Bonds rated B have a 25% chance of default in five years. Bonds rated CCC have a 40% chance of default in five years. A portfolio consists of 30% B and 70% CCC-rated bonds. If a randomly selected bond defaults in a five-year period, what is the probability that it was a B-rated bond?
According to Bayes' formula: P(B / default) = P(default and B) / P(default).
P(default and B )= P(default / B) × P(B) = 0.250 × 0.300 = 0.075
P(default and CCC) = P(default / CCC) × P(CCC) = 0.400 × 0.700 = 0.280
P(default) = P(default and B) + P(default and CCC) = 0.355
P(B / default) = P(default and B) / P(default) = 0.075 / 0.355 = 0.211 |