Capital Valuation Adjustment (KVA)

KVA comes up when trades are priced, when their profitability is judged, and when the capital cost of a portfolio is managed¹.

What KVA expresses

When a bank holds derivatives, it must hold at least the amount of capital set by regulation to cover counterparty default and market moves. That capital has to earn a return for shareholders. KVA (Capital Valuation Adjustment) expresses, as a present value, the cost of the regulatory capital needed to keep a trade until maturity²³. It is one of the XVA adjustments and lowers the value from the risk-free value. The five adjustments, their directions, and their formulas are covered in Overview of XVA.

Consider a hypothetical example in which a trade requires 100 million yen of regulatory capital for one year and the required return on capital (the hurdle rate) is 10%. The capital cost for that year is 10 million yen. If the capital stayed the same for five years, the cost would be 10 million yen each year for five years. In practice the capital for a trade changes over time, so you estimate the capital year by year and add the results. This estimate of how capital changes over time is the capital profile.

How KVA is calculated

KVA is found by multiplying the expected required capital at each future date by the hurdle rate, discounting, and summing to maturity. A 2017 paper by Yuichi Saito of the Bank of Japan's Institute for Monetary and Economic Studies (the Saito paper, below) gives the following formula⁴.

$$\mathrm{KVA} = \int_t^T h(u)\, \mathbb{E}^{Q}\!\left[D(u)\, K(u)\right] du$$

$h(u)$ is the return required on capital. It corresponds to the Saito paper's cost of capital $\gamma_K$. The Saito paper calls $\gamma_K$ in equation (2) the "cost of capital"; here $h$ is called the hurdle rate. $K(u)$ is the required capital (regulatory capital) at time $u$, $D(u)$ is the discount factor, and $\mathbb{E}^{Q}$ is the expectation under the risk-neutral probability. Because $h$ and $K$ are multiplied in the formula, KVA doubles if either $h$ or the capital doubles.

When you calculate by hand over intervals, let $K_i$ be the expected (average) required capital in interval $i$, $D(t_i)$ the discount factor at the end of the interval $t_i$, and $\Delta t_i$ the length of the interval in years. Then sum as follows⁵.

$$\mathrm{KVA} \approx \sum_{i=1}^{n} h \times K_i \times D(t_i) \times \Delta t_i$$

Deep diveThe KVA formula in the semi-replication model of Green and others

The paper by Green, Kenyon, and Dennis derives KVA by adding a capital term to the semi-replication framework of Burgard and Kjaer, which is used for CVA and FVA⁶. In the form where KVA is separated from FVA, credit spreads enter the discounting.

$$\mathrm{KVA}' = \int_t^T h(u)\, e^{-\int_t^u \left(r(s)+\lambda_B(s)+\lambda_C(s)\right)ds}\, \mathbb{E}_t\!\left[K(u)\right] du$$

$r$ is the risk-free rate, and $\lambda_B$ and $\lambda_C$ are the credit spreads of the institution itself and of the counterparty. The paper by Green and others defines the quantity with a negative sign; it is restated here as an amount of 0 or more. If a portion $\phi$ of the capital can be used for funding, the cost of capital falls by that portion. The paper takes $\phi = 0$ as the base case and also includes a $\phi = 1$ case in its numerical examples. It also writes that tying the issuance or buyback of capital to individual trades or strategies appears to be explicitly prohibited by Basel III⁷.

What determines the capital profile

On the capital adequacy ratio side, the required capital $K(u)$ is the total regulatory capital attached to the trade. Green and others add three pieces: the capital for counterparty default (CCR capital), CVA capital, and the capital for market risk⁸. On the leverage ratio side, KVA is found separately and compared with the capital adequacy ratio result, and the larger one is taken. The Saito paper covers three: CCR capital, CVA capital, and the leverage ratio. It leaves out market risk capital, treating it as offset by hedge trades⁹.

CCR capital is RWA multiplied by the required capital level $c$. The Saito paper writes RWA as $12.5 \times w \times \mathrm{EAD}$, where 12.5 is the reciprocal of 8% and converts an amount of capital into RWA. The RW here corresponds to the Saito paper's $12.5 \times w$. The Saito paper's $c$ is the Basel minimum of 8.0% with the various capital buffers added¹⁰. Methods for finding EAD include SA-CCR. The basics of RWA and the capital adequacy ratio are covered in Risk-Weighted Assets (RWA).

All of these capital amounts change over time. Both the standardized-approach EAD and the leverage exposure are functions of the future market values of derivatives and collateral, so you simulate future values to find the capital at each date¹⁴. Market risk capital also changes as the remaining term shortens¹². To know the cost over the life of a trade, you need the capital over its whole life, not just the current capital¹⁵.

Whether the capital adequacy ratio or the leverage ratio capital dominates is handled in the Saito paper by calculating both KVAs and taking the larger¹⁶.

$$\mathrm{KVA} = \max\left(\mathrm{KVA}_{\mathrm{CCR}} + \mathrm{KVA}_{\mathrm{CVA}},\ \mathrm{KVA}_{\mathrm{Lev}}\right)$$

In the Saito paper's numerical examples, RWA capital is larger under the standardized approach, and leverage ratio capital becomes larger when CCR capital is small under the internal model method (IMM). Which one dominates is considered to depend on how regulatory capital is measured and on the institution's portfolio¹⁷.

Worked example: a 5-year uncollateralized trade with Company K

Consider a hypothetical example in which a bank enters into a 5-year interest rate swap with Company K with no collateral agreement. A simple capital profile is assumed that declines toward maturity.

The assumptions are as follows. The annual average EAD starts at 750 million yen and falls by 150 million yen a year. The RW used in the RWA calculation is 50%, and the required capital level $c$ is the minimum of 8%. CVA capital is half of CCR capital, and market risk capital is one sixth of CCR capital. The hurdle rate is 10%, and discounting uses 1% a year compounded, with the discount factor taken at the end of each interval.

For 0–1 year, RWA is $750 \times 50\% = 375$ million yen, and CCR capital is $8\% \times 375 = 30$ million yen. The CCR capital in the other intervals comes from the same formula. The discount factor is $D(t_i) = 1.01^{-i}$, rounded to six decimal places.

Figure description

Required capital falls in steps, by 10 million yen each year. The discounted values follow the same steps, slightly below required capital. The values are the same as in the tables above. KVA is the sum of the discounted values multiplied by the hurdle rate.

KVA is $10\% \times 146.5688 = 14.6569$ million yen, or about 14.66 million yen. The value of the trade with Company K is about 14.66 million yen lower than the risk-free value. By type of capital, the CCR capital part is about 8.7941 million yen, the CVA capital part is about 4.3971 million yen, and the market risk capital part is about 1.4657 million yen.

The leverage ratio capital is also compared. Suppose the leverage exposure falls from 1 billion yen by 100 million yen a year, and $c_L$ is the minimum of 3%. The capital is then 30 million, 27 million, 24 million, 21 million, and 18 million yen. At the same hurdle rate, the KVA on the leverage ratio side is about 11.6772 million yen. The left term of the max in the previous section, excluding market risk, is about 13.1912 million yen (8.7941 + 4.3971). This is larger than the leverage ratio's 11.6772 million yen. The larger one is on the capital adequacy ratio side. Market risk capital is outside the paper's scope, so it is left out of this comparison.

Next, change the inputs.

KVA is proportional to both the hurdle rate and the capital. If CVA capital halves, the CVA capital part of about 4.3971 million yen becomes about 2.1985 million yen, and KVA is about 12.46 million yen. This calculation does not include the cost of the means used to halve CVA capital. The KVA figures in the table are rounded to two decimal places, and each difference is the difference of the unrounded values, rounded to two decimal places.

Pricing it in or using it as a benchmark

The formula for KVA was presented in a 2014 paper. The 2017 Saito paper, citing Sherif and Chambers, says there was no market consensus on how to treat it. It describes some institutions as reflecting a certain share of KVA in prices and others as not doing so. Reporting KVA in accounts drew more caution than reporting FVA or MVA¹. Even where KVA was not used in pricing, it was used as a profitability benchmark for trades (the hurdle rate) and as a tool for managing the cost of capital of a portfolio¹.

When KVA is included in the price, the trade's price changes by the amount of KVA. When it is not, the price stays the same, and KVA is used inside the bank as a benchmark for whether the trade is profitable. In the Company K example, suppose the present value of the revenue the bank earns from the trade, before considering KVA, is 18 million yen.

If the present value of revenue exceeds KVA, the cost of capital is covered with something left over; if it is lower, the cost is not covered. Even for the same trade, the conclusion differs with how regulatory capital grows and with the internal hurdle rate. The present value of revenue − KVA is $18 - 14.6569 = 3.3431$, $18 - 17.5883 = 0.4117$, and $18 - 21.9853 = -3.9853$, rounded to two decimal places in the table.

The hurdle rate is not always visible from outside. Green and others describe the cost of capital as an internal parameter that the board of directors sets in consultation with shareholders, and say that the return on equity (ROE) target a bank states is a close proxy. The numerical examples of Green and others use 10%¹⁸. The numerical examples in the Saito paper assume a cost of capital of 6.0% on RWA, with reference to the ROE of large Japanese financial institutions¹⁹. The 10% in the worked example is a value chosen to make the calculation easy to follow.

CVA adjustment, CVA capital, and double counting

CVA (the valuation adjustment) and CVA capital are different things. CVA is the adjustment that reflects changes in the counterparty's credit quality in the value. CVA capital is the capital that regulation requires to cover changes in CVA¹¹. The required capital $K(u)$ in KVA includes CVA capital, so any means of reducing CVA also affects KVA.

When CVA is hedged with a regulatory eligible hedge, the CVA capital charge falls, and so does the KVA on that CVA capital. In that case, some argue that including both CVA and KVA in the price at inception overstates the cost of hedging counterparty risk²¹. The calculation in the example above, where halving CVA capital lowers KVA by about 2.20 million yen, expresses this relationship in numbers.

A hedge trade does not only reduce capital. In the example by Green and others, a bank enters an uncollateralized 10-year interest rate swap with a customer and a collateralized 10-year swap with a market counterparty to hedge market risk. In that case, the hedge trade also carries a small amount of CCR capital and CVA capital, while the market risk capital for the whole falls substantially²². A trade that hedges KVA also requires capital²².

KVA changes when regulation changes

KVA is found from the expected regulatory capital, so its value changes when the regulatory calculation method changes. At the time of the Saito paper (2017), changes to the calculation method were planned for each of these rules. The standardized approach for CCR capital was scheduled to change from the current exposure method (CEM) to SA-CCR. A calculation method based on SA-CCR had also been proposed for the leverage exposure. CVA capital was heading toward a revision of its framework²³. The CVA framework that the BCBS later finalized revised the measurement into two approaches, the standardized approach (SA-CVA) and the basic approach (BA-CVA), plus a simplified method²⁴.

The difference in KVA between methods can be large. The Saito paper compares KVA of uncollateralized trades in a numerical example of a 5-year yen interest rate swap (payer) using market data of December 30, 2015. The conditions are that the institution and the counterparty are both rated A with credit spreads of 50bp and the cost of capital is 6.0%. The KVA under SA-CCR was about twice the KVA under the internal model method (IMM). When variation margin was exchanged, the IMM KVA came close to zero, but the SA-CCR KVA did not fall that far. The paper also considers a proposal to apply a floor based on the standardized approach to IMM. It gives a simple estimate: if the floor is 75% of the standardized approach, the IMM KVA becomes 1.5 times as large²⁵. These figures are results under the Saito paper's conditions and are not levels after the rules were finalized.

Green and others also say that the dates and content of regulatory implementation are often uncertain, which is one reason capital is hard to estimate¹⁵.

Assumptions and limits

The size of KVA depends on two estimates. One is the capital profile, which is determined by the simulation of future values and collateral and by the regulatory method used. The other is the hurdle rate, which is set internally. If either differs, the KVA of the same trade changes. The worked example on this page assumes a simple profile that declines toward maturity. The shape of an actual profile varies with the type of trade, the collateral agreement, and the movements of interest rates and other variables.

The form of the KVA formula changes with the valuation framework. In the formula of Green and others, credit spreads enter the discounting. The cost of capital also changes depending on whether part of the capital is regarded as usable for funding. Green and others say that how capital is used for funding is determined in part by the institution's internal policy⁷.

Hedging KVA has the same problems as other hedging. You can view the extra capital required by market moves as being covered by the profit earned on the trade, but the hedge trades also need capital²². As of 2017, there was also no market consensus on whether to include KVA in prices or how to report it in accounting¹.

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 113) (in Japanese). The passage on price reflection, accounting reports, and use as a hurdle rate cites Sherif and Chambers [2015]. ↩ ↩² ↩³ ↩⁴

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 5 (3) (page 130) (in Japanese). The paper describes KVA as the expected discounted present value of the cost of raising (the cost of capital for) the regulatory capital needed until the trade matures. Section 1 (page 110) says that KVA has begun to be considered as a valuation adjustment for the cost of raising regulatory capital (the cost of capital). ↩

3. Andrew Green, Chris Kenyon, and Chris Dennis, "KVA: Capital Valuation Adjustment" (arXiv:1405.0515v3, version of October 24, 2014), Section 3.1 (page 10). The paper describes the cost of capital as the return paid to shareholders on regulatory capital, in line with their decision to commit capital to derivatives trading. ↩

4. 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), equation (2) (pages 115–116) (in Japanese). The Saito paper defines KVA with a negative sign; here it is restated as an amount of 0 or more. $\gamma_K$ corresponds to $h$ and $DF(u)$ to $D(u)$; see also Section 5 (2) (page 125). ↩

5. The sum over intervals is a simple form written for the worked example. The Saito paper's formula is an integral, and how the value within an interval is taken and the timing of discounting vary by implementation. ↩

6. Section 2 of the paper by Green and others, equations (27)–(34) (pages 8–9). Equation (34) corresponds to KVA′, and equation (32) is the form in which the capital term includes the credit spreads. ↩

7. The paper by Green and others. The share $\phi$ of capital used for funding is in Section 1 (pages 2–3). That the share depends on internal policy is on page 3, and the fall in the cost of capital is just after equation (65) (page 18). ↩ ↩²

8. Section 4 of the paper by Green and others, equation (36) (page 10). The paper splits capital into three parts: $K = K_{MR} + K_{CCR} + K_{CVA}$. ↩

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 (April 2017), Section 5 (1) (pages 124–125) and Table 2 (in Japanese). The paper excludes market risk capital because hedge trades reduce it substantially. ↩

10. The same source, Section 5 (2) A, equations (10) and (11) (page 125). The paper says the required capital level $c$ is the Basel minimum of 8.0% with the various capital buffers taken into account. The 12.5 in the paper's RWA is the reciprocal of 8%, and the RW here corresponds to the Saito paper's $12.5 \times w$ ($w$ is the paper's risk weight). ↩ ↩²

11. The same source, Section 5 (1) (page 124). The paper says Basel III (introduced from 2013) added a capital charge for the risk of CVA changes, separately from CCR capital. The 2010 publication is in 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), page 2 (in Japanese). ↩ ↩²

12. Section 4.1 of the paper by Green and others (pages 10–12). Market risk capital is calculated on a net basis for the whole portfolio (page 11), and under the standardized approach it is a function of trade characteristics such as maturity, coupon, and notional (page 12). ↩ ↩²

13. 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 5 (2) C (pages 129–130), equation (19) (in Japanese). The required level is the minimum of 3.0% plus a certain add-on, and the paper says the add-on for G-SIBs was under discussion at the BCBS (as of the paper). ↩

14. The same source, Section 5 (3) (page 130). The standardized-approach EAD and the leverage exposure are expressed as functions of the market values of derivatives and collateral, and the IMM KVA is complex because computing the integrand itself needs an expected exposure simulation. American Monte Carlo is in Section 5 (5). ↩ ↩²

15. Section 1 of the paper by Green and others (page 2). Among the reasons capital is hard to price, the second is that the cost of capital over the whole life, not just the current capital, is needed, and the fourth is that the dates and content of regulatory implementation are often uncertain. ↩ ↩²

16. 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 (April 2017), Section 6 (1), equation (30) (page 137) (in Japanese). The settings of the numerical example, the costs of capital and required levels, are on the same page. ↩

17. The same source, Section 6 (2) B (page 140). Under the standardized approach RWA dominates, and under IMM the leverage ratio dominates (in the A-rated case). Which one dominates is considered to depend on how regulatory capital is measured and on the institution's portfolio. ↩

18. Section 3.1 of the paper by Green and others (page 10). The cost of capital is an internal parameter that the board of directors sets in consultation with shareholders and may not be visible from outside, and a close proxy is each bank's ROE target. A report by Roland Berger and Nomura (Reboul and others, 2014) gave 10% as the ROE target of a typical group. The numerical examples use 10% for $\gamma_K$. ↩

19. 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 5 (6) A (page 135) (in Japanese). The paper assumes a cost of capital on RWA of 6.0%, with reference to the ROE of large Japanese financial institutions, and a required level of 13.5%, which is the minimum requirement with the various capital buffers taken into account. ↩

20. Section 3 (page 9) and Section 4.1 (page 11) of the paper by Green and others. Regulatory capital is a portfolio-level requirement, and pricing a derivative requires looking at the effect of the trade together with its hedge trades, not the new trade alone. Because market risk capital is calculated on a net basis, capital has to be allocated from the requirement for the whole portfolio. ↩ ↩²

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 (April 2017), Section 2 (2) (page 114) (in Japanese). The point about double counting CVA and KVA cites Gregory [2015] and Sherif and Chambers [2015]. ↩

22. Section 3 of the paper by Green and others (page 9). The paper states that hedge trades require capital and gives the 10-year interest rate swap example. The hedge trade's capital is usually a small share (written as O(1/10)), so the requirement converges quickly. Hedging KVA also requires capital. ↩ ↩² ↩³

23. 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 5 (1) (page 124) (in Japanese). ↩

24. 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), "３．CVAリスクの見直しの概要" (3. Overview of the revision of CVA risk), page 4 (in Japanese). The revision into two measurement approaches and one simplified method follows this document's explanation. ↩

25. 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 (April 2017), Section 5 (6) (pages 134–136) (in Japanese). The settings are on pages 134–135 and the results on page 136. The 75% floor is in footnote 31 on page 136. ↩