Book 2. Credit Risk

FRM Part 2

CR 10. Credit Value at Risk

Presented by: Sudhanshu

Module 1. Defining Credit VaR

Module 2. Credit VaR Models

Module 1. Defining Credit VaR

Topic 1. Market VaR vs. Credit VaR

Topic 2. Factors for Calculating Credit VaR

Topic 1. Market VaR vs. Credit VaR

  • What is Credit VaR?
    • Credit value at risk (VaR) is the credit loss over a certain time period that won't be exceeded at a given confidence level. It's used by banks to figure out how much economic and regulatory capital they need. Credit VaR models can account for losses from defaults, downgrades, and changes in credit spreads.
  • Key Differences: Market VaR vs. Credit VaR: While similar, there are a few important distinctions between market and credit VaR calculations.

    • Time Horizon: Market VaR is typically calculated with a one-day time horizon, whereas credit VaR often uses a one-year time horizon.

    • Calculation Tools: Market VaR primarily uses historical simulation. However, credit VaR calculations often require more complex modeling tools.

Practice Questions: Q1

Q1. Which of the following statements most accurately reflects the time horizons typically used for market VaR and credit VaR calculations?

A. Both are calculated for one day.

B. Both are calculated for one year.

C. Market risk VaR is calculated over a longer time period than credit risk VaR.

D. Credit risk VaR is calculated over a longer time period than market risk VaR.

Practice Questions: Q1 Answer

Explanation: D is correct.

Market risk VaR is usually calculated over a one-day time horizon, while credit
risk VaR will often use a one-year time horizon.

Topic 2. Factors for Calculating Credit VaR 

  • Credit Correlation: Credit VaR models need to account for  credit correlation, which recognizes that defaults for different companies aren't independent.

    • During a strong economy, companies generally benefit, and default risk is lessened.

    • During a poor economy, companies are negatively impacted, and defaults become more common.

    • As credit correlation rises in an economic downturn, financial institution risks also increase.

  • ​ Rating Transition Matrices: Rating transition matrices are used by financial institutions to determine credit VaR. These matrices are based on historical data and show the likelihood of a company moving between different rating categories over a specific period.

    • The highest probabilities are for a company keeping its rating by year-end.

    • To analyze a period longer than one year, you can multiply the matrix by itself. For example, a three-year matrix is the one-year matrix raised to the third power.

    • As the time horizon lengthens, default probabilities get higher, and the chances of maintaining the same rating get lower.

    • As the time horizon shortens, default probabilities are lower, and the chances of maintaining the same rating are higher.

Practice Questions: Q2

Q2. During a period of slowing economic growth, an analyst will likely identify which of the following trends regarding credit correlation and financial institution risks?

A. A decrease in credit correlation and defaults.

B. An increase in credit correlation and defaults.

C. An increase in credit correlation and a decrease in defaults.

D. A decrease in credit correlation and an increase in defaults.

Practice Questions: Q2 Answer

Explanation: B is correct.

When the economy is slowing, companies are negatively impacted, and defaults will become more prominent. At the same time, credit correlation (which captures the lack of independence between defaults for different companies) increases as well.

Practice Questions: Q3

Q3. Historical data shows that over a one-year period, there is a 91.93% chance that a company rated BBB will keep its rating. What is the percentage chance this rating will remain unchanged over a four-year period?

A. 36.77%.

B. 67.72%.

C. 71.42%.

D. 95.97%.

Practice Questions: Q3 Answer

Explanation: C is correct.

If 91.93% is the likelihood that the company keeps its rating over a one-year period, then there is a 71.42% chance it keeps that rating over a four-year period.
 

0.9193^4=0.7142=71.42\%

Module 2. Credit VaR Models

Topic 1. Vasicek’s Model

Topic 2. Credit Risk Plus Model

Topic 3. CreditMetrics Model

Topic 4. Correlation Model

Topic 5. Credit Spread Risk

Topic 1. Vasicek’s Model

  • Vasicek's Gaussian copula model is a method used for a portfolio of loans to calculate high percentiles of the default rate distribution. This model is used to estimate capital requirements under the Basel II internal-ratings-based (IRB) approach.
  • Key Components: The model outputs the worst-case default rate (WCDR), which is the worst-case default rate during a time period T at the Xth percentile of the default rate distribution. The model connects the probability of default (PD) to the credit correlation     and the time period T.

    • Formula:​

    •  

    • Credit Correlation       :  ​This should be roughly equal to the correlation between the companies' returns on assets (ROA) or returns on equities (ROE).​
  • Calculating Portfolio Loss: For a large portfolio with n small loans, the Xth percentile of the loss distribution can be approximated as:
  •  
  •  

    • EAD: Exposure at Default

    • LGD: Loss Given Default

(\rho)
WCDR(T,X)=N[\frac{N^{-1}(PD)+\sqrt{\rho}N^{-1}(X)}{\sqrt{1-\rho}}]
(\rho)
\sum_{i=1}^n WCDR_i (T, X) \times EAD_i \times LGD_i

Practice Questions: Q1

Q1. Company A has an ROE of 8%, while Company B has an ROE of 6%. The average correlation between the two is 0.24, and both companies are publicly traded. The credit correlation     most likely used in Vasicek’s Gaussian copula model will be closest to:

A. 0.12.

B. 0.24.

C. 0.48.

D. 0.76.

(\rho)

Practice Questions: Q1 Answer

Explanation: B is correct.

The average correlation between company ROEs can be used to determine   . Because the average correlation is given as 0.24, that is the most likely correlation used in Vasicek’s model.

\rho

Topic 2. Credit Risk Plus Model

  • Overview: Developed by Credit Suisse Financial Products, the Credit Risk Plus (also known as CreditRisk+) model is a credit VaR calculation methodology.
  • Default Probability: Assuming independent defaults, a binomial distribution can be used to estimate the number of defaults.

    • The probability of m defaults with n loans and a probability of default q for each loan is:

    • For a small probability of default and a large number of loans, a Poisson distribution can be used:

    • The expected number of defaults (qn) is assumed to follow a gamma distribution, which transforms the Poisson distribution into a negative binomial distribution.​ 

      • When the standard deviation      decreases, the negative binomial distribution will follow the same probability distribution as the Poisson distribution.

      • When the standard deviation       increases, the likelihood of a large number of defaults increases.

  • Limitations: Both Vasicek's model and Credit Risk Plus only account for  defaults, not downgrades.
P\text{(m defaults)}=\frac{n!}{m!(n-m)!}q^m (1-q)^{n-m}
P\text{(m defaults)}=\frac{e^{(-qn)}m!}{m!}
(\sigma)
(\sigma)

Topic 3. CreditMetrics Model

  • Overview: The CreditMetrics model, from JPMorgan, was designed to account for both  defaults and downgrades. It uses a rating transition matrix, with ratings from internal bank data or external credit rating agencies.
  • Monte Carlo Simulation: Monte Carlo simulation is used for one-year credit VaR calculations for portfolios with multiple counterparties.

    • Each simulation trial determines the counterparty credit ratings at the end of one year and calculates the credit loss for each counterparty.

    • If a default occurs, the loss equals the exposure at default (EAD) multiplied by the loss given default (LGD).

    • If there is no default, the credit loss is the value of all transactions with that counterparty at year-end.

    • The term structure of credit spreads for each rating category is needed for these calculations.​

  • Loss Distribution: The CreditMetrics Monte Carlo simulation produces a probability distribution for total credit losses from defaults and downgrades across all counterparties. The credit VaR is then derived from this distribution.

Practice Questions: Q2

Q2. If an analyst wants a credit risk model that accounts for both defaults and downgrades, she will most likely use which of the following models?

A. CreditMetrics.

B. Credit Risk Plus.

C. Vasicek’s model.

D. Monte Carlo simulation.

Practice Questions: Q2 Answer

Explanation: A is correct.

The CreditMetrics model is used to account for both defaults and downgrades, whereas Vasicek’s model and the Credit Risk Plus model do not. Monte Carlo simulation is an underlying technique applied to various models.

Topic 4. Correlation Model

  • Correlation Model

    • This model assumes that credit rating changes for different counterparties are related, not independent.

    • A joint probability distribution of rating changes can be built using a

      Gaussian copula model.

    • The correlation between rating transitions for two companies is equated to the correlation between their equity returns.

Topic 5. Credit Spread Risk

  • The value of credit-sensitive products depends on credit spreads, so credit VaR calculations must assess potential credit spread changes.

  • A CreditMetrics approach can be used, where a rating transition matrix is developed over a specific period. Historical data provides a probability distribution for credit spread changes.

  • A Monte Carlo simulation can then be used to determine the credit spread for each rating category.

  • To include credit correlation, you can use a Gaussian copula model for different company rating change correlations or assume that rating category credit spread changes have very high correlations.

  • Rebalancing Strategies

    • Constant level of risk strategy: A company sells bonds that no longer hold a specific rating and replaces them with bonds that do. This strategy generally results in a smaller credit VaR.

    • Buy-and-hold strategy: A company holds the original bond for a period of time before selling. This strategy generally produces larger losses from big downgrades and defaults compared to the constant level of risk strategy.

Practice Questions: Q3

Q3. When accounting for credit spreads and potential bond losses for a bond currently rated A, an analyst will likely assign the:

A. biggest spreads to situations where the bond rating increases.

B. lowest probability to situations where the bond rating increases.

C. lowest spreads to situations where the bond maintains its rating.

D. highest probability to situations where the bond maintains its rating.

Practice Questions: Q3 Answer

Explanation: D is correct.

An analyst will likely assign the highest probability to situations where the bond maintains its rating. The biggest spreads will be for situations where the bond rating decreases, and the lowest spreads will be for situations where the bond rating increases. The lowest probability will likely be for a bond default.