View source for Credit valuation adjustment

{{Short description|Economics term}}
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* The mathematical concept as defined below;
* A part of the regulatory Capital and RWA ([[risk-weighted asset]]) calculation <ref>[[Basel Committee]] (2020). [https://www.bis.org/basel_framework/chapter/MAR/50.htm Credit valuation adjustment framework]</ref> introduced under [[Basel 3]];
* The CVA [[trading desk|desk]] of an investment bank, whose purpose is to:
** hedge for possible losses due to counterparty default;
** hedge to reduce the amount of capital required under the CVA calculation of [[Basel 3]];
* The "CVA charge". The hedging of the CVA desk has a cost associated to it, i.e. the bank has to buy the hedging instrument. This cost is then allocated to each business line of an investment bank (usually as a contra revenue). This allocated cost is called the "CVA Charge".
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A '''Credit valuation adjustment''' ('''CVA'''), 
{{efn|
A good introduction can be found in a paper by Michael Pykhtin and Steven Zhu.<ref name="Pykhtin & Zhu"/>
Karlsson et al. (2016) present a numerical efficient method for calculating expected exposure, potential future exposure and CVA for interest rate derivatives, in particular Bermudan [[swaptions]].<ref>Patrik Karlsson, Shashi Jain. and Cornelis W. Oosterlee (2016). [https://dx.doi.org/10.1080/1350486X.2016.1226144Counterparty "Credit Exposures for Interest Rate Derivatives using the Stochastic Grid Bundling Method"]. ''Applied Mathematical Finance''.</ref>
}}
in [[financial mathematics]], is an "adjustment" to a [[Derivative (finance)|derivative's]] price, as charged by a bank to a [[Counterparty#Financial services sector|counterparty]] to compensate it for taking on the [[counterparty credit risk|credit risk of that counterparty]] during the life of the transaction. 
"CVA" can refer more generally to several related concepts, as delineated aside.
The most common transactions attracting CVA involve [[interest rate derivative]]s, [[foreign exchange derivative]]s, and combinations thereof. 
CVA has a specific [[Capital requirement|capital charge]] under [[Basel III]], and may also result in earnings volatility under [[IFRS 13]], and is therefore managed by a specialized desk.
CVA is one of a family of related valuation adjustments, collectively [[xVA]];  for further context here see {{slink|Financial economics|Derivative pricing}}.

==Calculation==
{{further|XVA#Valuation adjustments}}

In [[financial mathematics]] one defines CVA as the difference between the risk-free portfolio value and the true [[Portfolio (finance)|portfolio value]] that takes into account the possibility of a [[counterparty]]'s default. 
In other words, CVA is the [[market value]] of [[Counterparty risk|counterparty credit risk]]. 
This price adjustment will depend on counterparty [[Credit spread (bond)|credit spread]]s as well as on the market risk factors that drive derivatives' values and, therefore, exposure. 
It is typically calculated under [[Monte Carlo methods in finance|a simulation framework]].<ref name="Hull_White">{{Cite web |author=John Hull |author-link=John C. Hull (economist) |date=May 3, 2016 |url=https://fincad.com/blog/valuation-adjustments-1 |title=Valuation Adjustments 1 |website=fincad.com}}</ref>
<ref name="Stein"/>
<ref>CVA calculation example: [https://web.archive.org/web/20191222022328/http://www.pricederivatives.com/en/derivatives-cva-example-monte-carlo-python/ Monte-Carlo with Python]</ref>
(Which can become computationally intensive; see {{efn|
According to the [[Basel Committee on Banking Supervision]]'s July 2015 consultation document regarding CVA calculations, if CVA is calculated using 100 timesteps with 10,000 scenarios per timestep, 1 million [[Monte Carlo methods in finance|simulations]] are required to compute the value of CVA. Calculating CVA risk would require 250 daily market risk scenarios over the 12-month stress period. CVA has to be calculated for each market risk scenario, resulting in 250 million simulations. These calculations have to be repeated across 6 risk types and 5 liquidity horizons, resulting in potentially 8.75 billion simulations.<ref>{{cite web|url=http://fixglobal.com/home/the-triple-convergence-of-credit-valuation-adjustment-cva/|author=Alvin Lee|title=The Triple Convergence Of Credit Valuation Adjustment (CVA)|publisher=Global Trading|date=17 August 2015|access-date=19 August 2015|archive-date=11 September 2015|archive-url=https://web.archive.org/web/20150911065450/http://fixglobal.com/home/the-triple-convergence-of-credit-valuation-adjustment-cva/|url-status=dead}}</ref>
}}.)

===Risk-neutral expectation===
Unilateral CVA is given by the [[risk-neutral measure|risk-neutral]] expectation of the discounted loss. The risk-neutral expectation can be written 
<ref name="Pykhtin & Zhu">{{cite web|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1032522 | title=A Guide to Modeling Counterparty Credit Risk|publisher=[[Global Association of Risk Professionals#History|GARP Risk Review]]|date=July 2007|author = Pykhtin, M.|author2=Zhu, S.}}</ref>
<ref>{{cite web|url=https://eba.europa.eu/documents/10180/950548/EBA+Report+on+CVA.pdf | title=EBA Report on CVA|publisher=EBA|date=25 February 2015| archive-url=https://web.archive.org/web/20150607093159/https://eba.europa.eu/documents/10180/950548/EBA+Report+on+CVA.pdf| archive-date=2015-06-07|author=[[European Banking Authority]]}}</ref>
as

: <math> \mathrm{CVA(T)} = E^Q[L^*] = \int_0^T E^Q\left[\left.LGD\frac{B_0}{B_t} E(t)\;\right|\;t=\tau\right] d\mathrm{PD}(0,t) </math>

where <math>T</math>&nbsp; is the [[Maturity (finance)|maturity]] of the longest transaction in the portfolio, <math> B_t </math> is the future value of one unit of the [[base currency]] invested today at the prevailing interest rate for maturity <math>t</math>, <math>LGD</math> is the [[loss given default]], <math>\tau</math> is the time of default, <math>E(t)</math> is the exposure at time <math>t</math>, and <math> \mathrm{PD}(s,t)</math> is the risk neutral probability of counterparty default between times <math>s</math> and <math>t</math>.
These probabilities can be obtained from the term structure of [[credit default swap]] (CDS) spreads.

===Exposure, independent of counterparty default===

Assuming independence between exposure and counterparty's [[credit quality]] greatly simplifies the analysis. Under this assumption this simplifies to

: <math> \mathrm{CVA} = LGD \int_0^T \mathrm{EE}^*(t)~d\mathrm{PD}(0,t) </math>

where <math>\mathrm{EE}^*</math> is the risk-neutral discounted expected exposure (EE):

: <math>\mathrm{EE}^*(t) = \mathbb{E}\left\lbrack{\frac{B_0}{B_t}~E(t)}\right\rbrack </math>

=== Approximation ===
The full calculation of CVA, as above, is via a [[Monte-Carlo simulation]] on all risk factors; this is computationally demanding. 
There exists a simple [[approximation]] for CVA, sometimes referred to as the "net current exposure method".<ref name="Stein"/>
This consists in: buying default protection, typically a [[Credit Default Swap]], netted for each counterparty; and the CDS price may then be used to [[Solver|back out]] the CVA charge.
<ref name="Stein">Harvey Stein (2012). [https://ieor.columbia.edu/files/seasdepts/industrial-engineering-operations-research/pdf-files/Stein_H_FeSeminar_Sp12.pdf "Counterparty Risk, CVA, and Basel III"]</ref>
<ref>{{cite web | url=http://www.pricederivatives.com/en/simple-cva-calculation-example-credit-valuation-adjustment-excel/ | title=Example calculation | date=7 October 2013 | archive-url=https://web.archive.org/web/20230603160413/http://www.pricederivatives.com/en/simple-cva-calculation-example-credit-valuation-adjustment-excel/ | archive-date=2023-06-03 }}</ref>

=== Accounting treatment ===
The CVA charge may be seen as an [[Adjusting entries|accounting adjustment]] made to [[reserve (accounting)|reserve]] a portion of profits on uncollateralized financial derivatives.
These reserved profits can be viewed as the [[net present value]] of the [[credit risk]] embedded in the transaction. 
Thus, as outlined, under [[IFRS 13]] changes in counterparty risk will result in earnings volatility; see {{slink|XVA#Accounting impact}} and next section.

==Function of the CVA desk==
{{further|XVA#Accounting impact|Financial risk management#Banking}}
In the course of trading and investing, Tier 1  [[Investment banking|investment banks]]  generate counterparty ''EPE'' and ''ENE'' (expected positive/negative [[potential future exposure|exposure]]). 
Whereas historically, this exposure was a concern of both the [[front office|Front Office]] trading desk and [[Investment banking#Middle office|Middle Office finance teams]],
increasingly CVA pricing and hedging is under the "ownership" of a centralized CVA [[trading desk|desk]].
<ref>Kenneth Kapner and Charles Gates (2016). [https://www.gfmi.com/articles/long-short-overview-xva/ "The Long and Short of It: An Overview of XVA"]. ''GFMI''</ref>
<ref>James Lee (2010). [https://web.archive.org/web/20120417014447/https://www.boj.or.jp/announcements/release_2010/data/fsc1006a5.pdf Counterparty credit risk pricing, assessment, and dynamic hedging], [[Citigroup Global Markets Japan|Citigroup Global Markets]]</ref>

In particular, this desk addresses volatility in earnings due to the abovementioned [[IFRS 13]] [[accounting standard]] requiring that CVA be considered in [[mark-to-market]] accounting.
The [[hedge (finance)|hedging]] here focuses on addressing changes to the counterparty's [[Credit spread (bond)|credit worthiness]], offsetting [[potential future exposure]] at a given quantile.
Further, since under [[Basel III]], banks are required to hold specific [[regulatory capital]] on the net CVA-risk,<ref name="Stein"/> the CVA desk is responsible also for managing (minimizing) the capital requirements under Basel.

== See also ==
* [[Financial derivative]]
* [[Potential future exposure]]
* [[XVA]]

==Notes==
{{Notelist}}

==References==
{{Reflist}}
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{{refend}}

==External links==
* [[Corporate Finance Institute]] (N.D.). [https://corporatefinanceinstitute.com/resources/knowledge/trading-investing/credit-valuation-adjustment-cva/ Credit Valuation Adjustment (CVA)]
*Laura Ballotta, Gianluca Fusai and Marina Marena (2016). [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2816355 "A Gentle Introduction to Default Risk and Counterparty Credit Modelling"]. {{ssrn|281635}}
[[Category:Actuarial science]]
[[Category:Mathematical finance]]
[[Category:Credit risk]]
[[Category:Monte Carlo methods in finance]]