Introduction

• The measurement and management of counterparty credit risk (CCR) has evolved rapidly since the late 1990s. Despite this quick progress in the evolution of statistical measures of CCR, stress testing of CCR has not evolved nearly as quickly.
• The fast pace of change in CCR modelling can be seen in the progression of statistical measures used to gauge counterparty credit risk. First, potential exposure models were developed to measure and limit counterparty risk. Second, the potential-exposure models were adapted to expected positive-exposure models that allowed derivatives to be placed in portfolio credit risk
• models similar to loans. These two types of models are the hallmark of treating CCR as a credit risk.
• The treatment of CCR as credit risk was the predominant framework for measuring and managing CCR from 2000 to 2006 and was established as the basis for regulatory capital as part of Basel II (BCBS 2005). The definitions of the exposure measures given in the next slide follow those in BCBS (2005).

Definitions

• Current exposure is the larger of zero and the market value of a transaction or portfolio of transactions within a netting set, with a counterparty that would be lost upon the default of the counterparty, assuming no recovery on the value of those transactions in bankruptcy. Current exposure is often also called replacement cost.
• Peak exposureis a high-percentile (typically 95% or 99%) of the distribution of exposures at any particular future date before the maturity date of the longest transaction in the netting set. A peak exposure value is typically generated for many future dates up until the longest maturity date of transactions in the netting set.
• Expected exposure is the mean of the distribution of exposures at any particular future date before the longest-maturity transaction in the netting set matures. An expected exposure value is generated for many future dates up to the longest maturity date of transactions in the netting set.
• Expected positive exposure (EPE)is the weighted average over time of expected exposures where the weights are the proportion that an individual expected exposure represents of the entire time interval. When calculating the minimum capital requirement, the average is taken over the first year or over the time period of the longest-maturity contract in the netting set.
• An unusual problem associated with CCR, that of wrong-way risk, has been identified (Levin and Levy 1999; Finger 2000). Wrong-way risk occurs when the credit quality of the counterparty is correlated with the exposure, so that exposure grows when the counterparty is most likely to default. When exposure is fixed as is the case for a loan, this does not occur, so adaptation of techniques used in other areas of risk management is more difficult.

Treatment Of CCR As Credit And Market Risk

• Although CCR has been historically classified under credit risk, but the treatment of CCR as a market risk was also developing since 1990s. It’s treatment as market risk was largely relegated to pricing in a credit valuation adjustment (CVA), prior to the financial crisis of 2007-2009.
• The complexities of risk-managing this price aspect of a derivatives portfolio did not become apparent until the crisis. Prior to the crisis, credit spreads for financial institutions were relatively stable and the CVA was a small portion of the valuation of banks’ derivatives portfolios. During the crisis, both credit spreads and exposure amounts for derivative transactions experienced wide swings, and the combined effect resulted in both large losses and large, unusual gains.
• Financial institutions are just now beginning to develop their frameworks to risk-manage CVA. The regulatory capital framework has adopted a CVA charge to account for this source of risk (BCBS 2011).
• The treatment of CCR as a credit risk or CCR as a market risk has implications for the organization of a financial institution’s trading activities and the risk-management disciplines. Both treatments are valid ways to manage the portfolio, but adoption of one view alone leaves a financial institution blind to the risk from the other view.
  • If CCR is treated as a credit risk, a bank can still be exposed to changes in CVA. A financial institution may establish PFE limits and manage its default risk through collateral and netting, but it still must include CVA in the valuation of its derivatives portfolio. Inattention to this could lead to balance-sheet surprises.
  • If CCR is treated as a market risk, dynamically hedging its CVA to limit its market risk losses, it remains exposed to large drops in creditworthiness or the sudden default of one of its counterparties. A derivatives dealer is forced to consider both aspects.
• The view of CCR has implications for how the risk is managed as well.
  • The traditional credit risk view is that the credit risk of the counterparty can be managed at inception or through collateral arrangements set up in advance, but there is little that can be done once the trades are in place. At default the financial institution must replace the trades of the defaulting counterparty in the market all at once in order to rebalance its book. A large emphasis is placed on risk mitigants and credit evaluation as a result.
  • The view of CCR as a market risk allows that its counterparty credit risk can be hedged. Instead of waiting until the counterparty defaults to replace the contracts, the financial institution will replace the trades with a counterparty in the market before it defaults by buying the positions in proportion to the counterparty’s probability of default. Thus a counterparty with a low probability of default will have few of its trades replaced in advance by the financial institution, but, as its credit quality deteriorates, a larger proportion of those trades will be replaced by moving them to other counterparties. At default, the financial institution will have already replaced the trades and the default itself would be a non-event.

Implications For Stress Testing

• The dual nature of CCR leads to many measures that capture some important aspects of CCR.
  • On the credit risk side, there are the important measures of exposure: current exposure, peak exposure and expected exposure.
  • On the market risk side there is the valuation aspect coming from CVA, and there is the risk generated by changes in the CVA, as measured by VaR of CVA, for example.
• This creates a dazzling array of information that can be difficult to interpret and understand at both portfolio and counterparty levels.
• As far as stress testing is concerned, we can compare the number of stresses that a bank may run on its market risk portfolio with the number of similar stresses a bank would run on its counterparty credit risk portfolio.
  • In market risk, running an equity crash stress test may result in one or two stress numbers: an instantaneous loss on the current portfolio and potentially a stress VaR loss. A risk manager can easily consider the implications of this stress.
  • In contrast, the CCR manager would have to run this stress at the portfolio level and at the counterparty level, and would have to consider CCR as both a credit risk and a market risk. The number of stress-test results would be at least twice the number of counterparties plus
    one. The number of stress-test results would at least double again if the risk manager stressed risk measures in addition to considering instantaneous shocks.
• Despite this array of potential stress results, a risk manager must stress-test counterparty exposures to arrive at a comprehensive view of the risk of the financial institution’s
  portfolio.

Stress Testing Of Exposures

• The most common stress tests used in counterparty credit are stresses of current exposure. To create a stressed current value, the bank assumes a scenario of underlying risk-factor changes and reprices the portfolio under that scenario. Generally, a financial institution applies these stresses to each counterparty. It is common practice for banks to report their top counterparties with the largest current exposure to senior management in one table, and then follow that table with their top counterparties, with the largest stressed current exposure placed under each scenario in separate tables.
• Example-
  • Current Exposure Stress Test: Equity Crash
  • Scenario: Equity Market Down 25%

($MM)	Rating	MtM	Collateral	Current Exposure	Stressed Current Exposure
Counterparty A	A	0.5	0	0.5	303
Counterparty B	AA	100	0	100	220
Counterparty C	AA	35	0	35	119
Counterparty D	BBB	20	20	0	76
Counterparty E	BBB	600	600	0	75
Counterparty F	A	-5	0	0	68
Counterparty G	A	-10	0	0	50
Counterparty H	BB	-50	0	0	24
Counterparty I	A	35	20	15	17
Counterparty J	BB	24	24	0	11

Source: Table 17-1 2019 Financial Risk Manager Exam Part II: Credit Risk Measurement and Management Seventh Edition by Global Association of Risk Professionals

Limitations Of Stress Testing Of Exposures

• This type of stress testing is quite useful, and financial institutions have been conducting it for some time. It allows the bank to identify which counterparties would be of concern in such a stress event, and also how much the counterparty would owe the financial institution under the scenario. However, stress tests of current exposure have a few problems :
  • Aggregation of the results is problematic- While the individual counterparty results are meaningful, there is no meaningful way to aggregate these stress exposures without incorporating further information. If we were to sum the exposures to arrive at an aggregate stress exposure, this would represent the loss that would occur if every counterparty defaulted in the stress scenario. Unless the scenario were the Apocalypse, this would clearly be an exaggeration of the losses. Other attempts to aggregate these results are also flawed. For example, running the stressed current exposure through a portfolio credit risk model would also be incorrect, since expected exposures, not current exposures, should go through a portfolio credit risk model.
  • It does not account for the credit quality of the counterparties- it accounts only for the value of the trades with the counterparty and not the counterparty’s willingness or ability to pay. This is an important deficiency since a US$200 million exposure to a start-up hedge fund is very different from a US$200 million exposure to an AAA corporate.
  • It provides no information on wrong-way risk- stress tests of current exposure provide little insight into wrong-way risk. As a measure of exposure that omits the credit quality of the counterparty, these stress tests without additional information cannot provide any insight into the correlation of exposure with credit quality.
• Hence we can say that, stresses of current exposure are useful for monitoring exposures to individual counterparties, but do not provide either a portfolio outlook or incorporate a credit quality.

Stress Testing The Loan Equivalent

• Expected loss for any one counterparty is the product of the probability of default, \(PD_i\), where this may depend on other variables, exposure at default, \(EAD_i\), and loss-given default, \(LGD_i\). The expected loss for the pool of loan counterparties is given by :

\(EL = \sum_{i=1}^{N} PD_i \times EAD_i \times LGD_i\) …(A)

• A stress test could take exposure at default and loss-given default as deterministic and focus on stresses where the probability of default itself is subject to a stress. In this case, the probability of default is taken to be a function of other variables; these variables may represent an important exchange rate or an unemployment rate, for example. In this case, the stressed expected loss is calculated conditional on some of the variables affecting the probability of default being set to their stressed values; the stressed probability of default is denoted ps.; and the stressed expected loss is:

\(EL_s = \sum_{i=1}^{N} PD_i^s \times EAD_i \times LGD_i\) …(B)

Stress Testing The Loan Equivalent And Derivatives Portfolio

• The stress loss for the loan portfolio is𝐸𝐿௦െ𝐸𝐿. A financial institution can generate stress tests in this framework rather easily. It can simply increase the probability of defaults, or it can stress the variables that these probabilities of defaults depend on. These variables are typically macroeconomic variables or balance-sheet items for the counterparty. The stress losses can be generated for individual loan counterparties as well as at an aggregate level.
• This framework can be adapted for CCR treated as a credit risk. In this case the probability of default and loss given default of the counterparty are treated the same, but now exposure at default is stochastic and depends on the levels of market variables. EPE multiplied by an alpha factor is the value that allows CCR exposures to be placed in a portfolio credit model along with loans and arrive at a high-percentile loss for the portfolio of exposures (both loan and derivatives). The same procedure is applied here and EPE is used in an expected-loss model. In this case we have

\(EL = \sum_{i=1}^{N} PD_i \times \alpha \times EPE_i \times LGD_i \quad \text{and} \quad EL_s = \sum_{i=1}^{N} PD_i^s \times \alpha \times EPE_i \times LGD_i\)

Stress Loss On Derivatives Portfolio

• Stress losses on the derivatives portfolio can be calculated similarly to the loan portfolio case. A financial institution can stress the probability of default similarly to the loan case by stressing probability of default or the variables that affect probability of default, including company balance-sheet values, macroeconomic indicators and values of financial instruments. It can also combine the stress losses on the loan portfolio and the stress losses on its derivatives portfolio by adding these stress losses together.
• In addition, a financial institution has a new set of variables to stress. EPE depends on market variables such as equity prices and swap rates, and so, these variables can also be stressed to observe their impacts. It is not clear whether a stress will, in aggregate, increase or decrease expected losses. This will depend on factors like directional bias of the bank’s portfolio, which
  counterparties are margined, and which have excess margin. This is in marked contrast to the case of loan portfolios where stresses of the probabilities of default are considered. Stresses to the variables affecting the probability of default generally have similar effects and the effects are in the same direction across counterparties. When conducting stresses to EPE, a bank need not consider aggregation with its loan portfolio. Loans are insensitive to the market variables and thus will not have any change in exposure due to changes in market variables.
• There are a many stresses that can be considered. Typically a financial institution will use an instantaneous shock of market variables. In principle, we could shock these variables at some future point in their evolution or create a series of shocks over time, but the more common approach shocks to current exposure are the norm. In the performance of these instantaneous shocks, the initial market value of the derivatives is shocked prior to running the simulation to calculate EPE. How this shock affects EPE depends on the degree of collateralization and the “moneyness” of the portfolio, among other things.
• A financial institution can consider joint stresses of credit quality and market variables as well. Conceptually, this is a straightforward exercise, but, in practice, deciding how changes in macroeconomic variables or balance-sheet variables are consistent with changes in market variables can be daunting. There is little connection between the variables . Equity-based approaches come close to providing a link; however, it remains unclear how to link an instantaneous shock of exposure to the equity-based probability of default. While exposure can and should react immediately, it is unclear whether equity-based probabilities of default should react so quickly.
• It’s also difficult to capture the connection between the probability of default and exposure that is often of concern in CCR. There are many attempts to capture the wrong-way risk, but most are ad hoc. At present the best approach to identifying wrong-way risk in the credit framework is to stress the current exposure, identify those counterparties that are most exposed to the stress and then carefully consider whether the counterparty is also subject to wrong-way risk.
• Stress tests of CCR as a credit risk allow a financial institution to advance beyond simple stresses of current exposure. They allow aggregation of losses with loan portfolios, and also allow consideration of the quality of the counterparty. These are important improvements that allow a financial institution to better manage its portfolio of derivatives. Treating CCR as a market risk allows further improvements (notably, the probability of default will be inferred from market variables), and it will be easier to consider joint stresses of credit quality and exposure.

Stress Testing CVA

• When stress testing CCR in a market risk context, we are usually concerned with the market value of the counterparty credit risk and the losses that could result due to changes in market variables, including the credit spread of the counterparty.
  • Generally, financial institutions consider the unilateral CVA for stress testing. Here, the financial institution is concerned with the fact that its counterparties could default under various market scenarios.
  • If financial institutions consider the possibility of their own default to their counterparties, then bilateral CVA should be considered for stress testing.
• Initially we just consider stress testing the unilateral CVA.
• All the key components of formula (D) depend on values of market variables.
  • \(PD_n^*(t_{j-1}, t_j)\) is derived from credit spreads of the counterparty
  • \(LGD_n^*\) is generally set by convention or from market spreads
  • \(EE_n^*(t_j)\) depends on the values of derivative transactions with the counterparty.
• To calculate a stressed CVA we would apply an instantaneous shock to some of these market variables. The stresses could affect \(EE_n^*(t_j)\) or \(PD_n^*(t_{j-1}, t_j)\). Stressed CVA is given by

\(CVA^s = \sum_{i=1}^{N} LGD_n^* \times \sum_{j=1}^{T} EE_n^s(t_j) \times PD_n^s(t_{j-1}, t_j)\)

And the stress loss is \(CVA^s – CVA\)

• Stressing current exposure, as described previously, has similar effects. An instantaneous shock will have some impact on the expected exposure calculated in later time periods, so all of the expected exposures will have to be recalculated. Stresses to the marginal probability of default are usually derived from credit spread shocks.
• Similarities can be seen between stress testing CCR in a credit risk framework and doing so in a market risk framework. There is a reliance in both cases on expected losses being the product of loss-given default, exposure and the probability of default.
• However, these values will be quite different, depending on the view of CCR as a market risk or credit risk. This can be because of the following reasons:
  • Use of risk-neutral values for CVA as opposed to physical values for expected losses.
  • CVA uses expected losses over the life of the transactions, whereas expected losses use a specified time horizon.
  • The model for determining the probability of default is market-based in CVA.

Stress Testing CVA and DVA

• Using a market-based measure for the probability of default provides some benefits. It is possible in these circumstances to incorporate a correlation between the probability of default and the exposure. Hull and White (2012) demonstrate an important stress test of the correlation between exposure and the probability of default. They show that the correlation can have an important effect on the measured CVA. Since there is likely to be a high degree of uncertainty around the correlation, a financial institution should runstress tests to determine the impact on profit and loss if the correlation is wrong.
• To capture the full impact of various scenarios on CVA profit and loss, a financial institution should include the liability side effects in the stress as well. This part of the bilateral CVA (BCVA), often called DVA, captures the value of the financial institution’s option to default on its counterparties.

Stress Testing DVA

• The formula for DVA is similar to the formula for CVA except for two changes. First, instead of expected exposure, we have to calculate the negative expected exposure (NEE). This is expected exposure calculated from the point of view of the counterparty. Second, the value of the option to default for the financial institution is dependent on the survival of the counterparty, so the probability that the counterparty has survived (given by𝑆ூ) must enter into the calculation,. A similar change must be made to the CVA portion, since the loss due to the counterparty defaulting now depends on the financial institution not defaulting first. Hence, the bilateral CVA (BCVA) formula can be written as :

\(BCVA = \sum_{i=1}^{N} LGD_n^* \times \sum_{j=1}^{T} EE_n^*(t_j) \times PD_n^*(t_{j-1}, t_j) \times S_i^*(t_{j-1})\)

\(\sum_{i=1}^{N} LGD_l^* \times \sum_{j=1}^{T} NEE_n^*(t_j) \times PD_l^*(t_{j-1}, t_j) \times S_n^*(t_{j-1})\)

where the subscript I refers to the financial institution.

• The survival probabilities also depend on CDS spreads and now the losses depend on the firm’s own credit spread. This may lead to counterintuitive results such as losses occurring because the firm’s own credit quality improves. When looking at stress tests from a bilateral perspective, the financial institution will also have to consider how its own credit spread is correlated with its counterparties’ credit spread. Stress losses can be calculated in a similar way as for CVA losses by calculating a stress BCVA and subtracting the current BCVA.
• BCVA allows CCR to be treated as a market risk. This means CCR can be incorporated into market risk stress testing in a coherent manner. The gains or losses from the BCVA stress loss can be added to the firm’s stress tests from market risk. As long as the same shocks to market variables are applied to the trading portfolio and to the BCVA results, they can be aggregated by simple addition.

Common Pitfalls In Stress Testing CCR (Counterparty Credit Risk)

• The methodologies to conduct stress tests beyond stressing current exposure, are only just being developed. It is rare for CCR to be aggregated with either stress tests of the loan portfolio or with trading-position stress testing results in a consistent framework.
• It is tempting to use stresses of current exposure when combining the losses with loans or trading positions. But, according to analysis, expected exposure or expected positive exposure should be used as the exposure amount, instead of current exposure. The use of current exposure instead of expected exposure can lead to substantial errors. This can be shown using a normal approximation (Gregory 2010) to expected exposures, which is accurate for linear derivatives with no intermediate payments.

Common Pitfalls In Stress Testing CCR

• This figure plots current exposure and expected exposure after a million-dollar shock to the market value of the derivative. For at-the-money exposures, the difference between current exposure and expected exposure is almost half the value of the shock.
• Use of delta sensitivities to calculate changes in exposures is also especially problematic for CCR, since it is highly nonlinear. While this can save on computational resources, the errors introduced are not obvious and the linearization can be highly misleading. At-the-money portfolios with large price moves applied to the portfolio are especially prone to errors from using delta approximations.

Source: Figure 17-1 2019 Financial Risk Manager Exam Part II:
Credit Risk Measurement and Management Seventh Edition by
Global Association of Risk Professionals