 ### Credit and Debt Value Adjustments

In this chapter, the motivation and challenges of pricing counterparty risk will be discussed. We shall also describe credit value adjustment ($$CVA$$) in detail and compute it together with the $$CVA$$ spread without wrong way risk, netting, or collateralization. Moreover, the effect of changes in credit spread and recovery rate assumptions on $$CVA$$ shall be measured. Incorporation of netting into the computation of $$CVA$$ is also another aspect that will be explained.

Furthermore, the section seeks to define and compute incremental and marginal $$CVA$$, and provide an explanation of the conversion of $$CVA$$ into a running spread. The effect of collateralization incorporation into the computation of $$CVA$$ will also be explained. Finally, debt value adjustment ($$DVA$$) and bilateral credit value adjustment ($$BCVA$$) will be described and computed.

## CVA Formula

The standard $$CVA$$ calculation formula is:

$$CVA=LGD\sum _{ i=1 }^{ m }{ EE\left( { t }_{ i } \right) \times PD\left( { t }_{ i-1 },{ t }_{ i } \right) }$$

The following are components relied upon by $$CVA$$: Loss given default ($$LGD$$), Expected Exposure ($$EE$$), Default Probability ($$PD$$). The formula includes a dimension of time since time must be integrated to accounting for the precise $$EE$$ and $$PD$$ distribution.

In the above formula, default enters the expression solely through the likelihood of default. This is a merit as it is not necessary for default events to be simulated despite the need for a simulation system for $$CVA$$ calculation.

Assume that $$CVA$$ is to be expressed as a spread instead of calculating it as a stand-alone value. Therefore, a division of $$CVA$$ by the risky annuity value is the simplest computation. The assumption in the formula is that $$EE$$ is constant over time and is equal to the average value ($$EPE$$) yielding the following $$EPE$$ bases estimation:

$$CVA\approx EPE\times Spread$$

With the expression of $$CVA$$ being in the same unit as the credit spread, which would for the transaction’s maturity.

## Exposure and Discounting

The assumption in the $$CVA$$ formula given is that the $$EE$$ is discounted hence providing a generally better solution than a separate expression of discount factors.

In case of necessity in explicit discounts factors, care must be taken. A quantification of the underlying exposure, via the T-forward formula, can technically account for the convexity effect.

This ensures the dependence on the expected future interest rate values by discount factors rather than their distribution, hence the success in moving the discount factor out of the expectation term. There may be a convenience in working with separate discount factors.

## Risk Neutrality

Where practical, computation of $$CVA$$ with risk-neutral parameters is the general approach since the approach is relevant for pricing as the price definition is based on hedging instruments supporting the concept of exit price necessary for accounting standards.

As compared to their real-world equivalents, risk-neutral parameters may generally be higher. Risk-neutral parameters are used for the purpose of pricing exposures. However, the following are drawbacks of using risk-neutral likelihood of default:

1. High likelihood of risk-neutral default compared to the real-world scenario;
2. Due to lack of liquid single-name credit default swaps referencing defaults, they can generally lack hedging; and
3. The business model of banks is generally to warehouse credit risk and therefore is only exposed to real-world risk of default.

$$CVA$$ computation may today apply the following scenarios of historical likelihood of default:

1. Smaller regional banks, having derivatives business that is less significant, arguing that their exit price would be with a local competitor who would also price the $$CVA$$ with historical likelihoods of default; or
2. Regions where banks are subject to IFS 13 accounting standards.

$$CVA$$ may be viewed as an actuarial reserve, by banks in situations such as the ones listed above, rather than a risk-neutral exit price.

## $$CVA$$ Semi-Analytical Methods

A position that can only have a positive value has its $$CVA$$ as the first simple example. Therefore:

$$CVA\approx -LGD\times PD\left( 0,T \right) \times V$$

Where $$T$$ is the transaction in question’s maturity, $$V$$ is the current standard value, and $$PD\left( 0,T \right)$$ is the likelihood of a counterparty defaulting at any during the transaction’s lifetime.

# Impact of Credit Assumptions

There are several aspects to consider with respect to the effect of the likelihood of default and $$LGD$$ on $$CVA$$.

$$CVA$$ is increased by the credit spread increase. However, there is non-linearity in this effect as the likelihoods of default are bounded by 100%.

A zero jump to default risk is another way of understanding this, because no loss will be theoretically caused by an immediate counterparty default as the current value is zero.

With a deterioration of the counterparty’s credit quality, there will be a decrease in the $$CVA$$ which will be followed by an increase when the counterparty is close to default.

The $$CVA$$ is changed by around 10% when going from an upwards-sloping to a flat curve.

## Recovery Impact

When the equation for the probability of default is substituted in the $$CVA$$ formula, the following is the resulting equation, where the actual and settled $$LGD$$s are explicitly referenced:

$$CVA=-{ LGD }_{ actual }\sum _{ j=1 }^{ m }{ EE\left( { t }_{ i } \right) \times \left[ exp\left( -\frac { { S }_{ { t }_{ i-1 } },{ t }_{ i-1 } }{ { LGD }_{ Settled } } \right) -exp\left( -\frac { { S }_{ { t }_{ i-1 } },t_{ i } }{ { LGD }_{ Settled } } \right) \right] }$$

# CVA Allocation and Pricing

$$CVA$$ is reduced by risk mitigants like collateral and netting but the computation of this is through the computation at the netting set level. An important consideration, therefore, is the allocation of $$CVA$$ to the transaction level for the purposes of pricing and valuation. The result is the consideration of the numerical issues that involves the rapid running of large-scale computations.

## Netting and Incremental CVA

For netting set:

$${ CVA }_{ NS }\ge \sum _{ i=1 }^{ n }{ { CVA }_{ i } }$$

Under the netting agreement, $${ CVA }_{ NS }$$ is the total $$CVA$$ for all transactions, and $${ CVA }_{ i }$$ is the stand-alone $$CVA$$ for transaction $$i$$. The allocation of netting benefits to each individual transaction can be achieved by the use of incremental the $$CVA$$ concept, analogously to the incremental $$EE$$.

The incremental effect of this transaction on the netting set is the basis of computing transaction $$i$$’s $$CVA$$:

$${ CVA }_{ i }^{ incremental }={ CVA }_{ NS-i }-{ CVA }_{ NS }$$

The order of execution of transactions affects $$CVA$$, but does not change because of the transactions. The following incremental $$CVA$$ formula can be derived:

$${ CVA }_{ i }^{ incremental }=-LGD\sum _{ i=1 }^{ m }{ { EE }_{ i }^{ incremental }\left( { t }_{ i } \right) \times PD\left( { t }_{ i-1 },{ t }_{ i } \right) }$$

Because of the netting effects benefits, the incremental $$CVA$$ can be negative leading to a $$CVA$$ being positive thus making it a benefit instead of an expense.

In the presence of netting, the incremental $$CVA$$ will never be exceeded by the stand-alone $$CVA$$ without netting. This fact can be attributed to the properties of $$EE$$ and $$netting$$.

$$CVA$$ treatment is made complex by the treatment of netting and thus becoming a multidimensional challenge.

## Marginal CVA

Adding the marginal $$EE$$ to the $$CVA$$ formula is the simplest definition of marginal $$CVA$$ which is useful in breaking down a $$CVA$$ for any number of netted transactions into a transaction level contribution summing up the total $$CVA$$.

Alternatively, the appropriate way of allocating a $$CVA$$ to a transaction level contribution at a given level is through marginal $$CVA$$. Different decompositions of $$CVA$$ can lead to different results.

Furthermore, a transaction’s timing affects the amount of charged $$CVA$$ hence posing challenges and causing gaming behaviors.

The conversion of an upfront $$CVA$$ into a running spread $$CVA$$ is another crucial point of consideration in $$CVA$$ pricing as charging a $$CVA$$ to a client will be facilitated.

When a spread is included in a contract, the challenge is nonlinear as the $$CVA$$ will be affected by the $$CVA$$. Recursively computing the accurate value ensures the $$CVA$$ embedded in the contract is offset by the $$CVA$$.

The relative size of a new transaction affects the netting benefits witnessed in a new transaction’s incremental $$CVA$$. The netting benefit is lost due to an increase in transaction size hence the approach to a stand-alone value by the $$CVA$$. The $$CVA$$ quote in basis points is only valid for a particular transaction size.

## Numerical Issues

Speeding up the underlying pricing functionality is an obvious way of improving the efficiency of computing $$CVA$$. The following are methods of achieving this:

1. Common functionality being stripped and is not dependent on the underlying market variables at a particular point;
2. Pricing functions’ numerical optimization;
3. Grids or approximations application; and
4. Parallelization.

The choice of whether to apply pathwise or direct simulation in the calculation of $$CVA$$ is another aspect that should be considered. The evaluation of pathwise simulations is not best for $$CVA$$ purposes as compared to $$PFE$$ purposes.

$$CVA$$ computation with respect to computed $$EE$$ at discrete points in time can be challenging for certain path-dependent derivatives based on a the continuous sampling of quantities.

# CVA With Collateral

Collateral only changes $$EE$$. Therefore, the same formula may be applied with the $$EE$$ based on collateralization assumption. For computation of $$CVA$$, a flat credit curve of 500 bps and 0.4 $$LGD$$ can be assumed.

## Impact of Margin Period of Risk

A simple estimation may be used to approximate the reduction of $$CVA$$ directly. The implicit assumption of a zero minimum transfer amount is responsible for a small absolute value the estimation provides. The risk from posting collateral is not considered thus producing the ballpark figure for the $$CVA$$ reduction.

At zero $$MPR$$, a small $$CVA$$ is witnessed which then decreases towards the uncollateralized value. A $$CVA$$ that is almost half the uncollateralized $$CVA$$ is witnessed at a margin period of risk of 30 calendar days.

## Thresholds and Initial Margins

An initial margin can be taken to be a negative threshold. Determining an accurate initial margin is extremely subjective, despite $$CVA$$ being reduced by an increased initial margin.

The liability component in credit exposure can be included in the counterparty risk pricing as the $$DVA$$ component. Bilateral $$CVA$$ ($$BCVA$$) is made up of $$CVA$$ and $$DVA$$. $$DVA$$ use was combined with credit spread use rather than historical default probabilities. One important $$DVA$$ characteristic is that it creates price symmetry where parties can theoretically agree on prices.

## Bilateral CVA formula

Under $$BCVA$$, the $$CVA$$ considered by a party is computed assuming that the party and its counterparty may default.

$$BCVA$$ is a simple sum of $$CVA$$ and $$DVA$$ components:

$$BCVA=CVA+DVA$$

$$CVA=-{ LGD }_{ C }\sum _{ i=1 }^{ m }{ EE\left( { t }_{ i } \right) \times { PD }_{ C }\left( { t }_{ i-1 },{ t }_{ i } \right) }$$

$$DVA=-{ LGD }_{ P }\sum _{ i=1 }^{ m }{ EE\left( { t }_{ i } \right) \times { PD }_{ P }\left( { t }_{ i-1 },{ t }_{ i } \right) }$$

The party making the computation is indicated by the suffix $$P$$ and $$C$$ indicates the counterparty. The $$DVA$$ term is the mirror image with respect to the negative expected exposure ($$NEE$$), the likelihood of default of the party, and $$LGD$$. The $$NEE$$ sign makes $$DVA$$ to be positive hence opposing the $$CVA$$ as a benefit.

For ease of understanding of the price symmetry, an obvious extension including $$CVA$$ is:

$$BCVA=-EPE\times { Spread }_{ C }-ENE\times { Spread }_{ P }$$

$$ENE$$ is the expected negative exposure and is the opposite of $$EPE$$. Therefore:

$$BCVA\approx -EPE\times \left( { Spread }_{ C }-{ Spread }_{ P } \right) .$$

In order for weaker counterparties to trade with stronger ones based on the differential in credit quality, they have to pay for their weaker credit quality, leading to a pricing agreement.

## Close-Out and Default Correlation

The following are crucial interconnected concepts ignored by the $$BCVA$$ formula given:

1. Survival: The $$CVA$$ and $$DVA$$ equation fails to include the likelihood of survival by the non-defaulting party.
2. Default correlation: The default correlation between the party and their counterparty is related to survival, and is not included. A positive correlation implies a high default probability, closer together, hence affecting the $$CVA$$ and $$DVA$$.
3. Close-out: The assumption is that at the time default, the underlying transactions will be settled at their mark-to-market values, which does not comply with the close-out reality. However, the consideration that the surviving party is not risk-free makes the close-out assumptions to be very relevant.

The $$BCVA$$ computation that possibly relies on a risk-free valuation fails to consider that either party can potentially gain their $$DVA$$ in case their counterparty defaults. The specifics of the close-out specification are relied upon by the monetization of the said gain, for example, market quotations and close-out amounts.

Market participants will generally not follow an approach that is advanced and will simply include the likelihoods of survival directly.

## DVA and Own-Debt

Because the fair value of a party’s own bonds is considered as the price other parties are willing to pay for them, there is logic to the use of $$DVA$$ on own debt. The issue, however, is whether a party will be able to buy back their own bonds with no significant cost incurred.

$$DVA$$ was therefore removed by equity analysts from their assessment of a company’s ongoing performance since $$DVA$$ was a strange accounting effect.

# DVA in Derivatives

There are arguments that were made in support of $$DVA$$, proposed its monetization in the following ways: defaulting, unwinds and novations, close-out process, and hedging. However, most of these arguments are fairly weak. The inclusion of $$DVA$$ in pricing has made the market practice to be somewhat divided.

# Practice Questions

1) Fenton Associates is a trading firm from Norway that needs to have a very quick idea on a swap’s bilateral credit value adjustment ($$BCVA$$). The firm knows that the expected positive exposure ($$EPE$$) for a trade of this type is 10.5% with an expected negative exposure ($$ENE$$) of 8.2%. The counterparty credit spread is found out to be around 150 bps and the credit spread of the trader’s own institution is 121 basis points per year. What is the $$BCVA$$?

1. 9.189
2. -37.957
3. -18.113
4. -34.599

Recall that $$BCVA$$, which is an obvious extension of $$DVA$$, is calculated by using the following formula:

$$BCVA=-EPE\times { Spread }_{ C }-ENE\times { Spread }_{ P }$$

From the question we have that:

$$EPE=0.105$$,

$${ Spread }_{ C }=150$$,

$$ENE=0.082$$ and

$${ Spread }_{ P }=121$$

Therefore:

$$BCVA=-0.105\times 150-0.082\times 121$$

$$= -25.672$$