Default correlation is the Pearson correlation between two binary indicators of default over a specified horizon. Each indicator is 1 if its borrower defaults and 0 otherwise. It measures how the two default events occur together relative to independence. For a lender holding both borrowers' debt, this dependence can increase the chance of clustered losses even when each borrower's default probability stays unchanged.
Specify the default event, horizon and probability measure before comparing numbers. A one-year physical default model and a five-year pricing model describe different objects. Combining a physical marginal PD with a risk-neutral joint probability does not define a consistent correlation.
From a joint probability to a correlation
Let DA and DB be default indicators, with marginal probabilities pA and pB. Write q=P(DA=1,DB=1) for the joint probability of default. Since DADB is 1 only when both default,
This is dimensionless. If either PD is 0 or 1, that indicator is constant, its variance is zero and Pearson correlation is undefined. The joint probabilities themselves remain meaningful.
Twice the joint risk need not mean a large coefficient
Consider two initially non-defaulted fictional borrowers over the next year. Stipulate physical PDs of 10% each and a 2% probability that both default. These are synthetic model probabilities, not observed frequencies or issuer forecasts.
One-year outcome
Stipulated model
Independent comparison
Both default
2%
1%
A defaults, B survives
8%
9%
A survives, B defaults
8%
9%
Neither defaults
82%
81%
Total
100%
100%
Both columns give each borrower a 10% PD. Independence would make the joint default probability 0.10×0.10=0.01. In the stipulated model,
Two joint tables compare synthetic one-year default probabilities. Dependent model: both 2%, A only 8%, B only 8%, neither 82%. Independent comparison: both 1%, A only 9%, B only 9%, neither 81%. Both have 10% marginal PDs. Indicator correlation in the dependent model is 1/9, approximately 0.1111.
The same marginal PDs support different joint risks. All entries are stipulated one-year physical probabilities; the independent column changes only the dependence assumption.
The coefficient is about 0.1111, yet joint default is twice as probable as under independence. The related conditional question has another answer: P(DB=1∣DA=1)=0.02/0.10=20%. Neither the 2% joint probability nor the 20% conditional probability is the correlation.
For the number of defaults N=DA+DB, the expected count stays at 0.20 under both columns. Its variance rises from 0.18 under independence to 0.20 in the stipulated model, in squared-default-count units, because
Var(N)=Var(DA)+Var(DB)+2Cov(DA,DB).
To revisit that variance expansion, Expectation, Variance and Covariance is available in full with a free signed-in account. Translating a default count into monetary loss additionally requires exposures and loss given default.
Which correlation is being quoted?
In a threshold model, borrower i defaults when a continuous credit-quality variable Xi crosses a threshold. Asset correlation describes the Xi variables; default correlation describes the resulting 0/1 indicators. Equity-return correlation can be used as a proxy in some modelling approaches, but it is another object again. A Gaussian copula's latent correlation parameter is therefore not interchangeable with ρD. The mapping depends on the marginal PDs and the assumed joint model. The ECB's portfolio-credit-risk study, §3.2.2, explicitly distinguishes these correlations.
Positive default correlation can arise from shared economic drivers without one firm's default causing the other's. Bernoulli Mixture Models develops this mechanism through defaults that are independent conditional on a common factor, but dependent unconditionally (membership required for the full lesson; public preview available).
Feasibility matters too. A two-borrower table requires
max(0,pA+pB−1)≤q≤min(pA,pB).
With both PDs at 10%, these limits restrict default correlation to −1/9≤ρD≤1; a correlation of −1 is impossible. For two binary indicators, the marginal PDs and a feasible correlation determine the whole two-by-two table. For a larger portfolio, marginal PDs and pairwise correlations generally leave higher-order dependence unspecified. They cannot, by themselves, supply a complete portfolio loss distribution or a forecast of the next default cluster.
Sources
Alexander J. McNeil, Rüdiger Frey and Paul Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, revised ed., Princeton University Press, 2015, §11.1.1, pp. 427–428, equation (11.1); §11.1.2, p. 428; §11.3.1, pp. 450–451. Default-indicator correlation, its distinction from asset correlation, and model risk beyond the first two joint moments.
