FVA and MVA

Banks that hedge uncollateralized client trades with collateralized offsetting trades account for the funding cost of collateral and initial margin, as FVA and MVA, when they price trades and calculate valuation adjustments.

Funding cost of an uncollateralized trade and its offset

FVA (funding valuation adjustment) typically arises in the following case. A bank enters into a derivative with a client without collateral. At the same time, it enters into a collateralized offsetting trade on the same terms with a central counterparty or another financial institution¹. Call this bank J Corp. Consider a hypothetical example in which J Corp enters into an uncollateralized interest rate swap with a client and enters into the collateralized offsetting trade.

When the trade with the client has a positive value to J Corp, the offsetting trade has a negative value of the same amount. J Corp posts collateral on the offsetting trade but receives none from the client. If the value is 1 million yen, for example, J Corp posts 1 million yen of collateral and funds it at its own funding rate. This cost is FVA.

Figure description

The flows among the client, J Corp, and the hedge counterparty. The client and J Corp exchange interest rate swap payments without collateral. J Corp and the hedge counterparty exchange payments on an offsetting trade on the same terms, and J Corp posts variation margin (VM) and initial margin (IM) to the hedge counterparty. Payments run in both directions, and each arrow shows only the rough direction of the trade. J Corp posts VM when its trade with the client has a positive value to J Corp and receives VM when the value is negative. J Corp only posts IM. The cost to J Corp of funding these two kinds of collateral at its own funding rate is the source of FVA and MVA.

Collateral on a collateralized trade (variation margin) can be reused. Under the no-arbitrage condition, the discount rate is then the OIS rate, the risk-free rate. After the financial crisis, however, firms' funding rates no longer converged to LIBOR and came to differ with creditworthiness and other factors. It became standard to adjust the value of uncollateralized or partly collateralized trades using the firm's own funding rate².

The rate applied in FVA is the funding spread: the amount by which the firm's funding rate exceeds the risk-free rate. The risk-free part goes into discounting, so only the spread appears in the FVA formula³.

The cost of funding is considered separately for the side where the value is positive and the side where it is negative. The funding cost on the positive side is the FCA (funding cost adjustment), and the funding benefit on the negative side is the FBA (funding benefit adjustment)⁴. In the J Corp example, when the trade with the client has a negative value, J Corp receives collateral on the offsetting trade, and the ability to reuse it is a funding benefit. The value of FVA differs depending on whether it counts FCA only or FBA as well. This difference connects to the issue of double counting with DVA (see the section "Whether to include FVA in pricing and accounting"). Overview of XVA explains that FVA and MVA are the two funding-related adjustments among the five in XVA.

How FVA is calculated

FVA is found by taking, at each future date, the expected value of the positive part of the value that collateral does not cover (expected exposure), multiplying it by the funding spread, discounting, and adding up over the life of the trade. The continuous-time formula is as follows⁵.

$$\text{FVA} = \int_0^T s_F(u)\, E^Q\bigl[D(u)\,(V(u) - X(u))^+\bigr]\, du$$

Time 0 is the valuation date and $T$ is maturity. $s_F(u)$ is the funding spread, $V(u)$ is the risk-free value, and $X(u)$ is the collateral the firm holds (negative if it posts collateral). $D(u)$ is the discount factor, and $E^Q$ is the expectation under the risk-neutral probability. Finally, $x^+ = \max\{x, 0\}$, so only the positive part of the value minus collateral is taken.

In practice, the sum is taken over dates $t_1, \dots, t_n$. Let the expected exposure at date $t_i$ be $\mathrm{EE}(t_i) = E^Q[(V(t_i) - X(t_i))^+]$. Here, $\mathrm{EE}(t_i)$ is the value before discounting, and the discount factor $D(t_i)$ is multiplied separately. If the spread is constant over time, the formula is as follows.

$$\text{FVA} \approx \sum_{i=1}^{n} s_F \times \mathrm{EE}(t_i) \times D(t_i) \times (t_i - t_{i-1})$$

$t_i - t_{i-1}$ is the length of the interval in years. The approximation represents the funding cost of each interval by $\mathrm{EE}(t_i)$ and $D(t_i)$ at the end of the interval⁶. Where the CVA formula multiplies the default probability and the loss rate of each interval, the FVA formula multiplies the length of the interval and the spread.

The derivation of the continuous-time formula above starts from a hedging portfolio⁷ that combines the underlying, bonds issued by the firm and by the counterparty, and cash accounts corresponding to regulatory capital, initial margin, and collateral. The change in the value of the derivative and the change in the portfolio are then added. If the risk from the underlying's moves and from the counterparty's default is hedged, setting the remaining terms to zero gives a partial differential equation for the value of the derivative. Subtracting the equation that the risk-free value satisfies gives a partial differential equation that only the valuation adjustment terms satisfy, with a value of 0 at maturity. The Feynman-Kac theorem is applied to this equation. The hedging error at the firm's own default is determined by the funding strategy. Assuming a strategy of funding with the firm's own bonds with a different recovery rate, which does not fully hedge the firm's own default risk, the result takes the expected-value form of the formula above. The advantage of this method is that it can state clearly the hedging strategy and the funding strategy needed for the hedge. How the assumptions are treated is covered in the section "Assumptions and limits."

Deep diveDefinition of the discount factor

In the paper, the discount factor $D(u)$ is a factor that discounts at the risk-free rate plus the credit spread of the firm and of the counterparty. Some theories include the firm's own credit spread, but in practice it is sometimes left out, because a fall in the firm's own creditworthiness would then reduce CVA⁸. The examples below take the discount factor as a given value.

Worked example: FVA on a 5-year uncollateralized swap

Consider a hypothetical example in which J Corp enters into a 5-year uncollateralized interest rate swap with a client, with a collateralized offsetting trade. The EE for each year is the expected amount of collateral that J Corp funds over that year, and the values are figures from a simulation. ENE is the expected size of the negative part of the value, which corresponds to the collateral J Corp receives on the offsetting trade. The funding spread is 0.50% a year, and the discount factor uses 2% annual compounding (the reciprocal of 1.02 to the power of $t$), shown to six decimal places. The FCA for each year is 0.50% × EE × discount factor × 1 year.

The FCA for year 1 is 0.50% × 60 million yen × 0.980392 = 294 thousand yen⁹. The FCA total is 1.422 million yen, so if FVA counts FCA only, J Corp's value is 1.422 million yen lower than the risk-free value.

FVA is proportional to the spread. At a spread of 0.25%, FCA is 711 thousand yen, and at 1.00% it is 2.845 million yen. If J Corp's funding becomes cheaper and the spread narrows, FVA falls and the valuation gains. If funding becomes more expensive and the spread widens, FVA rises and the valuation loses.

FBA is found in the same form as FCA, applied to ENE.

$$\text{FBA} \approx \sum_{i=1}^{n} s_F \times \mathrm{ENE}(t_i) \times D(t_i) \times (t_i - t_{i-1})$$

With the ENE in the table, FBA is 521 thousand yen. If FBA is counted, FVA is 1,422 − 521 = 901 thousand yen, which is 521 thousand yen less than if it is not counted¹⁰.

MVA: funding cost of initial margin

MVA (margin valuation adjustment) is the adjustment that reflects, in value, the cost of funding the initial margin the firm posts. The idea is close to FVA, and the only change is that the amount multiplied is initial margin instead of the positive part of the value¹¹. Under the margin rules, initial margin is held by a trust bank or a custodian and generally cannot be reused. Unlike variation margin, it therefore earns no investment benefit¹². MVA is simply the accumulated cost of funding this amount¹³.

$$\text{MVA} = \int_0^T s_F(u)\, E^Q\bigl[D(u)\,\mathrm{IM}(u)\bigr]\, du$$

$\mathrm{IM}(u)$ is the initial margin posted to the counterparty at time $u$. The only difference from the FVA formula is that initial margin takes the place of the difference between value and collateral. With the same approximation as for FVA, and with the expected initial margin at date $t_i$ written $\mathrm{EIM}(t_i) = E^Q[\mathrm{IM}(t_i)]$, the formula is as follows.

$$\text{MVA} \approx \sum_{i=1}^{n} s_F \times \mathrm{EIM}(t_i) \times D(t_i) \times (t_i - t_{i-1})$$

Under the margin rules, initial margin is an amount that covers the largest change in value over the margin period of risk. Specifically, it is set by a value at risk with a 99% confidence level and a 10-day holding period. The rules do not specify the calculation method in detail, so SIMM is used as the industry-standard method¹⁴. The calculation of initial margin itself is covered in ISDA SIMM: The Standard Model for Initial Margin. Below, initial margin at each future date is taken as a given quantity.

What MVA needs is not today's initial margin but its expected value at each future date, that is, the profile. For an interest rate swap, future initial margin can be found by simulating the sensitivities to each tenor at future dates¹³.

Consider a hypothetical example in which the expected initial margin that J Corp posts on the offsetting trade declines each year. The values are as follows. The spread and discount factors are the same as in the FVA example.

The MVA for year 1 is 0.50% × 40 million yen × 0.980392 = 196 thousand yen. The MVA total is 573 thousand yen. Together with the FVA (FCA only) of 1.422 million yen, J Corp's value falls by 1.995 million yen. MVA is also proportional to the spread: it is 287 thousand yen at 0.25% and 1.146 million yen at 1.00%.

Exchanging initial margin reduces CVA, which covers the counterparty's default, and the cost of capital (KVA), but it creates MVA as a new cost¹⁵. A 2017 paper estimated the total XVA for a 5-year yen interest rate swap with an A-rated counterparty under different collateral terms. When the funding spread was small, at 15bp, the total was smaller the more initial margin was exchanged, up to the full SIMM amount. When the spread was large, at 30bp, the funding burden could outweigh the reduction in CVA and KVA, and the amount of initial margin that minimized the total was smaller than the full amount. With an AA-rated counterparty, in contrast to the A-rated case, exchanging the full amount could make the total larger¹⁶. The paper concludes that the best counterparty and collateral terms can be judged numerically from the totals for each combination of rating, collateral agreement, and spread.

Whether to include FVA in pricing and accounting

Whether to include FVA in value is decided separately for accounting and for pricing. The situation around 2017 was as follows.

Whether including MVA in prices would become established was expected to settle as the margin rules were introduced.

Reflecting FVA in the valuation adjustment makes the valuation move with changes in expected exposure and funding rates. When the gap between LIBOR and OIS widens, funding costs can rise and FVA can increase.

Whether FVA should be included in prices has some aspects on which there is no market consensus.

FVA and DVA require care so that the gain from the firm's own default is not counted twice¹⁹. In the J Corp example, FVA came to 901 thousand yen or 1.422 million yen depending on whether FBA was counted. One method does not count the funding benefit when the value is negative (FBA) in FVA and reflects the gain from the firm's own default through DVA alone. The 2017 paper takes this method²¹. There are other ways to calculate FVA, and the size of FVA changes with the one used.

Assumptions and limits: model risk

The values of FVA and MVA are estimates calculated from simulations of future exposure and initial margin and from funding assumptions. If the assumptions or the calculation method change, the values change.

The paper's appendix adopts a setting that does not fully hedge the firm's own default risk (semi-replication). Fully hedging it would require a strategy that dynamically adjusts the firm's own bonds with a different recovery rate. The appendix explains that Burgard and Kjaer regarded this as hard to carry out and proposed this funding strategy⁷.

The effect of valuation adjustments on hedge accounting effectiveness is summarized in the table in the section "Whether to include FVA in pricing and accounting."

What this article does not cover

Developments after 2017 are not covered on this page.

Notes

1. 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), pages 123–124 (in Japanese). The paper explains the case with the practice of the time, using LIBOR for the uncollateralized trade with the client and the OIS rate for the collateralized offsetting trade. ↩

2. 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). That the discount rate for collateralized trades is the OIS rate is shown in the paper under the no-arbitrage condition. ↩

3. The same source, Section 2 (2), pages 113–116, formula (2). $s_F$ corresponds to $s_B$ in the paper. "The amount by which the funding rate exceeds the risk-free rate" is wording read from the definitions of the symbols. ↩

4. The same source, footnote 7 on page 116, and Section 1 (page 110). The statement that the benefit to the side that reuses collateral corresponds to FBA combines these two places. ↩

5. The same source, Section 2 (2), formula (2), pages 115–116. The paper's FVA carries a negative sign, so the sign is reversed to give an amount of 0 or more. The paper's $t$ is set to 0, and $DF(u)$ is replaced by the given discount factor $D(u)$. ↩

6. The discretization, and multiplying $D(t_i)$ separately, are set to match formula (2). The $\mathrm{EE}$ in the CVA formula in "Overview of XVA" is a value discounted to today, so the treatment of discounting differs. ↩

7. The same source, Section 2 (2), pages 114–115, and Appendix 1, pages 145–150. In the order of the sentences in the derivation paragraph, the hedging portfolio is formula (A-5), the partial differential equation for the valuation adjustment terms is formula (A-18), and the funding strategy is formula (A-20); semi-replication is in footnote 33 on page 149. ↩ ↩²

8. The same source, footnote 6 after formula (2) in Section 2 (2), page 116. The definition of the paper's $DF(u)$ is on page 116. ↩ ↩²

9. Each row of the table is rounded to the nearest thousand yen. Totals are the sums of the unrounded values (FCA 1,422.34, FBA 521.23, and MVA 573.08 thousand yen), rounded to the nearest thousand yen. The total can therefore differ by 1 thousand yen from the sum of the rounded rows (the rows of FBA sum to 520). ↩

10. The FBA formula is not in the paper. The paper gives only the formula for FVA without FBA and notes in a footnote that there is a method that counts FBA (footnote 7 on page 116). Here the formula is a simplification in which $\mathrm{EE}$ in the FCA formula is replaced by $\mathrm{ENE}$. ↩

11. 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), pages 112–113 (in Japanese). The paper writes that MVA is the cost of funding initial margin and is similar in idea to FVA, that some financial institutions are considering reporting MVA in financial reporting as part of FVA, and that whether including it in prices becomes established is thought to settle as the margin rules are introduced. ↩ ↩²

12. The same source, Section 4 (1), page 121. From the viewpoint of bankruptcy remoteness, initial margin is required to be held by a trust bank or a custodian, and reuse of the collateral is in principle prohibited. ↩

13. The same source, Section 4 (3), page 123. The valuation formula for MVA is formula (2) of Section 2 (2). The paper treats MVA as always a negative valuation adjustment. It writes that future initial margin for an interest rate swap can be found by simulating the sensitivities to each tenor. ↩ ↩² ↩³ ↩⁴

14. The same source, Section 4 (1), pages 121–122. The paper writes that the margin rules do not specify the value-at-risk calculation method in detail, so the industry body ISDA has published SIMM as the industry-standard model. ↩

15. The same source, Section 6 (1) (b), page 138. The paper writes that a high level of initial margin reduces CVA and KVA but creates MVA as a new funding cost. ↩

16. The same source, Section 6 (2) (a), pages 138–139. The target is a 5-year yen interest rate swap. As in figure 5, the paper compares the total XVA at funding spreads of 15bp and 30bp while varying the credit limit relative to initial margin. The difference between A and AA ratings is on page 139. ↩ ↩²

17. The same source, Section 2 (1), page 112. The paper writes that FVA is not required to be included in fair value in accounting but many financial institutions already recognize it, and cites a survey in which 24 companies reported FVA in their 2014 financial reports. ↩

18. The same source, Section 7, page 141. The paper writes that leading banks reflect CVA, DVA, and FVA in prices and are starting to estimate MVA as well, and that no agreement has been reached on including KVA in prices. ↩

19. The same source, Section 2 (2), page 114. The paper writes that what to mark to market and hedge needs separate consideration, that there was no market consensus on some aspects of which valuation adjustments to build into prices, and that care is needed not to double count the gain from the firm's own default between DVA and FVA. ↩ ↩²

20. 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 25 (Section 5), page 147 (in Japanese). The paper first notes that recognizing FVA is not required in accounting, and then describes the effect when it is reflected. ↩

21. The same source, footnote 7 on page 116. As one way to avoid double counting the gain from the firm's own default, the paper takes the method that does not count FBA. It writes that there are other ways to calculate FVA, citing the explanation in Adachi [2015]. ↩ ↩²

22. 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). ↩