Question

A local bank reviewed its credit-card policy with the intention of recalling some of its credit cards. In the past, approximately 5% of cardholders defaulted, leaving the bank unable to collect the outstanding balance. The bank also found that the probability of missing a monthly payment is 0.20 for customers who do not default.

part a) Given that a customer missed a monthly payment, compute the posterior probability that the customer will default.

part b) The bank would like to recall its credit card if the probability that a customer will default is greater than 0.20. Should the bank recall its credit card if the customer misses a monthly payment? Why or why not?

Answer 1

a) We are given here that:
P( missed payment | default ) = 1
P( missed payment | do not default ) = 0.2

Also, we are given here that:
P( default ) = 0.05, therefore P(do not default ) = 1 - 0.05 = 0.95

Therefore, using law of total probability we have here:
P( missed payment ) = P( missed payment | default ) P(default ) + P( missed payment | do not default ) P(do not default )

P( missed payment ) = 1*0.05 + 0.2*0.95 = 0.24

Given that a customer missed a monthly payment, probability that the customer will default is computed using Bayes theorem here as:
P( default | missed payment) = P( missed payment | default ) P(default ) / P( missed payment )

P( default | missed payment) = 1*0.05 / 0.24 = 0.2083

Therefore 0.2083 is the required probability here.

b) As the probability computed above is 0.2083 > 0.2, therefore Yes the bank should recall its credit card if the customer misses a monthly payment.