Question

A local bank reviewed its credit card policy with the intention of recalling some of its credit cards. In the past approximately 8% of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of 0.08 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is 0.20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1.

a. Given that a customer missed monthly payments, compute the posterior probability that the customer will default (to 3 decimals).
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

Using Bayes' Theorem, the posterior probability of a customer defaulting given they missed a monthly payment is 0.303. Since this probability is above the bank's threshold of 0.20, the bank should recall the credit card.

Step-by-step explanation:

To calculate the posterior probability of default given that a customer missed a payment, we need to use Bayes' Theorem, which is expressed as:

P(A|B) = (P(B|A) * P(A)) / P(B)

Where:

• P(A|B) is the posterior probability of defaulting given a missed payment.
• P(B|A) is the probability of missing a payment given that the customer will default, which is 1.
• P(A) is the prior probability of defaulting, which is 0.08.
• P(B) is the total probability of missing a payment.

First, we need to calculate P(B), which is the total probability of missing a payment:

P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)

P(B) = (1 * 0.08) + (0.20 * 0.92)

P(B) = 0.08 + 0.184

P(B) = 0.264

Now we can calculate the posterior probability:

P(A|B) = (1 * 0.08) / 0.264

P(A|B) = 0.08 / 0.264

P(A|B) = 0.303 (to three decimals)

Given that the bank wants to recall a credit card if the probability of defaulting is greater than 0.20, and the calculated posterior probability is 0.303, the bank should recall the credit card as the condition is met.