Market Risk VaR Versus Credit Risk VaR

• Definitions –
  • Market Risk VaR: It is typically defined as the maximum loss in market value over a set period that will not be exceeded with a certain confidence level. It is primarily concerned with losses due to changes in market variables like stock prices, interest rates, and foreign exchange rates.
  • Credit Risk VaR: It is similar to market risk VaR in its basic definition – representing a loss over a certain time period that will not be exceeded with a given confidence level. However, it extends beyond just defaults to include potential losses from credit downgrades or changes in credit spreads.
• Time Horizon –
  • Market Risk VaR: Generally, it has a short time horizon, often as brief as one day. This shorter duration reflects the high liquidity and the more immediate impact of market changes, allowing for quick adjustments.
  • Credit Risk VaR: It is generally calculated with a longer time horizon, usually one year. This is due to the nature of credit events like defaults or downgrades, which are less frequent and can have longer-term implications compared to market events.
• Tools for Measurement –
  • Market Risk VaR: Historical Simulation is one of the main tools for calculating market risk VaR. This approach uses actual historical changes in market risk factors to simulate potential future outcomes, helping assess risk under similar conditions.
  • Credit Risk VaR: Given the complexities of credit events and factors like credit correlations, tools for measuring credit risk VaR are generally more elaborate. These models need to account for the interconnectedness of different credit entities, how they are affected by economic conditions, and the longer-term nature of credit risk.

Rating Transition Matrices

• The topic of ratings transition matrices is covered extensively in the FRM curriculum, including in FRM Part 1. Given its familiarity to candidates, a concise explanation will be provided here.
• Financial institutions use ratings transition matrices to calculate credit VaR. These matrices are based on historical data and can reflect either internal ratings or those from external agencies like Moody's, S&P, and Fitch. S&P's one-year matrix tracks changes in company ratings from 1981 to 2020, showing probabilities for rating upgrades, downgrades, and defaults within a year.

  Initial Rating	Rating at Year-end
  	AAA	AA	A	BBB	BB	B	CCC/C	D
  AAA	89.85	9.35	0.55	0.05	0.11	0.03	0.03	0.00
  AA	0.50	90.76	8.08	0.49	0.05	0.06	0.06	0.02
  A	0.03	1.67	92.61	5.23	0.27	0.12	0.12	0.05
  BBB	0.00	0.10	3.45	91.93	3.78	0.46	0.46	0.17
  BB	0.01	0.03	0.12	5.03	86.00	7.51	7.51	0.70
  B	0.00	0.02	0.08	0.17	5.18	85.09	85.09	3.81
  CCC/C	0.00	0.00	0.12	0.20	0.65	14.72	50.90	33.42
  D	0.00	0.00	0.00	0.00	0.00	0.00	0.00	100.00

  Source: S&P

  Table CR10-1 : One-Year Ratings Transition Matrix, 1981-2020, with Probabilities Expressed as Percentages and Adjustments for Transition to the NR (not rated) Category

• To estimate changes over longer periods, such as two or more years, the matrix is multiplied by itself accordingly, revealing that the likelihood of maintaining the same rating decreases while default probabilities increase as the timeframe extends. For instance, considering the 92.61% probability that an A-rated company maintains its rating over one year, there is approximately a 73.24% chance it will retain its rating over four years
• Alternatively, shorter time periods can be considered. For a three-month period, the calculation involves taking the fourth root of the percentages displayed. For instance, with a 92.61% one- year probability, there is a 98.13% chance that the company will maintain its A rating over three months, or 0.25 years . As expected, the probability that a company's credit rating remains unchanged over a shorter period, such as three months, is typically very high.
• Hence, for longer time periods, default probabilities increase, and the likelihood of maintaining the same credit rating decreases. Conversely, shorter time periods see lower default probabilities and higher chances of maintaining the same credit rating.
• It is important to note that the assumption that credit rating changes in one period are independent of those in another isn't entirely accurate. If a company has been recently downgraded, it is more likely to be downgraded again in the near future – a phenomenon often referred to as ratings momentum. Nonetheless, the independence assumption can still be reasonable for certain analytical purposes.

Vasicek Model

• Vasicek's Gaussian copula model is used to calculate high percentiles of the default rate distribution for a loan portfolio. In this context, WCDR(T,X) represents the percentile of the default rate distribution over a period T, where WCDR stands for worst case default rate. Vasicek's model links WCDR(T,X) to the probability of default within time T, denoted as PD, and a parameter , which describes credit correlation. The formula for this relationship is formally established as

  

• Individual Loan Loss Calculation:
  • EAD (Exposure at Default): This is the total exposure at the time the borrower defaults.
  • LGD (Loss Given Default): This represents the percentage loss if the default occurs.
  • For an individual loan, the potential loss at the percentile is given by the formula:

    

• Portfolio Loss Distribution:
  • In a portfolio with n loans, where each loan is a minor part of the total, the of the total loss distribution is approximated by summing up the individual losses:

    

• Regulatory Capital Requirements:
  • As per Chapter 25, regulatory capital for the banking book is set based on the above sum, tailored for specific conditions (like T = 1 year and X = 99.9%).
  • Sometimes, a maturity adjustment factor is added to account for potential deterioration over periods longer than a year without defaulting.
• Correlation Factor ( ):
  • Structural models show that , the correlation parameter, should be roughly equal to the asset return correlation between two companies, and can be approximated by the equity return correlation.
  • For a portfolio, can be calculated by averaging the return correlations on equities for the companies involved, or using similar metrics from publicly traded companies if the original companies are not publicly traded.
• Model Limitations and Alternatives:
  • Vasicek's model, while widely used, has limitations regarding tail correlation, meaning it might underestimate risk in extreme conditions.
  • Alternative models, possibly involving different types of copulas, can better capture tail risk.

Credit Risk Plus

• The Credit Risk Plus (CreditRisk+) model, developed by Credit Suisse Financial Products in 1997, is a sophisticated credit risk management tool designed to estimate the probability distribution of losses due to defaults. It also incorporates the assessment of default correlations among borrowers within a portfolio.
• Framework and Methodology –
  • Actuarial Approach: CreditRisk+ adopts an actuarial approach to credit risk, inspired by methodologies used in the insurance industry. This model is primarily frequency-based, focusing on the number of defaults rather than on the timing of defaults.
  • Probability Distribution: It calculates the probability distribution of losses by treating the number of defaults as random variables which follow a Poisson distribution, with adjustments to accommodate varying risk levels across different segments of a loan portfolio.
• Modeling Default Rates –
  • Binomial Distribution: For a portfolio with ‘n’ loans, each with an independent default probability ‘q’, the expected number of defaults is ‘qn’. The probability of observing ‘m’ defaults is determined using the binomial formula:

    

  • Poisson Approximation: In scenarios where ‘q’ is small and ‘n’ is large, the Poisson distribution provides a close approximation:

    

    where is the expected number of defaults.

• Adjusting for Uncertainty –
  • Gamma Distribution: Given the fluctuation in default rates, it is assumed that the expected number of defaults (‘qn’) follows a gamma distribution, leading to a negative binomial distribution for more accurate modeling of defaults.
  • Monte Carlo Simulation: This method randomizes data to simulate overall default rates and connects specific obligor categories to these rates through regression analysis, calculating total losses by sampling losses given default for each category.
• Modeling Default Correlation and Implications of Uncertainty – Although the model does not explicitly account for correlation, it arises indirectly through shared exposure to fluctuating default rates. Increased uncertainty in default rates heightens default correlation, potentially leading to more defaults and a positively skewed loss distribution.
  • Sector Concentration: Instead of directly modeling individual correlations between borrowers, CreditRisk+ incorporates correlation by recognizing that defaults in similar economic sectors or geographic regions may be correlated.
  • Common Risk Factors: The model assigns weights to different sectors or risk factors, acknowledging that common external shocks to these factors can lead to simultaneous defaults, thus implicitly modeling correlation.
• Value at Risk (VaR) Calculation –
  • Loss Distribution: By repeatedly simulating the default and loss process, a comprehensive probability distribution for total loss is constructed.
  • VaR Computation: VaR is then calculated using this distribution, providing a quantitative measure of potential losses in extreme scenarios.
• Advantages and Applications –
  • Simplicity and Flexibility: One of the key strengths of CreditRisk+ is its conceptual simplicity and flexibility in application, allowing it to be tailored to specific portfolio characteristics.
  • Risk Management: It is widely used in risk management for calculating economic capital, stress testing, and scenario analysis.
  • Low Default Portfolios: The model is effective in environments with low default probabilities and where precise estimates of correlation are challenging to determine directly.
• Limitations –
  • Assumption of Independence: One limitation is the assumption that default events are independent within the same risk factor, which may not always hold true in real-world scenarios.
  • Lack of Time Dynamics: CreditRisk+ does not directly model the time dynamics of defaults, such as recovery rates or changes in credit conditions over time, which can be relevant in a fluctuating economic landscape.

Credit Metrics

• The CreditMetrics model, developed by J.P. Morgan in 1997, is a sophisticated framework used to estimate the probability distribution of losses arising from credit risk, particularly defaults, and to model default correlation in a credit portfolio. It distinguishes itself from other models like Vasicek's model and Credit Risk Plus by incorporating both credit downgrades and defaults into its risk assessment framework.
• Model Overview –
  • Focus: CreditMetrics is designed to measure the credit risk associated with individual loans, bonds, or other credit-sensitive assets in a portfolio by assessing the impact of changes in credit quality, including default events.
  • Methodology: It uses the changes in credit ratings as a primary metric to estimate the potential losses and incorporates the migration of credit ratings over time. It utilizes a rating transition matrix that can be based on either the bank's internal historical data or externally sourced ratings from agencies.
  • Market Value Impact: CreditMetrics quantifies the impact of these rating changes on the market value of each asset, calculating the changes in value that would result from each potential rating change at the end of a specified time horizon.
• Monte Carlo Simulation for Credit VaR –
  • Simulation Process: To calculate the one-year credit Value at Risk (VaR), the model employs a Monte Carlo simulation where each trial involves projecting the credit ratings of all counterparties by the end of one year.
  • Credit Loss Calculation:
    • If the year-end credit rating does not indicate default, losses are calculated based on the value of all transactions with the counterparty at the one-year mark.
    • If default is indicated, the credit loss is estimated as the exposure at default (EAD) multiplied by one minus the recovery rate.
• Term Structure of Credit Spreads –
  • Market Observations: A basic assumption might be that the term structure of credit spreads reflects current market conditions.
  • Credit Spread Index: Alternatively, a model might assume a specific probability distribution for a credit spread index, with all credit spreads linearly related to this index, providing a structured approach to estimate future credit conditions.
• Extension to Derivatives Transactions –
  • Derivatives and CVA: For transactions involving derivatives, the methodology extends to include calculations similar to Credit Valuation Adjustment (CVA), as described in related financial literature.
  • Interval Calculations: The model divides the time from the present to the end of the longest transaction into smaller intervals, estimating the default probability for each interval to more accurately reflect the credit risk over time.
• Applications and Limitations –
  • Applications: CreditMetrics is widely used for setting risk limits, capital allocation, and strategic planning by financial institutions. By providing a nuanced view of potential future credit changes, CreditMetrics aids in strategic portfolio management and risk mitigation. It allows financial institutions to more accurately gauge the potential impact of both credit downgrades and defaults on their portfolios.
  • Limitations: While sophisticated, it requires extensive data on credit rating transitions and correlations, and the accuracy of its predictions depends heavily on the quality and relevance of the historical data used.

Credit Metrics Correlation Model

• This method reflects real-world complexities in credit risk management.
• Using Gaussian Copula for Dependency –
  • Non-Independence of Rating Changes: Rating changes among counterparties are assumed to be dependent, contrary to simpler models that assume independent transitions.
  • Gaussian Copula: This mathematical tool is used to model the joint probability distribution of rating changes. The copula allows for the modeling of dependencies between the credit rating changes of different entities.
• Correlation Setup –
  • Equity Return Correlation: The correlation between rating transitions is typically equated to the correlation found between equity returns of the relevant companies. This approach is supported by financial models such as the Capital Asset Pricing Model (CAPM), which provides a framework for correlating returns.
• Monte Carlo Simulation –
  • SimulationTrials: In each Monte Carlo simulation trial, variables are sampled from standard normal distributions where they are assumed to have a predefined correlation.
  • Determining Ratings Based on Simulation: The new ratings of companies after one year are determined by these variables, following the transition probabilities outlined in a transition matrix.
• Default Probabilities –
  • Threshold for Default: Default events are also modeled using thresholds derived from the inverse normal distribution corresponding to the cumulative probabilities near 1 (indicating near certainty of default within the next period).
• Implications for Credit Risk Management –
  • Dynamic Risk Assessment: This method allows for a dynamic and nuanced assessment of credit risk by accounting for the interconnectedness of different counterparties’ financial health.
  • Policy and Decision Making: Insights from such simulations are crucial for setting risk limits, managing credit exposures, and optimizing capital allocation.

Credit Metrics Correlation Model – Example

• To simulate the rating change over one year for companies rated A and B, using the transition matrix provided in Table CR10- 1, assume a correlation of 0.2 between their equity returns. Each simulation trial involves drawing two correlated standard normal variables, , reflecting the changes in ratings for the A-rated and B-rated companies, respectively.
• From the transition matrix, the new ratings are determined based on specific cutoffs derived from the inverse cumulative distribution function (CDF), , corresponding to the cumulative probabilities. For example, the A-rated company
  • becomes AAA if
  • becomes
    
  • stays
    

  • defaults if x_A > 3.2905

• Similarly, the B-rated company
  • becomes
  • becomes
  • becomes
    and so on, and

  • defaults if x_B > 1.7744

• The full set of transitions are shown in Figure CR 10-1 here

  

  Figure CR10-1 : The CreditMetrics correlation model: Transition of A-rated and B-rated companies to a new rating after one year.

Credit Metrics vs Credit Risk Plus

• It is interesting to note that when the CreditMetrics and Credit Risk Plus models start with the same assumptions, they should predict similar total losses over a long period. However, the key distinction lies in the timing of recognizing these losses.
  • For example, let's assume that a loan in a portfolio experiences a downgrade from A to BBB in the first year, BBB to B in the second year, and ultimately defaults in the third year.
  • Using the Credit Risk Plus model, no losses might be accounted for until the third year when the default occurs.
  • Conversely, the CreditMetrics model would involve calculating revaluation losses at each downgrade stage – in years 1, 2, and 3.
  • Theoretically, the cumulative losses recognized by CreditMetrics over the three years should align with the total loss recognized in the third year under the Credit Risk Plus model.

Credit Spread Risk

• The value of credit-sensitive products in a trading book is highly dependent on credit spreads. Therefore, calculating Value at Risk (VaR) or Expected Shortfall (ES) for portfolios containing these products requires an analysis of potential changes in credit spreads. One common method involves using historical simulation to calculate a one-day 99% VaR. This short-term VaR can then be scaled up by a factor to estimate the 10-day 99% VaR.
• The process starts by gathering data on credit spread changes for each company the bank has exposure to over the last 500 days or another chosen period. In this method, each day’ s changes are used as a separate scenario: the first scenario uses day one’s changes, the second uses day two’s, and so forth. However, there are notable limitations to this approach:
  • Survivorship Bias: If a company has not defaulted in the historical period reviewed, the model assumes no probability of default in the future. This oversight can lead to an underestimation of risk.
  • Data Quality Issues: Credit spreads may not be updated daily for all companies, which means that the historical data might be incomplete or of low quality, further complicating accurate risk assessments.
• An alternative method to the historical simulation approach for calculating Value at Risk (VaR) in credit-sensitive portfolios involves adapting the CreditMetrics framework. The following are the main steps in this method:
  • Creation of a 10-Day Ratings Transition Matrix: This matrix is derived from historical data on credit rating changes, outlining the likelihood of transitions between different ratings over a 10- day period.
  • Monte Carlo Simulation: In each Monte Carlo simulation trial, the ratings transition matrix is used to determine whether each company retains its rating, moves to a different rating, or defaults. Alongside rating transitions, a probability distribution for credit spread changes associated with each rating category at the end of the 10-day period is sampled. This helps estimate the new credit spreads corresponding to each possible rating outcome.
  • Calculation of Portfolio Value: Using the outcomes from the Monte Carlo trials, the portfolio value at the end of 10 days is calculated for each scenario, allowing the computation of VaR.
• Introducing Credit Correlation –
  • Gaussian Copula Model: This statistical model is employed to correlate the credit rating change among different companies within the portfolio. By capturing the dependencies in rating changes, the model adds a layer of realism and complexity to the risk estimation.
  • Correlation of Credit Spread Changes: It may also be assumed that credit spread changes are either perfectly or nearly perfectly correlated across different rating categories. This assumption means that if credit spreads for A-rated instruments increase, then spreads for instruments in other rating categories will likely increase in a similar manner.
• Advantages of this Approach:
  • Dynamic Risk Assessment: This method allows for a more dynamic and responsive assessment of credit risk by factoring in the potential for credit rating transitions and their impact on credit spreads.
  • Comprehensive Risk Modeling: By incorporating correlations between credit changes of different entities, the approach provides a more comprehensive view of systemic risks in the portfolio.
  • Scenario-Based Planning: The use of Monte Carlo simulation helps in understanding the range of possible outcomes and the potential extremities (or tail risks) in portfolio valuation.

Credit Spread Risk – Example

• Consider a company that possesses a single two-year zero-coupon bond with a $1,000 principal. The risk-free rate stands at 3% while the current credit spread is 200 basis points, making the bond yield 5% (with annual compounding). Hence, total Bond Yield is 5% (sum of risk-free rate and credit spread). This sets the bond's current price at $907.03, calculated as Currently, the bond is rated BB.
• Over the next month, it has a 0.6% probability of upgrading to BBB, a 98.5% chance of maintaining its BB rating, a 0.8% chance of downgrading to B, and a 0.1% chance of defaulting. If the bond defaults, its value will drop to $400. For each rating scenario, there are three potential credit spreads, assumed to be equally likely. These are 80, 100, and 120 basis points for BBB; 160, 200, and 240 for BB; and 400, 450, and 500 for B.
• The most severe loss occurs if the bond defaults, a 0.1% likelihood, resulting in a financial hit of $507.03 ($907.03 – $400).
• Another significant loss scenario is if the bond’s rating drops to B with a credit spread of 500 basis points. This event has a probability of 0.8%/3 = 0.267% and would decrease the bond’s value, considering its remaining maturity of approximately 1.917 years (23 months),and the total yield of 8% (risk free rate of 3% plus 500 bps). The new value of the bond will become leading to a loss of $44.17. The details of all potential outcomes along with their respective probabilities is provided in Table CR10-2.
• Value at Risk (VaR) Calculations –
  • Confidence Level > 99.9%: VaR = $507.03 (reflecting the loss in the event of default)
  • Confidence Level between 99.9% and 99.633%: VaR = $44.17 (reflecting the worst non-default scenario, a downgrade to B at the highest spread)
  • Confidence Level at 99%: VaR = $2.92 (reflecting more minor losses from less drastic spread changes within the BB category)

Rating	Spread (bp)	Probability	Bond Value ($)	Loss ($)
Default		0.100%	400.00	507.03
B	500	0.267%	862.85	44.17
B	450	0.267%	870.56	36.47
B	400	0.267%	878.38	28.65
BB	240	32.833%	904.11	2.92
BB	200	32.833%	910.72	−3.70
BB	160	32.833%	917.41	−10.38
BBB	120	0.200%	924.17	−17.14
BBB	100	0.200%	927.58	−20.55
BBB	80	0.200%	931.01	−23.98

Table CR10-2 : All Outcomes for Losses