Credit Risk In Derivatives Transactions

• Under an International Swaps and Derivatives Association (ISDA) Master Agreement, an event
  of default occurs if a party fails to meet financial obligations, such as missing payments or not
  posting required collateral, or declares bankruptcy. This allows the non-defaulting party to
  terminate all transactions. Losses for the non-defaulting party can occur in two ways:
  • if the exceeds the collateral posted by the defaulting party, or
  • if the non-defaulting party has posted collateral that exceeds the value of transactions to
    the defaulting party, making them an unsecured creditor in either scenario.

CVA and DVA

• For a bank
  • Credit Valuation Adjustment (CVA) represents the current value of potential losses from a
    counterparty’s default, and
  • Debt (Debit) Valuation Adjustment (DVA) reflects the current value of potential costs to a
    counterparty due to the bank’s default.
• The bank benefits from the possibility of its own default, as it reduces the likelihood of
  fulfilling payment obligations on derivatives, thus increasing the value of its derivatives
  portfolio. The value of transactions considering defaults is calculated as:

  \(f_{nd} – CVA + DVA\)

  where
  \(f_{nd}\) is the value assuming no defaults on either side

• CVA and DVA are calculated over the life of the derivatives (up to 𝑇years), split into 𝑁
  intervals, as\(CVA = \sum_{i=1}^{N} q_i V_i\)

  \(DVA = \sum_{I=1}^{N} q_i^* v_i^*\)

  where

  • \(q_i\) is the risk-neutral probability of the counterparty defaulting during the \(i_{th}\) interval,
  • \(v_i\) is the present value of the expected loss to the bank if the counterparty defaults at the
    midpoint of the \(i_{th}\) interval,
  • \(q_i^*\) is the risk-neutral probability of the bank defaulting during the \(i_{th}\) interval, and
  • \(v_i^*\) is the present value of the expected loss to the counterparty (gain to the bank) if the
    bank defaults at the midpoint of the \(i_{th}\) interval.
• As deduced in FRM Part 1 and in one of the earlier chapters in FRM Part 2 (based on the idea
  of hazard rate approximation formula), the probability of the counterparty defaulting during the \(i_{th}\) interval (i.e., between \(t_i\) and \(t_{i-1}\)) is given by

  \(q_i = e^{\left( \frac{-s(t_{i-1}) t_{i-1}}{1 – R} \right)} – e^{\left( \frac{-s(t_i) t_i}{1 – R} \right)}\)

  where \(s(t_i)\) is the counterparty’s credit spread at time \(t_i\) , and R is the recovery rate if counterparty defaults

• When calculating \(v_i\), a Monte Carlo simulation is used to model the no-default value of transactions over time in a risk-neutral setting. The process involves simulating market variables from time 0 to T, and at each trial, determining the bank’s exposure to the counterparty at the midpoint of each interval. Exposure is calculated as max(V,0). where V is the total value of the transactions to the bank. If V is negative, there is no exposure; if they have a positive value, and the exposure is V if it is positive. The variable \(v_i\) is set equal to the present value of the average exposure across all simulation trials multiplied by (1 – R). The variable \(v_i^*\) is calculated similarly from the counterparty’s exposure to the bank.
• When collateral is involved, \(v_i\) becomes more complex to compute. It requires simulations to estimate the collateral each party holds at the midpoint of each interval, assuming that the counterparty will stop posting or returning excess collateral 𝑐 days before defaulting. This 𝑐, which is typically 10 or 20 days is known as the cure period or margin period of risk. The value of transactions 𝑐 days prior to the midpoint needs to be calculated to accurately determine the collateral held during a default. An basic example has been illustrated in the next page. The present value of the expected loss \(v_i\) is calculated from the average exposure across all simulation trials as in the no-collateral case. A similar analysis of the average exposure of the counterparty to the bank leads to \(v_i^*\).
• EXAMPLE – In a two-way zero-threshold collateral agreement between a bank and its counterparty, each side posts collateral equal to the maximum value of the outstanding transactions to the other side. The cure period is set at 20 days.Consider several scenarios during a Monte Carlo simulation:
  • If the value of transactions to the bank at midpoint (τ) is $60 and was $55 twenty days earlier, the bank assumes it has $55 in collateral during a default at (τ). The bank’s exposure is the uncollateralized value of $5.
  • If the value at (τ) is $60 but was $65 twenty days earlier, the bank is assumed to have full collateral, reducing its exposure to zero.
  • If the transaction value at (τ) is -$60 and was -$50 twenty days prior, the bank is considered to have posted $60 in collateral, resulting in zero exposure in the event of a default.
  • If the value at (τ) is -$60 but was -$70 twenty days earlier, it is assumed that the bank has overcollateralized by $70. In the event of a default, the bank’s exposure would be $10, representing the excess collateral posted.
• The quick calculation of the impacts of new transactions on Credit Valuation Adjustment (CVA) and Debit Valuation Adjustment (DVA) is facilitated by banks’ practice of storing all sampled paths from simulations of market variables. Determining the incremental effects on CVA and DVA from new transactions is straightforward. A new transaction that is positively correlated with existing transactions will likely increase CVA and DVA, while a negatively correlated transaction will likely decrease them.
• It is assumed in the CVA calculation method that the counterparty’s probability of default is independent of the bank’s exposure, a premise considered reasonable in many contexts. However, traders acknowledge “wrong-way risk” when default probability and exposure are positively correlated, and “right-way risk” when they are negatively correlated. More complex models have been developed to account for these kinds of correlations.
• Each counterparty to a bank is associated with one CVA and one DVA, both considered derivatives that fluctuate in value with changes in market variables, counterparty credit spreads, and bank credit spreads. The risks in CVA, and occasionally in DVA, are managed similarly to other derivatives through calculations using Greek letters, scenario analyses, etc.

Credit Risk Mitigants

• Banks utilize several strategies to minimize credit risk in bilaterally cleared transactions, including netting. For example, if a bank has three uncollateralized transactions with a counterparty valued at +$20 million, +$40 million, and -$30 million, treating them as independent transactions results in exposures of $20 million, $40 million, and $0, totaling $60 million. By applying netting, these transactions are consolidated into a single transaction worth $30 million, reducing the total exposure from $60 million to $30 million.
• Collateral agreements serve as a key method for mitigating credit risk in financial transactions. Collateral can consist of cash, which typically accrues interest, or marketable securities, which might be discounted (haircut) to determine their cash equivalent value for collateral purposes. Regulatory measures now govern these agreements among financial institutions. In derivatives transactions, collateral agreements provide an advantage during defaults, allowing the non- defaulting party to retain any collateral posted by the defaulting side. This arrangement typically obviates the need for costly and prolonged legal processes.
• Downgrade triggers are a credit risk mitigation tactic employed by financial institutions. These triggers are clauses included in the Master Agreement between a bank and a non-financial counterparty. They stipulate that if the counterparty’s credit rating drops below a specified threshold, such as BBB, the bank may demand collateral or terminate all outstanding derivatives transactions at their current market value.
  • However, downgrade triggers do not guard against significant, abrupt deteriorations in
    credit ratings, such as a change from A to default.
  • Furthermore, their effectiveness diminishes when they are widely used by a counterparty
    across various derivatives dealers, potentially offering limited protection, which happened in the case of AIG in 2008. AIG had secured numerous derivatives transactions, guaranteeing the performance of AAA-rated tranches of ABS CDOs. These transactions included downgrade triggers, which exempted AIG from posting collateral as long as its credit rating stayed above AA. However, on September 15, 2008, AIG was downgraded below AA by major rating agencies including Moody’s, S&P, and Fitch. Concurrently, the performance of the guaranteed tranches deteriorated significantly, leading to numerous collateral calls from counterparties. AIG was unable to fulfill these demands for collateral, and ultimately, a large-scale government bailout was necessary to prevent its bankruptcy.

Default Correlation

• Default correlation is a fundamental concept in credit risk management, representing the likelihood that two or more companies will default simultaneously or within a close timeframe. This correlation occurs due to several interconnected factors that influence the financial stability of companies, especially those within the same industry or geographic area.
  • Industry and Geographic Influences: Companies operating in the same industry or geographic region often face similar economic, regulatory, and market conditions. For instance, a significant downturn in the automotive industry or regulatory changes affecting telecommunications can impact all companies within those sectors, increasing the likelihood of simultaneous defaults.
  • Economic Cycles: Economic conditions naturally fluctuate, influencing business operations across various sectors. During economic downturns or recessions, default rates generally increase as companies struggle with reduced consumer demand, tightening credit conditions, and declining profits. Conversely, in booming economic conditions, default rates typically decrease.
  • Credit Contagion: This phenomenon occurs when the default of one company affects the financial health of other companies. This can happen through direct business relationships, such as if a major supplier goes bankrupt, affecting the supply chain, or through financial ties, such as interlinked debt obligations. Credit contagion can lead to a cascade of defaults within a network or sector.
• The two implications of default correlation are –
  • Limitations on Diversification: The presence of default correlation implies that diversifying a
    credit portfolio across different companies or sectors may not entirely mitigate risk. If default probabilities are correlated, adverse conditions affecting one borrower could affect others, making it difficult to fully diversify away the risk of large-scale defaults.
  • Impact on Risk‐neutral vs. Real‐world Probabilities: Risk-neutral probabilities, used in financial modeling and pricing of risky assets like credit derivatives, are typically higher than real- world probabilities. This is because risk-neutral measures incorporate a risk premium for bearing uncertainty and are adjusted for the market’s risk aversion. Default correlation increases these probabilities because it represents an additional layer of systemic risk that must be priced into risk-neutral evaluations.
• Estimating default correlation for credit portfolios is a critical task in financial risk management, particularly for institutions that handle diverse credit exposures. Understanding these correlations helps financial institutions assess and manage the systemic risks associated with their credit portfolios. Estimating default correlation is significant because of the following reasons:
  • Portfolio Risk Management: Estimating default correlations is essential for accurately assessing
    the risk of a credit portfolio. High correlations between the default probabilities of portfolio constituents can significantly increase the likelihood of simultaneous defaults, exacerbating portfolio volatility and risk.
  • Diversification Strategy: Default correlation plays a key role in shaping diversification strategies. In credit risk management, diversification is used to reduce risk by spreading exposure across various borrowers or sectors. Knowing the default correlations helps in selecting the right mix of assets to achieve effective diversification. For instance, if two assets are highly correlated in terms of default risk, adding them to a portfolio won’t significantly reduce the risk.
  • Pricing of Credit Derivatives: In the pricing of credit derivatives such as Credit Default Swaps (CDS) and Collateralized Debt Obligations (CDOs), default correlation is a crucial input. The pricing models for these instruments often require a clear understanding of the likelihood that multiple entities will default together, affecting the risk levels and yields of different tranches within structured financial products.
  • Regulatory Compliance and Capital Adequacy: Financial institutions are required by regulatory frameworks such as Basel III to hold capital reserves proportionate to the risks they carry, including correlated default risks. Accurately estimating these correlations allows banks to allocate capital more effectively and meet regulatory requirements for capital adequacy, ensuring they are resilient against potential losses from defaults.
  • Stress Testing and Scenario Analysis: Estimating default correlation is vital for conducting stress tests and scenario analyses. These tests simulate extreme economic conditions to determine how such scenarios would impact the portfolio. Understanding default correlations under stress conditions helps institutions prepare for and mitigate the effects of economic downturns.
  • Tail Risk Assessment: High default correlations can lead to tail risk, or the risk of rare but severe losses. In credit portfolios, such events can occur when many or all credit holdings default simultaneously, often triggered by a common economic shock. Assessing default correlation helps in measuring and managing this tail risk.
  • Strategic Decision Making: For portfolio managers, understanding default correlations can influence decisions on credit limits, the selection of new credit exposures, and the management of existing ones. It helps in crafting strategies that balance returns against the potential risk of concentrated defaults.

Types Of Default Correlation Models

• There are two main types of default correlation models:
  • Reduced Form Models: These models do not rely on a firm’s asset structure but use stochastic processes to model hazard rates that are typically influenced by broader economic factors. These models suggest that if one company’s hazard rate is high, there is an increased likelihood that another’s will be high as well, leading to correlated default risks.
    • These models are mathematically appealing because they capture how economic
      cycles can influence default correlations among companies.
    • However, a significant limitation of reduced form models is their restricted ability to
      depict a wide range of default correlations effectively. For instance, even if two companies have perfectly correlated hazard rates, the actual probability that they will both default within a very short timeframe remains remarkably low.
    • This limitation becomes particularly problematic in scenarios where companies operate in the same industry and region, or where one company’s financial health significantly depends on the others. In such cases, a higher default correlation might be necessary to accurately assess risk.
    • To address this issue, one solution is to modify the reduced form model to allow for larger jumps in the hazard rate, thus potentially increasing the predicted default correlation to a level that better reflects real-world risks.
  • Structural Models: These models, like the one pioneered by Merton, operate under the principle that a company defaults when the value of its assets falls below a predefined threshold. This approach to assessing default risk is grounded in the firm’s balance sheet, where the asset values are treated as stochastic processes.
    • To model the likelihood of joint defaults between two companies, say Company A
      and Company B, structural models introduce correlations between the stochastic processes governing their respective assets. By doing so, these models can capture the interconnectedness of companies’ financial health, a feature particularly useful in evaluating sectors or economies where the fortunes of firms are closely linked.
    • One of the primary advantages of structural models over reduced form models is their flexibility in adjusting correlation levels. In structural models, the correlation between the asset values of two companies can be explicitly modeled and adjusted to any degree, allowing for precise calibration of default correlation. This makes structural models highly adaptable to different scenarios, providing a more tailored risk assessment.
    • However, a notable disadvantage of structural models is their computational intensity.
      Due to the complexity involved in solving and simulating these models, especially when dealing with a large number of firms or when the asset processes are particularly complex, they can be computationally slow. This can make them less practical for real-time or large-scale applications compared to the typically more streamlined reduced form models.
• Both model types play a pivotal role in financial risk management by allowing institutions to estimate the likelihood and impact of potential default scenarios, thus aiding in strategic decision-making related to credit risk exposure and mitigat

The Gaussian Copula Model

• The Gaussian Copula Model for Time to Default is a widely used approach to model the default correlation of credit instruments such as loans or credit derivatives. The model operates under the assumption that all companies will ultimately default and seeks to measure the correlation between the probability distributions of default times across multiple companies.
• The model uses a copula function to couple the marginal distributions of the time until default for each entity in a portfolio. The copula allows for the modeling of default times with a multivariate normal distribution, where each entity’s time to default is influenced by a common systematic factor and an idiosyncratic factor. This common factor can be thought of as representing economic conditions affecting all entities, while idiosyncratic factors represent entity-specific conditions.
• The Gaussian copula model allows for the integration of real-world and risk-neutral probabilities into the estimation of the times to default:
  • Real-world probabilities are typically estimated using historical data from rating agencies.
  • Risk-neutral probabilities are derived from market data, such as bond prices.
• In the one-factor Gaussian copula model, the default time for each entity is assumed to depend on:
  • • A common systematic factor modeled as a standard normal variable.
    • An idiosyncratic factor unique to each entity, also modeled as a standard normal variable.

  The default of an entity occurs if the linear combination of these factors falls below a certain threshold, which is linked to the entity’s creditworthiness.

• The assumption can be modeled as

  \(x_i = a_i F + \sqrt{1 – a_i^2} Z_i\)

  where,

  • F is a common factor affecting defaults for all companies
  • \(Z_i\) is a factor affecting only company 𝑖.
  • the variable F and the variables \(Z_i\) have independent standard normal distributions.
  • the \(a_i\) are constant parameters between -1 and +1, and
  • the correlation between \(x_i\) and \(x_j\) is \(a_ia_j\).
• Conditional on the value of the factor F, the probability of default can be derived as

  \(Q_i(T|F) = N \left( \frac{N^{-1}[Q_i(T)] – a_i F}{\sqrt{1 – a_i^2}} \right)\)

• A special case of the one-factor Gaussian copula model is where the probability distributions of default are the same for all 𝑖 and the correlation between \(x_i\) and \(x_j\) is the same for all 𝑖 and 𝑗. Hence if \(Q_i(T) = Q(T)\) for all 𝑖 and that the common correlation is 𝜌, so that \(a_i = \sqrt\rho\) for all 𝑖. The above equation becomes

  \(Q(T|F) = N \left( \frac{N^{-1}[Q(T)] – \sqrt{\rho} F}{\sqrt{1 – \rho}} \right)\)

Credit VaR Using Gaussian Copula

• • Credit Value at Risk (Credit VaR) can be understood in a similar way to Value at Risk (VaR) used for market risks. Specifically, a Credit VaR with a 99.9% confidence level over a one-year period represents the maximum credit loss that is not expected to be exceeded with 99.9% certainty within that year.
  • Let’s consider a hypothetical scenario involving a bank with a large portfolio of loans that are similar in nature. For simplicity, we assume that each loan has the same probability of default and that the default correlation between any two loans is consistent across the portfolio. When applying the Gaussian copula model for time to default, given X% certainty, the percentage of losses over T years on a large portfolio will be less than V(X,T), where
    

    \(V(X,T) = N \left( \frac{N^{-1}[Q(T)] + \sqrt{\rho} N^{-1}(X)}{\sqrt{1 – \rho}} \right)\)

    where,

    • Q(T) is the probability of default by time T and
    • \(\rho\) is the copula correlation between any pair of loans.
  • A basic calculation of the credit VaR, using a confidence level of X% and a time horizon of T
    years, can be approximated as \(L \times (1 – R) \times V(X, T)\)where L represents the total size of the loan portfolio and
    R is the recovery rate.

Example

• Suppose that a bank has a total of $100 million of retail exposures. The 1-year probability of default averages 2% and the recovery rate averages 60%. The copula correlation parameter is estimated as 0.1. In this case,
  

  \(V(0.999,1) = N \left( \frac{N^{-1}(0.02) + \sqrt{0.1} N^{-1}(0.999)}{\sqrt{1 – 0.1}} \right) = 0.128\)

  showing that the 99.9% worst case default rate is 12.8%. The 1-year 99.9% credit VaR is therefore \(100 \times 0.128 \times (1 – 0.6) \ \text{or} \ \$5.13 \ \text{million}\).

Credit VaR Using Credit Metrics

• • • The CreditMetrics approach, developed by JPMorgan, offers a method to estimate Credit Value at Risk (Credit VaR) by considering not only defaults but also rating downgrades, which are treated as potential losses (and upgrades as potential gains).

Step-by-Step Estimation of Credit VaR Using CreditMetrics

• • • Setting Up Monte Carlo Simulation:
      • The core of the CreditMetrics method involves running a Monte Carlo simulation, which is a computational technique that uses repeated random sampling to obtain numerical results.
      • In each trial of the simulation, the credit ratings of all counterparties within a portfolio are projected over a set time period, typically one year. This projection takes into account potential changes in credit ratings, including both upgrades and downgrades, and defaults.
    • Sampling Credit Rating Changes:
      • The simulation estimates potential credit rating changes for each counterparty using a
        predefined transition matrix derived from the bank’s historical data or external ratings.
      • It is important to note that the rating changes for different counterparties are not assumed to be independent. Instead, dependencies between credit changes of counterparties are modeled using a Gaussian copula model. This model creates a joint probability distribution of rating changes, similar to how dependencies in default times are modeled.
    • Calculating Credit Losses:
      • Once the simulation determines the new credit ratings for each counterparty, the outstanding contracts are revalued to reflect potential losses due to credit downgrades or defaults within that year.
      • If a counterparty defaults, the credit loss is calculated as the total exposure at the time of default adjusted by the recovery rate. If there is a downgrade (or upgrade), the loss (or gain) is assessed based on the new valuation of the contracts.
    • Estimating Credit VaR:
      • After conducting a large number of simulation trials, a probability distribution of potential
        credit losses is generated.
      • Credit VaR is then calculated from this distribution at a specific confidence level, such as
        99.9%. This value represents the worst expected loss under normal market conditions over a one-year period, which the bank believes will not be exceeded with 99.9% certainty.

Integration of Credit Mitigation Strategies

• • • The model also allows for the incorporation of credit mitigation techniques. For example, clauses from contracts that might affect exposure, such as collateral agreements or other risk- reducing measures, can be factored into the simulation, adjusting the potential loss calculations.

Practical Considerations

• • While computationally intensive due to the need for numerous simulation trials and complex dependency modeling via the Gaussian copula, this approach provides a comprehensive view of potential credit losses, accounting for a wide range of credit events beyond mere default. This thoroughness makes CreditMetrics a valuable tool for banks in managing their credit risk and ensuring compliance with regulatory capital requirements.
  • In summary, CreditMetrics provides a detailed and nuanced approach to estimating Credit VaR by considering the entire spectrum of credit changes, using advanced statistical methods to
    simulate and predict the impact of these changes on a financial portfolio.