V ( ) E ( ) ( ) { 0 V ( ) H V ( ) } D δv < + δ + { 0 < H + δv ( ) < V ( ) } (38) and of he hreshold eˆ ( ) = H E ˆ D( ) H { 0 < H + δ V ( ) < V ( )} (39) ow we need o allocae bac he hreshold conrbuon, Equaon (39), o he ndvdual rades. Followng he ype A allocaon scheme n he prevous secon, eˆ H ( ) s allocaed o ndvdual rades n proporon o he expecaon of he dscouned rade values when 0 < H + δv ( ) < V ( ). Ths resuls n he rade allocaons gven by 3
15 eˆ ˆ ˆ ( ) = E D( ) V ( ) E { 0 ( ) ( ) D( ) δv ( ) + < V H + δv } { 0 < H + δv ( ) < V ( ) } Ê D( ) V ( ) { 0 < H + δv ( ) < V ( ) } + H Ê D( ) { 0 < H + δv ( ) < V ( ) } Ê D( ) V ( ) { 0 < H + δv ( ) < V ( ) } (40) In he ype B allocaon, we brng he weghng scheme nsde he expecaon operaor. Ths gves eˆ ˆ ˆ ( ) = E D( ) V ( ) E { 0 ( ) ( ) D( ) δv ( ) + < V H + δv } { 0 < H + δv ( ) < V ( ) } V ( ) + H Ê D( ) V ( ) { 0 < H + δv ( ) < V ( ) } (4) I s sraghforward o verfy ha he lagged EE conrbuons degenerae o he nsananeous EE conrbuons when δ = 0. Subsung δ V ( ) = 0 no Equaons (40) and (4), we oban Equaons (30) and (32), respecvely. 6. Calculang CVA Conrbuons by Smulaon Bans commonly use Mone Carlo smulaon n pracce o oban he dsrbuon of counerpary-level exposures. Based on hese smulaons a ban can also compue he counerpary-level CVA. In hs secon, we show how he calculaon of EE conrbuons can be easly ncorporaed o he Mone Carlo smulaon of he counerpary-level exposure ha bans already perform. 6. Exposure Independen of Counerpary s Cred Qualy Consder frs he case where he exposures are ndependen of he counerpary s cred qualy. In general, bans mplcly assume ha each counerpary s exposure s ndependen of ha counerpary s cred qualy when exposures are smulaed separaely. Le us now mae hs assumpon explcly. Then, condonng on τ = n he expecaons n Equaons (9), (29), (30) and (32) become uncondonal, and hese condonal expecaons can be replaced by he uncondonal ones. The smulaon algorhm for calculang counerpary-level CVA can be exended o calculae CVA conrbuons. For he ease of exposon, we assume ha all he rades wh he counerpary are neable and ha collaeral (f here s any) can be descrbed by he nsananeous model. Frs, he counerpary-level CVA can be calculaed n a Mone Carlo smulaon as follows: 4
16 . Generae a mare scenaros j (neres raes, FX raes, ec.) for each of he fuure me pons 2. For each smulaon me pon and scenaro j : ( j a. For each rade, calculae rade value V ) ( ) = ( j) ( j) b. Calculae porfolo value V ( ) V ( ) = c. If here s margn agreemen, calculae collaeral ( j) ( j) C ( ) = max{ V ( ) H,0} avalable a me. d. Calculae counerpary-level exposure ( here s no margn agreemen, C j) ( ) 0). E ( ) = max{ V ( ) C ( ),0} (f ( j) ( j) ( j) 3. Afer runnng large enough number M of mare scenaros, compue he dscouned EE by averagng over all he mare scenaros a each me pon: 4. Fnally, compue CVA as M ( j) ( j) ( ) ( ) ( ) j= e = D E. M, CVA = ( R) e ( )[ P( ) P( )] where, as before, R denoes he (consan) recovery rae and P( ) s he uncondonal cumulave probably of defaul up o me. The calculaon of EE and CVA conrbuons can be ncorporaed o hs algorhm as follows. Consder, for example, he EE conrbuons gven by Equaon (32). The followng calculaons are added o Seps 2-4: Sep 2: For each rade, calculae he rade s exposure conrbuon for scenaro j ( j E ) ( ), whch s equal o ( V j) ( ) > H, and zero oherwse. ( j V ) ( ) f ( j) 0 ( ) < V H, H V ( ) V ( ) f ( j) ( j) Sep 3: For each rade, compue he dscouned EE conrbuon by averagng over all he mare scenaros a each me pon: M M ( j ) ( j ) ( ) ( ) ( ) j= e = D E. Sep 4: CVA conrbuons are compued as 5
17 CVA = ( R) e ( )[ P( ) P( )]. 6.2 Exposure Dependen on Counerpary s Cred Qualy The algorhm above assumes ndependence beween he exposure and he counerpary s cred qualy. More generally, here may be dependence beween hem whch can come from wo sources: Rgh/wrong-way rs. The rs s called rgh-way (wrong-way) f exposure ends o decrease (ncrease) when counerpary qualy worsens. Srcly speang, rgh/wrongway rs s always presen, bu s usually gnored o smplfy exposure modelng. However, here are cases when rgh/wrong way rs s oo sgnfcan o be gnored (e.g., cred dervaves, commody rades wh a producer of ha commody, ec.). Exposure-lmng agreemens ha depend on he counerpary cred qualy. One example such agreemens s a margn agreemen wh he hreshold dependen on he counerpary s cred rang. Anoher example s an early ermnaon agreemen, under whch he ban can ermnae he rades wh he counerpary when he counerpary s rang falls below a pre-specfed level. Boh ypes of dependence of exposure on he counerpary s cred qualy can be ncorporaed n he EE and CVA calculaon f he rade values and cred qualy of he ban s counerpares are smulaed jonly. If a ban s counerpary rs smulaon envronmen s capable of such jon smulaon, he calculaon of EE and CVA conrbuons s also sraghforward. Le us nroduce a sochasc defaul nensy process λ ( ) whou specfyng s underlyng dynamcs. Ths nensy can be used as a measure of counerpary cred qualy: hgher values of he nensy correspond o lower cred qualy. The counerpary-level exposure E( ) may depend eher on he nensy value λ ( ) a me, or on he enre pah of he nensy process λ( ) from zero o. We can use he nensy process o conver he expecaon condonal on defaul a me n Equaon (2) o an uncondonal expecaon so ha he condonal EE conrbuon becomes eˆ ( ) E = λ( )exp λ( s) ds D( ) E ( ) P ( ) (42) 0 where P ( ) s he frs dervave of he cumulave PD P( ). A shor dervaon of Equaon (42) s gven n Appendx. For an overvew of sochasc defaul nensy and he assocaed Cox process, see Chaper 5 n Schönbucher (2003). 6
18 As descrbed for uncondonal EE conrbuons, he calculaons for condonal EE conrbuons can be performed durng a Mone Carlo smulaon of exposures. In hs case, gven he dependence of exposures on he counerpary cred qualy, he nensy process λ( ) needs o be smulaed jonly wh he mare rs facors ha deermne rade values. Ths jon smulaon s done pah-by-pah: smulaed values of he nensy and of he mare facors a me are obaned from he correspondng smulaed values a he earler me pons (, 2,... ). 2 Assumng ha we have already smulaed he mare facors and he nensy for mes j for all j <, he algorhm for compung CVA conrbuons for me can be expressed as follows:. Jonly smulae mare rs facors and nensy λ ( ) a me 2. For each rade, calculae s rade value V ( ) = 3. Calculae he porfolo value V ( ) V ( ) = 4. For each rade, updae he EE conrbuon couner a. If here s no margn agreemen, hen V >, add λ ( ) λ j = f ( ) 0 ( ) exp ( j )( j j ) D ( ) V ( ) P ( ) b. If here s a margn agreemen wh an nensy-dependen hreshold h[ λ( )], hen <, add λ ( ) λ j = f 0 V ( ) h[ λ( )] f V ( ) h[ λ( )] ( ) exp ( j )( j j ) D ( ) V ( ) P ( ) >, add λ ( λ ) j = ( ) V( ) exp ( j )( j j ) D( ) h[ λ( )] P ( ) V ( ) Afer runnng large enough number of mare scenaros, EE conrbuons are obaned by dvdng he EE conrbuon couner by he number of scenaros. 7. Analycal CVA Conrbuons under a ormal Approxmaon I s also useful n pracce o esmae EE and CVA conrbuons qucly ousde of he smulaon sysem. To faclae such calculaons, we derve analycal EE conrbuons, for he case when rade values are normally dsrbued. For smplcy, and o avod dealng wh sochasc dscounng facors, we assume ha, a me, he dsrbuon of rade values s gven 2 For a dscusson on pah-by-pah vs. drec jump o smulaon dae, see Pyhn and Zhu (2007). 7
19 under he forward (o me ) probably measure. Under hs measure, he dscouned condonal EE n Equaon (9) can be wren as and he dscouned uncondonal EE s eˆ ( ) = B(0, )E E( ) τ = B(0, ) eˆ ( ) (43) [ ] e ( ) = B(0, )E E( ) B(0, ) e( ) (44) where B(0, ) s he me-zero prce of he rs-free zero-coupon bond maurng a me. Ths change of measure allows us o wor wh undscouned EEs and EE conrbuons. Assume ha he value V( ) of rade a each fuure me s normally dsrbued wh expecaon µ and sandard devaon σ under he forward o probably measure: V ( ) = µ ( ) + σ ( ) X (45) where X s a sandard normal varable. Correlaons beween hese sandard normal varables (and, herefore, beween he dscouned rade values) are denoed by r j. Snce he sum of normal varables s also normal, he dscouned porfolo value V ( ) s normally dsrbued: V ( ) = µ ( ) + σ ( ) X (46) where X s anoher sandard normal varable, and he mean and sandard devaon of he porfolo value are gven by =, = µ ( ) µ ( ) ( ) = r ( ) ( ) (47) 2 σ j σ σ j = j = Denoe by ρ ( ) he correlaon beween he value V( ) of rade and he porfolo value V ( ). We can calculae hs correlaon as follows: ρ ( ) cov[ ( ), ( )] cov[ V ( ), V ( )] V V σ j( ) rj (48) σ ( ) σ ( ) σ ( ) σ ( ) σ ( ) j = j = = = j = Usng hs correlaon, we can represen X as X = X + Z (49) 2 ρ( ) ρ ( ) 8
20 where Z s a sandard normal random varable ndependen of X (and dfferen for each rade). 7. Exposure Independen of Counerpary s Cred Qualy We frs calculae counerpary-level EE and EE conrbuons assumng ndependence beween exposures and counerpary cred qualy. We oban he resuls for he general case of a neng agreemen wh a margn agreemen. The smpler case, wh no margn agreemen, s obaned as he lmng case when he hreshold goes o nfny. For he clary of exposon, we assume he nsananeous collaeral model. In he presence of a margn agreemen, he counerpary-level sochasc exposure s gven by Equaon (22). Subsung Equaon (46) no Equaon (22) and ang he expecaon, we oban e( ) = E [ µ ( ) + σ ( ) X ] + H E { 0 < µ ( ) + σ ( ) X < H } { µ ( ) + σ ( ) X > H } H µ ( ) σ ( ) [ ] = µ ( ) + σ ( ) x φ( x) dx + H φ( x) dx µ ( ) H µ ( ) σ ( ) σ ( ) where φ( ) s he probably densy of he sandard normal dsrbuon. Evaluang he negrals yelds an analycal formula for he EE: µ ( ) µ ( ) H e( ) = µ ( ) Φ Φ σ ( ) σ ( ) µ ( ) µ ( ) H µ ( ) H + σ ( ) φ φ + HΦ σ ( ) σ ( ) σ ( ) (50) where Φ( ) s he sandard normal cumulave dsrbuon funcon. Le us consder now he EE conrbuons gven by Equaon (32) (ype B allocaons). Removng he dscounng and subsung Equaons (45) and (46) no Equaon (32), we oban µ ( ) + σ ( ) X e( ) = E ( µ ( ) σ ( ) X ) + H E { 0 < µ ( ) + σ ( ) X < H } + µ ( ) + σ ( ) X { µ ( ) + σ ( ) X > H } (5) Appendx 2 shows ha, afer some analycal manpulaon, hs EE conrbuon can be wren as 9
21 µ ( ) µ ( ) H µ ( ) µ ( ) H e( ) = µ ( ) Φ Φ + σ ( ) ρ( ) φ φ σ ( ) σ ( ) σ ( ) σ ( ) + H H µ ( ) σ ( ) µ ( ) + σ ( ) ρ( ) x φ( x) dx µ ( ) + σ ( ) x (52) where he remanng negral can be easly evaluaed numercally. One can verfy ha EE conrbuons gven by Equaon (52) sum up o he counerpary-level EE, Equaon (50). Smlarly, we oban he EE conrbuons correspondng o ype A allocaons as µ ( ) µ ( ) H µ ( ) µ ( ) H e( ) = µ ( ) Φ Φ + σ ( ) ρ( ) φ φ σ ( ) σ ( ) σ ( ) σ ( ) µ ( ) H µ ( ) H µ ( ) Φ σ ( ) ρ ( ) φ µ ( ) H σ ( ) + σ ( ) + H Φ σ ( ) µ ( ) H µ ( ) H µ ( ) Φ + σ ( ) φ σ ( ) σ ( ) (53) In conras o Equaon (52), he EE conrbuons gven by Equaon (53) are gven n closed form, and do no requre numercal negraon. For he case whou a margn agreemen, we ae he lm H of Equaons (50)- (53). Ths leads o he counerpary-level EE µ ( ) µ ( ) e( ) = µ ( ) Φ + σ ( ) φ σ ( ) σ ( ) (54) and he EE conrbuons µ ( ) µ ( ) e( ) = µ ( ) Φ + σ ( ) ρ( ) φ σ ( ) σ ( ) (55) 7.2 Rgh/Wrong-Way Rs We now lf he ndependence assumpon o accommodae rgh/wrong-way rs. oe ha he EE conrbuons obaned n he prevous secon are conrbuons of he rades n he porfolo o he counerpary-level uncondonal dscouned EE. We need o modfy he approach o oban he conrbuons o he counerpary-level EE condonal on he counerpary defaulng a he me when he exposure s measured. An obvous approach s o defne an nensy process and compue he condonal EE conrbuons as he expecaon over all 20
22 possble pahs of he nensy process (See Appendx ), bu hs requres a Mone Carlo smulaon. In hs secon, we develop an alernave, smpler approach ha resuls n closed form expressons for he condonal EE conrbuons. For hs purpose, we defne a ormal copula 3 o model he codependence beween he counerpary s cred qualy and he exposures. 4 Thus, we frs map he counerpary s defaul me τ o a sandard normal random varable Y: Y = Φ [ P( τ )] (56) where Φ ( ) s he nverse of he sandard normal cumulave dsrbuon funcon. The counerpary-level condonal EE s gven by whle he condonal EE conrbuon of rade s gven by eˆ( ) = E E( ) τ = (57) eˆ ( ) = E E( ) τ = (58) where E( ) s he counerpary-level exposure and E( ) s he sochasc exposure conrbuon of rade a me. Snce he counerpary s cumulave probably of defaul P( ) s a monoonc funcon, each possble defaul me s mapped o a unque value of Y. Thus, we can replace he condonng on τ n Equaons (57) and (58) wh he condonng on Y and wre he counerpary-level condonal EE and he EE conrbuons as (59) eˆ( ) = E E( ) Y =Φ [ P( )] (60) eˆ ( ) = E E( ) Y =Φ [ P( )] We model he rgh/wrong-way rs by allowng rade values o depend on Y. More specfcally, we assume ha he sandard normal rs facor X, whch drves he value of rade, depends on Y accordng o X = by + b X (6) 2 ˆ 3 The ormal copula framewor was frs proposed by L (2000) o model correlaed defaul mes for prcng porfolo cred dervaves. 4 See for example Garca Cespedes e al. (2009) for he applcaon of such a copula approach o compue counerpary cred capal. 2
23 where X ˆ s a sandard normal varable ndependen of Y. The parameer b drves he rgh/wrong-way rs. When b = 0, he value of rade s ndependen of he counerpary cred qualy. Wrong-way rs occurs when < 0 (he value of rade ends o ncrease when Y b declnes). Smlarly, here s rgh-way rs when 0 < b (he value of rade ends o decrease when Y declnes). As he magnude of b, ncreases, so does he codependence beween he rade value and he counerpary cred qualy. If he porfolo conans rades wh non-zero b, he sandard normal rs facor X, whch drves he porfolo value, also depends on Y: X = β ( ) Y + β ( ) X (62) 2 ˆ wh ˆX a sandard normal varable ndependen of Y. The porfolo facor loadng β ( ) can be compued from he ndvdual rade facor loadngs b as follows: cov[ V ( ), Y ] cov[ V( ), Y ] σ ( ) β ( ) = cov[ X, Y ] = = = b σ ( ) σ ( ) σ ( ) (63) = = ow we have all he ngredens o derve he counerpary-level EE and EE conrbuons n he presence of rgh/wrong-way rs. One approach may be o calculae condonal expecaons n he same manner as we have calculaed he uncondonal ones n he prevous Secon. However, n Appendx 2 we show how hs can be done n a faser and more elegan way. In parcular, he condonal exposure model can be formulaed he n exacly he same mahemacal erms as he uncondonal model. The only dfference s ha nsead of he uncondonal expecaons, sandard devaons and correlaons ha specfy he behavor of he rade values, we now use he condonal ones. Therefore, we can use all he resuls of Subsecon 7. (Equaons (50)-(55)) afer pung has on he parameers: 5 ˆ µ ( ) E[ V ( ) τ ] µ ( ) σ ( ) b [ P( ) ] = = + Φ (64) ˆ σ ( ) SDev[ V ( ) τ = ] = σ ( ) b (65) 2 ˆ( µ ) E[ V ( ) τ = ] = µ ( ) + σ ( ) β ( ) Φ P( ) (66) [ ] 5 Ths concluson s conssen wh he resuls n Redon (2006). Usng a dfferen model of rgh/wrong-way rs, hey show ha he condonal counerpary-level uncollaeralzed EE s descrbed by he same expresson as he uncondonal EE, afer replacng he uncondonal expecaons and sandard devaons of he rade values wh he condonal ones. 22
24 2 ˆ( ) SDev[ V ( ) ] ( ) ( ) σ τ = = σ β (67) ˆ ρ ( ) = ρ ( ) b β ( ) 2 2 ( b )[ β ( )] (68) 7.3 Remars on he Analycal Formulae In hs secon we brefly commen on he properes and nerpreaon of he analycal conrbuons derved n hs secon. eng & no margn Equaon (55) can be undersood from he ncremenal vewpon of he CMC mehod. Accordng o Equaon (7), he EE conrbuon of rade s deermned by he nfnesmal change of he counerpary-level EE resulng from an nfnesmal ncrease of he wegh of rade n he porfolo. The effec of an ncrease of he wegh of a rade on he porfolo value dsrbuon can be vewed as he sum of wo effecs: o a unform shf of he dsrbuon o a change of wdh of he dsrbuon Le us consder hese wo effecs separaely. If he wegh of rade s ncreased by δ, he expecaon of porfolo value changes by δ µ ( ). Le us frs gnore he change of he sandard devaon and consder how a unform shf of he enre dsrbuon by δ µ ( ) affecs he counerpary-level EE. Scenaros wh posve porfolo value conrbue he same amoun δ µ ( ) o he exposure change, whle scenaros wh negave porfolo value conrbue nohng. Therefore, he ncremen of he EE wll be gven by he produc of he magnude of he shf δ µ ( ) and he probably of he porfolo value beng posve. I s sraghforward o verfy ha Pr[ V ( ) > 0] = Φ [ µ ( ) / σ ( )]. Thus, he frs erm n he rgh-hand sde of Equaon (55) descrbes he ncremen of he counerpary-level EE resulng from he nfnesmal unform shf of he porfolo value dsrbuon assocaed wh an ncrease of he wegh of rade. The second erm of Equaon (55) descrbes he change of he wdh of he porfolo value dsrbuon. The change of he sandard devaon of he porfolo value resulng from ncreasng he wegh of rade by δ can be calculaed as 2 2 ( ) ( ) SDev[ V ( ) + δv ( )] SDev[ V ( )] = var[ V ( )] + 2δ cov[ V ( ), V ( )] + δ var[ V ( )] σ ( ) V O = σ ( ) + 2 δ ρ ( ) σ ( ) σ ( ) + δ var[ ( )] σ ( ) = δ ρ ( ) σ ( ) + ( δ ) 23
25 2 where O( δ ) denoes he erms of he second order and hgher ha can be gnored. Thus, he porfolo value dsrbuon s wdenng f he correlaon ρ ( ) s posve, and narrowng f he correlaon s negave. A wdenng dsrbuon (wh no accompanyng shf) always ncreases he counerpary-level EE, whle narrowng always decreases. Indeed, for a gven realzaon of he porfolo value V ( ), he change of exposure assocaed wh he change of he sandard devaon of he porfolo value from σ ( ) o σ ( δ, ) = σ ( ) + δ ρ ( ) σ ( ) s gven by 6 V ( ) µ ( ) E( δ, ) E( ) = δ ρ( ) σ ( ) (69) { V ( ) 0 ( ) } σ > The second erm of Equaon (55) can be obaned by ang he expecaon of he rgh-hand sde of Equaon (69). I appears ha Equaon (55) has smple lnear dependence on µ ( ) and he produc ρ ( ) σ ( ). However, hs s only par of he rue dependence. Snce rade s par of he porfolo, µ ( ) depends on µ ( ) and σ ( ) depends on σ ( ) and he correlaon of rade wh he res of he porfolo. Moreover, correlaon ρ ( ) s he correlaon beween he values of rade and he porfolo ha ncludes rade self. Because of hs, ρ ( ) depends on he rao σ ( ) / σ ( ) (see Equaon (48)). Thus, unless rade represens a neglgble fracon of he porfolo, he rue dependence of EE conrbuon on rade parameers s non-lnear. eng & margn In hs case, only he frs wo erms of Equaon (52) allow nerpreaon from he ncremenal vewpon of he CMC mehod: he frs erm can be explaned as he effec of he unform shf and he second erm as he effec of he wdenng or narrowng of he porfolo value dsrbuon. The hrd erm resuls from he allocaon of exposure when he porfolo value s above he hreshold. An aemp o use he CMC mehod would gve zero EE conrbuon from V ( ) > H scenaros. Equaon (52) can be re-wren as µ ( ) µ ( ) H φ( ξ ) dξ e( ) = µ ( ) Φ Φ + H σ ( ) σ ( ) H µ ( ) µ ( ) σ ( ) ξ + σ ( ) µ ( ) µ ( ) H φ( ξ ) ξdξ + σ ( ) ρ( ) φ φ + H σ ( ) σ ( ) H µ ( ) µ ( ) σ ( ) ξ + σ ( ) (70) 6 Ths can be mmedaely seen from Equaon (46). 24
26 As n he non-collaeralzed case, hs EE conrbuon appears o be lnear n µ ( ) and he produc ρ( ) σ ( ), bu he rue dependence on hese quanes s more complex due o he exra dependence of µ ( ) and σ ( ) on hese quanes. Rgh/wrong-way rs If he value of rade s correlaed wh he counerpary s cred qualy, s value dsrbuon a me condonal on he counerpary s defaul a me dffers from s uncondonal value dsrbuon. If he correlaon s posve (rgh-way rs), he dsrbuon shfs down; f he correlaon s negave (wrong-way rs), he dsrbuon shfs up. In boh cases, he dsrbuon becomes narrower. Under he normal approxmaon, he shf of he dsrbuon s descrbed by Equaon (64), and he narrowng s descrbed by Equaon (65). An neresng propery of Equaon (64) s s dependence on he counerpary s PD. To undersand hs, le us consder he ban enerng no he same rade wh an nvesmen-grade counerpary A and wh a speculave-grade counerpary B. We are neresed n he rade value dsrbuon condonal on he counerpary s defaul a he me of observaon. For he case of wrong (rgh) way rs, he deeroraon of he counerpary s cred qualy o he pon of defaul pushes rade values hgher (lower). Snce counerpary A s furher away from defaul han counerpary B, he deeroraon of cred qualy o he pon of defaul s larger for counerpary A. Therefore, rade values condonal on defaul of A are shfed more han rade values condonal on defaul of B. oe ha hs s no specfc o he normal approxmaon, bu s a general propery no relaed o any model. 8. Examples In hs secon, we presen some smple examples ha llusrae he behavor of exposure (and hence CVA) conrbuons. For ease of exposon, we assume ha rade values are ormal, as well as mare and cred ndependence. However, as dscussed earler n Secon 7.2, he conclusons apply equally o he case of wrong-way rs by smply usng condonal expecaons, volales and correlaons, nsead of uncondonal ones. We frs presen an example when here s no collaeral agreemen n place, and hen show he mpac of addng a collaeral agreemen o he porfolo. 8. Conrbuons for a on-collaeralzed Porfolo As a frs sep o undersand hs behavor, consder Equaons (54) and (55), whch gve he counerpary-level EE and he EE rade conrbuons, n he case when here s no margn agreemen n place: µ ( ) µ ( ) e( ) = µ ( ) Φ + σ ( ) φ σ ( ) σ ( ), µ ( ) µ ( ) e( ) = µ ( ) Φ + σ ( ) ρ( ) φ σ ( ) σ ( ) The EE conrbuon of nsrumen s a funcon of: 25
27 he mean value conrbuon, µ he volaly conrbuon, ρ σ he overall value of he rao µ/σ (for he enre porfolo) where we have dropped he me varable noaon for brevy. Boh he counerpary-level EE and he rade conrbuons can be seen as he sum of wo componens: a mean value componen (frs erm n he equaons), and a volaly componen (second erm). These componens wegh he mean value (or mean value conrbuons) and he volaly (volaly conrbuon), respecvely, by he ormal dsrbuons and densy evaluaed a he rao µ/σ (for he enre counerpary porfolo). Thus, he overall level of he counerpary porfolo s mean value and volaly deermne how he ndvdual nsrumen s mean and volaly conrbuon are weghed o yeld he EE conrbuons. Fgure plos hese weghs as a funcon of µ/σ. A low rao weghs he volaly conrbuon much hgher; whle a hgh rao weghs mean values much more. For example, f µ/σ = -2, he volaly componen wegh s 2.4 mes he mean value wegh. In conras, µ/σ = 2 resuls n mean values beng weghed 8 mes he volales. orm Ds.0 orm Densy Mu/Sgma Fgure. Volaly and mean exposure weghs for EE conrbuons. To llusrae he mpac of varous parameers on EE conrbuons, consder now he smple counerpary porfolo, whch comprses of 5 ransacons over a sngle sep. Table gves he ndvdual rade s mean value, varance and volaly (n dollar values and % conrbuons). The porfolo has a mean value and varance of 0. We assume ha rade values are ndependen. 7 In hs case, he porfolo s rao µ/σ = 3.6. P P2 P3 P4 P5 Toal 7 Ths assumpon s only made for smplcy, and bears no mpac on he analyss. Alernavely we can smply use he volaly conrbuons drecly for any correlaed model. 26
28 µ % 0% 0% 20% 30% 40% 00% σ % 40% 30% 20% 0% 0% 00% σ Table. Porfolo mean values, varances and volales. The porfolo s consruced so ha for each rade, s mean value and volaly are nversely relaed; hus he frs nsrumen, P, has he lowes mean (0) and larges volaly (2), whle poson 5 has he hghes mean value (4), and lowes volaly (0). Ths may no only be reasonably realsc, bu wll also help hghlgh some of he pons below. Usng Equaons (54) and (55), we compue he EE and conrbuons for he porfolo. The EE for he porfolo s 0.00, wh mos of hs arsng from he mean value componen (9.992). The rade conrbuons o EE are farly close o he conrbuons o he mean values n Table (0.03%, 0.02%, 20.00%, 29.98%, 39.97%). ow, we vary he overall mean value of he porfolo, µ, whle leavng nac he volaly, σ, as well as he percen conrbuons of each nsrumen o he mean exposure and volaly n able. Ths allows us o express he rade conrbuons n erms of how deep n- or ou-of-he-money he counerpary porfolo s (relave o s volaly). Fgure 2 plos he EE, as well as s mean and volaly componens, as funcons of he porfolo s µ/σ. For large negave porfolo mean values, he EE (red lne) s zero. In hs case, he mean value componen of EE s acually negave, and he volaly componen compensaes for hs o generae posve EEs. As µ/σ ncreases beyond zero, he volaly componen decreases and, once µ/σ > 2, he EE s compleely domnaed by he mean value. Expeced Exposure EE mu componen sgma componen Mu/Sgma 27
29 Fgure 2. EE as a funcon of he porfolo s rao µ/σ. Fgure 3 shows he EE conrbuons for each of he 5 rades as a funcon of µ/σ. There s a clear shf n domnance beween he mean and volaly componens as he porfolo s mean value ncreases. A one sde of he specrum, when he mean porfolo values are negave, rades 4 and 5, whch have he larges (negave) mean values and lowes volales, produce very large negave EE conrbuons. The oppose occurs for rades and 2 (wh low negave means and large volales). As he porfolo s µ/σ ncreases, rades 4 and 5 end up domnang he conrbuons, wh he EE conrbuon convergng o he mean value conrbuons hemselves. For hs parcular symmerc porfolo, every rade conrbues 20% of EE a µ/σ = % 80% 60% EE Conrbuons 40% 20% 0% % % -60% P P2 P3 P4 P5-80% Mu/Sgma Fgure 3. EE Conrbuons as a funcon of he porfolo s rao µ/σ. 8.2 Conrbuons for a Collaeralzed Porfolo We consder now he case when here s a margn agreemen, and demonsrae he mpac of he collaeral on he rade conrbuons. In very general erms: As he hreshold becomes very large, rade conrbuons converge o hose of uncollaeralzed exposures; Wh lower hresholds, he conrbuons of more volale exposures are dmnshed (as he hreshold caps he exposures), and conrbuons of hgher mean exposures (n-hemoney posons) ncrease. Consder he same porfolo n able, bu assume now ha here s a collaeral agreemen n place where margns are placed nsananeously. Frs, we characerze he mpac of he hreshold on he counerpary level EE. Fgure 4 shows he reducons n EE as a resul of he margn agreemen (as % of uncollaeralzed EE) as a funcon of µ/σ, and for varous levels of he (sandardzed) collaeral hreshold. 28