Understanding Default Correlation
1. Understanding Default Correlation: A Simple Example
If I were to give you two loans, A and B, both having the same probability of default (PD), then to calculate the joint probability of default, a simple starting point is to assume that defaults are independent.
Under this assumption, the joint probability of default is simply equal to the product of the respective probabilities of default:
$$ P(\text{both default}) = p_A \times p_B $$For example, if:
$$ p_A = p_B = 5\% $$Then:
$$ P(\text{both default}) = 5\% \times 5\% = 0.25\% $$This tells us that the event of both loans jointly defaulting in the same year is an event which happens approximately once in 400 years.
2. Defaults Are Not Independent in Practice
In practice, defaults are not independent. They are correlated. This is because the obligors for the two loans might:
- Belong to the same economy,
- Be exposed to the same interest rates,
- Operate within the same industry.
Hence, it becomes important to account for the default correlation between the two obligors.
3. Mathematical Definition of Default Correlation
Mathematically, default correlation is defined as:
$$ \rho_{AB} = \frac{p_{AB} – p_A p_B}{\sqrt{p_A (1 – p_A)} \sqrt{p_B (1 – p_B)}} $$This formula can be rearranged to solve for the joint probability of default while taking into account the default correlation:
$$ p_{AB} = \rho_{AB} \times \sqrt{p_A (1 – p_A)} \sqrt{p_B (1 – p_B)} + p_A p_B $$4. Numerical Example with Correlated Defaults
Suppose we assume that the default correlation between the two loans is 30%:
$$ \rho_{AB} = 30\% = 0.3 $$Using the formula above, we can calculate the new joint probability of default:
$$ p_{AB} \approx 1.68\% $$This tells us that under the assumption of correlated defaults, the event of both loans defaulting in the same year happens approximately once in 60 years — a much higher likelihood than the independent case.
5. Why Is Default Correlation Important?
Because if we were to purely go by the individual probabilities of default, which tend to be small, the loan portfolio might appear to be quite safe.
However, in the presence of correlated defaults, if a downturn strikes, defaults tend to arrive in clusters or waves. Under such conditions, even seemingly low-risk products — such as the senior tranche of a Collateralized Debt Obligation (CDO) — can suddenly start to take losses.
This is why understanding and modeling default correlation is a key element of credit risk management.
6. Watch the Video
For a full explanation and worked example, watch the video below:
7. Want to Master More Concepts Like This?
Join our FRM Prep Course and get expert guidance, detailed video lessons, practice questions, and more. Please head over to the: