Wrong-Way Risk

In risk management, credit management by counterparty and CVA calculations call for estimating whether, when the market moves, the counterparty's creditworthiness also worsens at the same time. The question is then how far this differs from an estimate that treats exposure and the counterparty's default separately.

Falling creditworthiness and growing exposure

Wrong-way risk is the risk that a counterparty's probability of default (PD) and the exposure to that counterparty move in the same direction¹. The more likely the counterparty is to default, the more you stand to lose if it does. What Is Counterparty Credit Risk? covers how a counterparty's default makes you lose the positive part of the market value.

Consider a hypothetical example in which a bank enters into a forward with Bank L, a bank in an emerging market, to buy US dollars and sell the currency of L's country. If that currency falls, the market value of the forward rises in the bank's favor and the exposure grows. A fall in the currency is, in general, also a time when L's creditworthiness declines². Because the scenarios in which the exposure grows overlap with those in which the counterparty is more likely to default, the loss can be larger than an estimate that treats the two separately.

The opposite case, in which the exposure to a counterparty becomes smaller as its creditworthiness declines, is called right-way risk. The exposure is small in the scenarios where the counterparty defaults, so CVA is smaller than when the two are unrelated³.

General and specific wrong-way risk

Wrong-way risk is divided into two types, according to what links the exposure to the counterparty's creditworthiness⁴.

General wrong-way risk arises when the counterparty's PD is positively correlated with general market risk factors. Specific wrong-way risk arises when, because of the nature of a transaction, the exposure to a particular counterparty moves in the same direction as that counterparty's PD¹.

Specific wrong-way risk can be contained by avoiding such transactions as far as possible. For this reason, it is often general wrong-way risk that poses a problem for risk management⁴. A relationship in which the value of collateral tends to fall at the same time as the counterparty is more likely to default is also a form of wrong-way risk⁵.

Examples that surfaced in the 2008 financial crisis

In the financial crisis of 2007–08, concerns about the creditworthiness of financial institutions that had sold CDS protection, such as the monoline insurers and AIG, spread. For institutions that had bought CDS from them, both the sellers' PD and the exposure on the trades rose sharply at the same time. CVA increased, and the market as a whole recorded huge losses in market value⁶. Both forms surfaced in the crisis: general wrong-way risk, in which a sudden market deterioration raises the counterparty's PD, and specific wrong-way risk, in which the exposure to monolines and similar entities rises together with their PD⁷.

According to a report by the Basel Committee on Banking Supervision (BCBS), only one third of the counterparty credit risk losses in this crisis came from direct losses on counterparty defaults. The rest were mark-to-market losses from higher CVA⁸.

The CVA formula without independence

The basic CVA formula takes, for each interval, the discounted expected exposure (EE), multiplies it by the probability of default in that interval and by the loss given default (LGD), and adds up the results. The formula assumes that the scenarios used to calculate EE are unrelated to the counterparty's default.

With wrong-way risk, this assumption fails. The EE used for the contribution of the interval $(t_{i-1}, t_i]$ is therefore replaced by its expected value conditional on the counterparty defaulting in that interval⁹.

$$\mathrm{EE}^{\mathrm{WWR}}(t_i) = \mathbb{E}\!\left[\,E(t_i) \mid \tau \in (t_{i-1}, t_i]\,\right]$$

Here $E(t_i)$ is the exposure at time $t_i$ in a given scenario (the positive part of the market value, net of collateral), and $\tau$ is the time of the counterparty's default. The contribution of the interval is as follows, where $D(t_i)$ is the discount factor and $\mathrm{PD}_i$ is the probability of default in the interval.

$$\mathrm{LGD} \times D(t_i) \times \mathrm{EE}^{\mathrm{WWR}}(t_i) \times \mathrm{PD}_i$$

If exposure and default are unrelated, $\mathrm{EE}^{\mathrm{WWR}}(t_i)$ equals the unconditional $\mathrm{EE}(t_i)$. If $\mathrm{EE}^{\mathrm{WWR}}(t_i)$ is larger than $\mathrm{EE}(t_i)$, CVA is larger by that amount, and if it is smaller (right-way risk), CVA is smaller⁹.

Worked example: CVA in two scenarios

Consider a hypothetical example in which a trade with Bank L is viewed one year from now, discounting is ignored, and LGD is 60%. The exposure differs between two scenarios, so the example can be followed by hand. With a 90% probability the market is "normal," and with a 10% probability it is "stressed" (a scenario in which the currency of L's country falls sharply). The exposure in one year is 1 billion yen in the normal scenario and 5 billion yen in the stressed scenario. The EE calculated from market factors alone is 0.9 × 1 + 0.1 × 5 = 1.4 billion yen.

Compare three cases in which L's PD is related to the scenario in different ways. In every case, the overall PD across the two scenarios is fixed at 2.0%.

The average exposure when L defaults is found by adding up "probability × PD in that scenario × exposure" over the scenarios and dividing by the overall PD. CVA is that value multiplied by LGD. For the wrong-way case, the intermediate values are as follows.

The average exposure at default is 46.0 million ÷ 0.02 = 2.3 billion yen, which is $\mathrm{EE}^{\mathrm{WWR}}$. CVA is 0.6 × 46.0 million = 27.6 million yen. The results for the three cases are as follows.

Even with the same overall PD of 2.0%, CVA in the wrong-way case is 10.8 million yen (about 64%) larger than with no relationship. In the right-way case it is 4.32 million yen (about 26%) smaller. In this example, even if the PD is derived from market spreads such as CDS, what you obtain is taken to be the overall PD of 2.0% across the scenarios¹⁰.

If EE is built from market factors alone and multiplied in, CVA comes to 16.8 million yen even in the wrong-way case, and the difference from 27.6 million yen drops out of the valuation.

The next figure shows CVA as the PD in the stressed scenario is varied while the overall PD is held at 2.0%. The PD in the normal scenario is set by the PD in the stressed scenario so that the overall PD stays at 2.0%.

Figure description

The line shows CVA as the PD in the stressed scenario moves from 0% to 8%. CVA is 12 million yen at 0% and 31.2 million yen at 8%, so the line slopes upward. Two vertical lines are drawn. The one at a stressed PD of 2.0% marks the no-relationship case, with CVA of 16.8 million yen, and the one at 6.5% marks the wrong-way case, with CVA of 27.6 million yen. In the right-way case the stressed PD is 0.2%, and its CVA of 12.48 million yen lies near the left end of the line. Even with the same overall PD, CVA grows the more the PD is concentrated in the scenarios where the exposure is large.

Deep diveTracing the example above with the scenario-weight method

The simplest way to estimate $\mathrm{EE}^{\mathrm{WWR}}$ by simulation is to collect the paths in which the counterparty defaults at the same point in time and average them, but this needs a huge number of paths. A lighter approximation is the scenario-weight method, which weights each path by its default probability¹¹. Generate $M$ scenarios for the market factors, and let $\mathrm{PD}^{(m)}$ be the default probability of path $m$ in the interval and $E^{(m)}(t_i)$ its exposure.

$$\mathrm{EE}^{\mathrm{WWR}}(t_i) \approx \sum_{m=1}^{M} w^{(m)} E^{(m)}(t_i), \qquad w^{(m)} = \frac{\mathrm{PD}^{(m)}}{\sum_{k=1}^{M} \mathrm{PD}^{(k)}}$$

Express the wrong-way example above with 10 paths. Nine normal paths each have a PD of 1.5% and an exposure of 1 billion yen, and one stressed path has a PD of 6.5% and an exposure of 5 billion yen. The PDs add up to 9 × 1.5% + 6.5% = 20%. Each normal path has a weight of 1.5 ÷ 20 = 0.075 (0.675 for the nine), and the stressed path has a weight of 6.5 ÷ 20 = 0.325. $\mathrm{EE}^{\mathrm{WWR}}$ is 0.675 × 1 + 0.325 × 5 = 2.3 billion yen, which matches the table.

How wrong-way risk is modeled

A model has to settle three things: a model of the counterparty's default, a model of the exposure, and a model of the dependence between them (including abrupt changes in that dependence)¹².

Default models include structural models, which treat the counterparty as defaulting when the value of its assets falls below its debt, and reduced-form models, which treat default as a Poisson event given from outside the model¹³. There are several ways to model the dependence. The main ones are as follows.

Besides models, there is a way to find general wrong-way risk with stress scenarios. Guidance issued by the US banking supervisors in June 2011 gives this as a standard way. The bank builds and applies stress scenarios that find wrong-way risk in the portfolio, records the counterparties with a large impact, and assesses whether those counterparties' creditworthiness also worsens in the same scenarios¹⁸.

Capital requirements and supervision

The notice issued by the Financial Services Agency that applies to Japanese banks (the version updated on August 24, 2026) provides as follows for banks that have received approval to use the expected exposure method (internal model method).

The last two rows are provisions that serve as the criteria the Commissioner of the Financial Services Agency uses when examining a bank for approval to use the expected exposure method²².

The α of 1.4 takes general wrong-way risk into account only in part²³. A January 2010 overview document from the Financial Services Agency and the Bank of Japan presents a proposal, made in response to the crisis, to add a framework that captures general wrong-way risk on top of this and a framework that requires capital for some forms of specific wrong-way risk²⁴.

The overall methods for calculating exposure at default are covered in Overview of the Regulatory Treatment of Counterparty Risk.

Assumptions and limits

As of the July 2016 paper, there is no standard wrong-way risk model agreed in the market. The scenarios in which wrong-way risk surfaces are rare, so there is little data to test models against. Parameters calibrated on data from calm periods cannot reproduce large losses, and the risk tends to receive little attention in economic expansions. The paper also points out that abrupt changes in dependence under stress are hard to infer from historical data²⁶.

Hedging has limits too. Factors including model uncertainty are hard to hedge with instruments traded in the market, so it is common to protect against them with capital²⁷. A June 2010 lecture by Yoshihiko Uchida says that the correlation between creditworthiness and variables such as interest rates is itself hard to hedge. The lecture adds that delta hedging with a model that assumes the correlation can hedge to some extent the fall in price caused by creditworthiness and interest rates moving together (cross-gamma)²⁸.

Receiving collateral does not remove wrong-way risk. In a model where the exchange rate and the counterparty's default intensity jump together, the exposure of a cross-currency swap can jump during the margin period of risk and exceed the amount of collateral. The result shows that exchanging variation margin may not reduce CVA enough, and that it needs to be complemented by measures such as initial margin²⁹.

Some analyses also leave out the lowering of CVA under right-way risk, out of regard for conservatism in risk management. The paper cited here limits its scope to general wrong-way risk³.

The worked example above is a simplification limited to two market scenarios. An actual valuation uses many scenarios, with the methods described above.

Notes

1. Office of the Comptroller of the Currency et al., "Interagency Supervisory Guidance on Counterparty Credit Risk Management" (June 2011), III.6 "Wrong-Way Risk" (page 10). ↩ ↩²

2. Tetsuya Adachi, 末重拓己, and Toshinao Yoshiba, "CVA における誤方向リスク・モデルの潮流" (Trends in wrong-way risk models for CVA, our translation), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (July 2016), Chapter 2 (2) "対象とする WWR" (WWR covered), page 39 (in Japanese). The source's example is a forward contract, between a US bank and a bank in an emerging market, to buy US dollars and sell the local currency. ↩

3. The same paper, Chapter 2 (2), page 39. The source calls the case where there is a positive interdependence between the exposure and the counterparty's credit level right-way risk, and excludes it from its analysis out of regard for conservatism in risk management. ↩ ↩²

4. The same paper, Chapter 2 (2), page 39. The source's example of specific WWR is a CDS contract in which the reference entity is an affiliate of the protection seller. ↩ ↩²

5. Yuichi Saito, "金融規制の複合的影響を考慮したXVA" (XVA taking into account the combined effects of financial regulations, our translation), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Chapter 3 (1) "ニ. 適格担保" (d. Eligible collateral) and footnote 10 (page 118) (in Japanese). ↩

6. Adachi et al., "CVA における誤方向リスク・モデルの潮流" (July 2016), Chapter 1 (page 36). ↩

7. Financial Services Agency and Bank of Japan, "バーゼル委市中協議文書 カウンターパーティ・リスクの取扱いの強化の概要" (Overview of the Basel Committee consultative document on strengthening the treatment of counterparty risk, our translation) (January 2010), "１０．誤方向リスク（wrong-way risk）への対応①（IMM行）" (10. Responding to wrong-way risk (1) (IMM banks)) (page 25) (in Japanese). ↩

8. Adachi et al., "CVA における誤方向リスク・モデルの潮流" (July 2016), footnote 1 (page 36). The source cites a BCBS report and writes that, of the counterparty credit risk losses in the financial crisis since the summer of 2007, direct losses from default were only one third of the total. ↩

9. The same paper, Chapter 2 (3), equations (3) to (6) (pages 41–42). In the source the discount factor is a random variable; here it is taken outside as a fixed value $D(t_i)$. The notation corresponds to $\mathrm{EPE}^{\mathrm{WWR}}$ in the source, and the relation between its size and CVA is on page 42. That it does not necessarily hold at every time point under a copula is in footnote 12 (page 42). ↩ ↩²

10. The same paper, Chapter 2 (1) "CVA の構成" (Components of CVA) (page 38). Accounting and capital regulation require the default probability to be calculated from variables observable in the market, such as CDS and corporate bond spreads. "What you obtain is the overall PD" is how this worked example sets things up. ↩

11. The same paper, Chapter 2 (4) "条件付期待エクスポージャーの計算方法" (How to calculate conditional expected exposure) (page 43), equation (7). Each path's exposure is weighted by its default probability divided by the sum of the paths' PDs. The symbol is $w^{(m)}$, as in the source. ↩

12. The same paper, Chapter 2 (3) (page 42). It summarizes the key points of modeling in three items. ↩

13. The same paper, Chapter 3 (1) "Cpty のデフォルト事象のモデル化" (Modeling the counterparty's default event) (page 44). ↩

14. The same paper, Chapter 5 (1) "デフォルト強度／確率に関するパラメトリック・アプローチ" (Parametric approach to default intensity and probability) and footnote 47 (page 62). The methods of Hull and White [2012] and Ruiz, Pachon, and del Boca [2015]. ↩

15. The same paper, Chapter 4 (1) (page 56) and footnote 12 (page 42). ↩

16. The same paper, Chapter 5 (2) "デフォルト時ジャンプ・モデル" (Jump-at-default model) and footnote 48 (pages 63 and 65). The methods of Redon [2006] and Mercurio and Li [2015]. ↩

17. The same paper, Chapter 5 (3) "ストレス・シナリオを反映した条件付期待エクスポージャー" (Conditional expected exposure reflecting stress scenarios) (page 68). The method of Turlakov [2013]. ↩

18. "Interagency Supervisory Guidance on Counterparty Credit Risk Management" (June 2011), footnote 16 (page 11). ↩

19. Financial Services Agency, "銀行法第十四条の二の規定に基づき、銀行がその保有する資産等に照らし自己資本の充実の状況が適当であるかどうかを判断するための基準" (Standards for determining whether a bank's capital adequacy is appropriate in light of the assets it holds, under Article 14-2 of the Banking Act, our translation; FSA Notice No. 19 of 2006, version updated on August 24, 2026), Article 79-3, paragraph 2, item 1 (page 206) (in Japanese). ↩

20. The same Notice, Article 79-3, paragraph 5, item 2 (page 209). ↩

21. The same Notice, Article 79-3, paragraphs 13 and 14 (page 211). ↩

22. The same Notice, the introductory part of Article 79-3-3 (page 213). It provides that, in approving the use of the expected exposure method, the Commissioner of the Financial Services Agency examines whether the bank's internal management framework is expected to meet the standards in each item. ↩

23. Financial Services Agency and Bank of Japan, "バーゼル委市中協議文書 カウンターパーティ・リスクの取扱いの強化の概要" (January 2010), "１０．誤方向リスク（wrong-way risk）への対応①（IMM行）", "現行の取扱い" (current treatment) (page 25), and Yuichi Saito, "金融規制の複合的影響を考慮したXVA" (April 2017), footnote 21 (page 127) (in Japanese). ↩

24. Financial Services Agency and Bank of Japan, "バーゼル委市中協議文書 カウンターパーティ・リスクの取扱いの強化の概要" (January 2010), "１０．誤方向リスク（wrong-way risk）への対応①（IMM行）", "今般の危機を踏まえた問題点" (problems in light of the recent crisis) (page 25) (in Japanese). This is a proposal in a consultative document. ↩

25. "Interagency Supervisory Guidance on Counterparty Credit Risk Management" (June 2011), III.6 "Wrong-Way Risk" (page 11) and footnote 17. Footnote 17 gives, as examples of a legal connection, a single-name credit derivative on a related reference entity and a financing transaction in which the counterparty posts an affiliate's securities as collateral. ↩

26. Adachi et al., "CVA における誤方向リスク・モデルの潮流" (July 2016), Chapter 1 (pages 36–37). ↩

27. The same paper, footnote 3 (pages 36–37). ↩

28. Yoshihiko Uchida, "カウンターパーティー・リスクの管理とCVAの活用" (Management of counterparty risk and the use of CVA, our translation) (lecture material for a workshop at the Financial System and Bank Examination Department's Center for Advanced Financial Technology, Bank of Japan, June 2010), "CVAを用いたCCRのヘッジ" (Hedging CCR with CVA) (page 21) (in Japanese). The author states that it reflects his personal views and not the official view of the Bank of Japan. ↩

29. Yuichi Saito, "金融規制の複合的影響を考慮したXVA" (April 2017), footnote 13 (page 121). It cites the results of Adachi, Sueshige, and Yoshiba [2016b]. ↩