XVA

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An X-Value Adjustment (XVA, xVA) is an umbrella term referring to a number of different “valuation adjustments” that banks must make when assessing the value of derivative contracts that they have entered into.[1][2] The purpose of these is twofold: primarily to hedge for possible losses due to other parties' failures to pay amounts due on the derivative contracts; but also to determine (and hedge) the amount of capital required under the bank capital adequacy rules. XVA has led to the creation of specialized desks in many banking institutions to manage XVA exposures.[3] [4]

## Contents

* 1 Context
* 2 Valuation adjustments
* 3 References
* 4 Bibliography

## Context

Historically,[5][6][7][8][9] (OTC) derivative pricing has relied on the Black–Scholes risk neutral pricing framework which assumes that funding is available at the risk free rate and that traders can perfectly replicate derivatives so as to fully hedge.[10] This, in turn, assumes that derivatives can be traded without taking on credit risk. During the financial crisis of 2008 many financial institutions failed, leaving their counterparts with claims on derivative contracts that were paid only in part. Therefore counterparty credit risk must also be considered in derivatives valuation,[11] and the risk neutral value is then adjusted correspondingly.

## Valuation adjustments

When a derivative's exposure is collateralized, 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 exposure is not collateralized then a credit valuation adjustment, or CVA, is subtracted from this value [5] (the logic: an institution insists on paying less for the option, knowing that the counterparty may default on its unrealized gain); this CVA is the discounted risk-neutral expectation value of the loss expected due to the counterparty not paying in accordance with the contractual terms. This is typically calculated under a simulation framework.[12]

Note that when transactions are governed by a master agreement that includes netting-off of contract exposures, then the expected loss from a default depends on the net exposure of the whole portfolio of derivative trades outstanding under the agreement rather than being calculated on a transaction-by-transaction basis. The CVA (and xVA) applied to a new transaction should be the incremental effect of the new transaction on the portfolio CVA.[12]

While the CVA reflects the market value of counterparty credit risk, additional Valuation Adjustments for debit, funding cost, regulatory capital and margin may similarly be added.[13][14] As with CVA, these results are modeled via simulation as a function of the risk-neutral expectation of (a) the values of the underlying instrument and the relevant market values, and (b) the creditworthiness of the counterparty. Note that the various XVA require careful and correct aggregation to avoid double counting.[6]

These adjustments include:[15]

* DVA, Debit Valuation Adjustment: analogous to CVA, the adjustment (increment) to a derivative price due to the institution's own default risk. If the default risk of both counterparties is properly taken into account in the CVA/DVA calculation, the CVA/DVA computed by one counterparty is equal to the DVA/CVA computed by the other counterparty, i.e. the price of the trade is unique and symmetric.
* FVA, Funding Valuation Adjustment, due to the funding implications of a trade that is not under Credit Support Annex (CSA), or is under a partial CSA; essentially the funding cost or benefit due to the difference between the funding rate of the bank's treasury and the collateral (variation margin) rate paid by a clearing house.[16]
* MVA, Margin Valuation Adjustment, refers to the funding costs of the initial margin specific to centrally cleared transactions. It may be calculated according to the global rules for non-centrally cleared derivatives rules.[17]
* KVA, the Valuation Adjustment for regulatory capital that must be held by the Institution against the exposure throughout the life of the contract (lately applying SA-CCR).

Other adjustments are also sometimes made including TVA, for tax, and RVA, for replacement of the derivative on downgrade.[13] 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. As mentioned, the various XVA require careful and correct aggregation to avoid double counting.

For a discussion as to the impact of xVA on banks overall balance sheets, return on equity, and dividend policy, see:[8]

## References

1. ↑ "X-Value Adjustment". Association of Corporate Treasurers. https://wiki.treasurers.org/wiki/X-Value_Adjustment.
2. ↑ "Valuation adjustments and their impact on the banking sector". PricewaterhouseCoopers. December 2015. https://www.pwc.com.au/pdf/xva-explained.pdf.
3. ↑ Butcher, Sarah (June 20, 2014). "CVA traders left stranded as XVA becomes big new acroynm". https://news.efinancialcareers.com/uk-en/175881/cva-traders-left-stranded-xva-becomes-big-new-acroynm.
4. ↑ International Association of Credit Portfolio Managers (2018). "The Evolution of XVA Desk Management"
5. ↑ 5.0 5.1 Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach, Didier Kouokap Youmbi, Bank of England – Prudential Regulation Authority
6. ↑ 6.0 6.1 XVAs: Funding, Credit, Debit & Capital in pricing. Massimo Morini, Banca IMI
7. ↑ Brigo, Damiano (November 2015), Nonlinear valuation and XVA under credit risk, collateral margins and Funding Costs, Université catholique de Louvain, [Course notes: Doctoral course, Université catholique de Louvain, 19-20 Nov 2015], https://cdn.uclouvain.be/public/Exports reddot/csam/documents/Damiano_Brigo_Nonlinear_Valuation_and_XVA_under_Credit_Risk_Collateral_Margins_and_Funding_Costs.pdf
8. ↑ 8.0 8.1 Claudio Albanese, Simone Caenazzo and Stephane Crepey (2016). Capital Valuation Adjustment and Funding Valuation Adjustment. Risk Magazine, May 2016.
9. ↑ Brigo, Damiano (November 5, 2011). "Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Netting, Collateral, Re-hypothecation, WWR, Basel, Funding, CCDS and Margin Lending". Department of Mathematics, King's College, London. http://frank-oertel-math.de/CCR_CVA_Basel_III_Dialogue_Damiano_Brigo_Nov_2011.pdf.
10. ↑ See Black–Scholes equation § Derivation of the Black–Scholes PDE; Rational pricing § The replicating portfolio
11. ↑ Kjølhede, Christian; Bech, Anders. "Post-Crisis Pricing of Swaps using xVAs". Aarhus University. http://pure.au.dk/portal-asb-student/files/96440392/Master_Thesis_Pure.pdf.
12. ↑ 12.0 12.1 John Hull (May 3, 2016). "Valuation Adjustments 1". https://fincad.com/blog/valuation-adjustments-1.
13. ↑ 13.0 13.1 XVA and Collateral: pricing and managing new liquidity risks. Andrew Green
14. ↑ XVA: About CVA, DVA, FVA and Other Market Adjustments, Discussion paper: Louis Bachelier Finance and Sustainable Growth Labex. Stephane Crepey
15. ↑ "XVAs Defined: The Profitability Puzzle". https://www.numerix.com/xvas-defined-profitability-puzzle.
16. ↑ "Funding Valuation Adjustment (FVA), Part 1: A Primer | Quantifi" (in en-US). 2014-03-20. https://www.quantifisolutions.com/funding-valuation-adjustment-fva-part-1-a-primer/.
17. ↑ ((Basel Committee on Banking Supervision)); ((Board of the International Organization of Securities Commissions)) (March 2015), Margin requirements for non-centrally cleared derivatives, Basel: Bank for International Settlements (BIS), ISBN 978-92-9197-063-6, https://www.bis.org/bcbs/publ/d317.htm

## Bibliography

* Andrew Green (2015). XVA: Credit, Funding and Capital Valuation Adjustments. John Wiley & Sons. ISBN 978-1-118-55678-8.
* Jon Gregory (2015). The xVA Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital (3rd ed.). 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.
* 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 (2021). 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 978-9813222748.
* 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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