It is essential to understand the risk of simultaneous default by several entities, particularly when it comes to credit derivatives such as basket credit
default swaps. The joint probability of default and its correlation are key to understanding this risk.
Correlation Formula
The correlation between two random variables \( X \) and \( Y \) is given by the formula:
\[ \text{correlation}(X, Y) = \frac{\text{covariance}(X, Y)}{\sigma_X \cdot \sigma_Y} \] Where:
- \( \text{covariance}(X, Y) \) is the covariance between \( X \) and \( Y \).
- \( \sigma_X \) and \( \sigma_Y \) are the standard deviations of \( X \) and \( Y \), respectively.
For default events, we often deal with indicator variables \( 1_A \) and \( 1_B \) which take the value 1 if default occurs and 0 otherwise. The covariance between these indicator variables can
be defined as:
\[ \text{covariance}(1_A, 1_B) = E[1_A \cdot 1_B] - E[1_A] \cdot E[1_B] \] Here:
- \( E[1_A] = p(A) \) is the probability of default of entity A.
- \( E[1_B] = p(B) \) is the probability of default of entity B.
- \( E[1_A \cdot 1_B] = p(A \text{ and } B) \) is the joint default probability of A and B.
Variance Calculation
For a default event A, where \( x_i \) can be 0 or 1:
\[ E[1_A] = 1 \cdot p(A) + 0 \cdot (1 - p(A)) = p(A) \] Variance is the expectation of the squared deviation from the mean:
\[ \text{Var}(1_A) = E[(1_A - p(A))^2] \] Since \( 1_A \) can be 0 or 1, we expand this as:
\[ \text{Var}(1_A) = p(A) \cdot (1 - p(A))^2 + (1 - p(A)) \cdot (0 - p(A))^2 \] Simplifying this, we get:
\[ \text{Var}(1_A) = p(A) \cdot (1 - p(A))^2 + (1 - p(A)) \cdot p(A)^2 \] \[ \text{Var}(1_A) = p(A) \cdot (1 - p(A)) \]
Correlation of Default
Using these definitions, the correlation of default can be expressed as:
\[ \text{corr}(A, B) = \frac{p(A \text{ and } B) - p(A) \cdot p(B)}{\sqrt{p(A) \cdot (1 - p(A)) \cdot p(B) \cdot (1 - p(B))}} \] And the joint default probability:
\[ p(A \text{ and } B) = \text{correlation}(A, B) \cdot \sqrt{p(A) \cdot (1 - p(A)) \cdot p(B) \cdot (1 - p(B))} + p(A) \cdot p(B) \]
Example Calculation
Consider an equal investment of $5 million in two bonds (B1 and B2) with the following parameters:
- Probability of default for both bonds \( p(A) = p(B) = 0.05 \) (5%).
- Default correlation \( \text{correlation}(A, B) = 0.3 \).
- Recovery rate = 0.
The joint default probability is calculated as:
\[ 0.3 \cdot \sqrt{(0.05 \cdot 0.95) \cdot (0.05 \cdot 0.95)} + 0.05 \cdot 0.05 = 0.1675 \] The expected loss is:
\[ p(A \text{ and } B) \cdot 10M = 0.1675 \cdot 10,000,000 = \$167,500 \]
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