Monitoring Liquidity
After completing this chapter, you should be in a position to: Distinguish between... Read More
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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:
When assessing the value of a bank’s outstanding derivatives transactions with a counterparty, three components come into play:
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:
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:
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:
$$h=\frac{ s(t_i)}{1-R}$$
Where \( s(t_i)\) is the credit spread for maturity \(t_i\) and \(R\) is the recovery rate.
$$ P(\text{No Default by time t}) =e^{ \frac{(-s(t_i) \cdot t_i)}{1 – R}} $$
$$ q_i = e^{-\frac{s(t_{i-1}) t_{i-1}}{1 – R}} – e^{-\frac{s(t_i) t_i}{1 – R}} $$
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:
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:
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:
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 incorrect because simply monitoring rating agencies does not constitute a methodical estimation of credit VaR based on credit transition matrices.
B is incorrect because aggregation without considering migrations and diversification disregards key dynamics that CreditMetrics aims to include in the credit VaR estimation process.
D is incorrect because 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.