###### Monitoring Liquidity

After completing this chapter, you should be in a position to: Distinguish between... **Read More**

**After completing this reading, you should be able to**:

- Assess the credit risks of derivatives.
- Define credit valuation adjustment (CVA) and debt valuation adjustment (DVA).
- Calculate the probability of default using credit spreads.
- Describe, compare, and contrast various credit risk mitigants and their role in credit analysis.
- Describe the significance of estimating default correlation for credit portfolios and distinguish between reduced form and structural default correlation models.
- Describe the Gaussian copula model for time to default and calculate the probability of default using the one-factor Gaussian copula model.
- Describe how to estimate credit VaR using the Gaussian copula and the CreditMetrics approach.

Credit risk is a concern in over-the-counter (OTC) derivatives transactions, where there isn’t a central counterparty to guarantee performance. Instead, the transactions are often governed by an International Swaps and Derivatives Association (ISDA) Master Agreement, which outlines the terms and conditions of the trades. Under this agreement, an event of default can occur if a party fails to make payments or post margins as required or declares bankruptcy. This can result in losses for the non-defaulting party, which are generally of two types:

**Positive Value Exposure**: When the non-defaulting party has a net positive position (greater than the collateral posted by the defaulting party), it becomes an unsecured creditor for the difference.**Excess Collateral**: When the non-defaulting party has posted collateral in excess of the value owed to the defaulting party, it becomes an unsecured creditor for the return of this excess collateral.

When assessing the value of a bank’s outstanding derivatives transactions with a counterparty, three components come into play:

- \(F_{\text{nd}}\): The no-default value to the bank of the derivatives transactions.
- CVA (Credit Valuation Adjustment): The adjustment for the risk of counterparty default.
- DVA (Debit Valuation Adjustment): The adjustment for the risk of the bank’s own default.

**CVA **represents the present value of the expected cost to a bank of a counterparty’s default and is subtracted from the no-default value of a derivative position. In other words, CVA is a measure of the risk of loss due to the counterparty’s potential default. It decreases the value of the bank’s derivatives portfolio because it represents a cost to the bank. The formula for CVA is as follows:

\[\text{CVA} = \sum_{i=1}^{N} q_i v_i \]

Where:

- \(q_i\) is the risk-neutral probability of the counterparty defaulting.
- \(v_i\) is the exposure at the ith time interval, the present value of the expected net exposure at the midpoint of the interval after collateral has been taken into account.

**DVA, **on the other hand**,** is the present value of expected cost to the counterparty of the bank’s default and is added to the no-default value from the bank’s perspective. DVA, therefore, increases the value of the counterparty’s derivatives portfolio. In other words, DVA is the cost to the counterparty in the event of the bank’s default. Its formula is symmetrical to CVA:

Where:

- \( q_i^*\) = Risk-neutral probability of the bank defaulting during the ith interval.
- \( v_i^*\) = Present value of the expected loss to the counterparty (which translates into a gain for the bank) if the bank defaults during the ith interval.

To integrate CVA and DVA into the valuation of derivatives, we calculate the adjusted value of these transactions, incorporating the possibility of defaults. This adjustment is given by: $$F_{\text{nd}} – \text{CVA} + \text{DVA}$$

Where:

\(F_nd\) is the no-default value of the derivatives, which is the value assuming no default occurs. This value can be calculated using pricing models like Black-Scholes-Merton, which do not consider credit risk.

**Calculating the Probability of Default using Credit Spreads**

Risk-neutral default probabilities, \( q_i \) and \( q_i^* \) can be derived from credit spreads. For the ith interval, we can estimate the risk-neutral default probabilities, \( q_i \) and \( q_i^* \).

The following steps provide a systematic approach for calculating the probability of default using credit spreads:

**Credit Spread**\( s(t)\): We start by determining the counterparty’s credit spread, \( s(t_i)\) for a specific maturity through interpolation followed by estimating the counterparty’s average hazard rate, \(h\) between time 0 and time \(t_i\):

$$h=\frac{ s(t_i)}{1-R}$$

Where \( s(t_i)\) is the credit spread for maturity \(t_i\) and \(R\) is the recovery rate.

**Survival Probability**: Compute the probability that the counterparty survives to time,\(t_i\) (not defaulting by time\( t_i\)):

$$ P(\text{No Default by time t}) =e^{ \frac{(-s(t_i) \cdot t_i)}{1 – R}} $$

**Default Probability**, \( q_i \): Calculate the desired probability of default over a specific interval ( i ), between \( t_{i-1}\) and \( t_i\), using the survival probabilities at the beginning and end of the interval.

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

The probability \(q_i^*\) can be calculated in a similar manner, but we consider the bank’s credit spread in this case.

Consider a corporate bond with a maturity of 2 years. Suppose the bond’s yield is 7%, the risk-free rate is 3%, and the recovery rate is 40%. What is the probability of default occurring in the second-year interval?

The bond’s yield is 7%, and the risk-free rate is 3%, yielding a credit spread of 4%.

We will first calculate the survival probability for the 1-year and 2-year maturities and then the probability of default for the second year.

The probability that no default occurs by the end of the first year is:

$$P(\text{No Default by year 1}) = e^{- \frac{0.04 \cdot 1}{1 – 0.4}} =0.9355$$

The probability that no default occurs by the end of the second year is:

$$P(\text{No Default by year 2}) = e^{- \frac{0.04 \cdot 2}{1 – 0.4}} =0.8752$$

Therefore, \( q_2\), the probability of default occurring in the second year interval, is:

$$ 0.9355-0.8752=0.0603$$

Credit risk mitigants are strategies or tools used to reduce the risk that a borrower will default on its financial obligations. Credit analysis involves evaluating the efficacy of these mitigants in protecting the lender from potential losses. Effective mitigants lower the lender’s exposure to credit risk and can influence the terms at which credit is extended.

Netting is a method where offsetting claims with the same counterparty are consolidated into a single net claim, thereby lowering exposure. For instance, if multiple transactions with a single counterparty result in claims of +$10 million, +$30 million, and -$25 million, rather than treating them as three separate exposures, netting would consolidate them into one net exposure of +$15 million.

These agreements involve the provision of collateral (cash or securities) to secure a financial obligation. During defaults in derivatives transactions, the non-defaulting party is entitled to keep any posted collateral, which can significantly reduce credit risk. The value of marketable securities may be adjusted (“haircut”) for the purpose of determining their cash equivalent as collateral.

A downgrade trigger is a clause in a financial agreement that requires action if the counterparty’s credit rating falls below a certain level. Actions may include the posting of additional collateral or the termination of outstanding transactions at market values. However, downgrade triggers may not provide protection against sudden significant deteriorations in credit quality and may be less effective if widely triggered throughout the market simultaneously.

Default correlation refers to the likelihood of simultaneous defaults within a portfolio of credit obligations. It is critical in portfolio credit risk because it affects the distribution of losses and the potential for large, unexpected financial setbacks.

**Diversification:** Estimating default correlations helps in understanding the true diversification benefits within a credit portfolio. Lower correlations imply higher diversification benefits, reducing portfolio risk.

**Risk Concentration:** A high default correlation indicates risk concentration, suggesting that adverse conditions affecting one obligor are likely to affect others, increasing the risk of concurrent defaults.

**Credit Derivatives Pricing:** Accurate estimation of default correlations is essential for pricing credit derivatives like collateralized debt obligations (CDOs), where tranches have different risk profiles based on the correlation of underlying assets.

**Stress Testing and Regulatory Capital:** Regulators require banks to consider default correlations in stress testing and determining regulatory capital, ensuring institutions are sufficiently capitalized against correlated defaults.

**Tail Risk:** Default correlation is vital in assessing the tail risk of a credit portfolio, as correlated defaults can lead to extreme outcomes or tail events.

There are two main types of default correlation models:

**Structural Models:**These are based on underlying economic factors that impact a firm’s assets. The probability of default is correlated through the dependency of assets on common factors (such as the market or economic conditions).**Reduced-Form Models:**These models directly specify the default correlation without linking to the assets’ value. They commonly use copulas to model dependency structures between defaults.

The Gaussian copula model is utilized to describe the joint default likelihood across various entities or instruments. This statistical concept hinges on the idea that the time to default of multiple entities can be correlated. The use of copulas allows for the modeling of these dependence structures beyond just linear correlations, capturing more complex interconnections.

In the context of credit risk, the Gaussian copula provides a way to:

- Model the probability of simultaneous defaults within a portfolio.
- Correlate default times across an array of debt instruments.
- Gauge overall portfolio risk and inform decisions on risk management and hedging strategies.

Instead of individually defining correlations between companies \(i\) and \(j\) in our

Gaussian copula framework, it’s efficient to use a single-factor model. The core idea is:

$$x_i=b_i F+\sqrt{1-b_i^2} y_i$$

Here, \(x_i\) represents the creditworthiness of company \(i\), influenced by a common factor \(F\) (impacting all companies) and a unique factor \(y_i\) (specific to company \(i\) ). \(F\) and each \(y_i\) are independently distributed, following a standard normal distribution. The parameters \(b_i\) range from -1 to +1 and dictate the influence of \(F\) on each company. The correlation between companies \(i\) and \(j\) can be described as \(b_i b_j\).

When considering the likelihood of a company \(i\) defaulting by a time \(T\), symbolized as \(P(T)\), the Gaussian copula model suggests a default occurs if \(N\left(x_i\right)<P(T)\), or equivalently:

$$x_i<N^{-1}(P(T))$$

From our original equation, this translates to

$$b_i F+\sqrt{1-b_i^2} y_i<N^{-1}(P(T))$$

or

$$y_i<\frac{N^{-1}(P(T))-b_i F}{\sqrt{1-b_i^2}}$$

Given a specific value of \(F\), the conditional default probability is:

$$P_N(T \mid F)=N\left(\frac{N^{-1}\left(P_i(T)\right)-b_i F}{\sqrt{1-b_i^2}}\right)$$

In a scenario where default probabilities and inter-company correlations are uniform ( \(P_i(T)=P(T)\) and all \(b_i\) are equal), assuming the common correlation is \(\rho\), the equation simplifies to:

$$ P(T \mid F) = N\left(\frac{N^{-1}(P(T)) – \sqrt{\rho} \cdot F}{\sqrt{1-\rho}}\right) $$

Credit VaR, similar to market risk VaR, indicates the potential loss in credit value. For instance, a \(99.9 \%\) Credit VaR over one year represents the loss level that is unlikely ( \(99.9 \%\) confidence) to be exceeded in that period.

Consider a bank with a large, homogenous loan portfolio. If defaults are equally probable across loans and correlations between them are consistent, the Gaussian copula model can approximate the percentage of defaults by time \(T\) as a function of \(F\). The factor \(F\) follows a standard normal distribution, and we can be \(X \%\) sure that its value exceeds \(N^{-1}(1-X)\). Thus, the maximum expected loss over \(T\) years in the portfolio is given by \(V(X, T)\), where:

$$v(x, \rho)=N\left(\frac{N^{-1}(P(T))+\sqrt{\rho} N^{-1}(X)}{\sqrt{1-\rho}}\right)$$

Credit VaR can be estimated as \(L(1-R) V(X, T)\), where \(L\) is the total loan amount and \(R\) is the recovery rate. The contribution of an individual loan of size \(L_i\) to the Credit VaR is \(L_i(1-R) V(X, T)\).

Credit Value at Risk (Credit VaR) is the measure of the potential extreme loss in the value of a credit portfolio due to adverse credit events, such as defaults, over a certain time period, given a specified confidence interval. In essence, it is the worst-case loss one can expect under normal market conditions with a high degree of confidence.

The Gaussian copula model’s primary strength lies in its ability to capture and represent the interdependencies between the default times of various entities within a portfolio. This is crucial for accurately gauging the collective risk that a portfolio faces, as it reflects the complex, intertwined nature of financial markets where the default of one entity can significantly influence the default risk of others.

The process of calculating Credit VaR using the Gaussian copula model involves the following steps:

**Pairwise default correlation and default probability assessment**: Initially, the model necessitates the determination of pairwise default correlations among entities, acknowledging that the financial fate of one entity can be intertwined with that of another. Alongside this, the individual probability of default for each entity within the portfolio needs to be established. These probabilities, often derived from credit ratings or historical default data, serve as foundational inputs for the model.**Application of the copula function**: Subsequently, a copula function is employed. In essence, the copula function binds the individual default risks into a comprehensive, interconnected risk profile.**Simulation of default distribution and loss level identification**: With the correlation structure and joint default distribution at hand, the next step involves simulating this distribution across the entire portfolio. This simulation unveils the various potential default scenarios, taking into account the intricate correlation patterns among entities. Following the simulation, the model pinpoints the specific loss level corresponding to a pre-defined confidence interval, such as 99.9%.

CreditMetrics, Vasicek’s model, and Credit Risk Plus are all approaches used to estimate the likelihood of losses due to defaults. Unlike the latter two, CreditMetrics, a concept pioneered by JPMorgan in 1997, also takes into account both downgrades and defaults. This method utilizes a rating transition matrix, drawing on either the bank’s own historical data or external ratings, to track these changes.

When calculating a portfolio’s one-year Credit VaR, CreditMetrics employs a Monte Carlo simulation. This simulation forecasts the credit ratings of each counterparty after one year. For those not in default at this point, the credit loss is determined by assessing the value of all related transactions at the end of the year. For those in default, the credit loss is the total exposure at the time of default, adjusted by the recovery rate.

The model requires the credit spread term structure for each rating category to perform these calculations. This can either mirror the current market observations or assume a specific credit spread index with its own probability distribution, affecting all credit spreads linearly.

## Practice Question

As part of the CreditMetrics methodology for estimating credit VaR, an analyst is considering the credit migration of various debt instruments in the portfolio over the next year. Given detailed credit transition matrices and the current market values of these instruments, what critical steps should the analyst take to estimate the portfolio’s credit VaR?

- Monitor the credit rating agencies for any potential downgrades and adjust the credit VaR accordingly based on expert judgment.
- Aggregate the individual credit VaRs of each instrument without accounting for potential migrations and diversification effects.
- Simulate the potential changes in credit ratings over time and determine the impact on the portfolio value to establish the credit VaR.
- Use the average credit spread movements to predict future creditworthiness and apply a static approach to calculate VaR.

The correct answer is C.With the CreditMetrics approach, the analyst utilizes credit transition matrices that capture the probabilities of migrating between different credit ratings over a specified time frame. By simulating these migrations, the analyst can assess how potential changes in creditworthiness are likely to affect the market value of the portfolio’s components. The aggregate impact of these changes on the portfolio’s value facilitates the estimation of credit VaR.

A is incorrectbecause simply monitoring rating agencies does not constitute a methodical estimation of credit VaR based on credit transition matrices.

B is incorrectbecause aggregation without considering migrations and diversification disregards key dynamics that CreditMetrics aims to include in the credit VaR estimation process.

D is incorrectbecause using average spread movements for future predictions is a rudimentary approach, whereas CreditMetrics employs a more dynamic and comprehensive simulation framework.

Things to Remember

- The CreditMetrics approach relies on the dynamic simulation of credit rating transitions to capture potential changes in portfolio value over time.
- Credit transition matrices underpin this method by providing probabilities for various credit events, which inform the credit VaR estimation.
- This approach acknowledges that credit risk is not static and integrates the probabilities of future rating migrations into VaR calculations.