Book 2. Credit Risk

FRM Part 2

CR 11. Portfolio Credit Risk

Presented by: Sudhanshu

Module 1. Credit Portfolios and Credit VaR

Module 2. Conditional Default Probabilities and Credit VaR with Copulas

Module 1. Credit Portfolios and Credit VaR

Topic 1. Default Correlation for Credit Portfolios

Topic 2. Credit Portfolio Framework

Topic 3. Credit VaR

Topic 1. Default Correlation for Credit Portfolios

  • What is Default Correlation?

    • Definition: Default correlation measures the probability of multiple defaults occurring within a credit portfolio consisting of multiple obligors.

    • Risks to Consider: When analyzing credit portfolios, you need to consider various risks like default probability, loss given default (LGD), risk of deteriorating credit ratings, spread risk, and loss from bankruptcy restructuring.

    • Modeling with Bernoulli Variables: Default correlation can be modeled using Bernoulli-distributed random variables, where 0 represents no default and 1 represents a default for a specific time horizon.

    • Calculating Default Correlation: For a two-firm portfolio, the default correlation         is calculated by dividing the covariance of the two firms' defaults by the product of their individual standard deviations.

      • E[xi​]=πi​ (mean)

      • Var(xi​)=πi​(1−πi​) (variance)

      • Cov(x1​,x2​)=π12​−π1​π2​ (covariance)

    • Formula:

    •  

(\rho_{12})
\rho_{12}=\frac{\pi_{12}-\pi_1 \pi_2}{\sqrt{\pi_1 (1-\pi_1)}\sqrt{\pi_2(1-\pi_2)}}

Practice Questions: Q1

Q1. Which of the following equations best defines the default correlation for a two-firm credit portfolio?

A.  

 

B.

 

C.

 

D.

 

\rho_{12}=\frac{\pi_{12}-\pi_1 \pi_2}{\sqrt{\pi_1 (1-\pi_1)}\sqrt{\pi_2(1-\pi_2)}}
\rho_{12}=\frac{\pi_{12}}{\sqrt{\pi_1 (1-\pi_1)}\sqrt{\pi_2(1-\pi_2)}}
\rho_{12}=\frac{\pi_{12}}{\sqrt{(1-\pi_1)}\sqrt{(1-\pi_2)}}
\rho_{12}=\frac{\pi_{12}-\pi_1 \pi_2}{\sqrt{\pi_1 }\sqrt{\pi_2}}

Practice Questions: Q1 Answer

Explanation: A is correct.

The default correlation for a two-firm credit portfolio is defined as:

 

 

\rho_{12}=\frac{\pi_{12}-\pi_1 \pi_2}{\sqrt{\pi_1 (1-\pi_1)}\sqrt{\pi_2(1-\pi_2)}}

Topic 2. Credit Portfolio Framework

  • Drawbacks of the Default Correlation Framework

    • Calculation Complexity: A significant drawback is the large number of calculations required.

      • For a portfolio with 'n' firms, there are     possible event outcomes. For example, a 10-firm portfolio has 1,024 event outcomes.

      • The number of pairwise correlations to calculate is n(n−1)/2.

    • Inadequate for Complex Instruments: This framework is too simplistic to model credit positions with option-like features such as guarantees, revolving credit agreements, and other contingent liabilities.

    • Limited Data: Estimating defaults is challenging because they are relatively rare events.

      • Estimated correlations can vary greatly depending on the time horizon and industry.

      • Most studies use an estimated correlation of 0.05, and the joint probability of two firms defaulting is even smaller.

2^n

Practice Questions: Q2

Q2. Suppose a portfolio manager is using a default correlation framework for measuring credit portfolio risk. How many unique event outcomes are there for a credit portfolio with eight different firms?

A. 10.

B. 56.

C. 256.

D. 517.

Practice Questions: Q2 Answer

Explanation: C is correct.

There are 256 event outcomes for a credit portfolio with eight different firms calculated as:               .

2^8=256

Topic 3.  Credit VaR

  • Impact of Correlation on Credit VaR

    • Definition: Credit Value at Risk (Credit VaR) is the quantile of the credit loss less the expected loss of a portfolio.

    • Correlation's Role: Default correlation significantly impacts the volatility and extreme quantiles of loss, but not the expected loss. Therefore, it affects a portfolio's Credit VaR.

    • Extreme Cases:

      • Correlation = 1: There are no credit diversification benefits, and the portfolio behaves as a single credit position. The loss distribution is binomial, with a total loss occurring with probability π and zero loss with probability (1-π).

      • Correlation = 0: The portfolio is treated as a binomial-distributed random variable because there is no correlation between firms. The number of defaults is binomially distributed with parameters n and π .

  • Effect of Granularity on Credit VaR: Granularity refers to increasing the number of credits in a portfolio, which in turn reduces the weight of each individual credit as a proportion of the total portfolio value.

    • Impact on Credit VaR: As a credit portfolio becomes more granular, its Credit VaR decreases.

    • Low Default Probability: If the default probability is low, increasing granularity does not impact the Credit VaR as much.

    • Large Portfolios: For very large portfolios with many independent credit positions, the probability that the actual credit loss equals the expected loss eventually converges to 100%.

Practice Questions: Q3

Q3. Suppose a portfolio has a notional value of $1,000,000 with 20 credit positions. Each of the credits has a default probability of 2% and a recovery rate of zero. Each credit position in the portfolio is an obligation from the same obligor, and therefore, the credit portfolio has a default correlation equal to 1. What is the credit value at risk at the 99% confidence level for this credit portfolio?

A. $0.

B. $1,000.

C. $20,000.

D. $980,000

Practice Questions: Q3 Answer

Explanation: D is correct.

With the default correlation equal to 1, the portfolio will act as if there is only one credit. Viewing the portfolio as a binomial distributed random variable, there are only two possible outcomes for a portfolio acting as one credit. The portfolio has a 2% probability of total loss and a 98% probability of zero loss. Therefore, with a
recovery rate of zero, the extreme loss given default is $1,000,000. The expected loss is equal to the portfolio value times π and is $20,000 in this example (0.02 × $1,000,000). The credit VaR is defiined as the quantile of the credit loss less the expected loss of the portfolio. At the 99% confiidence level, the credit VaR is equal to $980,000 ($1,000,000 minus the expected loss of $20,000).

Module 2. Conditional Default Probabilities And Credit VaR with Copulas

Topic 1. Conditional Default Probabilities

Topic 2. Conditional Default Distribution Variance

Topic 3. Credit VaR With a Single-Factor Model

Topic 4. Credit VaR With Simulation

Topic 1. Conditional Default Probabilities

  • Single-Factor Model: The single-factor model examines the impact of varying default correlations based on a credit position's beta

    • Each firm's asset return is defined by the equation:

      • m = market return

      •      = firm's beta correlation with the market

      •      = firm's idiosyncratic shock

    • The term              represents the firm's standard deviation of idiosyncratic risk.

    • The model assumes that a firm defaults if its asset return (ai ) falls below a certain default threshold (ki​).

    • A key property of this model is conditional independence, which means that once market asset returns are realized, default risks are independent of each other. This is because asset returns and risks are assumed to be correlated only with the market factor.

a_i=\beta_i m+\sqrt{1-\beta_i^2 \epsilon_i}
\beta_i
\epsilon_i
\sqrt{1-\beta_i^2}

Practice Questions: Q1

Q1. A portfolio manager uses the single-factor model to estimate default risk. What is the mean and standard deviation for the conditional distribution when a specific realized market value    is used?

A. The mean and standard deviation are equivalent in the standard normal distribution.

B. The mean is        and the standard deviation is                .

C. The mean is       and the standard deviation is      .

D. The mean is       and the standard deviation is 1.

\bar{m}
\beta_i \overline{m}
\sqrt{1-\beta_i^2}
\bar{m}
\bar{m}
\beta_i

Practice Questions: Q1 Answer

Explanation: B is correct.

The conditional distribution is a normal distribution with a mean of          and a standard deviation of  

\sqrt{1-\beta_i^2}
\beta_i \overline{m}

Practice Questions: Q2

Q2. Suppose a credit position has a correlation to the market factor of 0.5. What is the realized market value that is used to compute the probability of reaching a default threshold at the 99% confidence level?

A. −0.2500.

B. −0.4356.

C. −0.5825.

D. −0.6243.

Practice Questions: Q2 Answer

Explanation: D is correct.

A default loss level of 0.01 corresponds to −2.33 on the standard normal distribution. The realized market value is computed as follows:

 

 

 

 

 

 

 

\begin{aligned} & -2.33=\frac{-2.33-(0.5) \overline{\mathrm{m}}}{\sqrt{1-0.5^2}} \\ & -2.33(0.86603)=-2.33-(0.5) \overline{\mathrm{m}} \\ & -2.01785+2.33=-(0.5) \overline{\mathrm{m}} \\ & 0.31215=-(0.5) \\ & -0.62430=\overline{\mathrm{m}} \end{aligned}

Topic 2. Conditional Default Distribution Variance

  • Calculating Conditional Distribution Parameters

    • The unconditional default distribution is a standard normal distribution.

    • The ​conditional distribution is a normal distribution with a mean of

                      and a standard deviation of                       .

      • The conditional mean shifts based on the specific market value.

      • The conditional standard deviation is less than the unconditional standard deviation of 1.

    • The conditional probability of default will be greater or smaller than the unconditional probability of default as long as m or       are not zero.

    • The conditional cumulative default probability function is given by:

 

​

 

\beta_i \overline{m}
\sqrt{1-\beta_i^2}
\beta_i
\mathfrak{p}(\mathfrak{m})=\Phi\left(\frac{k_i-\beta_i \bar{m}}{\sqrt{1-\beta_i^2}}\right)

Topic 3. Credit VaR with a Single-Factor Model

  • Determining Unconditional Loss Distribution

    • The unconditional probability of a default loss is equal to the probability that the realized market return results in a default loss.

    • The process to determine the unconditional distribution for Credit VaR calculation is as follows:

      • Assume the default loss level,  x, is a random variable, not a simulated value

      • Determine the market factor,  m, at the probability of the stated loss level using the relationship:

      •  

      • ​The market factor is assumed to be standard normal.

      • Repeat these steps for each individual credit to determine the loss probability distribution.

    • ​​The credit position's correlation to the market, β, determines the dispersion of defaults based on the range of the market factor.
x(\mathfrak{m})=\mathfrak{p}(\mathfrak{m})=\Phi\left(\frac{k-\beta \bar{m}}{\sqrt{1-\beta^2}}\right)

Topic 4. Credit VaR with Simulation

  • Copula Methodology: Copulas are a mathematical approach used to determine how defaults are correlated with one another using simulated results.

  • The Credit VaR is calculated using the following four steps under the copula methodology:

    • Define the copula function.

    • Simulate default times for each credit.

    • Obtain market values and profit and loss data for each scenario using the simulated default times.

    • ​Compute portfolio distribution statistics by summing the simulated terminal value results.

CR 11. Portfolio Credit Risk

By Prateek Yadav

CR 11. Portfolio Credit Risk

  • 62

More from Prateek Yadav