Stress Loss For Loan Portfolios

• 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)

  The stress loss for the loan portfolio is determined as 𝐸𝐿_𝑠−𝐸𝐿. Financial institutions can conduct stress tests by adjusting default probabilities or stressing the variables affecting these probabilities, typically macroeconomic or counterparty balance-sheet indicators. These stress tests can be conducted for individual loan counterparts and at an aggregated level.

Stress Loss On Derivatives Portfolio

• EL and ELS calculations for derivatives portfolios resemble those for loans, using PD and LGD. However, instead of the stochastic EAD, which is influenced by market factors, it utilizes the Expected Positive Exposure (EPE) multiplied by an alpha factor (α). This adjustment integrates Counterparty Credit Risk into the portfolio credit model. Thus, the computation for EL and EL_s for derivatives portfolios is as follows:

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

• Stress losses for derivatives portfolios can be calculated where institutions can stress the probability of default or its influencing variables, such as company balance-sheet values, macroeconomic indicators, and financial instrument values, similar to loans. Combining stress losses from both portfolios is feasible by aggregating these losses. Moreover, there are additional variables to stress in derivatives portfolios, like EPE, which is  linked to market variables like equity prices and swap rates. The impact of these stresses on expected losses is uncertain and may vary based on factors like portfolio directionality, margined counterparties, and excess margin. This contrasts with loan portfolios, where stresses on default probabilities tend to have uniform directional effects across counterparties. Consequently, when stressing EPE, aggregation with loan portfolios isn’t necessary, as loans aren’t influenced by market variables and won’t change exposure due to market shifts.
• A financial institution usually employs instantaneous shocks on market variables, primarily focusing on shocks to current exposure. These shocks involve altering the initial market value of derivatives before calculating EPE. The impact of such shocks on EPE relies on factors like collateralization and portfolio moneyness.
• Moreover, institutions may apply simultaneous stresses on credit quality and market variables. Although conceptually straightforward, aligning changes in macroeconomic or balance-sheet variables with market shifts can be challenging. Equity-based approaches offer some connection, but the concurrence of an exposure shock with equity-based default probabilities remains uncertain.
• 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

• In stress testing CCR within a market risk framework, focus lies on the market value of the counterparty credit risk and potential losses stemming from market variable fluctuations, such as changes in the counterparty’s credit spread.
  
  • Typically, unilateral CVA is considered by financial institutions for stress testing, involving concerns about potential counterparty defaults across diverse market scenarios.
  • When considering the potential for their own default to counterparties, financial institutions should consider stress testing bilateral CVA.

• A common simplified formula for CVA to a counterparty that omits wrong-way risk, and aggregated across N counterparties is given as

  
  where
  \(LGD_n^*\) is the risk-neutral loss-given default for counterparty
  \(EE_n^*(t_j)\) is the discounted expected exposure during the 𝑗^th time period calculated under a risk-neutral measure for counterparty 𝑛.
  \(PD_n^*(t_{j-1}, t_j)\) is the risk-neutral marginal default probability for counterparty 𝑛 in the time interval from 𝑡_j-1  and 𝑡_j , and
  𝑇 is the final maturity
• The constituents of the previous formula are dependent on market variables. In particular,
  
  • \(EE_n^*(t_j)\) depends on the values of derivative transactions with the counterparty.
  • \(LGD_n^*\) is generally set by convention or from market spreads
  • \(PD_n^*(t_{j-1}, t_j)\) is derived from credit spreads of the counterparty
• The stressed CVA is computed by introducing instantaneous shocks to certain variables, impacting both the risk-neutral discounted expected exposure, \(EE_n^*(t_j)\), and the risk-neutral marginal default probability,\(PD_n^*(t_{j-1}, t_j)\). The formula for the 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
  \(CVA^s – CVA\)
• Comparing stress testing methodologies between CCR in credit risk and market risk frameworks reveals similarity in relying on expected losses, calculated as the product of loss-given default, exposure, and 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:
  • Usage 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 used to determine the probability of default leans towards market-based approaches in CVA. The market-based approach is advantageous because it enables correlation integration between exposure and PD, which significantly affects the CVA. However, due to the uncertainty of correlation, financial institutions should conduct stress tests to assess the impact of incorrect correlation assumptions on profit and loss.
• For a comprehensive assessment of the impact of different scenarios on CVA profit and loss, financial institutions should incorporate the liability side effects in stress tests. This component, known as DVA within bilateral CVA (BCVA), evaluates the value associated with the institution’s option to default on counterparties.

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 DVA formula parallels the CVA formula with two key modifications.
  
  • Firstly, instead of expected exposure, the calculation involves determining the negative expected exposure (NEE), which is computed from the counterparty’s standpoint.
  • Secondly, the financial institution’s option value to default is contingent upon the counterparty’s survival, requiring incorporation of the counterparty’s survival probability (denoted by S_I in the computation.
• Similarly, adjustments are made to the CVA segment, as the loss arising from the counterparty default relies on the financial institution not defaulting prior. 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.

• Survival probabilities are influenced by CDS spreads, and losses now hinge on the firm’s credit spread. This dynamic could yield unexpected outcomes, like losses occurring despite improvements in the firm’s credit quality.
• From a bilateral standpoint in stress testing, the financial institution must also assess the correlation between its own credit spread and those of its counterparties. Stress losses can be computed akin to CVA losses by determining a stress BCVA and deducting 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.