Question

An analyst estimates that the probability of default on a seven-year AA-rated bond is 0.57, while that on a seven-year A-rated bond is 0.43. The probability that they will both default is 0.41.

a. What is the probability that at least one of the bonds defaults? (Round your answer to 2 decimal places.)

b. What is the probability that neither the seven-year AA-rated bond nor the seven-year A-rated bond defaults? (Round your answer to 2 decimal places.)

c. Given that the seven-year AA-rated bond defaults, what is the probability that the seven-year A-rated bond also defaults? (Round your answer to 2 decimal places.)

Answer 1

P(AA rated bond defaults) = 0.57

P(A rated bond defaults) = 0.43

P(they will both default) = 0.41

a) P(A or B) = P(A) + P(B) - P(A & B)

P(at least one of the bonds defaults) = P(AA rated bond default) + P(A rated bond default) - P(they will both default)

= 0.57 + 0.43 - 0.41

= 0.59

b) P(neither the seven-year AA-rated bond nor the seven-year A-rated bond defaults) = 1 - P(at least one of the bonds defaults)

= 1 - 0.59

= 0.41

c) Bayes' Theorem: P(A | B) = P(A & B) / P(B)

P(A rated bond defaults | AA rated bond defaults) = P(both bonds default)/P(AA rated bond defaults)

= 0.41/0.57

= 0.72