Joint Default Probability Range

Question

There is a 50% probability that bond A will default next year and a 30% probability that bond B will default. What is the probability range for event that at least one bond defaults?

The author then goes on with:

To have the largest probability we can assume whenever A defaults, B does not default; and vice versa. A does not default -- so the maximum probability that at least one bond defaults is 50% + 30% = 80%.

OK, so he is talking about a scenario where A and B are independent. $$p(\text{at least one defaults}) = 1 - p(\text{none default})$$ A and B are Bernoulli RVs, so: $$p(\text{none default}) = 0.5 \cdot 0.7 = 0.35$$ $$p(\text{at least one defaults}) = 0.65$$

65% is nowhere close to 80%. What is going on?

Answer 1

"Whenever A defaults, B does not" is not independent at all. The description explains how B's default depends on A's.

Let's talk about this with dice. If we know that A is some event with a probability of 3/6 and B is some event with a probability of 2/6 but we don't know what those events are, what is the maximum probability that either A or B could happen? Well, we could imagine an interpretation where A = "roll a 1, 2, or 3" and B = "roll a 4 or 5", so the probability of at least one occurring under this interpretation is 5/6. It should be fairly evident that we could not do better.

Answer 2

To add on others answers, "Whenever A defaults, B does not" means A’s default and B’s default are mutually exclusive events, it does not imply independence.

Answer 3

The probability that either bond A or B defaults is
$$\Pr(A \text{ or } B) = \Pr(A) + \Pr(B) -\Pr(A \cap B)$$. The maximum probability that at least one bond deafults is when we take $\Pr(A \cap B) = 0$.