Two borrowers can each have a relatively low probability of default, but their probability of defaulting together can be materially higher when their underlying economic risks are correlated.
This calculator uses a one-factor Gaussian asset-value model. The joint default probability is calculated from the bivariate normal distribution using a tetrachoric (Hermite) series expansion.
When selected, the asset correlation is calculated from the borrower PDs using the standard Basel corporate exposure correlation relationship.
For two borrowers with different PDs, the two Basel-implied correlations are averaged to obtain a single correlation parameter for the bivariate model.
Independent JDP = PD1 x PD2.
Additional joint-default probability caused by positive dependence.
The model assumes that each borrower's latent asset value is driven by a common systematic factor plus an idiosyncratic factor.
The borrower defaults when its latent asset value falls below the threshold corresponding to its probability of default.
The bivariate normal CDF can then be expressed using the tetrachoric, or Hermite, series:
The first term is particularly useful from a credit-risk perspective. It is simply the joint default probability that would exist if the borrowers were independent. The remaining terms represent the incremental effect of correlation.
Asset correlation describes the relationship between the underlying continuous asset values. Default correlation describes the relationship between the two binary default events. They are therefore not the same quantity.
The transformation from asset correlation to default correlation depends strongly on the two PDs. This is one reason why relatively modest asset correlation can produce significant default dependence for low-PD borrowers.
Default is a binary event, so there is a mathematical limit to the possible correlation between two default indicators. For two borrowers with equal PD, perfect positive dependence allows every default by one borrower to coincide with a default by the other, giving a maximum default correlation of 100%. When the PDs differ, the maximum possible correlation can be lower.