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 ๐A and ๐B over some time horizon โ๐กโ, and a joint default probability of both defaulting over โ๐กโ being equal to ๐AB. This can be thought of as the distribution of the product of two Bernoulli-distributed random variables ๐ฅA and ๐ฅB, 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 |
ฯA โ ฯAB |
Only Firm B defaults |
0 |
1 |
0 |
ฯB โ ฯAB |
Both Firms Default |
1 |
1 |
1 |
ฯAB |
No Default |
0 |
0 |
0 |
1 โ (ฯA + ฯB โ ฯAB) |
- fble ๐ฅA is
\(\sigma^2_{x_A} = \pi_A(1 โ \pi_A)\)
- The variance of the Bernoulli distributed variable ๐ฅB is
\(\sigma^2_{x_B} = \pi_B(1 โ \pi_B)\)
- The covariance between ๐ฅA and ๐ฅB is given by
\(Cov_{x_A x_B} = E[x_A x_B] โ E[x_A]E[x_B] = \pi_{AB} โ \pi_A \pi_B\)
- Hence the default correlation is
\(\rho_{A,B} = \frac{Cov_{x_A x_B}}{\sigma_{x_A} \sigma_{x_B}}\)
= \(\frac{\pi_{AB} โ \pi_A \pi_B}{\sqrt{\pi_A(1 โ \pi_A)}\sqrt{\pi_B(1 โ \pi_B)}}\)
Drawbacks In The Credit Portfolio Framework
- Computationally intensive: If the number of credits in the portfolio is greater than 2 (i.e., if ๐ > 2), we have 2n different events, with \(n + 1 + \frac{n(n-1)}{2}\) conditions. In addition, we require ๐(๐-1) pairwise correlations. For example, if we have 20 firms, there will be 1,048,576 event outcomes with 211 conditions.
- Some kinds of positions donโt 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 โtechnical factors.โ 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 portfolioโs credit value at risk (credit VaR) is defined as the quantile of the credit loss less the expected loss of the portfolio.
- โGranularโ refers to reducing the weight of each credit as a proportion of the total portfolio by increasing the number of credits.
- We can consider ๐ credits each with a default probability of ๐ percent and a recovery rate of zero percent. Letโs assume the total value of the portfolio is $1,000,000,000
- Letโs start by setting ๐ = 1
ย |
ฯ = 0.005 |
ฯ = 0.02 |
ฯ = 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 โ 5m = -5m |
0 โ 20m = -20m |
1b โ 50m = 950m |
99 Percent Confidence Level |
Number of defaults |
0 |
1 |
1 |
Proportion of defaults |
0 |
100% |
100% |
Credit VaR |
0 โ 5m = -5m |
1b โ 20m = 980m |
1b โ 50m = 950m |
- Continuing with the same portfolio, letโs divide it into relatively smaller positions by setting ๐ = 50.
<
tr>
ย |
ฯ = 0.005 |
ฯ = 0.02 |
ฯ = 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 โ 5m = 15m |
60m โ 20m = 40m |
100m โ 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 โ 5m = 35m |
80m โ 20m = 60m |
140m โ 50m = 90m |
- Continuing with the same portfolio, letโs divide it further by setting n = 1000 (for self-practice).
|
ฯ = 0.005 |
ฯ = 0.02 |
ฯ = 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 positionโs beta. Consider a set of ๐ firms (๐ = 1, 2, โฆ) where each firm has its own:
- correlation \(\beta_i\) with the market ๐,
- standard deviation of idiosyncratic risk \(= \sqrt{1 โ \beta_i^2}\),
- idiosyncratic shock \(\varepsilon_i\)
Firm ๐โs individual return on assets is given by:
\(a_i = \beta_i m + \sqrt{1 โ \beta_i^2}\varepsilon_i\)
- We assume that ๐ and ๐๐ are standard normal variates, and are not correlated with one another. In addition, we assume the ๐๐s 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 ๐๐ is a standard normal variate.
- The correlation between pairs of individual asset returns between two firms ๐ and ๐ is ๐ฝ๐๐ฝ๐. The model assumes that firm ๐ 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 firmsโ 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 ๐ฝ๐\(\bar{m}\) to the left side results in:
\(a_i โ \beta_i \bar{m} = \sqrt{1 โ \beta_i^2} \varepsilon_i\)
Under this condition, the default distributionโs mean shifts based on the specific market value for any ๐ฝ๐ > 0. The default threshold, \(k_i\), does not change, but the standard deviation of the default distribution is reduced from 1 to \(\sqrt{1 โ \beta_i^2}\).
The following graphs assume ๐ฝ๐ = 0.4, k = -2.33 (which implies ฯ = 0.01, and m = -1.0). The unconditional default distribution is a standard normal distribution. The conditional default distribution is \(\text{N}\left(\beta_i \bar{m}, \sqrt{1 โ \beta_i^2}\right) = \text{N}(-0.4, 0.9165)\).
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 firmโs returns are independent of the state of the economy. Given ๐, a realization of ๐๐, less than or equal to \(k_i โ \beta_i \bar{m}\), triggers default. This expression is linear and downward-sloping in ๐. As we let ๐ 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 \(1 โ \beta_i^2\), 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 ๐ as:
\(p(m) = \Phi \left( \frac{k_i โ \beta_i m}{\sqrt{1 โ \beta_i^2}} \right)\)
This graph assumes k = -2.33 (which implies ฯ = 0.01)
Source: Figure 7-4, 2018 Financial Risk Manager Part II, Credit Risk Measurement and Management
Credit VaR Using A Single Factor Model
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.