Model independent WWR for regulatory CVA and for accounting CVA and FVA
MModel independent WWR for regulatory CVAand for accounting CVA and FVA
Chris Kenyon, Mourad Berrahoui and Benjamin Poncet ∗
02 March 2020Version 1.00
Abstract
Wrong way risk (WWR) is a consideration for regulatory capital forcredit valuation adjustment (CVA). WWR is also of interest for pricingand accounting and in these cases must include funding as well as ex-posure and default in CVA and FVA calculation. Here we introduce amodel independent approach to WWR for regulatory CVA and also foraccounting CVA and FVA. This model independent approach is extremelysimple: we just re-write the CVA and FVA integral expressions in termsof their components and then calibrate these components. This providestransparency between component calibration and CVA/FVA effect be-cause there is no model interpretation in between. Including funding inWWR means that there are now two WWR terms rather than the usualone. Using a regulatory inspired calibration from MAR50 we investigateWWR effects for vanilla interest rate swaps and show that the WWR ef-fects for FVA are significantly more material than for CVA. This modelindependent approach can also be used to compare any WWR model bysimply calibrating to it for a portfolio and counterparty, to demonstratethe effects of the model under investigation in terms of components ofCVA/FVA calculations.
Wrong way risk (WWR) for regulatory credit valuation adjustment (CVA) ispart of the Standardized Approach for CVA (MAR50) and WWR is also im-portant for pricing CVA including funding, and funding valuation adjustment ∗ Contacts: [email protected], [email protected], [email protected]. This paper is a personal view and does not represent theviews of MUFG Securities EMEA plc (MUSE). This paper is not advice. Certain informationcontained in this presentation has been obtained or derived from third party sources and suchinformation is believed to be correct and reliable but has not been independently verified.Furthermore the information may not be current due to, among other things, changes in thefinancial markets or economic environment. No obligation is accepted to update any such in-formation contained in this presentation. MUSE shall not be liable in any manner whatsoeverfor 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 andaccepts no legal responsibility to any party who directly or indirectly receives this material.This paper does not necessarily represent the view of Lloyds Banking Group. No guaranteesof any kind. Use at your own risk. a r X i v : . [ q -f i n . P R ] M a r FVA). WWR describes the situation where exposure and default tend to in-crease together. We focus on general WWR, i.e. where there is no direct (e.g.legal) connection between default and exposure.We present a model independent approach to both regulatory CVA wherethere are two elements (exposure and default) and to accounting CVA and FVAwhich have funding as a third element. Including exposure, default, and funding,is market standard for pricing CVA and FVA. Our model independent approachoffers transparency between parameter estimation and CVA/FVA effect, and analternative to the more than thirty WWR models for CVA appearing since 2010((Chung and Gregory 2019; Sakuma 2020) are two of the most recent), as wellas filling a gap in addressing WWR in accounting CVA and FVA.Our model independent approach is simply to re-write analytic expressionsfor CVA and FVA in terms of the two, or three, underlying elements and theircorrelations. We continue re-writing in terms of simpler elements until theycan be estimated directly from market, simulation, or historical data. Once wehave re-written the expressions the independent and dependent (WWR) partsof CVA and FVA are explicit, then we estimate the parameters within the re-written equations. There is no model between these estimated parameters andCVA/FVA, the parameters directly affect regulatory and accounting results sothe connection is transparent. We propose estimation that is inspired by recentregulations (MGN20) and use it to provide numerical results.
We develop WWR models by starting from an expression from CVA, or FVA,derived elsewhere (Green 2016) or from MAR50 and simply rewrite it usingbasic arithmetic to reveal the WWR structure. Once the WWR structure issufficiently simple that we can estimate (aka calibrate) the parameters we cancalculate the WWR impact on CVA and FVA.The model independent approach offers transparency between parametercalibration choices and their effects. We also obtain structural informationabout the connection between counterparty portfolios and their WWR.
We first describe the regulatory CVA model, then re-write it in elementaryterms and provide a list of the properties revealed. Regulatory CVA is CVAcalculated in accordance with MAR50 (BCBS 2017) , i.e. excluding any effect ofown creditworthiness like funding effects. Parameters must be market-impliedwhere possible and historically calibrated otherwise.
A standard expression for CVA meeting MAR50 requirements is (Green 2016):CVA = (cid:90) T E Q [ L GD ( t ) λ ( t ) D r,λ ( t ) max(0 , Π( t ))] dt (1) D r,λ ( t ) = exp( − ( r ( t ) + λ ( t ))) (2)Notation is given in Table 1. 2 ymbol Meaning Q Risk-neutral measure r ( t ) riskless short rate at time tr F ( t ) := r ( t ) + s F ( t ) funding short rate short rate at time t , equal to risklessplus funding spread λ ( t ) hazard rate for counterparty C at time tL GD loss given default of counterparty C Π , Π + value of portfolio with counterparty, and exposureVar( a ) Variance of the distribution of the random variable a SD( a ) Standard Deviation of a Table 1: Notation term structure expression source default-exposure correlation ρ Q t historicaldefault variance Var Q t ( L GD λD λ ) historicalexposure variance Var Q t ( D r Π + ) market implied, from CVAsimulationTable 2: Parameter estimation for regulatory CVA WWR, for details see text.Given two random variables a and b it is elementary that E [ ab ] = E [ a ] E [ b ] + ρ a,b SD( a )SD( b ) (3)Applying this to Equation 1 with a = L GD ( t ) λ ( t ) D λ ( t ) and b = D r ( t ) max(0 , Π( t ))and using Π + = max(0 , Π( t )) we get:CVA = (cid:90) T E Q t [ L GD λD λ ] E Q t [ D r Π + ] dt (cid:124) (cid:123)(cid:122) (cid:125) CVA, no WWR + (cid:90) T ρ Q t (cid:113) Var Q t ( L GD λD λ )Var Q t ( D r Π + ) dt (cid:124) (cid:123)(cid:122) (cid:125) CVA WWR only (4)For stochastic processes ρ t is the term structure of terminal correlation (not theinstantaneous correlation) see (Rebonato 2005) Section 5.3. Regulatory WWRfor CVA is summarized in Table 2.An objection at this point might be that we have done nothing but re-writeEquation 1. This is exactly the simplicity of the model independent approach,we express WWR directly in terms that are applicable to CVA. It is easy tounderstand the meaning of ρ Q t : correlation between forward default probabilitiesand forward exposures. There are three items to estimate: exposure; default; and the terminal corre-lation between them, summarized in Table 2. Regulations state that market-implied estimation must be used where available, and if not available then his-torical estimation is permitted. 3
Dynamics of underlyings for discounting and exposure: this is done asusual, with no change to standard (no-WWR) CVA setup. Note thatif there are any credit-dependent derivatives in the exposure then theirdynamics are unchanged and exposure calculated as usual. • Dynamics of counterparty default probability and loss given default: his-torical estimation. Counterparty portfolios can go out 20 or 30 years butthere are no liquid options on CDS to calibrate to so this must be histor-ical. Whilst there are some options on index CDS their maximum liquidmaturity is generally less than one year so are not useful for CVA.Hazard rates calibrated from CDS spreads are quite insensitive to dis-counting and this justifies the separation of hazard and (relatively fixedCDS) here exposure (Brigo and Mercurio 2007) here. • Term structure of default-exposure terminal correlation: this is estimatedhistorically (see Numerical examples section for details) using a constant(current) portfolio against historical market data. If there were standardtraded CVA contracts, e.g. CCDS, for each counterparty then this couldbe market implied, but there are none. • Equation 4 is exact: this is trivially true as it is simply an elementary re-writing of CVA. No assumptions are required on exposure, underlyings,default, or correlation beyond the existence of second moments. • The terminal correlation term structure is deterministic. ρ Q t is not stochas-tic. • The terminal correlation term structure is counterparty specific. • The re-writing cleanly separates the items that can be market implied(the expectations and exposure variances) from those that need historicalestimation (terminal correlation and default variance). • The re-write demonstrates that WWR (or RWR) is a basic, first order,property of CVA in the same way that E [ ab ] = E [ a ] E [ b ] + ρ a,b SD( a )SD( b ). Pricing CVA and FVA usually include funding spreads and costs, thus thereare three elements to consider: exposure; default; and funding, not two as inRegulatory CVA. It is not surprising that there will now be two WWR termsrather than one. We apply the same re-writing approach starting from CVAand FVA formulae analogous to the regulatory case:CVA = (cid:90) T E Q (cid:2) L GD ( t ) λ ( t ) D r F ,λ ( t )Π + ( t ) (cid:3) dt (5)FVA = (cid:90) T E Q [ s F ( t ) D r F ,λ ( t )Π( t )] dt (6)4iven three random variables { a, b, c } we can re-write the expectation of theirproduct: E [ abc ] = E [ a ] E [ bc ] + ρ a,bc SD( a )SD( bc ) (7)= E [ a ] E [ b ] E [ c ] + E [ a ] ρ b,c SD( b )SD( c ) + ρ a,bc SD( a )SD( bc ) (8)= E [ a ] E [ b ] E [ c ] + E [ b ] ρ a,c SD( a )SD( c ) + ρ b,ac SD( b )SD( ac ) (9)= E [ a ] E [ b ] E [ c ] + E [ c ] ρ a,b SD( a )SD( b ) + ρ c,ab SD( a )SD( ab ) (10)and taking SD( ab ) for example we can re-write it so as to separate functions of a from functions of b :Var( ab ) = ρ a ,b SD( a )SD( b ) + E [ a ] E [ b ] − ( ρ a,b SD( a )SD( b ) + E [ a ] E [ b ]) (11)There are now three equivalent expressions for CVA and FVA depending onwhich re-write equation, 8, 9, or 10 we pick. Note that by using Equation11 we achieve a clean separation between market-implied items (expectationsand exposure variances) and historically calibrated items (terminal correlations,default variances, and funding variances).CVA = (cid:90) T E Q t [ L GD λD λ ] E Q t [ D s F ] E Q t [ D r Π + ] dt (cid:124) (cid:123)(cid:122) (cid:125) CVA, no WWR + (cid:90) T ρ c1 t E Q t [ L GD λD λ ] (cid:113) Var Q t ( D s F )Var Q t ( D r Π + ) dt (cid:124) (cid:123)(cid:122) (cid:125) CVA WWR funding-exposure (12)+ (cid:90) T ρ c2 t (cid:113) Var Q t ( L GD λD λ )Var Q t ( D s F D r Π + ) dt (cid:124) (cid:123)(cid:122) (cid:125) CVA WWR default-funding+exposure
FVA = (cid:90) T E Q t [ D λ ] E Q t [ s F D s F ] E Q t [ D r Π] dt (cid:124) (cid:123)(cid:122) (cid:125) FVA, no WWR + (cid:90) T ρ f1 t E Q t [ D λ ] (cid:113) Var Q t ( s F D s F )Var Q t ( D r Π ) dt (cid:124) (cid:123)(cid:122) (cid:125) FVA WWR funding-exposure (13)+ (cid:90) T ρ f2 t (cid:113) Var Q t ( D λ )Var Q t ( s F D s F D r Π ) dt (cid:124) (cid:123)(cid:122) (cid:125) FVA WWR default-funding+exposure t ( D s F ( D r Π + )) = ρ c2.1 t SD( D s F )SD(( D r Π + ) ) + E [ D s F ] E [( D r Π + ) ] − (cid:0) ρ c t SD( D s F )SD( D r Π + ) + E [ D s F ] E [ D r Π + ] (cid:1) (14)Var t (( s F D s F )( D r Π)) = ρ f . t SD(( s F D s F ) )SD(( D r Π) ) + E [( s F D s F ) ] E [( D r Π) ] − (cid:16) ρ f t SD( s F D s F )SD( D r Π) + E [ s F D s F ] E [ D r Π] (cid:17) (15)We have suppressed most of the time indices for clarity. Note that each termwithin an expectation or variance operator becomes a deterministic term struc-ture after the expectation or variance is take.The terminal correlation term structures used for pricing/accounting CVAare: • ρ c1 t : D s F versus D r Π + • ρ c2 t : L GD λD λ versus D s F D r Π + • ρ c2.1 t : D s F versus ( D r Π + ) These are counterparty specific.The terminal correlation term structures used for pricing/accounting FVAare: • ρ f1 t : s F D s F versus D r Π • ρ f2 t : D λ versus s F D s F D r Π • ρ f2.1 t : ( s F D s F ) versus ( D r Π) These are counterparty specific.Note that ρ c1 t may be quite close to ρ c2.1 t when D s F is always non-negative.Similarly for ρ f1 t and ρ f2.1 t . This is a property of “reasonably behaved” randomvariables with positive support, which can be verified empirically.Note that the variances we require after using Equation 16 either involvecredit, funding spread, or exposure. We have arranged that no variances involv-ing cross terms are required so these term structures of terminal volatility canbe estimated independently: { Var Q t ( D s F ) , Var Q t ( D r Π + ) , Var Q t ( D r Π) , Var Q t ( L GD λD λ ) , Var Q t ( s F D s F ) , Var Q t ( D λ ) , Var Q t ( D s F ) , Var Q t ( D r Π + ) , Var Q t ( D r Π) , Var Q t ( s F D s F ) } (16) This follows the same pattern as for regulatory CVA. Expectations are market-implied as are exposure variances. Default variances, funding variances andterm structures of terminal correlation are historically calibrated.6 .2.2 Properties • Equations 12 and 13 for CVA and FVA respectively, are exact: this isa trivial property since we are only re-writing the original equations 5and 6. No assumptions are required on exposure underlyings, default, orcorrelation beyond the existence of second moment for underlyings andsquares of underlyings. • There are three term structures of terminal correlation that determine theprice of WWR for CVA, and another three for WWR of FVA. • The terminal correlation term structures are deterministic. • The model cleanly separates the items that may be market implied (theexpectations and exposure variances) from those that need historical es-timation (terminal correlations and variances of default and funding) asbefore. • Having two WWR terms for accounting CVA and FVA is a first orderproperty that is simply the result of having three underlyings: exposure;default; and funding.In the following numerical examples we will show in detail the historicalestimation of variances and correlations.
We first describe the setup, then calibration, and then numerical results. Thesetup is an example and its elements are not prescriptive, for example fundingcosts might be idiosyncratic, or derived from Totem.
We consider WWR for the experimental setup of a bank trading with an uncol-laterlized counterparty as follows asof trades vanilla EUR interest rate swaps with maturities of 5, 10, 20, and 30years. Receive float and receive fixed. counterparty CDS
Eur iTraxx crossover bank funding spread
Eur iTraxx Senior Financials correlation calibration period counterparty CDS volatility calibration period one year up to asof date swaption volatiliy
ATM Normal volatilities for both historical period andvaluation date. 7sof is a recent date. The trades maturities cover a normal range of lengthsfor counterparty IRS trades. The counterparty CDS spread is a choice thatrepresents counterparties that may have significant CVA and so are of interestfor WWR. We use Eur iTraxx Senior Financials to represent the funding spreadover riskless for the bank as an observable and related spread. The correlationcalibration period is inspired by MAR50 in that it covers five years and includesa period of stress for interest rate trades. There are no liquid single name CDSoptions, and liquid index CDS options have maximum expiry of 9 months, sowe use historical calibration for CDS volatility, picking a recent period, i.e. lastone year.Since we are working with vanilla IRS we have no need of simulation modelfor CVA and FVA, we simply use the sum-of-risky-swaptions approach. Onthe valuation date we do need the volatilities of the exposure levels and usethe Normal distribution from the market data to obtain this. We use low-discrepancy integration for the exposure and similar volatilities for efficiency,but there is effectively no numerical noise.
An outline of the correlation calibration process is shown in Figure 1. In thefigure the objective is to get the correlation between the default probability(PD) between τ to τ + dt and the corresponding positive expected exposure(EE). Working backwards from the single lower plot of PD against EE for τ to τ + dt we see that the points used to calculate the correlation of PD vs EE comefrom different dates. On each of the three dates shown we compute the EE andPD against time (aka portfolio maturity). For PD against EE for τ to τ + dt wetake one value from the EE graph on each date on on value from the PD graph.We repeat the process in Figure 1 for every forward maturity τ over thelife of the portfolio with the counterparty (“Time” in the Figure). This setof correlations versus Time provides the correlation term structure that goesinto the WWR calculation. A similar method is used for every correlation termstructure.Some details are important. The portfolio is kept constant as it is evaluatedagainst the different market data from different dates. This is similar to howVaR is calculated. Default probabilities are calculated asof their market datadates because CDS maturities are standardized on IMM dates. Fixings aretaken from the valuation date and inserted into the historical market data files.The term structure of default probability volatility, including LGD changes,is calculated in a similar way to the correlation term structure except that themost recent one year period is used, as it would be for pricing. We show a sample of the input data as well as all the WWR numbers for thetrades considered. Figure 2 shows the history of EUR 6m fixings and { } swap rates together with Eur iTraxx Xover and Eur Senior Financials.8 ime EE Time PD EE(d1, τ ,dt) PD(d1, τ ,dt) Date d1 Time EE Time PD EE(d2, τ ,dt) PD(d2, τ ,dt) Date d2 Time EE Time PD EE(d3, τ ,dt) PD(d3, τ ,dt) Date d3 PD( τ ,dt) EE( τ ,dt) ρ ( τ , dt ) X Date 1 X Date3 X Date 2
Figure 1: Outline of correlation calibration process. For details see text.
Figure 3 shows the exposure-default correlation term structure for 30Y ATMreceive float IRS in the Top Left plot. The wave-like shape is superficiallyunexpected but can be understood from the plot of forward 6m default proba-bilities (Top Right). Observe that when the CDS spreads increase in 2009 the1y forward default probability increases significantly, but the 9.5y forward de-fault probability decreases. Since the area under the forward default probabilitycurve must integrate to unity, an increase in one region must be balanced by adecrease in another region.We can compute the crossover tenor between increase and decrease of defaultprobabilities by equating the instantaneous forward default probabilities: λ e − λ t = λ e − λ t crossover point t = log( λ /λ ) λ − λ (17)Suppose we have a flat CDS curve and we increase it by 500bps, which is similarto the smallest increase for the iTraxx Xover during the crisis, this then givesthe brown curve on the Bottom Right of Figure 3.The second crossover tenor in the correlation term structure is also due tocurve movements over 2008–2012. Consider the interest rates as though theywere CDS, simply because an IRS pays a discounted rate which has a similarform to the instantaneous default probability λ exp( − tλ ). We can then applythe same logic as above using the blue curve in Bottom Right plot of Figure 39
008 2009 2010 2011 2012 2013012345 p e r ce n t EUR 6m fix and5y, 10y, 30y swap rates bp s iTraxx Xover ( ) yand Snr Fin ( ) y Figure 2: Jan 2008 – Dec 2012 market data for correlation calibration. LHSinterest rates. RHS CDS spreads: iTraxx crossover are in colour; Senior Finan-cials are in black.considering the decrease in EUR rates from around 4.5% to around 1.5% overthe period. This gives a predicted crossover around 20y which is similar to thatobserved in both receive float IRS (Top Left) and receive fixed IRS (BottomLeft). For the receive fixed IRS the interest rate move dominates the creditmove so there is no early crossover in the exposure-default correlation termstructure.Shorter IRS, and IRS with different strikes have similar patterns to thecorrelation term structures shown, but truncated in the case of shorter IRS.Table 3 shows independent accounting CVA and the regulatory WW com-ponent in bps of notional for a range of tenors and strikes. We see that WWRfor receive-float IRS (uncommon case) can be up to 25% of independent CVAbut is typically a few percent. For receive-fixed IRS (usual case with clients)the WWR is also a few percent. In both cases crisis levels of CDS volatilityreduce WWR CVA. 10 - p e r ce n t reg recflt p e r ce n t iTraxx Xover 1y and 9.5yforward 6m default probabilities - - - - p e r ce n t reg recfix Y ea r s Crossover tenor for500bps and 300bps change
Figure 3: Top Left: regulatory CVA exposure-default correlation term struc-ture for 30Y receive float IRS, different curves are different strikes (-25bpsand 2%). Top Right: iTraxx Xover 6m forward default probabilities. Bot-tom Left: regulatory CVA exposure-default correlation term structure for 30Yreceive fix IRS, different curves are different strikes (-25bps and 2%). BottomRight: crossover tenors for 500bps (brown) and 300bps (blue) changes in CDSor IRS levels. Arrows indicate predicted crossovers for receive IRS: comparewith Top Left. For receive fixed IRS (Bottom Left) the rates move dominatesthe credit move. 11 K CVA Indep WW WW+Crisis T K CVA Indep WW WW+Crisisyears percent RecFlt RecFlt RecFlt years percent RecFix RecFix RecFix5 -0.25 5 -0.1 0. 5 -0.25 3 -0.5 0.5 0 2 -0.1 0. 5 0 8 -0.7 0.5 0.5 0 0. 0. 5 0.5 19 -0.9 0.5 1. 0 0. 0. 5 1. 33 -1. 0.5 2. 0 0. 0. 5 2. 60 -1. 0.10 -0.25 80 -2.7 -0.2 10 -0.25 18 -3.1 -0.210 0 58 -2.4 -0.2 10 0 27 -3.9 -0.210 0.5 30 -1.9 -0.2 10 0.5 61 -5.4 -0.310 1. 14 -1.3 -0.1 10 1. 106 -6.6 -0.310 2. 2 -0.5 -0.1 10 2. 218 -7.6 -0.420 -0.25 480 -10.4 8.2 20 -0.25 77 -9.1 -8.20 0 397 -9.7 7.6 20 0 103 -11.3 -9.420 0.5 256 -8.1 6.2 20 0.5 182 -16.1 -12.320 1. 160 -6.3 5. 20 1. 305 -20.7 -15.20 2. 55 -3. 2.9 20 2. 639 -26.5 -19.330 -0.25 908 -12.1 28.6 30 -0.25 193 -13.5 2.530 0 756 -11.5 28.8 30 0 249 -17.3 -3.830 0.5 497 -9.7 26.9 30 0.5 406 -25.6 -17.830 1. 317 -7.3 23.4 30 1. 641 -33.4 -32.30 2. 117 -3.1 15. 30 2. 1272 -42.9 -56.9Table 3: Regulatory CVA for Receive-Floating (RecFlt) and Receive-Fixed (RecFix) vanilla EUR IRS for a range of tenors(T) and strikes (K). We show WWR for CVA under current conditions (CVA+WW) and under conditions with increasedCDS volatility taken from 2008 as WW+Crisis. Note that strikes are absolute (not relative to ATM). Units are bps ofnotional.
We now include funding, modeled using the iTraxx Senior Financials CDS indexpoints shown in black in Figure 2 RHS.Figure 4 shows the correlation term structures for accounting CVA { c1,c2,c2.1 } and FVA { f1,f2,f2.1 } for receive float IRS and receive fix IRS. Overall the cor-relation patterns are a combination of the previous regulatory case, and a newbehavior when there are no changes of sign of the correlation term structures.These behaviors are a reflection of the more complex nature of the WWR termsand that FVA is not option-like (uses net exposure), unlike CVA (positive ex-posure only).Table 4 gives the CVA and FVA details for the range of vanilla EUR receivefloat IRS. Table 4 provides the details for receive fixed IRS.We can first note that the second WWR term for both CVA and FVA isnegligible. This second WWR term is a mixture of all three factors mixed indifferent ways by the variance terms so, net, there is no directional contribution.It is possible that this observation will also hold for other instruments.In absolute terms, and compared to the regulatory view, accounting CVAWWR is small for receive fixed IRS, mostly below or of the order of a couple ofpercent. In contrast accounting CVA WWR for receive float IRS can be above10% although it is also generally below a couple of percent.FVA WWR is significant as it is often more than ten percent and can reach12 - p e r ce n t c1,c2,c2.1, recfix - p e r ce n t c1,c2,c2.1, recflt - p e r ce n t f1,f2,f2.1, recfix - p e r ce n t f1,f2,f2.1, recflt Figure 4: Top Left: accounting CVA { c1,c2,c2.1 } correlations for 30y receivefix IRS. Top Right: receive float IRS. Bottom Left: FVA { f1,f2,f2.1 } correlationsfor receive fix IRS. Bottom Right: receive float IRS. Blue and Green curves are*1 and *2.1 correlations, these have the same underlyings but squared in thesecond case. *2 are the yellow curves.more than fifty percent in some cases. The greater magnitude of FVA WWRcompared to accounting CVA WWR is mostly because it is not option-like sohas simpler correlation behavior in general.13 K CVA Indep WW1 WW2 FVA Indep WW1 WW2 CVA FVAyears percent RecFlt RecFlt RecFlt RecFlt RecFlt RecFlt RecFlt RecFlt5 -0.25 5 0 0 1 0 0 5 0.85 0 2 0 0 -3 0 0 2 -25 0.5 0 0.1 -0.0001 -9 1 0 0 -7.55 1 0 0.1 -0.0002 -15 2 0 0 -12.75 2 0 0.2 -0.0003 -27 4 0 0 -2310 -0.25 73 0.1 -0.0002 25 1 -0.0001 73 2610 0 55 0.2 -0.0002 12 2 -0.0001 55 1410 0.5 28 0.4 -0.0003 -13 4 -0.0001 29 -9.410 1 13 0.7 -0.0006 -38 6 -0.0001 14 -32.110 2 2 1.4 -0.0013 -89 13 -0.0001 3 -75.720 -0.25 425 1.1 -0.0004 166 4 0.0003 426 17020 0 352 1.4 -0.0004 121 5 0.0003 354 12620 0.5 230 2.4 -0.0005 30 9 0.0003 232 39.620 1 142 3.7 -0.0007 -60 16 0.0003 146 -44.520 2 48 7.2 -0.0019 -241 35 0.0003 55 -20630 -0.25 783 1.1 -0.0004 291 8 0.002 784 298.930 0 652 1.9 -0.0004 206 11 0.002 654 217.230 0.5 430 4 -0.0005 37 19 0.002 434 56.230 1 270 6.9 -0.0007 -131 31 0.0019 277 -100.530 2 96 14.6 -0.0018 -469 68 0.0019 111 -401.3Table 4: Accounting CVA and FVA for Receive-Floating (RecFlt) vanilla EUR IRS for a range of tenors (T) and strikes(K). Note that strikes are absolute (not relative to ATM). Units are bps of notional. Last two columns are total CVA andFVA. WW1 and WW2 are the first and second WWR terms in Equations 12 and 13.14 K CVA Indep WW1 WW2 FVA Indep WW1 WW2 CVA FVAyears percent RecFix RecFix RecFix RecFix RecFix RecFix RecFix RecFix5 -0.25 3 0 -0.0001 -1 0 0 3 -0.45 0 7 0 -0.0001 3 0 0 7 2.35 0.5 16 0 -0.0003 9 -1 0 16 7.75 1 27 -0.1 -0.0005 15 -2 0 27 13.15 2 51 -0.1 -0.0009 27 -3 0 50 23.810 -0.25 17 0.2 -0.0006 -25 3 -0.0001 17 -21.710 0 26 0.1 -0.0004 -12 2 -0.0001 26 -10.610 0.5 54 -0.1 -0.0007 13 -2 -0.0001 54 11.410 1 94 -0.3 -0.0015 38 -5 -0.0001 94 33.410 2 193 -0.8 -0.0029 89 -12 -0.0001 192 77.120 -0.25 68 2.1 -0.0025 -166 18 -0.001 70 -147.820 0 92 1.6 -0.0018 -121 14 -0.0011 94 -107.220 0.5 164 0.5 -0.0009 -30 4 -0.0012 164 -26.620 1 269 -0.5 -0.002 60 -7 -0.0012 269 53.120 2 562 -2.3 -0.0055 241 -31 -0.0012 560 210.930 -0.25 162 4.4 -0.0033 -291 32 -0.0004 167 -259.130 0 211 3.1 -0.0024 -206 23 -0.0007 214 -183.230 0.5 349 0.6 -0.001 -37 5 -0.0011 350 -32.830 1 549 -1.8 -0.0025 131 -16 -0.0014 547 115.830 2 1094 -5.5 -0.0074 469 -59 -0.0018 1088 409.5Table 5: Accounting CVA and FVA for Receive-Fixed (RecFix) vanilla EUR IRS for a range of tenors (T) and strikes (K).Note that strikes are absolute (not relative to ATM). Units are bps of notional. Last two columns are total CVA and FVA.WW1 and WW2 are the first and second WWR terms in Equations 12 and 13.15 Conclusions
The model independent WWR approach presented here is unique because itliterally requires no modeling, but is simply a re-writing of the expressions forregulatory CVA and accounting CVA and FVA. Because the approach is modelindependent any calibration is directly in terms of the parameters for CVA andFVA calculation themselves and so provides maximum transparency.WWR depends on correlations of factors and here we use the elements in theCVA and FVA calculations themselves: default; exposure; funding. Correlationcalibration is usually historical and this is what we do here. We have pickeda historical period inspired by the MAR50 regulation in that it encompasses astressed year (around the end of 2008) and is a total of five years.XVA usually incorporates funding costs which are not part of regulations,but funding costs are of immediate interest to accounting (aka pricing). Wewere able to extend the model independent approach naturally to include thethird factor (funding) with the same degree of transparency as for regulatoryCVA. The increase in complexity, three correlations rather than one and twoWWR terms rather than one, is also expected given we now have three factorsrather than two. However, our results indicate that the second WWR term maybe negligible thus reducing complexity.We observe that structural shifts during the calibration period (2008-2012)are reflected in the correlation term structures, notably in the sign changesof correlations. The effect on CVA and FVA is a complex interplay betweenthe portfolio (IRS here), credit and funding. Because our CVA and FVA in-puts are directly in terms of CVA and FVA we can both observe and numeri-cally/analytically explain these effects (see Figure 3 and Equation 17).Our results suggest that WWR in CVA strongly depends on the instrument,but may generally be low, i.e. of the order of a couple of percent. In contrast,WWR in FVA is significant for IRS, i.e. 10% to more than 50% and these resultstoo may hold for other instruments.Technically we have shown how to deal with two and three factors in WWR.This generalizes to any number of factors which can then be assessed, as herewith WW2, for materiality.Finally, our model independent approach can be used forensically to comparedifferent WWR models on the same terms. Models for comparison can simplybe simulated and then use as calibration for this model independent approach.In this way any WWR model that can produce simulated paths of exposure,default and funding can be directly compared in terms of parameters directlyrelevant for CVA and FVA. Just as different WWR models can be compared,so can different calibration assumptions for this model independent approachitself and other WWR models.
Acknowledgements
The authors would gratefully like to acknowledge feedback from participantsat the FIS Round-table (September 2019, Canary Wharf) and the 3rd AnnualDerivatives Funding and Valuation conference (September 2019, Singapore).Discussions with Lee Mcginty, Andrea de Vitis, Yousef El Otmani, Hayato Iida,Tom Cannon, and Nitin Adlakha were also useful.16 eferences
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