Bounds on Default Union Probability and Correlation

Given P(A defaults) = 0.50 and P(B defaults) = 0.30, explore every feasible value of P(A ∩ B).
Choose the overlap
x = P(A ∩ B), with 0 ≤ x ≤ 0.30
0.150
00.30
B defaults B survives
A defaults 0.150 0.350
A survives 0.150 0.350
P(A ∪ B)
0.650
= 0.80 − x
Corr(1A, 1B)
0.0000
= (x − 0.15) / √0.0525
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.8 − x Corr(1A,1B) = [P(A∩B) − P(A)P(B)] / √[Var(1A)Var(1B)] = (x − 0.15) / √(0.25 × 0.21)
Why can x only range from 0 to 0.30?
1
Lower bound: A and B can be completely disjoint, so x can be 0.
2
Upper bound: the overlap cannot exceed the smaller event: min(0.50, 0.30) = 0.30.
Endpoint picture
Disjoint: x = 0 Union = 0.80
Correlation ≈ −0.6547
B entirely inside A: x = 0.30 Union = 0.50
Correlation ≈ +0.6547
Final bounds
Union probability
0.50 ≤ P(A ∪ B) ≤ 0.80
Indicator correlation
−0.6547 ≤ ρ ≤ 0.6547
Important correction: correlation divides by the product of the standard deviations, so the denominator is √(0.25 × 0.21) = √0.0525 ≈ 0.2291, not 0.0525.