Two Bonds: Range of Joint Default Probability and Correlation
Bond A defaults within the next year with probability $50\%$, and bond B defaults within the next year with probability $30\%$. Nothing is known about how the two defaults are related.
(a) What is the range of possible values for the probability that at least one of the two bonds defaults?
(b) What is the range of possible values for the correlation between the two default events (the correlation of their indicator variables)?
Hints
- Write $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ and ask how small and how large $P(A \cap B)$ can be.
- $P(A \cap B)$ is at most $\min(P(A), P(B))$ and at least $\max(0, P(A) + P(B) - 1)$. Both bounds are attainable here.
- For indicators, $\rho = \dfrac{P(A \cap B) - P(A)P(B)}{\sqrt{P(A)(1-P(A))\,P(B)(1-P(B))}}$. Plug in the extreme values of $P(A \cap B)$.
Worked Solution
How to Think About It: With only the marginals given, the one free quantity is $p_{AB} = P(A \cap B)$. Both the union probability and the correlation are monotone functions of $p_{AB}$, so find the feasible interval for $p_{AB}$ (Frechet bounds) and map its endpoints through.
Quick Estimate: If defaults were independent, $P(\text{at least one}) = 1 - 0.5 \cdot 0.7 = 0.65$ and $\rho = 0$. Perfect nesting (B defaults only when A does) pushes the union down to $0.5$; mutual exclusion pushes it up to $0.8$. The correlation swings symmetrically around 0 to roughly $\pm 0.65$.
Formal Solution:
Part (a): Probability that at least one defaults
*Step 1 -- Bounds on the joint probability.* Let $p_A = 0.5$, $p_B = 0.3$, $p_{AB} = P(A \cap B)$. Since $A \cap B \subseteq B$ and $A \cap B \subseteq A$, $p_{AB} \le \min(p_A, p_B) = 0.3$. Since $P(A \cup B) \le 1$, $p_{AB} \ge p_A + p_B - 1 = -0.2$, i.e. $p_{AB} \ge 0$. Both endpoints are achievable: $p_{AB} = 0.3$ when $B \subseteq A$, and $p_{AB} = 0$ when $A$ and $B$ are disjoint (possible because $0.5 + 0.3 \le 1$).
*Step 2 -- Union.*
$$P(A \cup B) = p_A + p_B - p_{AB} = 0.8 - p_{AB} \in [0.5,\ 0.8].$$
Part (b): Default correlation
*Step 1 -- Indicator moments.* Let $I_A, I_B$ be the default indicators. $\text{Var}(I_A) = p_A(1 - p_A) = 0.25$, $\text{Var}(I_B) = p_B(1 - p_B) = 0.21$, $\text{Cov}(I_A, I_B) = p_{AB} - p_A p_B = p_{AB} - 0.15$.
*Step 2 -- Correlation as a function of $p_{AB}$.*
$$\rho = \frac{p_{AB} - 0.15}{\sqrt{0.25 \cdot 0.21}} = \frac{p_{AB} - 0.15}{\sqrt{0.0525}}.$$
*Step 3 -- Extremes.* At $p_{AB} = 0.3$: $\rho = 0.15/\sqrt{0.0525} = \sqrt{0.0225/0.0525} = \sqrt{3/7} \approx 0.6547$. At $p_{AB} = 0$: $\rho = -0.15/\sqrt{0.0525} = -\sqrt{3/7}$. So
$$\rho \in \left[-\sqrt{3/7},\ \sqrt{3/7}\right] \approx [-0.655,\ 0.655].$$
*Why not $\pm 1$.* Correlation $+1$ would require $I_B$ to be an increasing affine function of $I_A$, i.e. $A = B$ as events, impossible with different marginals; $-1$ would require $B = A^c$, impossible since $0.3 \ne 0.5$.
Answer: (a) $P(\text{at least one defaults}) \in [50\%, 80\%]$. (b) $\rho \in [-\sqrt{3/7}, \sqrt{3/7}] \approx [-0.655, 0.655]$.
Intuition
Everything hinges on the joint default probability, which can range from $0$ (the two never default together) to $30\%$ (B defaults only when A does). The "at least one" probability is $80\%$ minus that joint probability, so it lies in $[50\%, 80\%]$, and the correlation of two Bernoulli variables with different means cannot reach $\pm 1$: here it is capped at $\pm\sqrt{3/7} \approx \pm 0.655$. This is the Frechet-bound fact every credit or basket-derivatives desk lives with: default correlation of binary events is mechanically bounded by the marginal probabilities, so a quoted "correlation of 0.9" between a 50% and a 30% default is simply impossible.