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  XVA: About CVA, DVA, FVA and Other Market AdjustmentsPreprint of   Opinion and Debates  ∗  num. 5, June 2014 St´ephane Cr´epey † Laboratoire de Math´ematiques et Mod´elisation d’´Evry, Universit´e d’´Evry Val d’EssonneJune 10, 2014 Abstract Since the crisis, different adjustments are needed to account for counterparty riskand funding costs in the risk management of OTC derivatives, notably credit valuationadjustment (CVA), debt valuation adjustment (DVA) and funding valuation adjust-ment (FVA). These adjustments, which are, to some extent, interdependent and mustbe computed jointly, count today among the main P&L centers of investment banks.They touch on many areas: modeling, computation, pricing, risk management, regula-tion, economics, legal, lobbying, politics, often in conflicting perspectives. Banks haveto cope simultaneously with economic risk, accounting P&L and regulatory capital con-siderations. The current trend of the regulation is to push participants to negotiatecentrally via clearing houses or to bring strong guarantees in terms of collateralization.But this evolution poses liquidity and systemic risks issues. ∗ Discussion papers of the Louis Bachelier Finance and Sustainable Growth Labex. † This work benefited from the support of the Louis Bachelier laboratory and the “Chair Markets in Tran-sition”, a joint initiative of  ´Ecole polytechnique, Universit´e d’´Evry Val d’Essonne and F´ed´eration BancaireFran¸caise.  2 Introduction The global credit crisis followed by the European sovereign debt crisis have highlighted thenative form of credit (not to say financial) risk, namely counterparty risk. Counterparty riskis the risk of non-payment due to the default of a party in an OTC derivatives transaction.By extension this is also the volatility of the price of this risk, the CVA (credit valuationadjustment). An important aspect of the problem, especially with credit derivatives, is thewrong-way risk, i.e. the risk of adverse dependence between the size of the counterpartyrisk exposure and the default riskiness of the counterparty. Moreover, as banks themselveshave become risky, counterparty risk must now be understood in a bilateral perspective(not only CVA but also DVA, i.e. debt valuation adjustment), where the counterparty riskof the two parties are jointly accounted for in the modeling. In this context, the classicalassumption of a risk-free asset that can be used for financing purposes (lending or borrowingas needed) is not sustainable anymore. Hence, another adjustment is needed in the formof an FVA, i.e. funding valuation adjustment, or a more specific LVA (liquidity componentof the FVA, net of credit spread, in order to avoid double counting with DVA). As willbe seen, there is still another adjustment, called replacement cost (RC), correspondingto the mismatch between the economical value of the contract and its valuation by theliquidator at default time. And the list is not closed since people now talk about KVA forcapital valuation adjustment (in reference to the capital cost of CVA volatility), or AVAfor additional valuation adjustment (toward “prudent valuation”, accounting for model riskand credit spreads including own, recently opposed to fair value by Basel). An acronymXVA was even introduced to generically refer to this increasing list of adjustments (seeCarver (2013)). Finally, in August 2007 a new dimension of systemic counterparty risk hasappeared, with the emergence of spreads between quantities that were very similar before,such as OIS swap rates and LIBOR rates (again a consequence of banks’ counterparty risk,but at a macro level). Through its relation with the concept of discounting, this systemiccomponent of counterparty risk has impacted on all derivatives markets. All the aboveadjustments, which are interdependent and must be computed jointly, now count amongthe main P&L centers of investment banks. The current trend of the regulation is to pushparticipants to negotiate centrally via clearing houses or to bring strong guarantees in termsof collateralization. But this evolution poses liquidity and systemic risks issues.The basic counterparty risk mitigation tool is a credit support annex (CSA, sometimesalso-called ISDA agreement) specifying the valuation scheme that will be applied by the liq-uidator in case of default of a party, including the netting rules applicable to the underlyingportfolio. The CSA value process also serves of reference for the determination of mar-gin calls, similar to a margining procedure for futures contracts, except that the collateralposted through a CSA remains the property of the posting party, so that it is remunerated.However, wrong-way risk (see above) and gap risk (slippage of the value of the portfoliobetween default and liquidation) imply that collateralization cannot be a panacea. It alsoposes liquidity problems. Therefore counterparty risk cannot be simply mitigated, throughcollateralization, it also needs to be hedged against default and/or market risk. Eventually,the collateralized and hedged portfolio needs to be funded, which raises the controversialissue of DVA/FVA overlap or double counting.From the point of view of the organization of the bank, due to netting, counterpartyrisk and funding costs can only be assessed at the CSA portfolio level (or better, regardingfunding costs, at the level of the whole book of the bank). Therefore the trend is to havea central XVA desk in charge of valuing and hedging counterparty risk (funding costs are  3typically managed by the treasury or ALM of the bank). A “clean” price and hedge ignoringcounterparty risk and assuming that all trading strategies are funded at the risk-free rate(or OIS rate, see Box 7) is first computed by the different business trading desks. Then theXVA desk values the counterparty risk of the portfolio and channels this charge back to thevarious trading desks, after a desallocation and conversion of a global and upfront chargeinto streams of fixed coupons. The all-inclusive price-and-hedge of a contract are finallyobtained as the difference between the clean price-and-hedge and the price-and-hedge ad- justment provided by the XVA desk (also accounting for funding costs as provided by theALM). Review of the literature  Here is a brief review of books about counterparty risk andfunding 1 : ã  The first book one can mention is the collection of seminal CVA papers in Pykhtin(2005) (now of course a little outdated). ã  The book by Gregory (2009, 2012) is technically quite simple but explains basic CVA concepts in a clear way. It is good for managers and finance people who need to get ageneral grasp of CVA fundamentals with some elements about funding/discounting,without going too technical. ã  The book by Cesari, Aquilina, Charpillon, Filipovic, Lee, and Manda (2010) is ratherbasic from a modeling point of view but it also looks at the IT implications of buildinga CVA system and tries to solve a number of practical problems that deal with CVAfor realistically large portfolios. The focus is on the so-called American (or leastsquare) Monte Carlo technique, first introduced for CVA applications in Brigo andPallavicini (2007, 2008). ã  The book by Kenyon and Stamm (2012), although technically basic, is srcinal, inthat it tackles current and relevant problems such as multi-curve modeling and creditvaluation adjustments, closeout and especially goodwill (which depends on the cred-itworthiness of a firm and can therefore be used for hedging the DVA). Funding costs,hints at systemic risk, regulation and Basel III are also considered. ã  The book by Brigo, Morini, and Pallavicini (2013) is mostly based on Brigo’s workwith several co-authors in the period 2002-2012 and has been written to be widelyaccessible while being technically advanced. It deals with CVA, DVA and wrong-way risk across different asset classes, gap risk, collateral, closeout, rehypothecationand funding costs. There is also a final part on CVA restructuring through so-calledCCDS (“contingent CDS”), CDO tranches type structures, floating rate CVA andmargin lending. ã  The book by Cr´epey, Bielecki, and Brigo (2014) focuses on the mathematical de-pendence structure of the problem, using mainstream stochastic analysis: BSDEs inparticular, in line with Cr´epey (2013), to address the nonlinear recursive nature of thefunding issue in a systematic way, and dynamic copulas to reconcile bottom up and topdown perspectives in the study of counterparty risk embedded in credit derivatives. 1 This book review is essentially borrowed from Chapter 1 in Cr´epey, Bielecki, and Brigo (2014).  4In this article, we review counterparty risk and funding costs in their different aspectssuch as CVA, DVA, FVA, LVA, RC (Sect. 1 through 3), collateral, wrong-way risk (Sect. 4) and central clearing (Sect. 5). On our way, we discuss in box form a number of aside issuessuch as historical versus risk-neutral valuation (“ P  versus  Q ”), multiple curves and weprovide a brief informal introduction to BSDEs. The boxes are also to insist on importantmessages. 1 Credit Valuation Adjustment The CVA (credit valuation adjustment) is the price of counterparty risk. By extension,counterparty risk is also the risk of volatility of the CVA. In fact, during the financial crisis,it was said that roughly two-thirds of losses attributed to counterparty were due to CVAlosses and only about one-third were due to actual defaults (see Lambe (2011)).A debt towards a defaulted counterparty is due in full to the liquidator (a nonpayingparty would itself be declared in default). Therefore, the counterparty risk exposure is onlythe positive part of the value of a contract (or portfolio). It follows that for any swappedcontract (as opposed to, say, a bond position, which is always the same side of the money,lender or borrower), the loss associated to this risk has an optional feature (see Fig. 1). Thisis why counterparty risk cannot be simply handled by the application of a simple spreadin the discount rate (or this would be an implicit and stochastic spread). Moreover, dueto the netting rules that apply at the CSA portfolio level, where a CSA can cover tens of thousands of contracts, often across different classets of assets, in the end a CVA appearsnot only as a derivative, but as an exceedingly complex one, a kind of giant hybrid option.In fact, it is treated as such by risk control, who imposes market limits and monitors therisks of the CVA desk using VaR and sensitivities.There are two perspectives on the CVA, an accounting one and a regulatory one.Under last years’ advisory and regulatory developments, IFRS 13 allows a bank to accountfor CVA losses and DVA gains, whereas, since DVA does not act as a buffer against defaults,Basel III only reckons CVA into capital relief calculations. This is stated in Paragraph 75of  Basel Committee on Banking Supervision (2011) as“Derecognise in the calculation of Common Equity Tier 1, all unrealised gainsand losses that have resulted from changes in the fair value of liabilities that aredue to changes in the bank’s own credit risk.”(see also Basel Committee on Banking Supervision (2012) for a detailed discussion). 1.1 Cash Flows CVA is not such a well defined quantity. The payoffs of the corresponding “option” are notso clear. Indeed, regarding the CSA value process, ISDA documents use a “replacement”formulation which leaves it open between rather different possibilities, notably default-freeversus substitution valuation, where the meaning of “substitution” is not fixed: shouldsubstitution mean valuation accounting for the default risk of the surviving party only, orvaluation of the contract between the surviving and a new party “similar in every regard”(including credit risk) to the defaulted one? in the latter interpretation, what should be theassessment of the credit risk of the defaulted party: its credit risk right before its default?But what would be the exact meaning of “right before the default”? one day before? or oneweek? or one month? Another alternative discussed in Brigo, Buescu, and Morini (2012) is
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