Overview of XVA

People who price derivatives and people who manage counterparty credit risk use XVA to consider a trade's price, profit and loss, CVA hedging, and regulatory capital.

The five adjustments that make up XVA

XVA is the collective name for the adjustments that derivative valuation came to need after the 2007–08 financial crisis. After the crisis, adjusting prices for the counterparty's creditworthiness became market practice. Firms' funding rates also no longer moved in line with interbank rates. As capital and margin rules tightened, firms began to include the costs of funding and capital in valuation as well¹. The five main adjustments are as follows.

CVA is the difference between the value assuming the counterparty does not default and the value that takes the possibility of default into account². It grows as the counterparty's creditworthiness falls, and it lowers the value. It is similar in nature to a loan loss allowance³.

DVA applies the same idea to the firm's own default. When the value is negative and the firm is the paying side, a default of the firm would relieve it of part of its payments, so DVA raises the value⁴. Calculated with the same method and inputs, the firm's DVA has the same size as, and the opposite sign to, the CVA that the counterparty calculates on the firm⁵.

FVA reflects in the value the funding cost of a trade without collateral. A typical case is a bank that enters into an uncollateralized derivative with a client and a collateralized offsetting trade on the same terms with a central counterparty or another financial institution. The bank funds the collateral it posts on the offsetting trade at its own funding rate, and FVA reflects that cost⁶.

MVA is the cost of funding the initial margin the firm posts. Under margin rules, financial institutions whose OTC derivative business exceeds a set size must exchange initial margin on trades that are not cleared⁷.

KVA reflects the cost of holding regulatory capital (the cost of capital). At the time of a 2017 paper by the Bank of Japan's Institute for Monetary and Economic Studies, there was no market consensus on how far to include KVA in prices. Financial institutions that did not include it in prices still used KVA for the return threshold a trade must clear (the hurdle rate) and for managing the cost of capital⁸.

The size of each adjustment varies widely with the credit ratings of the counterparty and the firm, the collateral agreement, the spread the firm pays over base rates when it borrows (the funding spread), and other factors. If the collateral agreement reduces the value not covered by collateral, CVA and FVA become smaller, and the larger the funding spread, the larger FVA and MVA become⁹.

How the adjustments modify the risk-free value

XVA starts from the value found without considering credit risk or collateral (the risk-free value) and adds or subtracts the adjustments to it¹⁰. With each adjustment expressed as an amount of 0 or more, the adjusted value is as follows.

$$\hat{V} = V - \text{CVA} + \text{DVA} - \text{FVA} - \text{MVA} - \text{KVA}$$

$V$ is the risk-free value and $\hat{V}$ is the adjusted value. The adjustments that lower the value are subtracted, and the one that raises it is added¹¹.

The adjustments are calculated not trade by trade but per netting agreement (per netting set). This is because, if the counterparty defaults, the receivables and payables of the trades under the same agreement can be offset against each other. Collateral is also exchanged per netting agreement¹². For trades where both sides post collateral and the exposure is fully covered by collateral, no CVA or DVA adjustment is made. For trades without collateral, CVA and DVA are reflected on the part not covered by collateral¹³. How collateral terms relate to the discounting curve is covered in Collateral and Discounting.

Deep diveExpected-value formulas for the five adjustments

The 2017 paper by the Bank of Japan's Institute for Monetary and Economic Studies gives each adjustment as an expected value, on the assumption that all cash flows are replicated¹⁴. Let B be the firm and C the counterparty. The credit spreads of the counterparty and the firm are $\lambda_C$ and $\lambda_B$, and their recovery rates are $R_C$ and $R_B$. The risk-free value at time $u$ is $V(u)$, and the collateral the firm holds is $X(u)$ (negative if the firm has posted it). The discount factor is $DF(u)$, and the expectation under the risk-neutral probability is $E^Q$. Finally, $x^+ = \max\{x, 0\}$.

$$\text{CVA} = (1 - R_C) \int_t^T \lambda_C(u) E^Q[DF(u)(V(u) - X(u))^+] du$$ $$\text{DVA} = (1 - R_B) \int_t^T \lambda_B(u) E^Q[DF(u)(X(u) - V(u))^+] du$$ $$\text{FVA} = \int_t^T s_B(u) E^Q[DF(u)(V(u) - X(u))^+] du$$ $$\text{MVA} = \int_t^T s_B(u) E^Q[DF(u) I(u)] du$$ $$\text{KVA} = \int_t^T \gamma_K(u) E^Q[DF(u) K(u)] du$$

$s_B$ is the firm's funding spread, $I(u)$ is the initial margin posted to the counterparty, $\gamma_K$ is the cost of capital, and $K(u)$ is the required capital. CVA and FVA apply to the positive part of the value not covered by collateral, and DVA applies to the negative part. MVA and KVA are proportional to the size of the initial margin and of the required capital, respectively.

How to calculate CVA

CVA is found by estimating, for each future period, the amount the firm expects to lose if the counterparty defaults in that period, and adding up the results. The estimate for each period is the product of the expected exposure (EE), the probability that the counterparty defaults in that period (PD), and the loss given default (LGD). Assuming the counterparty's default and market moves are unrelated to each other, the formula is as follows¹⁵.

$$\text{CVA} = \text{LGD} \times \sum_{j=1}^{n} \text{EE}(t_j) \times \text{PD}(t_{j-1}, t_j)$$

$\text{EE}(t_j)$ is the expected value of the positive part of the value at time $t_j$, discounted to today. $\text{PD}(t_{j-1}, t_j)$ is the probability that the counterparty defaults between times $t_{j-1}$ and $t_j$. LGD is the share lost on default, equal to one minus the recovery rate.

EE is found by simulating the future paths of market factors such as interest rates and exchange rates over many scenarios and taking the average of the positive part of the netting set's future value¹⁶. Finding it precisely requires simulation by the Monte Carlo method¹⁷.

PD is estimated as a value under the risk-neutral probability from market spreads such as the counterparty's credit default swap (CDS) spread. The LGD assumed in that estimate is also used as the LGD in CVA¹⁶. When there is no liquid CDS referencing the counterparty, the spread on the counterparty's bonds or a CDS referencing a similar company is used instead. Some financial institutions build credit curves by rating, region, and industry and assign them to counterparties¹⁸. How to find probabilities from CDS spreads is covered in Pricing a CDS.

Worked example: a five-year uncollateralized trade with Company F

Consider a hypothetical example in which a bank has a five-year interest rate swap with Company F without a collateral agreement. EE takes one assumed value for each one-year interval, the probability that Company F defaults in each year is 1.0%, and LGD is 60% (a recovery rate of 40%).

CVA is 2.1 million yen, and the value with CVA reflected is 2.1 million yen lower than the risk-free value. Because PD and LGD are the same in every interval, multiplying the total EE of 350 million yen by 1.0% and 60% gives the same result.

CVA moves in proportion to its inputs. If Company F's creditworthiness falls and PD rises to 2.0% each year, CVA rises to 4.2 million yen, and the increase of 2.1 million yen is a valuation loss for the bank. Conversely, if PD falls to 0.5%, CVA becomes 1.05 million yen, and the decrease of 1.05 million yen is a valuation gain. Even if Company F's creditworthiness does not change, CVA changes when interest rates move and EE changes¹⁶. If the two sides start exchanging collateral and EE halves, CVA also halves to 1.05 million yen.

Accounting and regulatory treatment

In accounting, International Financial Reporting Standard (IFRS) 13, Fair Value Measurement, requires the fair value of a derivative to reflect the credit risk of the firm and the counterparty, which means including CVA and DVA¹⁹. US accounting standards (Topic 820) also require, in principle, that DVA be reflected in fair value²⁰. Fair value does not have to include FVA, but at the time of the 2017 paper, many financial institutions also recognized it for accounting purposes²¹.

In regulation, Basel III, published in 2010, introduced a capital charge for changes in the value of CVA, separate from the capital held against counterparty default. The Basel Committee on Banking Supervision (BCBS) agreed on the finalized framework in December 2017. The finalization revised the methods of measurement into two approaches, the standardized approach (SA-CVA) and the basic approach (BA-CVA), plus a simplified method for banks with small derivative business²². Under capital adequacy rules, DVA is deducted from Common Equity Tier 1 (CET1) to prevent moral hazard⁴.

Managing CVA in practice: the CVA desk and hedging

According to a 2018 article written by a former Financial Services Agency staff member and a Bank of Japan staff member, advanced financial institutions set up a CVA desk. The CVA desk is independent of the other trading desks and manages CVA centrally. Besides hedging CVA, it also handles risk management, pricing, and reviews of collateral agreements. If the firm revises its collateral agreements to exchange variation margin and initial margin, exposure falls and the amount of CVA can be kept down²³.

Changes in the counterparty's credit spread are hedged with the counterparty's CDS. When the CDS referencing the counterparty is illiquid, the desk hedges instead with index CDS or with CDS referencing companies in the same industry or region. Changes in EE caused by market factors are hedged with futures, swaps, and options²³.

To use the regulatory standardized approach (SA-CVA), a bank must have a CVA desk (or a similar function) responsible for CVA risk management and hedging. This is one of the minimum requirements²⁴.

Assumptions and limitations

The formula in "How to calculate CVA" assumes that the counterparty's default and market moves are unrelated to each other. In reality, exposure can grow just when the counterparty's creditworthiness falls, and this is called wrong-way risk. To capture wrong-way risk, EE must be replaced with an expected value conditional on the counterparty's creditworthiness, which makes accurate valuation difficult²⁶.

When including the adjustments in prices, care is needed not to double count between adjustments. Hedging CVA with eligible hedges reduces the CVA capital charge, so including both CVA and KVA in the price at the start of a trade counts the cost of hedging counterparty risk twice. DVA and FVA also need care so that the benefit from the firm's own default is not counted twice²⁷. There are several ways to calculate FVA, and some leave out the funding benefit when the value is negative (Funding Benefit Adjustment, FBA)²⁸. At the time of the 2017 paper, the market had not fully agreed on which adjustments to include in prices²⁷.

Notes

1. 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), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 1, page 110 (in Japanese). ↩

2. 前田壮一 and 長谷川達也, "ＣＶＡリスク計測の枠組みの見直し" (Revising the framework for measuring CVA risk), 金融財政事情 (February 12, 2018 issue), "信用評価調整（ＣＶＡ）とは" (What is credit valuation adjustment (CVA)?), page 32 (in Japanese). ↩

3. Takashi Kawakami, "ヘッジ取引におけるデリバティブ信用評価調整の影響についての考察" (A study of the effects of credit valuation adjustments of derivatives on hedging transactions), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2018), Section 2 (1) and (2), pages 123–124 (in Japanese). ↩

4. 前田壮一 and 長谷川達也, "ＣＶＡリスク計測の枠組みの見直し" (Revising the framework for measuring CVA risk), 金融財政事情 (February 12, 2018 issue), "信用評価調整（ＣＶＡ）とは" (What is credit valuation adjustment (CVA)?), pages 32–33 (in Japanese). ↩ ↩²

5. Takashi Kawakami, "ヘッジ取引におけるデリバティブ信用評価調整の影響についての考察" (A study of the effects of credit valuation adjustments of derivatives on hedging transactions), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2018), Section 2 (2), page 125 (in Japanese). ↩

6. The same document, Section 2 (1), pages 123–124. The paper explains this with an example from that time, using LIBOR for the uncollateralized trade with the client and the OIS rate for the collateralized offsetting trade. ↩

7. 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), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 2 (1), page 112 (in Japanese). The paper says margin rules were introduced in stages from September 2016. That exchanging margin became mandatory for trades that are not cleared is in Section 3 (1), page 116. ↩

8. The same document, Section 2 (1), page 113, and Section 7, page 141. ↩

9. 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), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), the abstract (page 109) and Section 7 (page 141) (in Japanese). How CVA, FVA, and MVA are proportional to the value not covered by collateral, the funding spread, and initial margin is in equation (2) in Section 2 (2) (pages 115–116). ↩

10. The same document, Section 2 (2), page 115, and Appendix 1, page 148. The paper calls the price without valuation adjustments the "risk-free price" and expresses the adjusted value as the sum of the risk-free value and the valuation adjustments. ↩

11. Equation (2) in Saito (2017) (pages 115–116) defines CVA, FVA, MVA, and KVA as values of 0 or less and DVA as a value of 0 or more, and adds all of them to the risk-free value. Because this page expresses each adjustment as an amount of 0 or more, the adjustments that lower the value carry a minus sign. ↩

12. 前田壮一 and 長谷川達也, "ＣＶＡリスク計測の枠組みの見直し" (Revising the framework for measuring CVA risk), 金融財政事情 (February 12, 2018 issue), "信用評価調整（ＣＶＡ）とは" (What is credit valuation adjustment (CVA)?), page 32 (in Japanese). Saito (2017), Section 2 (2), page 115, also says XVA is calculated per netting set. ↩

13. Takashi Kawakami, "ヘッジ取引におけるデリバティブ信用評価調整の影響についての考察" (A study of the effects of credit valuation adjustments of derivatives on hedging transactions), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2018), Section 2 (1), page 124 (in Japanese). ↩

14. 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), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 2 (2), pages 114–116 (in Japanese). The replication assumption is on page 114 and equation (2) is on pages 115–116. The sign is reversed to give values of 0 or more. ↩

15. Takashi Kawakami, "ヘッジ取引におけるデリバティブ信用評価調整の影響についての考察" (A study of the effects of credit valuation adjustments of derivatives on hedging transactions), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2018), the first term of equation (1) in Section 2 (2), page 125 (in Japanese). The paper's $EPE(t_j)$, $LGD_C$, and $PD_C(t_{j-1}, t_j)$ correspond to $\text{EE}(t_j)$, LGD, and $\text{PD}(t_{j-1}, t_j)$ on this page. The paper writes the adjustment with a negative sign, but this page expresses it as an amount of 0 or more and, treating LGD as the same in every period, takes it outside the sum. ↩

16. 前田壮一 and 長谷川達也, "ＣＶＡリスク計測の枠組みの見直し" (Revising the framework for measuring CVA risk), 金融財政事情 (February 12, 2018 issue), "実務におけるＣＶＡ管理" (Managing CVA in practice), page 33 (in Japanese). ↩ ↩² ↩³

17. Takashi Kawakami, "ヘッジ取引におけるデリバティブ信用評価調整の影響についての考察" (A study of the effects of credit valuation adjustments of derivatives on hedging transactions), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2018), Section 2 (2), pages 125–126 (in Japanese). ↩ ↩²

18. The same document, Section 2 (2), page 126. ↩

19. Takashi Kawakami, "ヘッジ取引におけるデリバティブ信用評価調整の影響についての考察" (A study of the effects of credit valuation adjustments of derivatives on hedging transactions), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2018), Section 1, page 120 (in Japanese). ↩

20. 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), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 1, page 110 (in Japanese). ↩

21. 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), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 2 (1), page 112 (in Japanese). ↩

22. Financial Services Agency and Bank of Japan, "「信用評価調整（CVA）リスクの最低所要自己資本」の概要" (Overview of "minimum capital requirements for credit valuation adjustment (CVA) risk") (February 2018), "１．これまでの経緯" (1. Background), page 2, and "３．CVAリスクの見直しの概要" (3. Overview of the revision of CVA risk), page 4 (in Japanese). That the finalization document was published in December 2017 is in 前田壮一 and 長谷川達也, "ＣＶＡリスク計測の枠組みの見直し" (Revising the framework for measuring CVA risk), 金融財政事情 (February 12, 2018 issue), page 32 (in Japanese). ↩

23. 前田壮一 and 長谷川達也, "ＣＶＡリスク計測の枠組みの見直し" (Revising the framework for measuring CVA risk), 金融財政事情 (February 12, 2018 issue), "実務におけるＣＶＡ管理" (Managing CVA in practice), page 33 (in Japanese). ↩ ↩²

24. The same document, "標準的方式（ＳＡ－ＣＶＡ）" (Standardized approach (SA-CVA)), page 36. ↩ ↩² ↩³

25. The same document, "適格ヘッジの範囲拡大" (Wider range of eligible hedges), page 35. Under the earlier rules, proxy hedges with CDS on similar companies and hedges of market factors were not recognized as eligible hedges, and hedges of market factors could even increase required capital (page 34). ↩

26. Takashi Kawakami, "ヘッジ取引におけるデリバティブ信用評価調整の影響についての考察" (A study of the effects of credit valuation adjustments of derivatives on hedging transactions), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2018), footnote 8 on page 125 (in Japanese). ↩

27. 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), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 2 (2), page 114 (in Japanese). ↩ ↩²

28. The same document, footnote 7 on page 116. ↩