CVA and DVA

CVA and DVA are used by the CVA desk, which handles the pricing and valuation of derivatives, and by those who monitor risk and swings in profit and loss, to build the counterparty's credit risk into the value of trades that are close to unsecured. CVA and DVA are also read to explain how profit and loss moves once the adjustment is booked.

The loss that CVA represents

CVA is a loss only when the value of the trade to the firm is positive at the moment the counterparty defaults. If the value is positive, the firm's claim is repaid only up to the recovery rate. If the value is negative, the firm owes the full amount¹. The loss is the positive part of the value at default times the loss given default (LGD, which is 1 minus the recovery rate).

Looking only at the case where the counterparty defaults first, this loss has the form of a sold call option with a strike of 0 on the value at default².

The definitions and directions of the five adjustments, an overview of the CVA desk, and a basic 5-year calculation example are in Overview of XVA. The symbols are the same as there: $\mathrm{EE}$ for the expected value of the positive part of the value, $D$ for the discount factor, $\mathrm{PD}_i$ for the default probability in an interval, and $\mathrm{LGD}$ for the loss given default.

Derivation of the CVA formula

Let $\tau$ be the counterparty's default time and $T$ the maturity, and take expectations under the risk-neutral probability. Write $S(t)$ for the probability that the counterparty survives to time $t$ and $\lambda(t)$ for the hazard rate, so that $S(t) = e^{-\int_0^t \lambda(s)\,ds}$. The probability that default occurs in the short period $dt$ just after time $t$ is $\lambda(t) S(t)\,dt$, which equals the fall in the survival probability, $-dS(t)$³.

If default occurs at time $t$, the firm's loss is $\mathrm{LGD} \times V(t)^+$, and its present value is found by multiplying by the discount factor $D(t)$. Here $V(t)$ is the value assuming the counterparty does not default (after deducting collateral, if any), and $V(t)^+ = \max\{V(t), 0\}$. Multiplying by the probability of default and integrating to maturity gives CVA.

$$\mathrm{CVA} = \mathrm{LGD}\int_0^T D(t)\,\mathbb{E}[V(t)^+ \mid \tau = t]\,\lambda(t) S(t)\,dt$$

If the counterparty's default and market moves are unrelated to each other, the condition on the default time drops out, and the expectation becomes the expected exposure $\mathrm{EE}(t) = \mathbb{E}[V(t)^+]$⁴.

$$\mathrm{CVA} = \mathrm{LGD}\int_0^T D(t)\,\mathrm{EE}(t)\,\lambda(t) S(t)\,dt$$

To compute the integral, place times $0 = t_0 < t_1 < \dots < t_n = T$ and assign the defaults in each interval $(t_{i-1}, t_i]$ to the end of the interval, $t_i$. The default probability in an interval is $\mathrm{PD}_i = S(t_{i-1}) - S(t_i)$.

$$\mathrm{CVA} = \mathrm{LGD}\sum_{i=1}^{n} D(t_i)\,\mathrm{EE}(t_i)\,\mathrm{PD}_i$$

If EE or the discount factor moves a lot within an interval, the intervals must be made finer, or the sum will differ noticeably from the integral. The continuous form corresponds to the formula in a 2017 paper from the Institute for Monetary and Economic Studies of the Bank of Japan, and the discrete form corresponds to the formula in a 2018 paper from the same institute⁵.

The credit curve and the hazard rate

The input $\mathrm{PD}_i$ comes from a curve of survival probabilities by time (the credit curve), built from the counterparty's credit spread. The credit spread is the yield on a corporate bond minus the yield on a government bond of the same maturity. It covers compensation for loss on default and also other factors, such as the lower market liquidity of corporate bonds⁶. The usual method is to estimate default probabilities under the risk-neutral probability from the spreads on corporate bonds or CDS, and build the curve from them. The most reliable instrument is a credit default swap (CDS) that references the counterparty directly⁷.

Here $s$ is the spread on a CDS, which is different from the credit spread as a yield difference on bonds. If the hazard rate is constant to maturity, it can be estimated from the CDS spread $s$ and the recovery rate $R$ with the formula below. The relationship follows from the condition that the present value of the spread paid on the CDS equals the present value of the expected payment of $1 - R$ on default, and it is called the credit triangle⁸.

$$\lambda = \frac{s}{1 - R}$$

Consider a hypothetical example in which the CDS spread is 150bp (1.5%) and the recovery rate is 40%. Then $\lambda = 1.5\% \div 0.6 = 2.5\%$. The probability of surviving 3 years is $e^{-0.025 \times 3} = 92.77\%$, and the probability of default within 3 years is 7.23%. This formula is a rule of thumb.

When several CDSs with different maturities are available, the hazard rate for each interval is found in order from the shortest maturity so that all the prices are reproduced (calibration of the curve)⁸. With a constant hazard rate in each interval, the default probability in interval $i$ is the survival probability to the start of the interval, $S(t_{i-1})$, times $1 - e^{-\lambda_i (t_i - t_{i-1})}$.

When there is no liquid CDS that references the counterparty directly, the spread on the counterparty's bonds or a CDS that references a similar company is used instead. Some financial institutions build curves by rating, region, and industry and assign them according to the counterparty's characteristics⁷. The LGD assumed when estimating the default probability is also used as the LGD in CVA⁹.

The hazard rate obtained from CDS is a value under the risk-neutral probability, and it differs from the probability that default actually occurs³.

Worked example: a 3-year unsecured trade with Company N

The procedure for computing CVA is as follows.

1. Collect the CDS spreads and the recovery rate.
2. Find survival probabilities from the hazard rate, and take their differences as the default probability (PD) of each interval.
3. Simulate market moves to get EE and the discount factor at each time.
4. For each interval, multiply LGD (found from the recovery rate), the discount factor, EE, and PD.
5. Add the interval values to get CVA.

Building credit and exposure separately and multiplying them only at the end is how the independence assumption shows up here (its content is in the section "Derivation of the CVA formula").

Consider a hypothetical example in which the firm has a 3-year trade with Company N with no collateral. The task is to find the firm's CVA. There are three times, at the ends of years 1, 2, and 3. LGD is 60% (recovery rate 40%). Company N's hazard rate rises by interval: 1%, 2%, and 3%. The survival probability at each year-end is found from the cumulative hazard rate as $S(t_i) = e^{-\sum \lambda_j}$, and the differences are the default probabilities of the intervals.

Consider a hypothetical example in which EE takes values that a simulation of interest rate and other moves in many scenarios might give, and the discount factors are 0.98, 0.96, and 0.94 at the ends of years 1, 2, and 3. The contribution of each interval is LGD × discount factor × EE × PD.

For example, the first interval is $0.6 \times 0.98 \times 60 \times 0.009950 = 0.351$ million yen. CVA is 2.063 million yen, so the value that reflects CVA is 2.063 million yen lower than the risk-free value¹⁰. In the example in the XVA overview, the default probability was the same every year. Here it differs by interval, so the calculation goes through survival probabilities.

What moves CVA

The factors that move CVA fall into three groups: the counterparty's credit spread, the EE of the derivative, and the correlation between the two¹¹. The table shows the results of moving one condition at a time in the example above, leaving the others unchanged.

With the credit triangle at a 40% recovery rate, a 0.5-percentage-point rise in the hazard rate corresponds to a 30bp rise in the CDS spread ($0.5\% \times 0.6$). If the counterparty's creditworthiness worsens, CVA increases, and the increase is a loss in the firm's valuation. If it improves, CVA falls, and the decrease is a gain. EE is proportional to CVA, so if exposure rises 50% because of moves in interest rates or other variables, CVA also rises 50%.

The change in value when the spread moves 1bp is called the CS01. Applying the same idea to CVA, the CS01 of CVA in this example is about 17,000 yen per 1bp. The hazard rate was raised by the spread increase divided by $1 - R$ (about 1.67bp for a 1bp rise)¹².

This value assumes that EE does not change when the spread moves. In practice, when interest rates or other variables move and EE changes, the CS01 of CVA changes too, so a CVA desk hedges dynamically with CDS and similar instruments as things move¹¹.

Cases where the independence assumption breaks down, such as exposure growing just when the counterparty's creditworthiness falls, are covered in Wrong-Way Risk.

One-sided and bilateral CVA, and DVA issues

Reflecting only CVA in the value is called one-sided CVA, and reflecting both CVA and DVA is called bilateral CVA¹³. The firm's own adjustment uses the part of the trade value that is negative (the firm's liability). Let $\mathrm{ENE}(t) = \mathbb{E}[\max\{-V(t), 0\}]$ be the expected value of the negative part, $\mathrm{PD}^{B}_i$ the firm's default probability, and $\mathrm{LGD}_B$ the firm's LGD. Under the independence assumption, DVA and the adjusted value are as follows¹³.

$$\mathrm{DVA} = \mathrm{LGD}_B\sum_{i=1}^{n} D(t_i)\,\mathrm{ENE}(t_i)\,\mathrm{PD}^{B}_i$$ $$\hat{V} = V - \mathrm{CVA} + \mathrm{DVA}$$

Consider a hypothetical example in which, continuing from above, the firm's own hazard rates are 0.2%, 0.4%, and 0.6% by interval, its LGD is 60%, and the expected value of the negative part of the value, ENE, is 20, 30, and 40 million yen. The discount factors are the same.

DVA is 0.226 million yen, and the bilateral adjustment is $-2.063 + 0.226 = -1.837$ million yen. If the firm's hazard rate doubles in all three intervals, DVA rises to 0.450 million yen and the value goes up by about 0.22 million yen. Conversely, if the firm's creditworthiness recovers, DVA falls and so does the value. What marks DVA out is that a worsening of the firm's own creditworthiness and a rise in value move in the same direction.

Deep diveThe bilateral CVA formula that handles which party defaults first

The discrete formulas above assume that the firm's and the counterparty's defaults are independent and do not explicitly model that only the party that defaults first ends the trade. The paper by Brigo and Capponi writes the firm's default time as $\tau_B$ and the counterparty's as $\tau_C$, and expresses the adjustments as follows, with the remaining trade settled at the value at the time of the first default¹⁴.

$$\mathrm{CVA} = \mathrm{LGD}_C\,\mathbb{E}\left[\mathbf{1}_{\{\tau_C \le \min(\tau_B, T)\}}\,D(\tau_C)\,V(\tau_C)^+\right]$$ $$\mathrm{DVA} = \mathrm{LGD}_B\,\mathbb{E}\left[\mathbf{1}_{\{\tau_B \le \min(\tau_C, T)\}}\,D(\tau_B)\,\big(-V(\tau_B)\big)^+\right]$$

$V(\tau)$ is the risk-free value of the remaining trade at the default time, and $\mathbf{1}$ is a function that equals 1 if the condition holds and 0 if it does not.

CVA arises only when the counterparty defaults first, and DVA only when the firm defaults first. If the two defaults are independent and the hazard rates are $\lambda_B$ and $\lambda_C$, the probability that the counterparty defaults first in the short period just after time $t$ is $\lambda_C(t) S_C(t) S_B(t)\,dt$, which includes the probability $S_B(t)$ that the firm survives.

In the formula of the 2017 paper from the Institute for Monetary and Economic Studies, the factor $DF$ that discounts at the risk-free rate plus the credit spreads of both parties includes this survival probability¹⁵.

There are several points to consider in deciding whether to put DVA into the value or into profit and loss.

The first is how easily the two parties can agree on a price. Because of the symmetry between CVA and DVA, bilateral CVA makes it easier for both sides of a trade to arrive at the same price. The firm's CVA is the counterparty's DVA, and the firm's DVA is the counterparty's CVA. From Company N's side in the example above, CVA is 0.226 million yen and DVA is 2.063 million yen, so the adjustment is $-0.226 + 2.063 = +1.837$ million yen (the firm's ENE corresponds to Company N's EE). Adding the firm's adjustment of $-1.837$ million yen gives 0, and if both parties value the trade on the same assumptions, the prices agree. In a trade between two parties with different creditworthiness, the price is set so that the party with the lower creditworthiness bears the amount of the adjustment¹³.

The second is that accounting and regulation treat it differently. IFRS 13 and US GAAP (Topic 820) in principle require DVA to be reflected in the fair value of the firm's own liabilities as well¹⁶. Capital adequacy regulation, on the other hand, deducts DVA from Common Equity Tier 1 to prevent moral hazard, and measures CVA on a one-sided basis (only the counterparty's credit risk)¹⁷.

The third is that the market has not agreed on whether to include it in the price, and that the reflection of the firm's own creditworthiness overlaps. As of the 2017 paper, there were parts of the question of which valuation adjustments to build into prices on which the market had reached no consensus. DVA and FVA can both include a gain from the firm's own creditworthiness, so care is needed not to count the gain from the firm's own default twice¹⁸. Discounting can also bring in the firm's own creditworthiness. If the firm's own credit spread goes into the discount factor $DF$ mentioned in the deep dive (the factor that discounts at the risk-free rate plus the credit spreads of both parties), a fall in the firm's creditworthiness reduces CVA. For that reason, some practitioners do not include the firm's own credit spread in discounting¹⁵.

The fourth is the difficulty of hedging. The 2018 paper from the Institute for Monetary and Economic Studies says that hedging directly by trading CDS that reference the firm itself is prohibited. Hedging with CDS on similar companies or with index CDS becomes the realistic alternative. Reducing the firm's own credit risk by buying back its own bonds is another way, but nimble trading is difficult in practice, and it is said not always to be an effective hedge¹⁹.

Assumptions and limits: model risk in CVA

The value of CVA is an estimate computed from models of future exposure and default probability. Changing the assumptions or the method of calculation changes the value.

The time step can be set by considering the close-out period²⁴. Because of this model risk, it is said to be difficult to hedge CVA completely²². On the regulatory side, the Basel Committee did not adopt an internal model approach in its final document, because CVA measurement is complex and carries high model risk. Use of the standardized approach (SA-CVA) requires supervisory approval²⁵.

Notes

1. Damiano Brigo and Agostino Capponi, "Bilateral counterparty risk valuation with stochastic dynamical models and application to Credit Default Swaps" (arXiv, 2018 version), Section 2 (around equations (2.4) and (2.5)). ↩

2. The same paper, Proposition 2.1 (equation (2.6), page 7). The paper writes the value of a trade subject to default risk as the value of the risk-free trade plus a bought put option with a strike of 0 (if the firm defaults first) and a sold call option with a strike of 0 (if the counterparty defaults first). ↩

3. OpenGamma Quantitative Research, "The Pricing and Risk Management of Credit Default Swaps, with a Focus on the ISDA Model," Section 3.3 (equations (6) and (7), and the statement that the probabilities do not represent real-world default probabilities). ↩ ↩²

4. 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). It says equation (1) holds under the condition that the correlation between market risk and credit risk is not considered. ↩

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), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 2 (2), the first of equation (2), pages 115–116, is the continuous form. The first term of equation (1), page 125, of the Kawakami paper cited above is the discrete form (both in Japanese). ↩

6. Bank of Japan, "日本の社債発行スプレッドの変動要因" (English title translated here; no official English title confirmed), Bank of Japan Working Paper Series (April 2010), Section 1 (printed page 2) (in Japanese). It defines the credit spread as the difference between the yield on a corporate bond and the yield on a government bond of the same maturity, and says it includes, besides the expected loss on default, premiums such as that for the lower market liquidity of corporate bonds compared with government bonds. ↩

7. 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 126 (in Japanese). ↩ ↩²

8. OpenGamma paper cited above, Section 4.6 (equations (30) to (32)) and Section 6.2 (around equation (54)). The formula for a constant hazard rate assumes that interest rates and the hazard rate are independent and that accrued interest is ignored, with continuous payments. The paper says it can serve as a rule of thumb for switching between spread and hazard rate and as a first estimate for calibrating the curve. ↩ ↩²

9. Soichi Maeda and Tatsuya Hasegawa, "ＣＶＡリスク計測の枠組みの見直し" (Review of the framework for measuring CVA risk), Kinzai Institute for Financial Affairs, Kinyu Zaisei Jijo (February 12, 2018 issue), "実務におけるＣＶＡ管理" (CVA management in practice), page 33 (in Japanese). ↩

10. The contributions in the table are calculated with PD to four decimal places (in %) and with EE and the discount factors as shown in the table, and rounded to three decimal places in million yen. The total is the sum of the contributions before rounding (2.0632 million yen) rounded to three decimal places, and it also equals the sum of the rounded contributions (0.351, 0.903, 0.809). ↩

11. 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 5, pages 147 and 149 (in Japanese). It lists the credit spread, the expected exposure, and the correlation between the two as the factors that make up CVA, and page 147 also describes the practice in which a CVA desk hedges dynamically with CDS and similar instruments as CVA and DVA change. ↩ ↩²

12. The definition of CS01 is in the OpenGamma paper cited above, Section 8. The paper defines it for the price of a CDS, and the same idea is applied here to CVA as a calculation for this site. ↩

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 (2), page 125 (in Japanese). Equation (1) is bilateral CVA when the defaults are independent and the correlation with market risk is not considered. The paper writes the adjustment and ENE as negative values, and the text expresses them as amounts of 0 or more. ↩ ↩² ↩³

14. Brigo and Capponi, cited above, Section 2 (equations (2.4) to (2.6)). The paper's symbols are rewritten in the symbols used here. ↩

15. 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) and note 6, page 116 (in Japanese). Note 6 says that putting the firm's own credit spread into the discount factor means a fall in the firm's creditworthiness reduces CVA, so in financial practice the firm's own credit spread is sometimes not included. ↩ ↩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, Institute for Monetary and Economic Studies, Bank of Japan (April 2017), Section 1, page 110 (in Japanese). ↩

17. Soichi Maeda and Tatsuya Hasegawa, "ＣＶＡリスク計測の枠組みの見直し" (Review of the framework for measuring CVA risk), Kinzai Institute for Financial Affairs, Kinyu Zaisei Jijo (February 12, 2018 issue), "信用評価調整（ＣＶＡ）とは" (What is credit valuation adjustment (CVA)?), page 33 (in Japanese). ↩

18. 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). ↩

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 5, page 149 (in Japanese). ↩

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), note 8 on page 125 (in Japanese). ↩

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

22. The same document, Section 5, page 149. ↩ ↩²

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

24. 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), note 12 on page 119 (in Japanese). Citing Gregory, the paper says that for CVA and DVA a time step reflecting the close-out period is used. ↩

25. Soichi Maeda and Tatsuya Hasegawa, "ＣＶＡリスク計測の枠組みの見直し" (Review of the framework for measuring CVA risk), Kinzai Institute for Financial Affairs, Kinyu Zaisei Jijo (February 12, 2018 issue), "ＦＲＴＢとの整合性" (Consistency with the FRTB), page 34, and "標準的方式（ＳＡ－ＣＶＡ）" (The standardized approach (SA-CVA)), page 36 (in Japanese). ↩