Bounds on joint default of two bonds

Question

Bond A defaults with probability $0.5$ and bond B with probability $0.3$ . Nothing else is known about how the two defaults are related. Over what range can (a) the probability that both default and (b) the correlation between the two default indicators lie?

Short answer

(a) $[0,\,0.3]$ ; (b) $[-\sqrt{3/7},\,\sqrt{3/7}]\approx[-0.65,\,0.65]$ . The joint probability $p$ is at least $\max(0,0.5+0.3-1)=0$ and at most $\min(0.5,0.3)=0.3$ . The indicator correlation is $(p-0.15)/\sqrt{0.5\cdot0.5\cdot0.3\cdot0.7}$ , which is linear in $p$ , so plug in the two endpoints.

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 4.5 Expected Value, Variance & Covariance, 'Joint default probability', p.97; question and answer rewritten by SelfMadeQuant.com.

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