Joint Default

Bond A defaults next year with probability $0.5$ and bond B with probability $0.3$ . Nothing is said about how the two defaults relate. What is the range of the probability that at least one defaults, and the range of the correlation between the two defaults?

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#At least one default

Write the union in terms of the overlap,

\PP(A \cup B) = \PP(A) + \PP(B) - \PP(A \cap B) = 0.8 - \PP(A \cap B). \tag{1}

The overlap is pinned by the Frechet bounds, $\PP(A \cap B) \in [\max(0,\,0.8-1),\, \min(0.5,0.3)] = [0,\,0.3]$ . The union therefore runs from $0.8$ when the defaults are disjoint down to $0.5$ when B sits entirely inside A,

\PP(A \cup B) \in [0.5,\ 0.8]. \tag{2}

The marginals are fixed, but the overlap is free. Disjoint defaults push the union to its largest and the correlation to its most negative; nesting B inside A pulls the union to its smallest and the correlation to its most positive.

#The correlation

Treat the defaults as indicators with $\PP(A) = 0.5$ , $\PP(B) = 0.3$ , so the variances are $0.5 \cdot 0.5 = 0.25$ and $0.3 \cdot 0.7 = 0.21$ . The covariance is $\PP(A \cap B) - \PP(A)\PP(B) = \PP(A \cap B) - 0.15$ , and

\Cor = \frac{\PP(A \cap B) - 0.15}{\sqrt{0.25 \cdot 0.21}} = \frac{\PP(A \cap B) - 0.15}{\sqrt{0.0525}}. \tag{3}

Sweeping $\PP(A \cap B)$ across $[0, 0.3]$ gives the endpoints, where $\tfrac{0.15^2}{0.0525} = \tfrac{3}{7}$ ,

\Cor \in \left[-\sqrt{\tfrac{3}{7}},\ +\sqrt{\tfrac{3}{7}}\right] \approx [-0.655,\ 0.655]. \tag{4}

The correlation never reaches $\pm 1$ because the unequal marginals forbid the two indicators from moving in perfect lockstep. The widest union, at $0.8$ , lines up with the most negative correlation, and the narrowest, at $0.5$ , with the most positive.