BBehavioural effects on XVA
Chris Kenyon and Hayato Iida ∗
09 Mar 2018Version 1.00
Abstract
Bank behaviour is important for pricing XVA because it links different counterpartiesand thus breaks the usual XVA pricing assumption of counterparty independence. Considera typical case of a bank hedging a client trade via a CCP. On client default the hedge (ef-fects) will be removed (rebalanced). On the other hand, if the hedge counterparty defaultsthe hedge will be replaced. Thus if the hedge required initial margin then the default prob-ability driving MVA is from the client not from the hedge counterparty. This is the oppositeof usual assumptions where counterparty XVAs are computed independent of each other.Replacement of the hedge counterparty means multiple CVA costs on the hedge side needinclusion. Since hedge trades are generally at riskless mid (or worse) these costs are paidon the client side, and must be calculated before the replacement hedge counterparties areknown. We call these counterparties anonymous counterparties. The effects on CVA andMVA will generally be exclusive because MVA largely removes CVA, and CVA is hardlyrelevant for CCPs. Effects on KVA and FVA will resemble those on MVA. We provide atheoretical framework, including anonymous counterparties, and numerical examples. Pric-ing XVA by considering counterparties in isolation is inadequate and behaviour must betaken into account.
Valuation adjustments today are typically computed without regard for behaviour, consideringeach counterparty in isolation (Burgard and Kjaer 2013; Green, Kenyon, and Dennis 2014; Greenand Kenyon 2015; Kenyon and Green 2015). In practice this is not the case, events with respect toone counterparty influence behaviour with respect to other counterparties. We consider a typicalexample of a bank hedging a client trade and show that XVA prices can be significantly differentwhen behaviour is taken into account: both higher and lower. In terms of Accounting, since weprice assuming behaviour that is typical of market participants, this fulfulls the requirements ofIFRS 13. ∗ Contacts: [email protected], [email protected]. This paper is a personal viewand does not represent the views of MUFG Securities EMA plc (MUSE). This paper is not advice. Certain infor-mation contained in this presentation has been obtained or derived from third party sources and such informationis believed to be correct and reliable but has not been independently verified. Furthermore the information maynot be current due to, among other things, changes in the financial markets or economic environment. No obliga-tion is accepted to update any such information contained in this presentation. MUSE shall not be liable in anymanner whatsoever for any consequences or loss (including but not limited to any direct, indirect or consequentialloss, loss of profits and damages) arising from any reliance on or usage of this presentation and accepts no legalresponsibility to any party who directly or indirectly receives this material. a r X i v : . [ q -f i n . P R ] M a r ank Street Client
Bank Street Client Bank Street
Bank
Street Client Bank Replace
Client
Bank Replace Client Bank Replace Client
Figure 1: Example of behaviour following either the default of a client, or the default of a hedgecounterparty. A hedge is closed (rebalanced) following Client default whereas a hedge is replacedfollowing hedge counterparty default. If the hedge counterparty again defaults then it will inturn be replaced if the client trade is still active.
Here we consider a typical example of bank (B) business, providing a trade to a client (C) andbuying a hedge from the street (H), see Figure 1. The hedge with the street will be collateralizedand the trade with the client may or may not be collateralized.On default of a bank counterparty we observe different behaviour depending on whetherthe hedge counterparty defaults or the client. If the client defaults then the hedge is typicallyclosed (Figure 1, RHS). If the hedge counterparty defaults then the hedge is replaced because theclient trade is still alive and the client typically has no interest in closing its trade as it has aneconomic purpose. This is a simplification as a hedge may be spread over several counterparties.But whenever a hedge counterparty defaults then the remaining hedges will be rebalanced.Considering both sides of the bank’s activity and the bank’s behaviour radically changes XVAprices. In this paper we ignore the possibility of changing the hedger dynamically to optimizepricing and XVA. That is tackled elsewhere (Kenyon and Green, Global Derivatives 2016).Qualitatively we can explain the effect on CVA as follows. If H defaults then B has noloss because B has calculated the CVA and charged it to C. The problem is that C has notdefaulted so B must replace the hedge, and this new hedge can default again. Typically thehedge trades are at mid so B cannot charge the CVA of H to H, contradicting the assumptionof independent counterparties that is usually made. The problem is that the CVA desk may notcharge the originating desk for multiple defaults of H and so suffer losses with respect to highlyrated clients. The source of loss here for the CVA desk is not the client side but the hedge. Inaddition, if the hedge is spread out over several counterparties (with equal default probabilities,and no connection between them) then the expected loss may be increased because there is no2etting across counterparties. However the hedge may net several different clients so the themulti-counterparty-hedge netting effect can be in either direction. The only way to be sure ofeffects is to calculate them. We do not consider any multi-counterparty-hedge netting effect herefor simplicity and because weexpect it to be a small fraction of the multi-CVA effect.When the hedge-side counterparty is replaced, the choice of new hedge counterparty can onlybe done at that time because any prior choice may itself have defaulted. Thus we must priceinto the multi-CVA effect counterparties that are not known at t = 0. We call these anonymouscounterparties and show that they can be handled relatively simply since we do not care aboutthe identity of the future counterparty but only their riskiness.Considering MVA we generally have the opposite effect dominating. That is, the default ofthe C means that B closes the hedge H so the default probablity driving the MVA cost is fromthe client side not from the hedge side. Since hedge counterparties are generally banks, they willoften have a lower hazard rate than clients. Thus the actual MVA is generally lower than whatwould be expected from looking at the counterparty in isolation.The effects on MVA and CVA will generally be exclusive because IM largely removes CVA,and the CVA is low in any case for collateralized counterparties. Hedge counterparties that donot require IM, e.g. because they are smaller, are attractive because of they do not require IMand so provide cheaper hedges.Effects on KVA and FVA will resemble those on MVA. Portfolio rebalancing due to clientdefault will impact KVA and FVA in a similar way to MVA as they are largely exposure profilebased. We consider two behavioural cases: firstly a client portfolio hedged with a (riskless) centralcounterparty (CCP) where the hedge is removed on client default; and secondly a client portfoliohedged with a sequence of defaultable counterparties up to client default. Given that interestrate portfolios are typical in these settings we model stochastic interest rates. Our focus is thechange in MVA and CVA from the bank’s behaviour.We assume that the default of the bank, client and counterparties are independent for sim-plicity. Our development extends (Burgard and Kjaer 2013; Green, Kenyon, and Dennis 2014;Green and Kenyon 2015). We price in the risk-neutral measure for simplicity despite the factthat most credit risk is difficult to hedge, see (Kenyon and Green 2015) on how to include othermeasures. The key technical innovations are 1) including the behaviour of the bank w.r.t. clientand hedge default thus breaking the usually-assume independence of counterparties; 2) intro-ducing anonymous counterparties in t = 0 XVA pricing. Anonymous counterparties are futurehedging counterparties decided in the future according to criteria at t = 0. Although unknownat t = 0 their presence can be included with some assumptions on the future state of the market,i.e. creditworthiness of future hedge counterparties (i.e. CDS spreads) conditional on default ofprevious hedge counterparties.Notation is given in Table 1. We assume that the CCP is riskless for simplicity in this section. The next section introducesrisky hedge counterparties. We separate the pricing problem of the adjusted client plus hedgecost (cid:98) V C+H into the two sides linked by behaviour transmitted via client default. (cid:98) V C+H := (cid:98) V C + (cid:98) V H (1)3 ymbol(s) Meaning V, (cid:98) V CCP , V
CCP riskless price, adjusted CCP price, CCP price (cid:98) V C+H , (cid:98) V C , (cid:98) V H adjusted price total, client-side, and hedge-side (cid:98) V to client adjusted price charged to client P r , P C , P B , P i bond prices: riskless, client, bank, anonymous r, r C , r B , r i risk-neutral bond drifts: riskless, client, bank, anonymous r I ; x,y interest rate on IM posted by x to yr X interest rate on VM collateral P σ , σ C , σ B , σ i bond volatilities: riskless, client, bank, anonymous R C , R B , R i recovery rates: client, bank, anonymous J C , J B , J { i } default (Poisson) processes: client, bank, anonymous dW Brownian motion process∆ ∗ differential w.r.t. jump ∗ q C , q i repo rate on client, anonymous, bonds λ C , λ B , λ i hazard rates for: client, bank, anonymous X H variation margin collateral on hedge-side I x,y initial margin posted by x to yα C , α B , α i hedge-portfolio bond holdings: client, bank, anonymous φ , K use of capital for funding and capital on hedge-side γ K cost of capital (rate) β ∗ cash accountsΠ H hedging portfolio to replicate hedge-side (cid:15) H hedging error on bank default on hedge-side g C , g B , g i closeout on client, bank, and anonymous counterparty default U H = (cid:98) V H − V H adjustments value on hedge-side s ∗ = r ∗ − r spread over riskless rate for ∗ τ { i } default time of counterparty iD ∗ discount factor using discounting ∗ Q rate matrix for counterparty-sequence state transitionTable 1: Notation.4here C stands for client and H for hedge. Note that this is not the separation of (client) pricingas (cid:98) V to client = (cid:98) V C + (cid:98) V H − V (2)because the CCP pricing is riskless, i.e. (cid:98) V CCP = V CCP = V (cid:54) = (cid:98) V H , i.e. there is no priceadjustment w.r.t. the CCP. The technical development for (cid:98) V C is standard so we only providethe development for (cid:98) V H to highlight the changes.The assets available have the following dynamics, where we assume that the bank has onlya single class of bonds available for funding for realism (i.e. we assume trading does not driveissuance). No CCP (hedge H ) bonds are included as the CCP is assumed riskless, but Client, C ,bonds are included for the hedge-side price replication as they signal when to close the hedge. dP r /P r = rdt + σ r dW (3) dP C /P C = r C dt + σ C dW − (1 − R C ) dJ C (4) dP B /P B = r B dt + σ r dW − (1 − R B ) dJ B (5)We assume zero bond-cds and bond-repo basis so, r C − q C = (1 − R C ) λ C (6) r B − r = (1 − R B ) λ B (7)Initial margin (IM) received cannot be rehypothecated (and CCPs do not post IM) so theself-financing condition only includes IM posted by the bank I x,y where x, y indicates the posterand counterparty respectively: (cid:98) V H − X H + I B,H + α B P B − φK = 0 (8)and the cash account changes (prior to rebalancing, but we omit the customary tildes for sim-plicity) are dβ r = − δrP r dt, dβ C = − α C q C P C dt, dβ K = − γ K Kdt (9) dβ X = − r X Xdt, dβ I ; x,y = ( r I ; x,y I x,y − r I ; y,x I y,x ) dt x, y ∈ { B, H } (10)Using Itˆo’s lemma we obtain the differential of value of the derivative portfolio towards the CCPand of the replicating portfolio Π H : d (cid:98) V H + d Π H = ∂ (cid:98) V H ∂t dt + 12 σ r P r ∂ (cid:98) V H ∂P r dt + ∂ (cid:98) V H ∂P r dP r + ∆ B (cid:98) V H dJ B + ∆ C (cid:98) V H dJ C (11)+ δdP r + α B dP B + α C dP C − α C q C P C dt (cid:124) (cid:123)(cid:122) (cid:125) dβ C − δrP r dt (cid:124) (cid:123)(cid:122) (cid:125) dβ r − r X Xdt (cid:124) (cid:123)(cid:122) (cid:125) dβ X − γ K Kdt (cid:124) (cid:123)(cid:122) (cid:125) dβ K − ( r I ; B,H I B,H − r I ; H,B I H,B ) dt (cid:124) (cid:123)(cid:122) (cid:125) dβ I ; B,H
Since this CCP hedge is cancelled on client default (e.g. by trading the opposite position then5ompressing), all three entities,
B, H, C appear in Equation 11. So, d (cid:98) V H + d Π H = (cid:34) ∂ (cid:98) V H ∂t + 12 σ r P r ∂ (cid:98) V H ∂P r + ∂ (cid:98) V H ∂P r rP r − δrP r + α B r B P B + α C r C P C − α C q C P C + δrP r − r X X − γ K K − r I ; B,H I B,H (cid:35) dt + (cid:34) ∂ (cid:98) V H ∂P r σ r P r + δσ r P r + α B σ B P B + α C σ C P C (cid:35) dW + (cid:104) ∆ B (cid:98) V H − α B (1 − R B ) P B (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) (cid:15) H dJ B + (cid:104) ∆ C (cid:98) V H − α C (1 − R C ) P C (cid:105) dJ C Setting the dW, dJ C brackets to zero resolves the values of α C in the dt bracket (there is nonet δ term in the dt bracket). The self-financing condition, Equation 8 sets the value of α B sothere is no degree of freedom to remove the hedge-side mismatch (cid:15) H on bank-default (nor on theclient side). Using the self-financing condition for α B r B P B we obtain for the dt bracket, settingcollateral received from the CCP to zero (CCPs do not post collateral):0 = ∂ (cid:98) V H ∂t + 12 σ r P r ∂ (cid:98) V H ∂P r + ∂ (cid:98) V H ∂P r rP r + r B ( − (cid:98) V H + X H − I B,H + φK )+ λ C (1 − R C ) P C − r X X − γ K K − r I ; B,H I B,H hence ∂ (cid:98) V H ∂t + 12 σ r P r ∂ (cid:98) V H ∂P r + ∂ (cid:98) V H ∂P r rP r − (cid:98) V H ( r B + λ C ) = ( r X − r B ) X + ( r I − r B ) I B,H − λ C g C + ( γ K − r B φ ) K Now when the client defaults, both the CCP and the bank are non-defaulting so the clientcloseout with the CCP is at the riskless (CCP) price so g C = V H and V H satisfies the Black-Scholes equation which we can subtract via the anzatz (cid:98) V H = V H + U H giving ∂U H ∂t + 12 σ r P r ∂ U H ∂P r + ∂U H ∂P r rP r − U H ( r B + λ C ) = s B ( V H − X H − I H ) + s X ; H X H + s I ; B,H I B,H + ( γ K − r B φ ) K This gives expressions for the valuation adjustments on the hedge (H, i.e. CCP) side as below6sing Feynman-Kac. CVA
CCP = 0 (12)FVA
CCP = − (cid:90) T E [ s B D r B + λ C ( V H − X H )] dt (13)ColVA CCP = − (cid:90) T E [ s X D r B + λ C X H ] dt (14)MVA CCP = − (cid:90) T E [( s B − s I ; B,H ) D r B + λ C I B,H ] dt (15)KVA CCP = − (cid:90) T E [( γ K − φr B ) D r B + λ C K H ] dt (16)These will be charged to the client side as they cannot be charged to the CCP, i.e.CVA to client = CVA CCP − (cid:90) T E (cid:2) L GD λ C D r B ,λ C ( V C − X C − I C,B ) + (cid:3) dt (17)FVA to client = FVA CCP − (cid:90) T E [ s B D r B + λ C ( V C − X C − I B,C )] dt (18)ColVA to client = ColVA CCP − (cid:90) T E [ D r B + λ C ( s X ; C X C + r I ; C,B I C,B )] dt (19)MVA to client = MVA CCP − (cid:90) T E [( s B − s I ; B,C ) D r B + λ C I B,C ] dt (20)KVA to client = KVA CCP − (cid:90) T E [( γ K − φr B ) D r F ,λ C K C ] dt (21)That is, the client charge is the sum of the valuation adjustments on the CCP side and on theclient side, and the valuation adjustments on the CCP side incorporate the client default. Thislast feature breaks the usual assumption of independence of counterparties for XVA calculations. When hedging with a non-CCP the hedge counterparty can default, but the bank will replacethe hedge for as long as the client is non-defaulting. Thus the bank will experience periodic costson hedge defaults. The hedge-side position is not a single trade (or portfolio) but a sequence oftrades with different counterparties. It is this hedge-side position that the bank must replicateto price the hedge position. The differential of the value of the hedge position is: d (cid:98) V H = ∂ (cid:98) V H ∂t dt + 12 σ r P r ∂ (cid:98) V H ∂P r dt + ∂ (cid:98) V H ∂P r dP r + ∆ B (cid:98) V H dJ B + ∆ C (cid:98) V H dJC + (cid:88) i ∆ i (cid:98) V i dJ { i } (22)where i runs over the sequence of hedge counterparties. We are replicating the cost of the hedgeposition (actually sequence of positions with different counterparties i ), not of individual hedgetrades with a single counterparty. This means that only one α i will be non-zero at any time.Thus the change in value of the hedge position on the default of a given hedge counterparty mustbe such that any losses are covered, i.e.∆ i (cid:98) V i = (1 − R i )( V H − X H − I i,B ) + (23)7he processes dJ { i } are the sequence of processes of hedge counterparty defaults. Apart fromthe first, the counterparties are anonymous. When hedging with counterparty i we do not knowwhich counterparty will be next ( i +1). This can only be decided at τ { i } when i th in the sequencedefaults as any prior choice may default prior as well.The replicating hedge portfolio now includes (shorted) bonds with the sequence of counter-parties i to generate these cashflows on default of each of the i , i.e.Π = . . . + (cid:88) i α i P i (24)where the bonds have dynamics, and cash accounts as dP i /P i = r i dt + σ i dW − (1 − R i ) dJ i dβ i = − α i q i P i dt (25)Following a similar development as in the previous section we get to0 = ∂ (cid:98) V H ∂t + 12 σ r P r ∂ (cid:98) V H ∂P r + ∂ (cid:98) V H ∂P r rP r + λ C ∆ C (cid:98) V H + (cid:88) i i λ i ∆ i (cid:98) V H − r X X − γ K K − ( r I ; B,H I B,H − r I ; H,B I H,B )Note that we do not distinguish between hedge counterparties for IM collateral rates and put H (instead of a sum over i with time periods) for simplicity. The 1 i is the indicator function thereis a trade with counterparty i . This is a reminder that only one hedge counterparty is used atany point for the position.This gives the following expressions for valuation adjustments on the hedge side taking intoaccount multiple hedge counterparty defaults.CVA H = − (cid:90) T E (cid:34)(cid:88) i i s i D r B + λ C ( V i − X i − I i,B ) + (cid:35) dt (26)FVA H = − (cid:90) T E (cid:34) s B D r B + λ C (cid:88) i i ( V H − X H − I B,i ) (cid:35) dt (27)ColVA H = − (cid:90) T E (cid:34) D r B + λ C (cid:88) i i ( s X ; C X C + r I ; i,B I i,B ) (cid:35) dt (28)MVA H = − (cid:90) T E (cid:34) s I ; B,i D r B + λ C (cid:88) i i I B,i (cid:35) dt (29)KVA H = − (cid:90) T E (cid:34) ( γ K − φr B ) D r B + λ C (cid:88) i i K i (cid:35) dt (30)These expectations appear challenging to compute because of the 1 i term which changes themfrom the usual one dimensional expression to (theoretically) infinitely dimensional. There is onedimension per hedging counterparty. In addition there is the implicit assumption that the CDSspreads of the sequence of counterparties are known, even if the actual counterparties are notknown.However, the situation is less challenging than it appears, consider Equation 26 where wecombine elements not affected by the CDS level, or replacement, into f ( t ) (cid:90) T E (cid:34)(cid:88) i i s i f ( t ) (cid:35) dt (31)8e know that exactly one of the 1 i is non-zero at any time, so if all the s i are equal (for whateverreason), to say s , we have (cid:90) T E (cid:34)(cid:88) i i s i f ( t ) (cid:35) dt = (cid:90) T E [ s f ( t )] dt (32)Now suppose instead that prior to any defaults we have s i all different, so we now need toconsider how many defaults have occured up to t to know which s i is being used. Thus the statetransitions between hedge-side counterparties form a semi-Markov process with rate matrix Q : Q = − λ λ ... − λ λ ... ... − λ n λ n ... (33)The full Q is infinite but can be truncated at n transitions without material error where n canbe calculated from the λ i for a given required accuracy. The time zero state vector is simply { , , . . . , } expressing that there is one hedge-side counterparty at the start and this is the firstone.We can now express Equation 31 fully generally as: (cid:90) T E (cid:34)(cid:88) i i s i f ( t ) (cid:35) dt = (cid:90) T E (cid:34)(cid:88) i (PDF SMP ( { , , . . . , } , Q, t, i ) s i ) f ( t ) (cid:35) dt (34)(35)Where PDF SMP ( { , , . . . , } , Q, t ) is the PDF of the semi-Markov process defined by the givenstarting vector, the rate matrix Q for state i , i.e. the i-th hedge-side counterparty. This can becalculated efficiently by matrix exponentiation.In summary, in both hedge cases the hazard rate of the hedge does not appear in the dis-counting of the valuation adjustments, but the hazard rate of the client does. Thus, in practice,usual hedging behaviour breaks the assumption of independence of counterparties. Here we consider a client trade hedged with the street. We consider street counterparties directlyand via a central counterparty (CCP). However, we do not compute the effect of multiple CCPdefaults since even a single CCP default is likely to be market-changing beyond what is capturedin our equations. This means that trading via a CCP is relevant for MVA but not the otherXVA. Althougth this is only one of the XVA, the effect of behaviour on MVA with a CCP ishighly significant precisely because of the low CCP default probability relative to clients of thebank. Effects on MVA and CVA will generally be exclusive as MVA largely removes CVA.We assume that hedge trades are at mid, i.e. the hedging counterparty does not charge thebank for the hedger’s MVA and vice versa.We consider exposures from interest rate positions. Looking at Equations 26—30 we notethat if earlier defaults do not change subsequent then multiple defaults simply change the defaultprobabilty, not the quantities involved. For non-credit portfolios this is the case assuming theusual independence of expsoure and default, i.e. no wrong-way risk and no right-way risk.Equations 26—30 are challenging to compute directly, but given that they converge quicklyfor multiple defaults, we make the following approximations:9 include only three possible defaults (not just the first default as usual). For the casesconsidered, i.e. maximum CDS spread of 5% three defaults up to 30Y captures 93% ofevents, and for 2.5% this captures 99% of events. Missed event will have a very smallimpact as they occur late in the time considerd, i.e. missing 7% of events does not have a7% impact but much less than 5%. • exposure profiles assumed from non-credit products and assume that these are unaffected byprevious defaults. We consider three exposure profiles, either from MVA or CVA dependingon the case – Triangular decreasing to zero at maturity – Flat – Triangular increasing from zero at startSince we consider percentage changes in MVA and CVA, these profile specificationsare sufficient. We also consider two cases for CVA effects • assume that all available hedge counterparties have the same undisturbed hazard rate • assume that each default increases available hedge counterparty hazard rates by a fixedmultiplier, 20%.We report results in terms of the changes on the hedge side as the client side is, by assumption,unaffected. The effects may be slightly underestimated in practice because hazard rates ofhedge counterparties are assumed independent of non-hedge entity defaults. That is, only directcontaigon is included, and at a relatively mild level (20% increase in CDS spread).Table 2 shows the relative effect on MVA when hedging client trades through a CCP, i.e.a hedge counterparty that is considered default free. The fact that hedges will be closed onclient default means that just looking at the hedge counterparty (where the MVA occurs) ismisleading. For reasonably risky clients (250bpds CDS spread) decreases of MVA of roughly20% occur for flat and increasing profiles at 10Y portfolio length. For 30Y portfolios there arevey significant decreases in MVA, 30% to 50% for decreasing-to-increasing MVA profiles. Forrisky counterparties (500bps) there are major decreases in MVA for 10Y portfolios and above.Trades with a CCP combine many different counterparties so the actual decrease in MVAwill be a mixture of the different client CDS results shown.The results for MVA with a CCP are also valid, exactly as they are, for non-CCPs wherehedge counterparties are replaced when they default.Table 3 and Table 4 shows the change in CVA from the combination of two effects: hedgeclosure on client default; and hedge replacement on hedge default. These effects act in oppositedirections and we consider ranges of client CDS level and hedge CDS level for different portfoliolengths from 5Y to 30Y. Table 3 shows results assuming that the next hedge CDS level availableon previous hedge default is the same level. This is a rather aggresive position as bank defaultstend to affect each other. Accordingly Table 4 shows the changes in CVA where we assume thatthe available hedge CDS level increases by 20% on each previous hedge default.Table 3 shows CVA changes where client CDS and hedge CDS effects become more asym-metric as the length of the portfolio considered increases. The asymmetry comes from theincreased possibility of multiple hedge replacements (multiple CVA charges) on the hedge-side.The symmetry of the effects is modified by the portfolio’s exposure profile. Decreasing profilesemphasising client default and increasing profiles emphasising hedge-replacement. For long, 30Y,trades highly significant ( > Considering actual bank behaviour in XVA significantly decreases client prices from MVA on thehedge side. Prices changes to the client from CVA on the hedge side, with hedge counterpartieswhere there is no IM or legacy portfolios, show significant increases and decreases depending onthe relative CDS levels of client and hedge. However, the main effect for CVA is any increase inavailable hedge CDS levels caused by the default of previous hedge counterparties. This leadsto significant increases in CVA on the hedge-side in most cases considerd. Pricing XVA byconsidering counterparties independently of each other is inadequate for both MVA on the hedgeside and CVA where there is no IM, bank behaviour needs to be included. Effects on KVA andFVA will resemble those on MVA. Portfolio rebalancing due to client default will impact KVAand FVA in a similar way to MVA as they are largely exposure profile based, so behaviour needsincluding for all XVAs. 11 o jump in next-hedge CDS spread on each previous-hedge default
Decreasing CVA Profile Flat CVA Profile Increasing CVA ProfileHedge CDS Hedge CDS Hedge CDS
50. 100. 250. bps
50. 100. 250. bps
50. 100. 250. C li e n t C D S
50. 0% 1% 5% 50. 0% 2% 8% 50. 0% 3% 11%100. -1% 0% 4% 100. -2% 0% 6% 100. -3% 0% 8%250. -5% -4% 0% 250. -8% -6% 0% 250. -10% -8% 0%500. -11% -10% -6% 500. -16% -14% -9% 500. -22% -19% -12%
50. 0% 7% 29% 50. 0% 10% 45% 50. 0% 16% 76%100. -7% 0% 20% 100. -9% 0% 31% 100. -14% 0% 52%250. -22% -17% 0% 250. -31% -24% 0% 250. -43% -34% 0%500. -40% -36% -23% 500. -52% -47% -31% 500. -69% -65% -46%Table 3: Relative percentage change in CVA on hedge-side from combined hedge replacementon hedge default and hedge closure on client default. Table shows effect of CVA profiles andportfolio maturities (in years) for different hedge-side CDS levels and client CDS levels (in basispoints). Profiles: decreasing = triangular decreasing to zero at maturity; flat = flat; increasing= triangular increasing from zero. Bank CDS is 100bps in all cases.
20% jump in next-hedge CDS spread on each previous-hedge default
Decreasing CVA Profile Flat CVA Profile Increasing CVA ProfileHedge CDS Hedge CDS Hedge CDS
50. 100. 250. bps
50. 100. 250. bps
50. 100. 250. C li e n t C D S
50. 0% 2% 7% 50. 0% 3% 11% 50. 1% 4% 15%100. -1% 1% 5% 100. -2% 1% 8% 100. -2% 1% 12%250. -5% -3% 1% 250. -7% -5% 2% 250. -10% -7% 3%500. -11% -9% -5% 500. -16% -14% -7% 500. -21% -18% -10%
50. 1% 10% 39% 50. 2% 15% 62% 50. 3% 23% 109%100. -5% 3% 29% 100. -8% 4% 46% 100. -11% 6% 79%250. -21% -15% 7% 250. -30% -21% 10% 250. -42% -30% 16%500. -40% -35% -19% 500. -52% -46% -26% 500. -69% -63% -39%Table 4: Changes in CVA on hedge-side as in Table 3 but now we assume that the next hedgeCDS level is increased by 20% on each previous-hedge default. We assume that the client is froma different sector so the client CDS is unchanged by hedge-side defaults. Note that the veryapproximate symmetry present in Table 3 is reduced.12 eferences
Burgard, C. and M. Kjaer (2013). Funding strategies, funding costs.
Risk 26 (12), 82.Green, A. and C. Kenyon (2015, May). MVA by Replication and Regression.
Risk 27 , 82–87.Green, A., C. Kenyon, and C. R. Dennis (2014). KVA: Capital Valuation Adjustment byReplication.
Risk 27 (12), 82–87.Kenyon, C. and A. Green (2015). Warehousing credit risk: pricing, capital, and tax.