# XVA

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An **X-Value Adjustment** (**XVA**, **xVA**) is a generic term referring collectively to a number of different “Valuation Adjustments” in relation to derivative instruments held by banks.^{[1]}^{[2]} The purpose of these is twofold: primarily to hedge for possible losses due to counterparty default; but also, to determine (and hedge) the amount of capital required under Basel III. For a discussion as to the impact of xVA on the bank's overall balance sheet, return on equity, and dividend policy, see: ^{[3]} XVA has, in many institutions, led to the creation of specialized desks.^{[4]} Note that the various XVA require careful and correct aggregation without double counting.^{[5]}

## Context[edit]

Historically,^{[6]}^{[5]}^{[7]}^{[3]} (OTC) derivative pricing has relied on the Black-Scholes' risk neutral pricing framework, under the assumptions of funding at the risk free rate and the ability to perfectly replicate derivatives so as to fully hedge; see Black–Scholes equation § Derivation; Rational pricing § The replicating portfolio. This, in turn, is built on the assumption of a credit-risk-free environment. Post financial crisis of 2008, therefore, counterparty credit risk must also be considered in the valuation,^{[8]} and the risk neutral value is then adjusted correspondingly. See Financial economics § Derivative pricing for further discussion.

## Valuation adjustments[edit]

These calculations in overview: When the deal is collateralized then the "fair-value" is computed as before, but using the overnight index swap (OIS) curve for discounting. (The OIS is chosen here as it reflects the rate for overnight secured lending between banks, and is thus considered a good indicator of the interbank credit markets.) When the deal is not collateralized then a credit valuation adjustment, or **CVA**, is added to this value; ^{[6]} essentially, the discounted risk-neutral expectation of any loss due to the counterparty not performing - typically calculated under a simulation framework.^{[9]}

Note that when transactions are governed by a master agreement that includes netting, then the expected loss from a default depends on the whole portfolio, and cannot be calculated on a transaction-by-transaction basis. The CVA (xVA) applied to a new transaction should be the incremental effect of the new transaction on portfolio CVA.^{[9]}

While the CVA reflects the market value of counterparty credit risk, *additional* Valuation Adjustments for Debit, Funding, regulatory capital and margin may similarly be added.^{[10]}^{[11]} As for CVA, these results are modeled via simulation as a function of the risk-neutral expectation of (a) values of the underlying instrument, and the relevant market values, and (b) creditworthiness of the counterparty. As above, the various XVA require careful and correct aggregation without double counting.

These adjustments:^{[12]}

**DVA**, Debit Valuation Adjustment: analogous to CVA, an adjustment (reduction) to a derivative price due to the institution's own default risk.**FVA**, Funding Valuation Adjustment, due to the funding implications of a trade that is not under a perfect Credit Support Annex (CSA); essentially the difference between the rate paid for the collateral to the bank's treasury, and the rate paid by the clearinghouse.**KVA**, the Valuation Adjustment for regulatory capital through the life of the contract (lately applying SA-CCR).**MVA**, Margin Valuation Adjustment, refers to the costs specific to centrally cleared transactions, such as adjustments for initial margin and variation margin. It may also be calculated per the global Non-centrally cleared derivatives rules.^{[13]}

Other adjustments are also sometimes made including TVA, for tax and RVA, for replacement of the derivative on downgrade. ^{[10]} FVA may be decomposed into FCA for receivables and FBA for payables - where FCA is due to self funded borrowing spread over Libor, and FBA due to self funded lending. Relatedly, LVA represents the specific liquidity adjustment, while CollVA is the value of the optionality embedded in a CSA to post collateral in different currencies. CRA , the collateral rate adjustment, reflects the present value of the expected excess of net interest paid on cash collateral over the net interest that would be paid if the interest rate equaled the risk-free rate.

## References[edit]

**^**X-Value Adjustment Association of Corporate Treasurers**^**XVA explained. PWC, December 2015- ^
^{a}^{b}Claudio Albanese, Simone Caenazzo and Stephane Crepey (2016). Capital Valuation Adjustment and Funding Valuation Adjustment.*Risk Magazine*, May 2016. **^**CVA traders left stranded as XVA becomes big new acroynm, efinancialcareers.com. Sarah Butcher, 2014.- ^
^{a}^{b}XVAs: Funding, Credit, Debit & Capital in pricing. Massimo Morini, Banca IMI - ^
^{a}^{b}Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach, Didier Kouokap Youmbi, Bank of England – Prudential Regulation Authority **^**Nonlinear valuation and XVA under credit risk, collateral margins and Funding Costs. Prof. Damiano Brigo, Univ. Catholique de Louvain**^**Post-Crisis Pricing of Swaps using xVAs, Christian Kjølhede & Anders Bech, Master thesis, Aarhus University- ^
^{a}^{b}VALUATION ADJUSTMENTS 1. John Hull and Alan White - ^
^{a}^{b}XVA and Collateral: pricing and managing new liquidity risks. Andrew Green **^**XVA: About CVA, DVA, FVA and Other Market Adjustments, Discussion paper: Louis Bachelier Finance and Sustainable Growth Labex. Stephane Crepey**^**XVAs Defined: The Profitability Puzzle, numerix**^**Margin requirements for non-centrally cleared derivatives. BCBS and IOSCO

## Bibliography[edit]

- Andrew Green (2015).
*XVA: Credit, Funding and Capital Valuation Adjustments*. Wiley. ISBN 978-1-118-55678-8. - Jon Gregory (2015).
*The xVA Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital (3rd Edition)*. Wiley. ISBN 978-1-119-10941-9. - Chris Kenyon and Andrew Green (Eds) (2016).
*Landmarks in XVA: From Counterparty Risk to Funding Costs and Capital*. Risk Books. ISBN 978-1782722557.CS1 maint: extra text: authors list (link) - Roland Lichters, Roland Stamm and Donal Gallagher (2015).
*Modern Derivatives Pricing and Credit Exposure Analysis: Theory and Practice of CSA and XVA Pricing, Exposure Simulation and Backtesting*. Palgrave Macmillan. ISBN 978-1137494832. - Dongsheng Lu (2015).
*The XVA of Financial Derivatives: CVA, DVA and FVA Explained*. Palgrave Macmillan. ISBN 978-1137435835. - Ignacio Ruiz (2015).
*XVA Desks - A New Era for Risk Management*. Palgrave Macmillan UK. ISBN 978-1-137-44819-4. - Antoine Savine and Jesper Andreasen (2018).
*Modern Computational Finance: Scripting for Derivatives and XVA*. Wiley. ISBN 978-1119540786. - Donald J. Smith (2017).
*Valuation in a World of CVA, DVA, and FVA: A Tutorial on Debt Securities and Interest Rate Derivatives*. World Scientific. ISBN 9813222743. - Alexander Sokol (2014).
*Long-Term Portfolio Simulation - For XVA, Limits, Liquidity and Regulatory Capital*. Risk Books. ISBN 978-1782720959. - Osamu Tsuchiya (2019).
*A Practical Approach to XVA*. World Scientific. ISBN 978-9813272750.

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