A Unified Framework for CVA, DVA and FVA Applied to Credit Portfolios

Transcription

1 See he Disclaimer Appendix A Unified Framework for CVA, DVA and FVA Applied o Credi Porfolios Youssef Elouerkhaoui Managing Direcor, Markes Quaniaive Analysis youssef.elouerkhaoui@cii.com 19 March 2014 Ciigroup Global Markes Limied

2 Ouline Moivaion: CVA, DVA and FVA Risk Miigaion for Credi Maser Funding Equaion wih CVA, FVA and FVO Fundamenal Invariance Principle for Funding and CVA FVA vs FVO CVA in he Enlarged Filraion Credi Opions Revisied: Impac of he Choice of Filraion Wrong-Way Risk vs Gap Risk 2

3 Inroducion There are a lo of discussions wihin he indusry around bes pracices for marking, managing and miigaing counerpary risk charges and funding coss. The inclusion of FVA adjusemens for unsecured derivaives has been heavily debaed beween praciioners and academics; and he marking mehodology for DVA is sill raising many quesions. According o he E&Y 2012 CVA/FVA Survey, all banks record CVA and OCA (Own Credi Adjusmen) on liabiliies under FVO accouning; and mos repor DVA. (DVA reporing will be mandaory wih he inroducion of IFRS 13). The majoriy of dealers have moved o CSA discouning. Bu for uncollaeralized derivaives, only a handful record an FVA. Funding is usually priced in a rade incepion, bu here is no adjusmen 3

4 made o he FVA during he life of he rade for financial reporing purposes. Unsecured CVA will be less of a problem going forward as we move o mandaory clearing for derivaives, and cash-rich corporaes are incenivized o reduce heir CVA charges and earn CSA ineres on cash reserves. For credi, even Secured CVA can sill be sizeable because of he Gap risk induced by he enlarged filraions. In ha case, The Base PV will be funded by he CSA and discouned a he CSA rae. The CVA and DVA are funded via Treasury and will incur a funding charge (FVA). The main raionale for he inclusion of FVA in books and records reporing is based on he concep of exi price, i.e, he price ha would be paid beween marke paricipans o novae a ransacion. There is no consensus on how he FVA should be calculaed. To mark he OCA, more banks are moving away from CDS curves o using eiher primary issuance daa or arge funding curves.

5 The FVA Debae In a nushell, he FVA debae is abou wheher we should include FVA in derivaives pricing or no. Academics, Hull and Whie (2012a), argue ha we shouldn : I migh be argued ha he use of a risk-free discoun rae indicaes he valuaion is only appropriae when he bank can fund he derivaive a he riskfree rae. This is no rue. The risk-free rae is used for discouning because his is required by he risk-neural valuaion principle. Risk-neural valuaion is an arificial bu fanasically useful ool ha gives he correc economic valuaion for a derivaive, aking ino accoun all is marke risks. Anoher argumen agains FVA is a well-esablished principle in corporae finance heory ha pricing should be kep separae from funding. The discoun rae used o value a projec should depend on he risk of he projec raher han he riskiness of he firm ha underakes i. FVA is closely relaed o debi value adjusmen (DVA), bu i is imporan o avoid confusing he wo differen ypes of DVA. One is he DVA arising because a dealer may defaul on is derivaives porfolio (we will refer o his as DVA1). The oher (DVA2) is he DVA arising because a dealer may defaul on is oher liabiliies long-erm deb, shor-erm deb, and so on. 4

6 Pro-FVA Traders on he oher hand rejec he academic argumens and sae ha FVA is a marke realiy ha drives he profiabiliy of he business. The response from Laughon and Vaisbro (2012): They use he Black-Scholes-Meron (BSM) heory o argue derivaives should be valued on a risk-neural basis, independen of he cos o he rader of funding he posiion, bu he heory ress on he abiliy of marke paricipans o fully hedge all risk facors. In realiy, hey are no able o do so because markes are incomplee. As a resul, risk preference is reinroduced ino valuaions, and he law of one price no longer holds. In pracice, a bank borrows a he rae i can usually unsecured. The bank may ry o convince counerpars ha is borrowing coss should go down as Hull and Whie argue. Bu mos of is deb will already have been issued a a fixed rae, which is unlikely o be renegoiaed, even assuming he crediors believe he incremenal deb is acually risk-free. And i doesn maer o he rader a which rae he bank should be able o borrow, only he rae a which i can. models need o be amended o be useful o raders. They should remove he assumpion of he abiliy o borrow a he risk-free rae o finance an apparenly risk-free baske, and insead assume ha: a) he cos of borrowing for an insiuion is exogenous and unaffeced by a single rade; b) marke-makers give no value o heir expeced profi or loss upon heir own defaul. 5

7 The FVA Debae Coninues Hull and Whie (2012b) mainain in heir follow-up aricle: he Meron argumen is no he only jusificaion of BSM economic argumens, such as Fisher Black and Myron Scholes original conenion based on he capial asse pricing model, give he same resul and do no assume any risk-free borrowers. All hey require is ha, in equilibrium, a risk-free porfolio should earn he risk-free rae. We may no know exacly wha ha rae is, bu he concep is clear and here are excellen proxies for i. The key quesion in his debae appears o be wheher he debi valuaion adjusmen (DVA) ha accouns for he dealer s own defaul is a real benefi or an accouning quirk. DVA2 arises from he possibiliy ha he bank may defaul on is borrowings, and seems o be more conroversial. Accounans signalled accepance of DVA2 in he US Financial Accouning Sandards Board s direcive 159 in In he end, he reporing of DVA2 is jus a move o marke accouning. For hose who believe DVA2 is an unreal accouning absracion ha carries his oo far, FVA makes sense: i is equal bu opposie o DVA2, so can be regarded as a way of removing i from pricing. Bu we believe he DVA2 benefis are real and should accrue o he funding desk. This means FVA should no be included in prices. 6

8 CVA and FVA Lieraure Some of he key papers ha ackled CVA and FVA consisenly, include: D. Brigo, A. Pallavicini, D. Perini (2011): Funding Valuaion Adjusmen: A Consisen Framework Including CVA, DVA, Collaeral, Neing Rules and Re-hypohecaion. D. Brigo, A. Pallavicini, D. Perini (2012), Funding, Collaeral and Hedging: Uncovering he Mechanics and he Subleies of Funding Valuaion Adjusemns. C. Burgard, M. Kjaer (2010): PDE Represenaions of Opions wih Bilaeral Counerpary Risk and Funding Coss. C. Burgard, M. Kjaer (2012), A Generalised CVA wih Funding and Collaeral via Semi-Replicaion. S. Crepey (2011): A BSDE Approach o Counerpary Risk Under Funding Consrains. S. Crepey (2012), Counerpary Risk and Funding: The Four Wings of he TVA. 7

9 Our Approach: Key Principles There is no need for a new arbirage-free pricing heory. Use he same heory, wih he same risk-neural measure and he same money-marke accoun (risk-free) numeraire. The only difference is ha we price a more complex payoff which includes all he defaul coningen legs and all funding legs. There is no need o make any ad-hoc or a-priori assumpions on wha he final resul should be. The fundamenal resul is ha by pricing all he funding legs, since we are by consrucion funding fla (or funding neural), he (risk-free) money-marke accruals drops off and he dependence on he heoreical risk-free rae vanishes. See Crepey (2012a) for an aemp o re-build he arbirage-free pricing heory from firs principles. 8

10 Our Approach: Hedging Porfolio There is no need o include he hedging insrumens and he hedge porfolio. The hedging insrumens jus define which risk-neural probabiliy measure should be used. This defines he drif of he diffusions, and in he case of credi, defines he inensiy processes o be used. If we hedge wih bonds, we use he bond measure, and hence he inensiy would be he one infered by bond pricing which accouns for he bond-cds basis. If we hedge wih CDS, hen he inensiy would be derived by calibraing on he CDS prices. See Brigo e al. (2012) where hey include he hedge porfolio hen simplify he equaions and remove he dependence on he hedge by choosing he appropriae hedge risk measure. See Burgard and Kjaer (2012) where all he inensiies ha hey use are bond-implied inensiies since hey use bonds o hedge he defaul risk and neuralize he JTD of he hedged porfolio. 9

11 Bank Funding Srucure Source: Cii. Bank funding srucure for a derivaive conrac. 10

12 In his Presenaion We shall: Derive a Maser Funding Equaion wih CVA, FVA and FVO Develop a new echnique based on he Fundamenal Invariance Principle for Funding and CVA Use he Invariance Principle o solve he Maser equaion for he Desk CVA and Treasury FVO Derive he relaionship beween FVA and FVO and show ha FVA doesn disappear Apply o Credi CVA and link Gap risk and CDS Opion Skews hrough Filraion enlargemen echniques. Analyze he relaionship beween Gap risk and Wrong-Way risk for CDSs. 11

13 Se-up We work on a probabiliy space (Ω,G,P), where we have a se of defaul imes (τ 1,...,τ n ), represening he defauls of a reference porfolio on n obigors (i.e., he names in he neing se). The oher names in our credi universe are represened as ( τn+1,...,τ n+m ). We denoe by τ c, he defaul ime of he counerpary, and we denoe by τ b, he defaul ime of he bank. Their recovery raes are ( R1,..,R n+m ), Rc and R b respecively. And heir defaul indicaors are denoed by D i = 1 {τ i }, Dc = 1 {τ c }, Db = 1 {τ b } respecively. The enlarged filraion {G } ha we work wih conains boh he defauls filraion {H } = ( ni=1 H i ) ( n+m ) i=n+1 Hi H b H c and he background filraion {F }. 12

14 Arbirage-Free Pricing Theory We define a generic derivaive conrac by is cumulaive dividend process C. In general, he dividend process is considered o be of he form C = A B, where A and B are bounded increasing adaped righ-coninuous lef-limi (cadlag) processes. Risk-Neural Pricing. The value of he derivaive securiy C wihou defaul risk and funding is given by: V (r) = E [ T p (r),s dc s where p,s (r) is he risk-free discoun facor wih mauriy s, a ime, p,s (r) exp( s r u du). Noe. The risk-free rae r is he (heoreical) money-marke accoun ha we use in he classic arbirage-free pricing heory., 13

15 Marke Discoun Facor Raes We will also have wo (marke) discoun facors: p (rc ),s exp( s ru cdu), where rc on he collaeral. is he CSA rae paid or received ( r F ) p,s exp ( s ru Fdu), where r F is he (unsecured) funding rae we ge from reasury. Example. For a Sandard CDS conrac, wih paymen schedule (T 0 = 0,T 1,...T N ), δ i is he accrual fracion beween T i and T i 1, and S is he running spread. The dividend process is given by C s = D k s T i 1 {Ti s} Sδ i( 1 D k Ti ). 14

16 Close-Ou Value Bilaeral CVA. We consider he bank s own defaul probabiliy and he possibiliy of he bank defauling before he counerpary. The loss incurred when he counerpary defauls afer he bank is referred o as Asse CVA. The benefi gained when he bank defauls before he counerpary is called Liabiliy CVA. Le τ f denoe he firs-o-defaul ime of he counerpary and he bank τ f = min(τ b,τ c ). Assumpion. We assume ha we do no have simulaneous join defauls. Close-Ou Value. Upon erminaion of he conrac, he Close-Ou Value will be denoed by χ (τc ). In general, he close-ou value of he conrac is considered o be he value of he conrac wihou counerpary risk a he ime of defaul χ (τc ) = V τ c. 15

17 Close Ou Amoun. The ISDA Marke Review of OTC Derivaive Bilaeral Collaeralizaion Pracices (2010) summarizes: Upon defaul close-ou, valuaions will in many circumsances reflec he replacemen cos of ransacions calculaed a he erminaing pary s bid or offer side of he marke, and will ofen ake ino accoun he credi-worhiness of he erminaing pary. However, i should be noed ha exposure is calculaed a mid-marke levels so as no o penalize one pary or he oher.

18 ISDA Documenaion (2009) The relevan secions in he ISDA Close-ou Amoun Proocol (2009) are highlighed: Close-ou Amoun means, wih respec o each Terminaed Transacion and a Deermining Pary, he amoun of he losses or coss of he Deermining Pary... in replacing, or in providing for he Deermining Pary he economic equivalen of,... When considering informaion described in clause (i), (ii) or (iii), he Deermining Pary may include coss of funding, o he exen coss of funding are no and would no be a componen of he oher informaion being uilised... he Deermining Pary may in addiion consider in calculaing a Close-ou Amoun any loss or cos incurred in connecion wih is erminaing, liquidaing or re-esablishing any hedge relaed o a Terminaed Transacion... for he purpose of deermining a Close-ou Amoun, he Deermining Pary will: if obaining quoaions from one or more hird paries, ask each hird pary (A) no o ake accoun of he curren crediworhiness of he Deermining Pary or any exising Credi Suppor Documen and (B) o provide mid-marke quoaions 16

19 ISDA Documenaion (2002) The relevan secions in he ISDA 2002 Maser Agreemen are highlighed: he Calculaion Agen will deermine he Cash Selemen Amoun on he basis of quoaions (eiher firm or indicaive) for a replacemen ransacion supplied by Cash Selemen Reference Banks (bu he Calculaion Agen may no ake ino accoun any loss or cos incurred by a pary in connecion wih is erminaing, liquidaing or re-esablishing any hedge relaed o he Relevan Swap Transacion (or any gain resuling from any of hem)). 17

20 Collaeral and Funding We denoe by M he value of he margin (or collaeral) posed a ime : M 0 means ha we are receiving margin, M 0 means ha we are posing margin. We denoe by F he value of he funding from reasury a ime : F 0 means ha we are receiving funding, i.e., borrowing from reasury and paying ineres (his corresponds o V 0 where we pay he premium), F 0 means ha we are providing funding, i.e., lending o reasury and receiving ineres (his corresponds o V 0 where we receive he premium). 18

21 FVA, DVA and FVO We derive a general formula ha combines he effec of (inernal) reasury funding, (exogenous) CSA funding, and counerpary defaul risk. The formula is generic and does no depend on he ype of CSA ha we have in place or he deails of he margining agreemen. Funding Equaion. We sar wih he defaul-free case where we consider all he cash-flows of he ransacion including he CSA and he Funding swap. This is he case considered in Pierbarg (2010) in a Black-Scholes PDE framework, which is limied o diffusions and Brownian moion filraions. We do his in a general (probabilisic) se-up ha includes jumps and larger filraions needed for credi producs. We include boh he CSA Funding and Treasury Funding. 19

22 Collaeral and Funding Accouns CSA Funding. When we are funded via he CSA agreemen we should include he ineres paid or received on he collaeral accoun, which will change he value of he conrac wihou defaul risk and will have an impac on he CVA/DVA formulas. The value of he collaeral accoun beween he valuaion dae and he mauriy T, is given by he sum of he variaion margins and he ineres paid on he accoun: where we have A C = M +E [ T p (r),s dαc s, dα C = dm r c M d, By inegraion by pars, we have M + T M T = 0, a rade mauriy. p (r),s dm s = p (r),t M T T M s dp (r),s = T r s M s p (r),s ds. 20

23 Hence A C = E [ T (r s rs)m c s p,s (r) ds. Treasury Funding. When we are funded via Treasury we need o accoun for he ineres charge or benefi on he (cash) funding accoun. Using a similar reasonning for he funding accoun, we have dα F = df r F F d, F T = 0, a rade mauriy. F = V M, for all < T. which leads o: A F = E [ T ( rs r F s ) Fs p (r),s ds.

24 The Funding Equaion We need o value he package which conains he derivaive conrac, he CSA agreemen and he Funding (swap) wih Treasury. Proposiion 1 (The Funding Equaion). The value of he defaulfree rade wih funding from he CSA and Treasury will be: V = E [ T p (r),s dc s + T p (r),s (r s r c s)m s ds+ T where he funding accoun is given by F = V M. p (r),s (r s r F s ) Fs ds, 21

25 Equivalen Represenaions The funding equaion can also be re-wrien in oher useful forms: eiher using CSA discouning: V = E [ T p (rc ),s dc s + T p (rc ),s (r c s rf s ) Fs ds, or using (unsecured) funding discouning: V = E [ T ( r F ) p,s dc s + T ( r F ) p,s (rs F s) rc Ms ds. Boh represenaions have been used in Pierbarg (2010). 22

26 The Invariance Principle for Funding Funding equaion invariance principle saes ha we can discoun he cash-flows of he rade (inclusive of CSA and Treasury funding) wih any rae ha we choose. Proposiion 2 (Funding Invariance Principle). Le ( r 0 be any (posiive) ineres rae process, hen he funding equaion can be re-wrien equivalenly using he discouning wih ( r) -process, i.e., he value of he (defaul-free) rade wih CSA and Treasury funding is: V = E [ T p (r ),s dc s + T p (r ),s (r s r c s) Ms ds+ T where he funding accoun is given by F = V M. ) p (r ),s (r s r F s ) Fs ds, Proof. Define he ne cash-flow sream (of he hree conracs) (r) d C s dc s +(r s rs)m c s ds+ ( r s rs F ) Fs ds. 23

27 Then, we can wrie he Funding equaion as Define he process A (r) : V = E [ T A (r) p (r) 0, V + p,s (r) (r) d C s 0 p(r) 0,s. d C (r) s. Using he Funding equaion, we can show ha A (r) A (r) = p (r) 0, V + = p (r) 0, E = E [ T 0 p(r) 0,s [ T p,s (r) 0 p(r) 0,s d C (r) s d C (r) s + [ (r) d C = E s 0 p(r) 0,s A (r) T. is a maringale: d C (r) On he oher hand, differeniaing we ge he following SDE: da (r) = p (r) [ (r) 0, r V d+dv +d C. s

28 Similarly, if we inroduce he process A (r ), we have [ da (r ) = p (r ) 0, r V d+dv +d C (r ) Observing ha d C (r ) = dc s + ( r rc ) M d+ ( r rf (r) = d C + ( r r ) M d+ ( r r ) F d, we obain, for all < T, da (r ) = p (r ) 0, = p (r ) 0, = p (r ) 0, = p (r ) 0, [ [ r V d+dv +d C (r ). ) F d r V d+ ( r r) V +dv + + ( r r ) (M +F )d [ (r r ) [V (M +F )d r V d+dv +d ( r r ) = p(r ) 0, p (r) da (r). 0, V M F [ d C (r) } {{ } d+ da(r) =0 p (r) 0, C (r)

29 Hence, he process A (r ) is also a maringale, which gives he resul (recall ha V T is he pos-dividend price): A (r ) = E [ A (r ) T = E [ T 0 p(r ) 0,s d C (r ) s. Noe. In a deerminisic seing, we can ge he inuiion behind he maringale based proof by solving he deerminisic differenial equaion insead. Corollary. Seing r = rc or r = rf gives he equivalen funding equaions wih CSA discouning or (unsecured) funding discouning.

30 The Maser Equaion wih Funding and CVA Now, we add defaul risk o he equaion and we include all he cashflows from he rade, CSA funding, Treasury funding, and he defaul erms. Proposiion 3 (The Maser Equaion). The value of he risky rade will cover all he cashflows from he rade, he financing from he CSA and Treasury, and recovery paymens a he ime of defaul: V = E [ T +E [ T 1 {τf >s} p(r),s dc s 1 {τf >s} p(r),s (r s r c s)m s ds+ [ ( +E 1 {τf T} p(r),τ M τf f +1 {τf T} p(r),τ f T 1 {τf >s} p(r),s (r s r F s ) F s ds F τ R f F τ )+1 {τf f T} p(r),τ V R f τ f, where he funding accoun is given by F = V M, for all < τ f. 24

31 The recovery payoff (of he swap) will be afer neing wih he margin from he collaeral accoun ) + ) ) (R c (V τf M τ + (V τf M f τ f V R τ f = 1 {τf =τ c } +1 {τf =τ b }( ( V τf M τ f ) + +Rb (V τf M τ f ) ) where we use he noaions x + = max(x,0) and x = min(x,0)., The recovery payoff of he funding swap from reasury (if reallocaed back o he desk) F R τ f = F τ f + F + τ f 1 {τf =τ c } +RF b F + τ f 1 {τf =τ b }. Noe. The conrac V pays he cashflows dc s ; he CSA agreemen pays financing and exchanges he margin M τf a he ime of defaul;

32 he reasury funding pays financing and exchanges he funding accoun balance F τf a he ime of defaul. If he DVA of he Treasury funding posiion is re-allocaed back o he desk, he benefi from he funding leg loss is ξ F,b (τ f ) = F R τ f F τ f = ( 1 R F b ) F + τ f 1 {τf =τ b }. The recovery leg on he funding swap is he DVA of he funding leg. I is accouned for in he FVO for Deb CVA. This is referred o in Hull and Whie (2012) as DVA 2. The DVA of he funding leg should be on he aggregae posiions across all neing ses and all counerparies: F = k F k = k ( V k M k ). F R τ f = F τ f + k F + τ f 1 {τf =τ k } +RF b F + τ f 1 {τf =τ b }.

33 Collaeral and Funding Accouns wih Defaul Discree Se-up. On a discree ime grid { i }, he change in he funding accoun (or he value of each funding posiion) is given by A F i A F i 1 = F i 1 p (r), i 1 1 {τf > i 1 } ( 1+r F i 1 i ) F i 1 p (r), i 1 {τf > i } p (r),τ f F R τ f 1 {i 1 <τ f i }. Summing up all he individual funding posiions, we ge A F N = n i=1 n i=1 r F i 1 F i 1 i p (r), i 1 {τf > i } + n, i 1 F R τ f F i 1 p (r) p (r),τ f p (r) i=1,τ f 1 {i 1 <τ f i }. F i 1 p (r), i 1 {τf > i } p (r), i 1 p (r), i 1 25

34 Similarly, we have for he collaeral accoun A C N = n i=1 n i=1 r c i 1 M i 1 i p (r), i 1 {τf > i } + n, i 1 M τf M i 1 p (r) p (r),τ f p (r) i=1,τ f 1 {i 1 <τ f i }. M i 1 p (r), i 1 {τf > i } p (r), i 1 p (r), i 1

35 The Invariance Principle for Funding and CVA The invariance principle holds as well for he maser equaion wih CSA funding, Treasury funding and CVA. Proposiion 4 (Funding Invariance Principle). Le ( r 0 be any (posiive) ineres rae process, hen he maser funding equaion wih CVA can be re-wrien equivalenly using he discouning wih he ( r) -process, i.e., he value of he risky rade wih CSA and Treasury funding is: V = E [ T +E [ T 1 {τf >s} p(r ),s dc s 1 {τf >s} p(r ),s (r s rc s) Ms ds+ [ ( +E 1 {τf T} p(r ),τ M τf f +1 {τf T} p(r ),τ f T ) 1 {τf >s} p(r ),s (r s rf s ) F s ds F τ R f F τ )+1 {τf f T} p(r ),τ V R f τ f, where he funding accoun is given by F = V M, for all < τ f. 26

36 Proof. Define he ne payoff a defaul (of he hree conracs) Θ τf M τf + ( F R τ f F τ f )+ V R τ f, and he ne cash-flow sream (of he hree conracs) (r) d C s dc s +(r s rs)m c s ds+ ( r s rs F ) F s ds. Then, we have V = E [ T Define he process Â(r) : Â (r) p (r) 0, V + 1 {τf >s} p(r) (r),s d C s 1 0 {τ f >s} p(r) 0,s +E [ T d C (r) s + p,s (r) Θ s dd ( τ f ) s 0 p(r) 0,s Θ s dd ( τ f ) s ds Using he Maser funding equaion, we can show ha Â(r) maringale: Â (r) = E [ T 1 0 {τ f >s} p(r) (r) 0,sd C s + T 0 p(r) 0,s Θ s dd ( τ f ) s. is a

37 Similarly, we have for he process Â(r ) : dâ(r ) = r p(r ) 0, V +p (r ) 0, d V +1 {τf >} p(r ) 0, d C (r ) +p (r ) 0, Θ dd ( τ f ), where we subsiue d C (r ) dâ(r ) = p (r ) 0, ( r r ) = p(r ) 0, p (r) dâ(r). 0, Hence, he process Â(r ) wih d C (r) 1 {τf >}, and we obain, for all < τ f, V } M {{ F } d+ dâ(r) =0 p (r) 0, is also a maringale, which gives he resul.

38 The Bank s Balance Shee We need o separae he bank s balance shee ino he Desk balance shee and Treasury s balance shee, and allocae each erm in he (bank) oal economic value o he appropriae cos cenre. Bank Balance shee = Desk Balance shee + Treasury Balance shee. The full economic value of he derivaive and he funding posiion can be wrien as: V = V Desk + V Treasury. 27

39 The Desk Balance Shee On he desk balance shee, we only consider he funding icke from Treasury. The exernal DVA of he funding leg would show up on Treasury balance shee in he FVO accouning of he issued deb. V Desk F = = E +E +E [ T 1 {τf >s} p(r),s dc s +E [ 1 {τf T} p(r) T 1 {τf >s} p(r),s (r s rs)m c s ds } {{ } CSA Funding T 1 {τf >s} p(r),s (r s rs F ) F s ds } {{ } Treasury Funding V Desk M. τ f,τ M τf f +1 {τf T} p(r),τ V R f Treasury funds he base PV (or he unsecured porion of he base PV), he CVA P&L and he DVA P&L. 28

40 The Treasury Balance Shee On he Treasury balance shee, we will have: he back-o-back funding icke wih he desk, he exernal (marke) funding, and he FVO Deb CVA from he issued deb. Treasury s funding posiion wih FVO DVA is V Treasury = E T 1 {τf >s} p(r),s (r s rs F ) F s ds } {{ } Inernal Funding Ticke [ T +E 1 {τf >s} p(r),s (r s rs F ) ( F s ds+1 {τf T} p(r),τ F R f τ f F ) τ f } {{ } Exernal Deb Issuance. 29

41 CVA wih Funding and Margining To solve he CVA equaion wih Funding, we proceed in wo seps: Firs, we solve he pricing equaion wih CSA and Treasury funding wihou defaul-risk, i.e., we find he funded PV of he conrac, Second, we plug he funded PV ino he CVA equaion o derive he addiional CVA erms. We can separae he CVA and funding problem because: The value of he collaeral accoun M used for margining and he value of he swap recovery erm V τ R f are a funcion of he (funded) defaul-free PV. The Funding raes r F ha we pay and receive from reasury are symmeric. 30

42 Noe. Brigo e al. (2012) and Burgard and Kjaer (2012) consider asymmeric funding raes, which inroduce an implici dependence of he (base) PV funding charge on V, and he CVA funding charge on V. This level of generaliy is no necessary given he way banks raise long-erm funding. Banks need o pre-fund heir balance shees. They lock-in heir funding raes and warehouse he cash needed o finance he balance shee. For a given balance-shee size and mauriy profile, cash is raised in he deb markes a a fixed cos deermined a he ime of he deb issuance. The rading desks will be allocaed a porion of ha balance shee o use for heir rading needs. By neing asses and liabiliies, for each rading desk, we obain he ne balance shee usage ha hen ges charged a he locked-in issuance level. The only case where asymmeric funding raes may make sense is for shor-erm (rolling) funding raes.

43 Solving he CVA Funding Equaion Using our funding invariance principle for boh base PV and CVA, we can simplify he equaions by discouning wih r F, and dropping he funding adjusmen erms: 1. Funded (base) PV: V = E [ T ( r F ) p,s dc s + T ( r F ) p,s (r s F s) rc Ms ds, 2. Funded (margined) CVA: [ ( T V = E 1 {τf >s} p r F ) ( T,s dc s + 1 {τf >s} p r F ),s (rs F s) rc Ms ds [ ( +E 1 {τf T} p r F ) (,τ M τf f +1 {τf T} p r F ),τ V R f τ f. Noe. This works in he mos general case. Bu o proceed furher, we use he fac ha he funding raes (alhough hey can be sochasic) do no depend on he PV of rade. 31

44 Funded Margined CVA Proposiion 5 (Funded (margined) CVA) The value of he rade wih defaul risk, margining, CSA funding and (unsecured) Treasury funding, is given by V Desk = V CVA DVA, [ ( CVA = E 1 {τf T} 1 {τ f =τ c } p r F ) ( ) +,τ f (1 R c ) Vτf M τf, [ ( DVA = E 1 {τf T} 1 {τ f =τ b } p r F ) ( ),τ f (1 R b ) Vτf M τf, where he Defaul-Free Funded PV of he rade V is he soluion of he funding equaion [ ( T r F ) ( T r F ) V = E p,s dc s + p,s (r s F s) rc Ms ds. 32

45 ( r F ) ( r F r c) Proof. We denoe by V and V he defaul-free PV wih no CSA funding and he PV of he CSA funding cash-flow sream respecively ( r F ) [ ( T r F ) V E p,s dc s, V ( r F r c) E [ T ( r F ) p,s (rs F s) rc Ms ds The value of he funded defaul-free PV will be he sum of he wo erms: ( r F ) V = V +V ( r F r c). By inroducing he survival indicaor, we can re-wrie each one of he exinguishing erms as a funcion of he equivalen risk-free erm: [ ( T E 1 {τf >s} p r F ) ( r F ) [ ( r F ) ( r F ) = V E p E [ T 1 {τf >s} p ( r F ),s dc s,s (r F s rc s) Ms ds = V ( r F r c) E.,τ V f τ f [ ( r F ) p,τ f V 1 {τf T}, ( r F r c) 1 {τf T} τ f.

46 Summing up he wo exinguishing erms, we ge [ ( T E 1 {τf >s} p r F ) [ ( T +E 1 {τf >s} p r F ),s (rs F s) rc Ms ds ( r F ),s dc s = V E [p,τ V τf f 1 {τf T} Plugging ino he expression of he risky PV wih CSA funding gives he CVA and DVA formulas. Noe. To compue he defaul-free PV, for a fully collaeralized rade, we need o discoun he cash-flows wih he CSA rae, o compue he CVA value we need o discoun he CVA payoff wih he unsecured funding rae. The rade P&L is funded via he CSA, bu he CVA P&L is no and should herefore be discouned a he unsecured rae.

47 Close-Ou wih Funding Using he Close-ou amoun based on he 2009 ISDA documenaion, he CVA and DVA formulas should be based on V b τ f and V c τ f, respecively, i.e., he funded PV using eiher he bank s cos of funding r F,b s, or he counerpary s cos of funding r F,c s. Wih asymmeric Close-ou, we ge CVA = E [ 1 {τf T} 1 {τ f =τ c } p ( r F,b ),τ f (1 R c ) [ ( DVA = E 1 {τf T} 1 {τ f =τ b } p r F,b ),τ f (1 R b ) [ ( +E 1 {τf T} 1 {τ f =τ b } p r F,b ),τ f ( V b τf V c τ f ) ( ) V b + τf M τf, ( V c τf M τf ) where he funded PVs are given by he soluion of he funding, 33

48 equaions for each counerpary wih is own funding coss V b = E V c = E [ T [ T ( r F,b ) p,s dc s + ( r F,c ) p,s dc s + T T p p ( r F,b ),s ( r F,b ( r F,c ),s ( r F,c s r c s) Ms ds s r c s) Ms ds,. Noe. This will be more relevan for dealers and financial insiuions han for corporae end-users.

49 FVO Deb CVA Typically, FVO Deb CVA is compued for he Bank s Srucured Deb and is Primary Deb Issuance, bu Funding is charged on an Accrual Accouning basis. Gross DVA is defined as he PV of he bond cashflows wih oday s credi spreads (including liquidiy basis, i.e., Bond-CDS basis). The Ne DVA is he difference beween he Gross DVA a oday s spreads and he Gross DVA a issuance spreads. FVO is based on he Ne DVA. For bonds (deb issuance), we use a Recovery of Par assumpion. This is also consisen wih he recovery assumpion used for CDSs. Oher recovery assumpions are also possible: e.g., Recovery of Treasury or Recovery of Marke Value. 34

50 Deb Issuance Spreads We use he following noaions for shor raes and forward raes: p,s (r) ( s ) [ ( s ) = exp f,u du = E exp r u du, Q,s (b) ( s ) ( s ) = exp λ b,u du = E [exp λ b u du. We issue deb wih mauriy T 0, a par, wih a coupon ( r F 0,) 1 = 0 T. We have a he ime of deb issuance 0 T 0 p (r) 0,s Q(b) 0,s rf 0,s ds+p(r) 0,T Q(b) 0,T +RF b Inegraing by pars, we ge: which gives 1 = 0 = T T T 0 p (r) 0,s Q(b) 0,s λb 0,s ds. p (r) 0,s Q(b) 0,s (f 0,s +λ b ) (r) 0,s ds+p 0,T Q(b) 0,T, 0 0 p (r) 0,s Q(b) 0,s (r F 0,s f 0,s ( 1 R F b ) λ b 0,s ) ds. 35

51 Hence, he par (break-even) issuance rae is given by r F 0,s = f 0,s + ( 1 R F b ) λ b 0,s.

52 Gross DVA The Gross DVA is he bond value wih oday s spreads: B = T = 1+ = 1+ p,s (r) Q(b) T T,s rf 0,s ds+p(r),t Q(b),T +RF b T p (r),s Q(b),s (r F 0,s f,s ( 1 R F b p (r),s Q(b),s (f 0,s f,s + ( 1 R F b p (r),s Q(b),s λb,s ds ) λ b,s ) ds )( )) λ b 0,s λb,s ds. I can also be re-wrien as a funcion of he new (break-even) funding rae: B = 1+ T p,s (r) Q(b) r F,s = f,s ( 1 R F b,s (r F 0,s,s) rf ds, ) λ b,s. 36

53 Ne DVA Similarly, we have oday s bond value wih he issuance spreads B = T = 1+ = 1+ p,s (r) Q(b) T T 0,s rf 0,s ds+p(r),t Q(b) 0,T +RF b p (r),s Q(b) 0,s (r F 0,s f,s ( 1 R F b p (r),s Q(b) 0,s (f 0,s f,s ) ds. Thus, he FVO Deb CVA is given by T p (r),s Q(b) ) λ b 0,s 0,s λb 0,s ds ) ds FVO = B B T = p (r),s Q(b),s (r F 0,s r,s F ) T ds p (r),s Q(b) 0,s (f 0,s f,s ) ds. 37

54 FVA vs FVO The funding leg (for a uni noional) of my balance shee is FVA = = T = Q (b),s p(r),s (f,s r F 0,s) ds T Q,s (b) p(r) T,s (f,s r F,s ) ds+ T Q (b),s p(r),s sf,s ds FVO + Q,s (b) p(r) T,s (r F,s rf 0,s) ds p (r),s Q(b) 0,s (f 0,s f,s ) ds. FVO Deb CVA does no exclude he credi risk from he valuaion, i merely marks he bond o marke. If we issue he bond a par, radiional accouning pre-fas 157, will keep he bond on he balance shee marked a Par. Wih he FVO adjusemen, i marks he bond back o marke. The change in he bond price is no solely due o credi spreads, i can also move because of ineres raes and liquidiy spread changes. 38

55 FVO Bond Mahs Using radiional bond mahs and a coninuous (implied) yield o conver he bond cash flows o price, we can wrie 1 = p (y) 0,s = exp We also have by inegraion [ T p (y) ) 0,s( r F 0,s ds 0 ( ) s 1 = [ T 0 y 0,udu 0 p (y) 0,s y 0,sds +p (y) 0,T, where. +p (y) 0,T, hence, he yield will be equal o he break-even (par) issuance rae y 0,s = r F 0,s. If he new bond price is B, hen he new implied yield will be B = [ T p (y),s ( ) r F 0,s ds +p (y),t, 39

56 and he FVO adjusmen is FVO = [ T p (y),s ( r F 0,s = B B = B 1. ) ds+p (y),t [ T p (y) ) 0,s( r F (y) 0,s ds+p 0,T If we are marking an FVO adjusmen, we need o change he funding rae from he lock-in funding rae o he curren funding rae, and charge he desk accordingly. In essence, his is equivalen o buying back he issued dae and locking-in he gain or he loss, hen re-issuing he same ousanding deb noional a he new prevailing issue levels. The new levels can move because of: a) base ineres raes, b) credi spreads, or c) liquidiy basis. The curren mehodology for compuing FVO includes all hree changes in he bond credi spreads. Bu Treasury also hedges he ineres rae risk on he issued deb, which would offse he change in he bond price due o ineres raes.

57 The choice of ineres rae curve o use is arbirary, i could be driven by he inres rae hedges. If we hedge wih swaps, hen i s Libor. If we hedge wih Treasuries, hen i s reasury raes. Anoher alernaive is o use Asse Swaps insead. The asse swap would be collaeralised, hence discouned a he CSA rae, bu he cashflows would be Libor + spread, in exchange for he bond cashflows.

58 CVA wih Gap Risk and Funding In his secion, we focus on pricing he unilaeral margined CVA wih Gap risk and Funding. We consider a neing se wih one CDS posiion o highligh some of key feaures relevan for credi. A large neing se of CDSs can be compressed ino a CDO wih a large number of names, and a similar approach can hen be applied o price he CVA on he CDO. The margined CVA is given by CVA 0 = (1 R c )E p0,τ c (V τc M τc ) + 1 {τc T}, where V is he defaul-free value of he rade wih funding. [ ( r F ) If we have a fully collaeralized rade, he cash-flows will be discouned a he CSA funding rae V = E [ T p (rc ),s dc s, 40

59 and he margin will be equal o he value of he derivaive prior o defaul M τc = V τc, where is he ime lag beween ime and he las margin dae before ime. The CVA will be given by he value of he jump in MTM a he ime of defaul CVA 0 = (1 R c )E = (1 R c ) [ p ( r F ) ( +1{τc 0,τ Vτc c V τc ) T} ( T p r F ) 0 0,τ c E[ (Vτc V τc ) + τc = P(τ c d). This is a series of clique (or rache) opion condiional on defaul. Their value is mainly driven by Gap risk. If we have a cure period, hen here will be an addiional impac of he forward volailiy on he value of his opion, bu he gap risk will sill be he dominan facor. To evaluae his opion, we need o compue he condiional forwards firs, hen overlay he vol-induced diffusion on op.

60 ) + = 1{τi >τ c } ) + = 1{τi >τ c } Forwards. Saring wih he forwards, we have ( Vτc V τc ( Vτc V τc ( ) Vτc V τc. The pricing boils down o compuing he jumps in he expeced loss V Di T = E [ D i T G, for each ime horizon V Di T τ c = V Di T τ c V Di T τ c = E τc [ D i T Eτc [ D i T, for all T τc,

61 CVA in Own Naural Filraion Single-Name Case. We sar wih he single-name case and show how he jump upon defaul is compued. We use he Dellacherie formula o derive he condiional expecaions before defaul and afer defaul. Own Filraion. The filraion here conains he defaul of he single name τ i and he counerpary τ c : G i = F H i Hc. Working in own filraion, we have [ ( E 1 D i T = P τi > T G i ) = 1 {τi >} 1 P(τ i > T,τ c > F ) {τ c >} P(τ i >,τ c > F ) +1 {τi >} 1 P(τ i > T F σ(τ c )) {τ c } P(τ i > F σ(τ c )). 41

62 CVA in he Enlarged Filraion Enlarged Filraion. We use a op-down approach and assume ha he {G }-filraion can be approximaed as G = F H c = G i σ ( n+m H i i=1 L ( i) ) = F H c Hi = G i σ( L ), n+m H j j=1 j i i.e., we only need o condiion on realizaions of he (macro) loss variable L (of all he oher names in our credi universe) and we do no need he (micro) informaion on he individual single-name defauls. 42

63 In his case, we will have E [ 1 D i T = P(τi > T G ) = 1 {τi >} 1 {τ c >}P ( τ i > T,τ c > F σ ( L P ( τ i >,τ c > F σ ( L )) +1 {τi >} 1 {τ c }P ( τ i > T F σ ( L ) σ(τc ) ) P ( τ i > F σ ( L ) σ(τc ) ). Noe. In he general case, o compue his for he whole porfolio, we work on he enlarged filraion and we use he op-down decomposiion above. The choice of he (macro) filraion condiioning should be general and should no depend on he names wihin he neing se. The pricing for each rade (or each name) in he neing se would depend on is own naural filraion and he (macro) condiioning filraion. The (macro) condiioning filraion defines he gap risk. If we are in a disressed sae of he economy, hen gap risk will be small as mos names would have already widened before defaul. If we are in a benign sae of he economy, hen gap risk will be large. ))

64 Pre-Defaul and Pos-Defaul Value We have he following resul. Lemma 6 (Pre and Pos Defaul Value Process) The value process is of he form E [ 1 D i T = 1{τc >} F1(,T,L ) +1{τc } F2(,T,L,τ c), where F 1(,T,L ) and F 2 (,T,L,τ c) are he value funcions predefaul and pos-defaul respecively F 1(,T,L ) F 2(,T,L,τ ) c = 1 {τi >}P ( τ i > T,τ c > F σ ( L P ( τ i >,τ c > F σ ( L )), = 1 {τi >}P ( τ i > T F σ ( L ) σ(τc ) ) P ( τ i > F σ ( L ) σ(τc ) ). )) Noe. In Elouerkhaoui (2012), we have used he filraion of he neing se, and a op-down approximaion of he filraion wih he loss variable of he neing se. 43

65 The new approach is a hybrid ha accouns for he micro defaul informaion of each name (or each rade) in he neing se, by condiioning on is own naural filraion, hen i is augmened wih a op-down approximaed loss variable ha would drive he gap risk. Hence, he PVs a he ime of defaul and prior o defaul will be given by E τc [ 1 D i T 1{τc =} = F2(,T,L,τ c) 1{τc =}, E τc [ 1 D i T 1{τc =} = F1(,T,L ) 1{τc =}.

66 Expecaion of he Jump a Defaul Proposiion 7 The expecaion of he jump a defaul is E [ V 1 Di T τ c τ c = = E [ 1 {τi >τ c } ( V 1 Di T τ c V 1 Di T τ c ) τ c = where he pos-defaul and pre-defaul expecaions are given by, 1. Pos-defaul E [ 1 {τi >τ c } V 1 Di T τ c τ c = = P(τ i > T τ c = ), 2. Pre-defaul E = x [ 1 {τi >τ c } V 1 Di T τ c τ c = P ( L = x,τ i > τ c = ) P ( τ i > T P ( τ i > L = x,τ c > ) L = x,τ c > ). 44

67 We need o compue he following objecs: 1. The CDS survival probabiliy condiional on defaul (wih WWR) P(τ i > T τ c = ), for all T, 2. The CDS survival probabiliy condiional on realizaion of L and survival P ( τ i > T L = x,τ c > ), for all T, 3. The join probabiliy of CDS survival and L defaul condiional on P ( L = x,τ i > τ c = ). To compue he marginal disribuions of he loss variable condiional on defaul or condiional on survival, we use he CDO-Squared correlaion model (see Elouerkhaoui (2012)).

68 Credi Opions Revisied The choice of filraion ha we work wih has a fundamenal impac on he value of CDS opions as well. Working in he CDS own naural filraion, we ge he sandard Black formula used in he marke. By enlarging he filraion, we have more informaion from he oher names in he credi universe, which in urn skews he forward disribuion of he CDS spread. This leads o faer ailed disribuions and an implied volailiy skew effec. We see a similar effec wih he margined CVA payoff, where he forward vol (disribuion) is driven by he condiioning macro loss variable. 45

69 CDS Opions in Own Naural Filraion Pricing Single Name CDS Opions. The Filraion in his case is G = F H i. We define he survival probabiliy and he condiional (forward) survival probabiliy of he reference eniy: Q i 0,T P(τ i > T), Q i,u,t P(τ > T F ) P(τ > U F ), which is used o define a sricly posiive a.s. (and {F }-adaped) forward annuiy and is associaed forward annuiy measure. The Break-even spread is a maringale under his measure, and we can use he sandard Black formula o compue his expecaion (e.g., see Brigo and Morini (2005) for echnical deails). 46

70 Proposiion 8 The price of a CDS opion in is own filraion G = F H i is given by O 0,,T = T j > p 0,Tj Q i 0,T j δ j Black ( S 0,,T,S 0 ),σ BS T, where he break-even spread and he annuiy are defined as S 0,,T = (1 R i) T p 0,s dq i 0,s T j >p 0,Tj Q i 0,T j δ j, A 0,,T = T j > p 0,Tj Q i 0,T j δ j.

71 CDS Opions in he Enlarged Filraion Enlarged Filraion. When we work on he enlarged filraion we have: G = F H i σ( L ). We use a op-down approach o approximae he defaul filraion wih he macro loss variable. Then, we ge a mixure of Black formulas by condiioning on realizaions of he macro loss variable. We define he Spo and Forward survival probabiliies of he reference eniy condiional on he (macro) loss variable: Q i,x,t P( τ i > T L = x),,u,t P ( τi > T F { L U = x}) P ( τ i > U F { L U =. x}) Q i,x Condiioning on {L = x}, he forward survival probabiliies define a (sricly posiive) condiional forward annuiy, which is used o derive a Black-Scholes price under he associaed forward annuiy measure. 47

72 Proposiion 9 The price of a CDS opion in he enlarged filraion G = F H i σ( L ) is given by a mixure of Black-Scholes prices: O 0,,T = x P( L dx) A x 0,,T Black( S0,,T x,s0,σbs x ) T, where he break-even spread and he annuiy are defined as S0,,T x = (1 R i) T p 0,s dq i,x,s T j >p 0,Tj Q i,x,,t δ j j A x 0,,T = p 0,Tj δ j Q i,x,t = p j 0,Tj δ j E [( 1 DT i ) j L = x. T j > T j >

73 CDS Opions Filraion Condiioning We analyze he impac of he Macro filraion condiioning on he implied Black volailiies. We se he Macro marke spread o S M = 500 bps, and he Marke correlaion o ρ M = 0.2. Source: Cii. Disribuion of he Marke condiioning variable L T. 48

74 CDS Opions Filraion Induced Volailiy Skew The Opion mauriy is 1 year. The CDS mauriy is 5 years. The CDS spread is 100 bps. The Base volailiy is 0.2. Source: Cii. Filraion condiioning implied volailiy skew. 49

75 Applicaions We analyze he impac of pos-defaul and pre-defaul correlaions on Credi CVA. Pos-defaul correlaion is driven by ρ c. Pre-defaul correlaion is driven by ρ c, ρ M and S M. The primary driver of pre and pos spread widening is correlaion. The condiioning Marke spreads have a milder impac. Valuaion dae is 18-Mar-13. The CDS rade mauriy is 20-Mar-16. Counerpary correlaion is se o ρ c = 0.5. Marke correlaion is ρ M = 0.5. Marke spread is S M = 500 bps. The counerpary spread is 200 bps. The reference CDS spread is 200 bps. SNAC Coupon is 100 bps. We have he following pre and pos value profiles. 50

76 Gap Risk vs Wrong-Way Risk Marke correlaion is se o ρ M = 0.5. Marke spread is se o S M = 500 bps. We vary he counerpary correlaion ρ c. Source: Cii. Pre and Pos-Defaul PV Profile for a counerpary correlaion of ρ c =

77 Wrong-Way Risk Correlaion As we change he counerpary correlaion ρ c boh he pre and pos PVs move. Source: Cii. Pre and Pos-Defaul PV Profile for a counerpary correlaion of ρ c =

78 Gap Risk Correlaion Counerpary correlaion is se o ρ c = 0.5. Marke spread is se o S M = 500 bps. We vary he Marke correlaion ρ M. Source: Cii. Pre and Pos-Defaul PV Profile as we change he Marke correlaion of ρ M. 53

79 Gap Risk Marke Spread Counerpary correlaion is se o ρ c = 0.5. Marke correlaion is se o ρ M = 0.5.We vary he Marke spread S M. Source: Cii. Pre and Pos-Defaul PV Profile as we change he Marke spread S M. 54

80 CVA Pricing We have he following resuls. 1. Marke correlaion is se o ρ M = 0.5. Marke spread is se o S M = 500 bps. We vary he counerpary correlaion ρ c. ρ c PV 26,625 26,625 26,625 26,625 NonMarginedPV 800 2,968 7,158 11,395 MarginedPV 0 1,090 3,737 7, Counerpary correlaion is se o ρ c = 0.5. Marke spread is se o S M = 500 bps. We vary he Marke correlaion ρ M. ρ M PV 26,625 26,625 26,625 26,625 NonMarginedPV 7,158 7,158 7,158 7,158 MarginedPV 6,646 5,788 3,737 1,004 55

81 3. Counerpary correlaion is se o ρ c = 0.5. Marke correlaion is se o ρ M = 0.5. We vary he Marke spread S M. S M PV 26,625 26,625 26,625 26,625 NonMarginedPV 7,158 7,158 7,158 7,158 MarginedPV 1,823 2,905 3,737 4,124

82 Conclusion We have addressed he problem of compuing CVA, FVA and FVO from a credi perspecive. We have developed a new echnique o solve he Maser Funding Equaion based on a Fundamenal (Funding) Invariance Principle. We have analyzed he relaionship beween FVA and FVO, and we have shown ha he FVA doesn disappear when you accoun for FVO as i is usually claimed in he lieraure. We have derived he CVA for a CDS by compuing he Gap risk in he Enlarged Filraion. And we have shown ha his is similar o a Filraion-Induced Volailiy Skew for CDS Opions. We have analyzed he impac of pre and pos defaul correlaions and marke condiioning on he Gap risk and Wrong-Way risk. 56

83 References D. Brigo, M. Morini (2005), CDS Marke Formulas and Models, Working Paper. D. Brigo, A. Pallavicini, D. Perini (2012), Funding, Collaeral and Hedging: Uncovering he Mechanics and he Subleies of Funding Valuaion Adjusemns, Working Paper. C. Burgard, M. Kjaer (2012), A Generalised CVA wih Funding and Collaeral via Semi-Replicaion, Working Paper. S. Crepey (2012a), Bilaeral Counerpary Risk Under Funding Consrains, Par I: Pricing, Working Paper. S. Crepey (2012b), Bilaeral Counerpary Risk Under Funding Consrains, Par II: CVA, Working Paper. S. Crepey (2012c), Counerpary Risk and Funding: The Four Wings of he TVA, Working Paper. C. Dellacherie (1972), Capacies e Processus Sochasiques, Springer-Verlag, Berlin, Deloie, Solum Parners (2013), Counerpary Risk and CVA Survey: Curren Marke Pracice Around Counerpary Risk Regulaion, CVA Managemen and Funding, February

84 Y. Elouerkhaoui (2012), From Funding o Gap Risk: A Consisen Mehodology for Credi CVA, ICBI Conference. Erns & Young (2012), Reflecing Credi and Funding Adjusmens in Fair Value, Spring M. Fujii, A. Takahashi (2010), Derivaives Pricing under Asymmeric and Imperfec Collaeralizaion and CVA, CARF Working Paper Series F-240. J. Hull, A. Whie (2012a), The FVA Debae, Risk, 25h Anniversary ediion, July J. Hull, A. Whie (2012b), The FVA Debae Coninues, Risk, Ocober S. Laughon, A. Vaisbro (2012), In Defence of FVA: a Response o Hull and Whie, Risk, Sepember V. Pierbarg (2010), Funding Beyond Discouning: Collaeral Agreemens and Derivaives Pricing, Risk, February 2010.

85 Sales and Trading Exernal Communicaions Disclaimer This communicaion has been prepared by individual sales and/or rading personnel of Ciigroup Global Markes Limied (CGML) or is subsidiaries or affiliaes (collecively Cii). CGML is auhorised and regulaed in he Unied Kingdom by he Financial Services Auhoriy (ogeher wih is successors, he UK Regulaor) and has is regisered office a Ciigroup Cenre, Canada Square, London E14 5LB. This communicaion is direced a persons (i) who have been or can be classified by Cii as eligible counerparies or professional cliens in line wih he rules of he UK Regulaor, (ii) who have professional experience in maers relaing o invesmens falling wihin Aricle 19(1) of he Financial Services and Markes Ac 2000 (Financial Promoion) Order 2005 and (iii) oher persons o whom i may oherwise lawfully be communicaed. No oher person should ac on he conens or access he producs or ransacions discussed in his communicaion. In paricular, his communicaion is no inended for reail cliens and Cii will no make such producs or ransacions available o reail cliens. The informaion conained herein may relae o maers ha are (i) no regulaed by he UK Regulaor and/or (ii) no subjec o he proecions of he Unied Kingdom s Financial Services and Markes Ac 2000 and/or he Unied Kingdom s Financial Services Compensaion Scheme. All maerial conained herein, including any proposed erms and condiions, is indicaive and for discussion purposes only, is subjec o change wihou noice, is sricly confidenial, may no be reproduced and is inended for your use only. I does no include a number of erms and condiions ha will be included in any acual ransacion and final erms and condiions are subjec o furher discussion and negoiaion nor does i purpor o idenify all risks (direc or indirec). This communicaion is no a commimen o deal in any produc, offer financing or ener ino any ransacion described herein. Cii is no acing as your agen, fiduciary or invesmen adviser and is no managing your accoun. The provision of informaion in his communicaion is no based on your individual circumsances and should no be relied upon as an assessmen of suiabiliy for you of a paricular produc or ransacion. I does no consiue invesmen advice and Cii makes no recommendaion as o he suiabiliy of any of he producs or ransacions menioned. Even if Cii possesses informaion as o your objecives in relaion o any ransacion, series of ransacions or rading sraegy, his will no be deemed sufficien for any assessmen of suiabiliy for you of any ransacion, series of ransacions or rading sraegy. Save in hose jurisdicions where i is no permissible o make such a saemen, we hereby inform you ha his communicaion should no be considered as a soliciaion or offer o sell or purchase any securiies, deal in any produc or ener ino any ransacion. You should make any rading or invesmen decisions in reliance on your own analysis and judgmen and/or ha of your independen advisors and no in reliance on Cii and any decision wheher or no o adop any sraegy or engage in any ransacion will no be Cii s responsibiliy. Cii does no provide invesmen, accouning, ax, financial or legal advice; such maers as well as he suiabiliy of a poenial ransacion or produc or invesmen should be discussed wih your independen advisors. Prior o dealing in any produc or enering ino any ransacion, you and he senior managemen in your organisaion should deermine, wihou reliance on Cii, (i) he economic risks or meris, as well as he legal, ax and accouning characerisics and consequences of dealing wih any produc or enering ino he ransacion (ii) ha you are able o assume hese risks, (iii) ha such produc or ransacion is appropriae for a person wih your experience, invesmen goals, financial resources or any oher relevan circumsance or consideraion. Where you are acing as an adviser or agen, you should evaluae his communicaion in ligh of he circumsances applicable o your principal and he scope of your auhoriy. The informaion in his communicaion, including any rade or sraegy ideas, is provided by individual sales and/or rading personnel of Cii and no by Cii s research deparmen and herefore he direcives on he independence of research do no apply. Any view expressed in his communicaion may represen he curren views and inerpreaions of he markes, producs or evens of such individual sales and/or rading personnel and may be differen from oher sales and/or rading personnel and may also differ from Cii s published research he views in his communicaion may be more shor erm in naure and liable o change more quickly han he views of Cii research deparmen which are generally more long erm. On he occasions where informaion provided includes exracs or summary maerial derived from research repors published by Cii s research deparmen, you are advised o obain and review he original piece of research o see he research analys s full analysis. Any prices used herein, unless oherwise specified, are indicaive. Alhough all informaion has been obained from, and is based upon sources believed o be reliable, i may be incomplee or condensed and is accuracy canno be guaraneed. Cii makes no represenaion or warrany, expressed or implied, as o he accuracy of he informaion, he reasonableness of any assumpions used in calculaing any illusraive performance informaion or he accuracy (mahemaical or oherwise) or validiy of such informaion. Any opinions aribued o Cii consiue Cii s judgmen as of he dae of he relevan maerial and are subjec o change wihou noice. Provision of informaion may cease a any ime wihou reason or noice being given. Commissions and oher coss relaing o any dealing in any producs or enering ino any ransacions referred o in his communicaion may no have been aken ino consideraion. Any scenario analysis or informaion generaed from a model is for illusraive purposes only. Where he communicaion conains forward-looking informaion, such informaion may include, bu is no limied o, projecions, forecass or esimaes of cashflows, yields or reurn, scenario analyses and proposed or expeced porfolio composiion. Any forward-looking informaion is based upon cerain assumpions abou fuure evens or condiions and is inended only o illusrae hypoheical resuls under hose assumpions (no all of which are specified herein or can be ascerained a his ime). I does no represen acual erminaion or unwind prices ha may be available o you or he acual performance of any producs and neiher does i presen all possible oucomes or describe all facors ha may affec he value of any applicable invesmen, produc or invesmen. Acual evens or condiions are unlikely o be consisen wih, and may differ significanly from, hose assumed. Illusraive performance resuls may be based on mahemaical models ha calculae hose resuls by using inpus ha are based on assumpions abou a variey of fuure condiions and evens and no all relevan evens or condiions may have been considered in developing such assumpions. Accordingly, acual resuls may vary and he variaions may be subsanial. The producs or ransacions idenified in any of he illusraive calculaions presened herein may herefore no perform as described and acual performance may differ, and may differ subsanially, from hose illusraed in his communicaion. When evaluaing any forward looking informaion you should undersand he assumpions used and, ogeher wih your independen advisors, consider wheher hey are appropriae for your purposes. You should also noe ha he models used in any analysis may be proprieary, making he resuls difficul or impossible for any hird pary o reproduce. This communicaion is no inended o predic any fuure evens. Pas performance is no indicaive of fuure performance. Cii shall have no liabiliy o he user or o hird paries, for he qualiy, accuracy, imeliness, coninued availabiliy or compleeness of any daa or calculaions conained and/or referred o in his communicaion nor for any special, direc, indirec, incidenal or consequenial loss or damage which may be susained because of he use of he informaion conained and/or referred o in his communicaion or oherwise arising in connecion wih he informaion conained and/or referred o in his communicaion, provided ha his exclusion of liabiliy shall no exclude or limi any liabiliy under any law or regulaion applicable o Cii ha may no be excluded or resriced. The ransacions and any producs described herein may be subjec o flucuaions of heir mark-o-marke price or value and such flucuaions may, depending on he ype of produc or securiy and he financial environmen, be subsanial. Where a produc or ransacion provides for paymens linked o or derived from prices or yields of, wihou limiaion, one or more securiies, oher insrumens, indices, raes, asses or foreign currencies, such provisions may resul in negaive flucuaions in he value of and amouns payable wih respec o such produc prior o or a redempion. You should consider he implicaions of such flucuaions wih your independen advisers. The producs or ransacions referred o in his communicaion may be subjec o he risk of loss of some or all of your invesmen, for insance (and he examples se ou below are no exhausive), as a resul of flucuaions in price or value of he produc or ransacion or a lack of liquidiy in he marke or he risk ha your counerpary or any guaranor fails o perform is obligaions or, if his he produc or ransacion is linked o he credi of one or more eniies, any change o he crediworhiness of he credi of any of hose eniies. Cii (wheher hrough he individual sales and/rading personnel involved in he preparaion or issuance of his communicaion or oherwise) may from ime o ime have long or shor principal posiions and/or acively rade, for is own accoun and hose of is cusomers, by making markes o is cliens, in producs idenical o or economically relaed o he producs or ransacions referred o in his communicaion. Cii may also underake hedging ransacions relaed o he iniiaion or erminaion of a produc or ransacion, ha may adversely affec he marke price, rae, index or oher marke facor(s) underlying he produc or ransacion and consequenly is value. Cii may have an invesmen banking or oher commercial relaionship wih and access o informaion from he issuer(s) of securiies, producs, or oher ineress underlying a produc or ransacion. Cii may also have poenial conflics of ineres due o he presen or fuure relaionships beween Cii and any asse underlying he produc or ransacion, any collaeral manager, any reference obligaions or any reference eniy. Any decision o purchase any produc or ener ino any ransacion referred o in his communicaion should be based upon he informaion conained in any associaed offering documen if one is available (including any risk facors or invesmen consideraions menioned herein) and/or he erms of any agreemen. Any securiies which are he subjec of his communicaion have no been and will no be regisered under he Unied Saes Securiies Ac of 1933 as amended (he Securiies Ac) or any Unied Saes securiies law, and may no be offered or sold wihin he Unied Saes or o, or for he accoun or benefi of, any US person, excep pursuan o an exempion from, or in a produc or ransacion, no subjec o, he regisraion requiremens of he Securiies Ac. This communicaion is no inended for disribuion o, or o be used by, any person or eniy in any jurisdicion or counry which disribuion or use would be conrary o law or regulaion. This communicaion conains daa compilaions, wriings and informaion ha are confidenial and proprieary o Cii and proeced under copyrigh and oher inellecual propery laws, and may no be reproduced, disribued or oherwise ransmied by you o any oher person for any purpose unless Cii s prior wrien consen have been obained. IRS Circular 230 Disclosure: Ciigroup Inc. and is affiliaes do no provide ax or legal advice. Any discussion of ax maers in hese maerials (i) is no inended or wrien o be used, and canno be used or relied upon, by you for he purpose of avoiding any ax penalies and (ii) may have been wrien in connecion wih he promoion or markeing of a ransacion (if relevan) conemplaed in hese maerials. Accordingly, you should seek advice based your paricular circumsances from an independen ax advisor. Alhough CGML is affiliaed wih Ciibank, N.A. (ogeher wih Ciibank N.A. s subsidiaries and branches worldwide, Ciibank), you should be aware ha none of he producs menioned in his communicaion (unless expressly saed oherwise) are (i) insured by he Federal Deposi Insurance Corporaion or any oher governmenal auhoriy, or (ii) deposis or oher obligaions of, or guaraneed by, Ciibank or any oher insured deposiory insiuion Ciigroup Global Markes Limied. Cii, Cii and Arc Design are rademarks and service marks of Ciigroup Inc. or is affiliaes and are used and regisered hroughou he world.