Default Correlation For Credit Portfolios

• Default correlation measures the likelihood of multiple defaults in a credit portfolio issued by multiple obligors.
• Consider two firms A and B with respective probabilities of default 𝜋_A and 𝜋_B over some time horizon ‘𝑡’, and a joint default probability of both defaulting over ‘𝑡’ being equal to 𝜋_AB. This can be thought of as the distribution of the product of two Bernoulli-distributed random variables 𝑥_A and 𝑥_B, which are set to 1 in case of default and set to 0 in case of no default. There can be a total of four possible outcomes.
  

  Outcome	xA	xB	xAxB	Probability
  Only Firm A defaults	1	0	0	πA – πAB
  Only Firm B defaults	0	1	0	πB – πAB
  Both Firms Default	1	1	1	πAB
  No Default	0	0	0	1 – (πA + πB – πAB)

• fble 𝑥_A is

  \(\sigma^2_{x_A} = \pi_A(1 – \pi_A)\)

• The variance of the Bernoulli distributed variable 𝑥_B is

  \(\sigma^2_{x_B} = \pi_B(1 – \pi_B)\)

• The covariance between 𝑥_A and 𝑥_B is given by

  \(Cov_{x_A x_B} = E[x_A x_B] – E[x_A]E[x_B] = \pi_{AB} – \pi_A \pi_B\)

• Hence the default correlation is
  

  \(\rho_{A,B} = \frac{Cov_{x_A x_B}}{\sigma_{x_A} \sigma_{x_B}}\)
  = \(\frac{\pi_{AB} – \pi_A \pi_B}{\sqrt{\pi_A(1 – \pi_A)}\sqrt{\pi_B(1 – \pi_B)}}\)
  

Drawbacks In The Credit Portfolio Framework

• Computationally intensive: If the number of credits in the portfolio is greater than 2 (i.e., if 𝑛 > 2), we have 2ⁿ different events, with \(n + 1 + \frac{n(n-1)}{2}\) conditions. In addition, we require 𝑛(𝑛-1) pairwise correlations. For example, if we have 20 firms, there will be 1,048,576 event outcomes with 211 conditions.
• Some kinds of positions don’t fit well into this portfolio credit risk framework:
  • Guarantees, revolving credit agreements, and other contingent liabilities behave much like credit options.
  • CDS basis trades are not essentially market- or credit-risk-oriented. They may be driven by “technical factors.” A dramatic example occurred during the subprime crisis where the CDS basis widened sharply due to the dire lack of funding liquidity.
  • Convertible bonds are both market- and credit-risk-oriented. Equity and equity vega risk can be as important in convertible bond portfolios as credit risk.
• There is very limited data for estimating defaults because firm defaults are rare events.

Effect Of Correlation On A Credit Portfolio And Credit VaR

• Default correlation has a tremendous impact on portfolio risk. But it affects the volatility and extreme quantiles of loss rather than the expected loss:
  • If default correlation in a portfolio of credits is equal to 1, then the portfolio behaves as if it consisted of just one credit. No credit diversification is achieved.
  • If default correlation is equal to 0, then the number of defaults in the portfolio is a binomially distributed random variable. Significant credit diversification may be achieved.
• A portfolio’s credit value at risk (credit VaR) is defined as the quantile of the credit loss less the expected loss of the portfolio.

Effect Of Granularity On Credit VaR Of Uncorrelated Portfolio

• “Granular” refers to reducing the weight of each credit as a proportion of the total portfolio by increasing the number of credits.
• We can consider 𝑛 credits each with a default probability of 𝜋 percent and a recovery rate of zero percent. Let’s assume the total value of the portfolio is $1,000,000,000
• Let’s start by setting 𝑛 = 1
  

  	π = 0.005	π = 0.02	π = 0.05
  Expected Loss	5,000,000	20,000,000	50,000,000
  95 Percent Confidence Level
  Number of defaults	0	0	1
  Proportion of defaults	0/1 = 0	0/1 = 0	1/1 = 100%
  Credit VaR	0 – 5m = -5m	0 – 20m = -20m	1b – 50m = 950m
  99 Percent Confidence Level
  Number of defaults	0	1	1
  Proportion of defaults	0	100%	100%
  Credit VaR	0 – 5m = -5m	1b – 20m = 980m	1b – 50m = 950m

• Continuing with the same portfolio, let’s divide it into relatively smaller positions by setting 𝑛 = 50.
  < tr>

  	π = 0.005	π = 0.02	π = 0.05
  Expected Loss	5m	20m	50m
  95 Percent Confidence Level
  Number of defaults	1	3	5
  Proportion of defaults	1/50 = 2%	3/50 = 6%	5/50 = 10%
  Credit VaR	20m – 5m = 15m	60m – 20m = 40m	100m – 50m = 50m
  99 Percent Confidence Level
  Number of defaults	2	4	7
  Proportion of defaults	2/50 = 4%	4/50 = 8%	7/50 = 14%
  Credit VaR	40m – 5m = 35m	80m – 20m = 60m	140m – 50m = 90m

• Continuing with the same portfolio, let’s divide it further by setting n = 1000 (for self-practice).
  

  	π = 0.005	π = 0.02	π = 0.05
  Expected Loss
  95 Percent Confidence Level
  Number of defaults
  Proportion of defaults
  Credit VaR
  99 Percent Confidence Level
  Number of defaults
  Proportion of defaults
  Credit VaR

Single Factor Model

• The single-factor model is used for assessing the impact of default correlations based on a credit position’s beta. Consider a set of 𝑖 firms (𝑖 = 1, 2, …) where each firm has its own:
  • correlation \(\beta_i\) with the market 𝑚,
  • standard deviation of idiosyncratic risk \(= \sqrt{1 – \beta_i^2}\),
  • idiosyncratic shock \(\varepsilon_i\)

  Firm 𝑖’s individual return on assets is given by:

  \(a_i = \beta_i m + \sqrt{1 – \beta_i^2}\varepsilon_i\)

• We assume that 𝑚 and 𝜀_𝑖 are standard normal variates, and are not correlated with one another. In addition, we assume the 𝜀_𝑖s are not correlated with one another.

  i.e. \(m \sim N(0, 1) ; \ \varepsilon_i \sim N(0, 1)\)

  \(Cov[m, \varepsilon_i] = 0 ; \ Cov[\varepsilon_i, \varepsilon_j] = 0\)

  Under these assumptions, each 𝑎_𝑖 is a standard normal variate.

• The correlation between pairs of individual asset returns between two firms 𝑖 and 𝑗 is 𝛽_𝑖𝛽_𝑗. The model assumes that firm 𝑖 defaults if \(a_i \leq k_i\), the logarithmic distance to the defaulted asset value that is measured by standard deviations.
• The single-factor model assumes conditional independence, which is the property that once a particular value of the market factor is realized, the asset returns, and hence default risks, are independent of one another. This results from the assumption that the firms’ returns are correlated only via their relationship to the market factor.
• If the market factor, \(m\), has a specific value of \(\bar{m}\), then substituting this in equation (A) and taking 𝛽_𝑖\(\bar{m}\) to the left side results in:

  \(a_i – \beta_i \bar{m} = \sqrt{1 – \beta_i^2} \varepsilon_i\)

  Under this condition, the default distribution’s mean shifts based on the specific market value for any 𝛽_𝑖 > 0. The default threshold, \(k_i\), does not change, but the standard deviation of the default distribution is reduced from 1 to \(\sqrt{1 – \beta_i^2}\).

  The following graphs assume 𝛽_𝑖 = 0.4, k = -2.33 (which implies π = 0.01, and m = -1.0). The unconditional default distribution is a standard normal distribution. The conditional default distribution is \(\text{N}\left(\beta_i \bar{m}, \sqrt{1 – \beta_i^2}\right) = \text{N}(-0.4, 0.9165)\).

  

Source: Figure 7-3, 2018 Financial Risk Manager Part II, Credit Risk Measurement and Management

• Specifying a realization \(m = \bar{m}\) does three things:
  • The conditional probability of default is greater or smaller than the unconditional probability of default, unless either \(\bar{m} = 0\) or \(\beta_i = 0\); that is, either the market factor shock happens to be zero, or the firm’s returns are independent of the state of the economy. Given 𝑚, a realization of 𝜀_𝑖, less than or equal to \(k_i – \beta_i \bar{m}\), triggers default. This expression is linear and downward-sloping in 𝑚. As we let 𝑚 vary from high (strong economy) to low (weak economy) values, a smaller (less negative) idiosyncratic shock will suffice to trigger default.
  • The conditional variance of the default distribution is \(1 – \beta_i^2\), which is less than the unconditional variance of 1.
  • It makes the asset returns of different firms independent.
• The conditional cumulative default probability function can be represented as a function of 𝑚 as:

  \(p(m) = \Phi \left( \frac{k_i – \beta_i m}{\sqrt{1 – \beta_i^2}} \right)\)

  

  This graph assumes k = -2.33 (which implies π = 0.01)

  Source: Figure 7-4, 2018 Financial Risk Manager Part II, Credit Risk Measurement and Management

Credit VaR Using A Single Factor Model

• The unconditional distribution used to calculate credit VaR is determined by the following steps:
  • The loss level is treated as a random variable 𝑋 with realized values of 𝑥. Here, 𝑥 is not simulated, but rather work is done analytically for each value of 𝑥 between 0 and 1.
  • Given a loss level of 𝑥, the value for the market factor, \(m\), is determined at the probability of the stated loss level. The loss level and the market factor return are related by \(x(m) = p(m) = \Phi \left( \frac{k – \beta m}{\sqrt{1 – \beta_i^2}} \right)\). The market factor return, \(m\), for a given loss level, \(x\), is determined by solving \(\Phi^{-1

    }(\bar{x}) = \frac{k – \beta \bar{m}}{\sqrt{1 – \beta^2}}\).

  • The market factor is assumed to be standard normal.
  • This process is repeated for each loss level to determine the loss probability distribution.
• As simple as the model is, we have several parameters to work with:
  • The probability of default 𝜋 sets the unconditional expected value of defaults in the portfolio.
  • The correlation to the market 𝛽² determines how spread out the defaults are over the range of the market factor. When the correlation is high, then, for any probability of default, defaults mount rapidly as business conditions deteriorate. When the correlation is low, it takes an extremely bad economic scenario to push the probability of default high.
• To understand the impact of the correlation parameter, start with the extreme cases:
  • 𝛽 → 1 (perfect correlation): Recall that we have constructed a portfolio with no idiosyncratic risk. If the correlation to the market factor is close to unity, there are two possible outcomes. Either \(m \leq k\), in which case nearly all the credits default, and the loss rate is equal to 1, or 𝑚 > 𝑘, in which case almost none default, and the loss rate is equal to 0.
  • 𝛽 → 0 (zero correlation): If there is no statistical relationship to the market factor, so idiosyncratic risk is nil, then the loss rate will likely be very close to the default probability 𝑝.

  In less extreme cases, a higher correlation leads to a higher probability of either very few or many defaults, and a lower probability of intermediate outcomes.

Credit VaR Using Copulas

• The following four steps are used to calculate Credit VaR using copulas:
  • A copula function is specified.
  • Default times are simulated.
  • Market values and P&L data are obtained for each scenario by applying the simulated default times.
  • Portfolio distribution statistics are computed by adding the results.