# LO 30.9: Calculate BCVA and BCVA spread.

LO 30.9: Calculate BCVA and BCVA spread.
Given a charge for counterparty risk that favors a stronger counterparty (typically a bank), CVA historically did not take into account that both counterparties could be subject to default risk. The 20072009 financial crisis changed risk parameters and perceptions drastically.
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Counterparty risk is now viewed as bilateral. Bilateral counterparty risk assumes that both counterparties may default. The formula for the credit value adjustment for a bilateral contract derives from the original CVA formula and assumes no simultaneous default (e.g., wrong-way risk).
The positive expression in the following bilateral credit value adjustment (BCVA) formula represents the CVA of the counterparty, C, and the negative expression represents the CVA of the financial institution, I. The CVA of the institution is also known as the debt value adjustment (DVA). The two terms in this expression are mirror images of one another. If the financial institution defaults first, it books a gain when the marked-to-market (MtM) exposure is negative. This is the case because the institution in default will only pay the counterparty the recovery amount of what they owe, which is a fraction of what they would have otherwise owed had they not defaulted. That difference is a gain to the defaulting party.
BCVA = CVA + DVA m
CVA = +L G D C x
2EE(ti) x PDC ( t ^ , t; ) J
DVA = LGDj x J 2 NEE(t; ) x PDj (t_ !, t; )
i=l m
i=l
where: NEE = negative expected exposure (EE from the counterpartys perspective)
Implications of the BCVA model include:
1. BCVA can be negative if the second expression is larger than the first, implying that the
risk value of a derivative is greater than its risk-free value. Stand-alone CVA may only be positive.
2. Two counterparties in agreement on the parameters of the BCVA equation will settle up owing to the equations symmetry. For example, Party 1 has BCVA of +X, then Party 2 has BCVA of X. Party 2 owes Party 1 +X due to Party 2s counterparty risk.
3. Netting with BCVA may be a disadvantage when the second expression dominates,
implying that the financial institution is riskier than its counterparty. Without netting, the institution may select contracts with a positive MtM settlement, discarding those with a negative MtM value as bankruptcy liabilities.
4.
If both parties agree on the parameters of the BCVA calculation, then counterparty risk in the marketplace (the sum of all BCVAs) is zero. However, this holds more in theory than in practice.
Professors Note: This BCVA formula excludes a survival probability, which considers the possibility that a financial institution may default before its counterparty. I f this is the case, the institution will not suffer a loss from the counterparty. The survival probability will be included in the BCVA equation in Topic 33, when we discuss stress testing the debt value adjustment.
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BCVA may be expressed as a spread or basis point charge to the weaker counterparty as follows:
BCVA(t,T)
X DS x EPE – X f s xE N E
where : XCDS _ tke institutions own CDS spread ENE = expected negative exposure (the opposite of EPE)
Here the BCVA can be represented as a running spread. The formula implies that the institution may account for its own default through reduction of the unilateral CVA charge by its own credit spread multiplied by the ENE.
The calculation of this formula is identical to that for unilateral CVA. It differs only in that there is an additional identical subtractive calculation to reflect the BCVA of the financial institution.
Example: Computing BCVA
A risk manager needs a quick calculation of the BCVA on a swap. Assume inputs are as follows: EPE = 3%, ENE =3%, counterparty credit spread = 300 bps, and financial institution credit spread = 200 bps. Compute BCVA from the perspective of the financial institution.
>From the perspective of the financial institution:
3% x 300 3% x 200 = 9 bps
This is what the financial institution may charge the counterparty for overall counterparty risk.
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K e y C o n c e p t s
LO 30.1
Motivations for pricing counterparty risk include (1) organization of responsibilities within the institution with respect to the pricing calculation and (2) determining whether a trade is sufficiently possible when factoring in counterparty risk charge.
LO 30.2
A credit value adjustment (CVA) is the price of counterparty risk. A positive value is a cost to the counterparty bearing the risk. The basic CVA formula assumes no wrong-way risk.
LO 30.3
CVA is calculated as follows:
m
CVA = L G D x y ^ x E E (ti)x P D (ti_1, t; )
i=l
CVA as a spread is CVA divided by the risky annuity for the maturity of the contract in question, producing an annual spread or charge expressed in basis points:
CVA(t,T)
CDS p r e m i u m (t>T)
X CDSx EPE
LO 30.4
Credit spread levels, the shape of the credit spread curve, the impact of the recovery rate, and the basis risk that arises from different recovery rate assumptions must all be considered when evaluating the impact of the default probability and recovery on CVA.
LO 30.3
Netting reduces the CVA price because it nets exposure when trades are settled.
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LO 30.6
Incremental CVA is used to calculate the cost of a new trade versus an existing one to determine the effect that the new trade has on CVA. Standalone CVA cannot do this. The formula for the incremental CVA calculation is identical to that for standalone CVA, except for the incremental expected exposure.
Marginal CVA is used for trade level attribution (i.e., to discover the determinants of the CVA). The formula for the calculation of marginal CVA is identical to that for standalone CVA, except for the substitution of marginal expected exposure for expected exposure.
LO 30.7
Collateralization reduces the CVA, changing only the counterpartys expected exposure.
LO 30.8
Bilateral CVA is a collateral value adjustment that takes into account the possibility that both counterparties could default, though not simultaneously. The CVA of the financial institution is also known as the debt value adjustment (DVA).The BCVA is the sum of CVA and DVA components.
BCVA = CVA + DVA m
CVA = +L G D C x
2EE(t;) x PDC (ti_1,ti) ^
DVA = LGDj x J2 NEE(t; )x P D ,(ti_ 1,ti)
i=l m
i=l
Implications of the BCVA model include:

BCVA can be negative. Stand-alone CVA may only be positive (representing a cost). Parties in agreement on the BCVA settle in accordance with the BCVA equations symmetry.
Netting may be disadvantageous where the financial institutions counterparty risk
exceeds that of the counterparty. Without it, the institution can pick which contracts to settle. Parties in agreement will have in theory all BCVAs net out to zero due to the symmetrical nature of the BCVA formula.

LO 30.9
The BCVA formula as a credit spread is:
BCVA(t,T)
xDS x EPE – xfDS x ENE
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To price BCVA: (credit spread of counterparty A x EPE) (credit spread of counterparty B x ENE) = either positive number that stronger counterparty charges the weaker one or negative number that the stronger counterparty may owe the weaker one if its ENE is greater than the counterpartys EPE.
Negative BCVA: The counterparty has a higher chance of defaulting and will owe money (C VA cost).
Positive BCVA: the counterparty has a lower chance of defaulting and will be owed money (recipient of CVA fee).
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C o n c e p t C h e c k e r s
1.
2.
3.
4.
5.
Which of the following statements is not a motivation for pricing counterparty risk? A. Accurate pricing should only account for the cost of the trade. B. Counterparty risk pricing should account for risk mitigants. C. Best practices organize pricing responsibilities in the organization. D. Pricing bilateral derivatives contracts.
With respect to the CVA calculation, which of the following statements is correct when a risk manager wishes to understand which trades have the greatest impact on a counterpartys CVA? The manager would use: A.
incremental CVA because it accounts for the change in CVA once the new trade is priced, accounting for netting.
B. marginal CVA because he could break down netted trades into trade level
C.
contributions. incremental CVA because he could break down netted trades into trade level contributions.
D. marginal CVA because it accounts for the change in CVA once the new trade is
priced, accounting for netting.
A trader wants to know the approximate CVA for a counterparty in a swap transaction. The counterpartys expected potential exposure (EPE) is 7%, and its credit spread is 473 basis points. What is the CVA as a running spread? A. 0.33%. B. 1.48%. C. 2.23%. D. 9.75%.
Regarding the impact of changes in the credit spread and recovery rate assumptions on the CVA, which of the following statements is true? A. A decrease in the credit spread will most often increase the CVA. B. For an upward-sloping curve, the CVA will be higher compared to a downward-
sloping curve.
C. Increasing the recovery rate will reduce the CVA. D. If the actual recovery rate is higher than the settled recovery rate, the CVA will most likely be higher compared to a situation where both recovery assumptions are the same for both rates.
When incorporating netting and collateralization into the CVA calculation, which of the following statements is incorrect?
I. Netting increases the CVA price because it reduces exposure when trades are
settled.
II. Collateralization does not change the CVA because it only changes the
counterpartys expected exposure.
A. I only. B. II only. C. Both I and II. D. Neither I nor II.
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C o n c e p t C h e c k e r An s w e r s
1. A Accurate pricing should account for not only the cost of the trade, but also the cost of
counterparty risk.
2. B Understanding which trades have the greatest impact on a counterpartys credit value
adjustment requires use of the marginal CVA. Incremental CVA, by contrast, is useful for pricing a new trade with respect to an existing one.
3. A Calculation of the CVA as a running spread entails multiplying the counterpartys EPE by its