After completing this reading, you should be able to:

• Explain the motivation for and the challenges of pricing counterparty risk.
• Describe credit value adjustment (CVA).
• Calculate CVA and CVA as a spread with no wrong-way risk, netting, or collateralization.
• Evaluate the impact of changes in the credit spread and recovery rate assumptions on CVA.
• Explain how netting can be incorporated into the CVA calculation.
• Define and calculate incremental CVA and marginal CVA and explain how to convert CVA into a running spread.
• Explain the impact of incorporating collateralization into the CVA calculation, including the impact of the margin period of risk, thresholds, and initial margins.
• Describe debt value adjustment (DVA) and bilateral CVA (BCVA).
• Calculate DVA, BCVA, and BCVA as a spread.
• Describe wrong-way risk and contrast it with right-way risk.
• Identify examples of wrong-way risk and examples of right-way risk.
• Discuss the impact of collateral on wrong-way risk.
• Identify examples of wrong-way collateral.
• Discuss the impact of wrong-way risk on central counterparties.
• Describe the various wrong-way modeling methods, including hazard rate approaches, structural approaches, parametric approaches, and jump approaches.
• Explain the implications of central clearing on wrong-way risk.

## The Motivation for and the Challenges of Pricing Counterparty Risk

#### Commodity Forward

A commodity entered with a consumer who seeks to hedge the price of a good can yield wrong-way risk.

Consider the case of an airline wary of an increase in the oil market. The airline decides to use derivatives to hedge all of its consumption by entering into a forward contract. As per the contracts, the dealer agrees to sell oil in the future at $30 per barrel. As maturity approaches, the price rises to$50 per barrel. This is a win for the airline because they will now be buying cheap. But that also means there’s a big loss for the dealer. The problem gets worse if the dealer has concentrated (many) short positions because there will be a flood of claims tabled by various parties (possibly other airlines). This will put intense pressure on the credit quality of the dealer

In this case, the positive value (exposure) from the perspective of the airline coincides with an increase in the dealer’s probability of default. This increases overall counterparty risk and produces wrong-way risk.

### Examples of Right-way Risk

Commodity Forwards may also yield right-way risk when entered into with a producer who seeks to hedge the price of their goods.

Consider the case of an oil producer wary of a decline in the oil market. The producer decides to use derivatives to hedge 50% of their production by entering into a forward contract. As per the contracts, the dealer agrees to buy oil in the future at $50 per barrel. As maturity approaches, the price declines to$30 per barrel. In this case, the producer (the counterparty) is likely to default, and they are losing money on the unhedged part of their production. But with the price of oil so low, the value of the forward contracts to the dealer is also low. In this case, the derivatives have a positive value (exposure) to the oil producer and a negative value to the derivatives dealer.

As such, the dealer’s exposure is likely to be low when the oil producer (counterparty) defaults.

#### Call Option

Consider a situation where Smart Tech buys a three-month call option from Cortana Inc., with Alpha stock as the underlying.

Strike price: $50; Type: European; Expiry: Day 90; Underlying: Alpha Stock At expiry, Alpha stock has a price of$55, and the call is, therefore, ITM with a value of \$5 (ignoring the premium). During the same period, Cortana’s stock rallied after a landmark lawsuit wins against another firm.

In this case, the credit exposure of Smart Tech to Cortana Inc. has increased, but this has coincided with an improvement in Cortana’s creditworthiness.

### General vs. Specific Wrong-way Risk

• Specific wrong-way risk(SWWR) arises due to the specific characteristics of the counterparty or the transaction,
• That includes things like a rating downgrade, poor earnings, or litigation.
• General wrong-way risk(GWWR)—also known as conjectural wrong-way risk, is driven by macro-economic relationships
• That includes things like interest rates, pandemics, political unrest, or inflation in a particular region.
• The two types of risk differ as follows:
###### Figure 2-General WWR vs. Specific WWR

$$\small{\begin{array}{c|c} {\textbf{GWWR}} &\textbf{SWWR} \\ \hline \text{Should be priced and managed correctly} &{\text{Should in general be avoided, as it may} \\ \text{be extreme}} \\ \hline {\text{Relationships may be detectable using} \\ \text{historical data}} &{\text{Hard to detect except by a knowledge of} \\ \text{the relevant market, counterparty and the} \\ \text{economic rationale behind transaction}} \\ \hline {\text{Can potentially be incorporated into} \\ \text{pricing models}} &{\text{Difficult to model and risky to use naïve}\\ \text{correlation assumptions; should be addressed} \\ \text{qualitatively via methods such as stress-testing} }\\ \end{array} }$$

### Challenges Associated with Quantifying Wrong-way Risk

To quantify WWR, the risk manager has to model the relationship between credit, collateral, funding, and exposure.

The modeling process is complex due to a number of issues:

#### Uninformative historical data

Although historical data may contain some information on WWR, extracting the underlying relationships is challenging. Time series analysis or correlation may not help

#### Misspecification of relationship

Correlation can be zero, but it does not imply independence! Rather than exhibiting a correlation, the relationship between two events may be a cause-and-effect type of relationship.

#### Direction

The direction of WWR is not always clear. For example, low-interest rates usually indicate a recession and tough credit conditions, but high-interest rates could well trigger the same conditions

## The Impact of Collateral on Wrong-way Risk

Collateral is a good way to reduce exposure. WWR can cause the exposure to increase significantly, and therefore it is important to consider the impact of collateral on WWR.

• When the exposure is increasing gradually, collateral does a good job of minimizing the impact of wrong-way risk because derivative agreements have clauses that require parties to post collateral after a certain threshold is exceeded. This extra collateral is therefore easy to request and receive.
• But in cases where there’s a sudden jump in exposure, collateral does very little in the way of mitigating WWR
• For example, if there happens to be a 20% currency devaluation that coincides with sovereign default, collateral is rendered almost useless because it may not be received at all

## The impact of wrong-way risk on central counterparties

CCPs may be particularly prone to WWR because they rely so much on collateral as protection. CCPs tend to closely manage and monitor membership by only admitting parties with a certain credit quality

Moreover, all members have to provide an initial margin and also make contributions to the default fund that serves as an extra layer of cushion. But there’s a problem: These initial margins and default fund contributions are based on the portfolio’s market risk. As such, this separation of credit risk and market risk may culminate in a situation where collateral requirements ignore WWR!

For CDS in particular, CCPs are faced with an uphill task trying to quantify the WWR component when setting initial margins and default fund contributions. That’s because higher credit quality can increase WWR (when things take a turn for the worse, previously highly rated credits can get hit quite hard!).

The collateral posted carry WWR; members may post highly risky or illiquid securities.

## Wrong-Way Collateral

Wrong-way collateral occurs when an increase in exposure to a counterparty is accompanied by a decrease in the value of the collateral. In other words, it arises when the exposure to a counterparty is adversely (negatively) correlated with the value of collateral posted by that counterparty.

Ideally, collateral should reduce wrong-way risk by ensuring that if the counterparty defaults, there’s a “fallback” asset that can be seized to offset at least part of the loss.  But there’s trouble if an increase in exposure coincides with a decrease in the value of collateral. In such circumstances, the collateral may still offer a certain level of protection against loss, but this may be much lower than initially envisioned.

### Examples of Wrong-way Collateral

#### Payer interest rate swap collateralized by a government bond

Assume a bank has entered into an interest rate swap as the payer, i.e., the party paying the fixed interest rate throughout the life of the swap. Assume further that the counterparty has offered a high-quality government bond as collateral.

Now remember that bonds and interest rates have an inverse relationship: when the interest rate rises, bond prices fall, and vice versa. So, if interest rates rise, it means the bank will be gaining (winning), and its exposure to the counterparty will increase. However, the value of collateral will decline. This would present a classic case of wrong-way collateral.

#### Cross-currency swap collateralized by cash in one of the two underlying currencies

Assume an American bank has entered into a cross-currency swap with a Brazilian company. The bank will pay interest denominated in Brazilian reals (BRL) and receive in USD. Further, assume that the Brazilian company has offered collateral in Brazilian reals. In the scenario of a sovereign default in Brazil, the local currency (BRL) will be devalued. This will cause the bank’s exposure to spike and reduce the value of collateral at the same time.

## Wrong way-risk modeling approaches

### Hazard rate approaches

This approach views the credit spread as a stochastic process and then goes ahead to establish the correlation between the credit spread (“hazard rate”) and the other variables used to model exposure.

This approach can be implemented relatively tractably, as credit spread paths can be generated first, and exposure paths only have to be simulated in cases where a default is observed. The required correlation parameters are observed directly via historical time series of credit spreads and other relevant market variables.

On the downside, simple hazard rate approaches generate only very weak dependency between exposure and defaults. In other words, these approaches generate very weak wrong-way effects.

As an example, a simple hazard rate approach can be approached to model wrong-way risk in an interest rate swap. Credit spreads are modeled through a lognormal process and then correlated to interest rates. For the fixed rate payer, a positive correlation with credit spreads implies higher interest rates in default scenarios, leading to a higher positive exposure and higher wrong-way risk. For a negative correlation, the negative exposure increases (right-way risk).

###### Figure 2- Interest rate simulations conditional on counterparty default (at some point in the five-year period) for the hazard rate model.

Figure 2 above shows paths of interest rates generated given defaults incase correlation is negative. This is due to the fact that defaults occur where credit spread are wider and interest rates are lower (implying the negative correlation).

###### Figure 3: Future values for a receiver interest rate swap conditional on counterparty default for a hazard rate approach with negative correlation.

From figure 3 above, we can see that swap values are more in -the-money in case of default. This is an impact of WWR thus leading to a higher CVA when the correlation is negative and vise-versa.

#### Drawback of the approach

This approach generates only weak dependency between exposure and default. The WWR effects are still not strong even when a strong correlation is used.

### Structural Approaches

This is a more-simple approach which requires that the dependency between the counterparty default time and exposure distribution be specified.

Exposure and default distribution are mapped separately into a bivariate distribution. In case of a WWR, positive dependency will lead to an early default time being coupled with a higher exposure. The reverse is true for the right- way-risk.

The advantage of this approach is that the original unconditional values are sampled directly and hence there is no need to recalculate the exposures. The exposure distributions computed initially are used and WWR is added on top of the existing methodology.

When plotted against correlation, it can be seen that CVA is reduced(increased) by negative(positive) correlation due to the effects right way risk (WWR) The effect is stronger and the CVA doubles when correlation is about 0.50. See Figure 4 below.

#### Drawbacks of the approach

• Calibrating the correlation being talked about, is difficult since the correlation is not clear.
• It is not appropriate to assume that all the required information for defining the WWR is contained in the original unconditional exposure distribution.

### Parametric Approaches

This is a more direct approach that involves parametrically linking default probability to the exposure by simple functions as proposed by Hull and White (2011). An intuitive calibration based on what-if scenario or using historical data to calibrate the relationship may be applied.

When calibration is achieved through historical data, portfolio value for dates in the past need to be calculated and then the relationship between this and the counterparty’s credit spread is examined. If the portfolio is found to have high values with credit spread that exceed the average, is a clear indication of the WWR. The historical data needs to show a meaningful relationship. Furthermore, the current portfolio of trades with the counterparty needs to be of the same nature to that of used in the historical calibration.

In the Hull and White WWR model, the single parameter (b) drives the relationship, which has an impact similar to the correlation in the structural model.

From figure 5 below, it can be seen that, a positive value will result into a WWR accompanied by a higher CVA. The reverse is true for a negative value.

### Jump approaches

These approaches can apply best in cases where we have specific WWR. The best example here is the FX case. Ehlers and Schonbucher (2006) have paid keen interest on the effect of default on FX rates and considered cases where the hazard rate approach is not suitable to explain empirical data. This is an indication that there is an additional jump in the FX rate at default.

The model proposed by Levy and Levin (1999), to model FX exposures with WWR assumes relevant counterparty FX rate jumps at the counterparty default time.

The jump factor is commonly known as the residual value (RV) factor of the currency and it is assumed that currency depreciates by an amount (1-RV) at the default time of the counterparty and appropriate FX rate jumps.

###### Figure 6: Illustration of the currency jump approach to WWR for FX products.

An empirical estimate of the magnitude of the jump through the residual value (RV). The sovereign defaults’ currency is made on the basis of 92 historical defaults and is shown in Table 1 below. Sovereigns with better ratings have a larger RV. This can is probably because their default requires a more severe financial shock, and therefore, the conditional FX rate needs to be increased. The same approach may also be applied to other counterparties. For instance, when a large corporate defaults, we expect it to greatly affect the domestic currency.

###### Table 1:  Residual currency Values (RV) upon sovereign default as a function of the sovereign rating prior to default

$$\small{\begin{array}{|c|c|}\hline\textbf{Rating} & \textbf{Residual Value}\\ \hline\text{AAA} & 17\%\\ \hline\text{AA} & 17\%\\ \hline\text{A} & 22\%\\ \hline\text{BBB} & 27\%\\ \hline\text{BB} & 41\%\\ \hline\text{B} & 62\%\\ \hline\text{CCC} & 62\%\\ \hline\end{array}}$$

The effect of the conditional expected exposure for the devaluation WWR approach is fairly uniform contrary to the previous conclusions that the impacts of WWR vary greatly with time horizon. It can be assumed that in the short-term, immediate sovereign default may result in a large currency jump (a small RV), while a later default may result in a smaller currency jump (larger RV in the medium/long term.)

###### Figure 7: Illustration of the conditional expected exposure for the devaluation WWR approach for an FX forward assuming a residual value factor RV = 80%. The FX volatility is assumed to be 15%.

The devaluation approach also applies in CDS markets. Largely, CDS are quoted in US dollars, with a small percentage quoted in other currencies.

Consider table 2 below,

###### Table 2: CDS Quotes (Mid-Market) on Italy in Both US Dollars and Euros from April 2011

$$\small{\begin{array}{|c|c|c|}\hline\textbf{Maturity} & \textbf{USD} & \textbf{EUR} \\ \hline\text{1Y} & 50 & 35\\ \hline\text{2Y} & 73 & 57\\ \hline\text{3Y} & 96 & 63 \\ \hline\text{4Y} & 118 & 78\\ \hline\text{5Y} & 131 & 91\\ \hline\text{7Y} & 137 & 97\\ \hline\text{10Y} & 146 & 103\\ \hline\end{array}}$$

The only difference between CDS contracts is the currency of cash payment on default. Euro-denominated have a large “quanto” effect and as such, CDS are cheaper by about 30% for all maturities. This matches a RV of about 69% in the event that Italy defaults using five-year quotes (91/131). The RV is time-homogeneous which is in line with the approach discussed above.

## Central Clearing and Wrong-Way Risk

Central Counterparties (CCPs) especially those that clear CDS products, are prone to WWR since they rely on collateral as their protection. The common feature of a CCP is that the defaulter pays. i.e., Losses resulting from a clearing member shall be included in the resources dedicated by that clearing member to compensate for the resulting losses. When such losses are imposed on members, the CCP puts itself at the risk of becoming insolvent.

CCPs tend to not associate credit quality and exposure. In order to be a clearing member, the CCP requires that parties must have a certain credit quality and not external credit ratings. However, initial margins and default fund contributions will then be charged, driven primarily by the market risk of their own portfolio. As a result, these CCPs are at risk of implicitly ignoring the WWR.

The problem associated with WWR transactions such as CDS and CCPs is that it is difficult to quantify the WWR component in defining initial margins and default funds. In addition, WWR increases proportionally with increasing credit quality.

It has also been argued that large dealers represent more WWR than smaller credit quality counterparties. As such, it is clear that CCPs require greater initial margins and default fund contributions from better credit quality members.

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