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2021-11-13

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. Hence, management established a prior probability of .05 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is .20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1.
1. Given that a customer missed one or more monthly payments, compute the posterior probability that the customer will default.
1. The bank would like to recall its card if the probability that a customer will default is greater than .20. Should the bank recall its card if the customer misses a monthly payment?
Why or why not?

Richard Cheatham

Step 1
(a). Compute the posterior probability that the customer will default given that the customer missed one or more monthly payments:
Denote the event that the cardholder will default as D.
Denote the event that the cardholder will not default as ${D}^{c}$.
It is given that, the prior probability that any cardholder will default is 0.05.
That is, .
Denote the event that the cardholder missing a monthly payment as M.
It is given that, the probability of missing a monthly payment is 0.2 for customers who do not default. That is, $P\left(M\mid {D}^{c}\right)=0.2$.
It is given that, the probability of missing a monthly payment is 0.2 for customers who default. That is, $P\left(M\mid D\right)=1$.
The posterior probability that the customer will default given that the customer missed one or more monthly payments is obtained as 0.2083 from the calculation given below:
$P\left(D\mid M\right)=\frac{P\left(M\mid D\right)×P\left(D\right)}{P\left(M\mid D\right)×P\left(D\right)+P\left(M\mid {D}^{c}\right)×P\left({D}^{c}\right)}$
$=\frac{1×0.05}{1×0.05+0.2×0.95}$
$=\frac{0.05}{0.24}$
$=0.2083$
Thus, the posterior probability that the customer will default given that the customer missed one or more monthly payments is 0.2083.
Step 2
(b). Comment whether the bank should recall its card if the customer misses a monthly payment:
It is given that, the bank would recall its card if the probability that a customer will default is greater than 0.2.
Here, the probability that the customer will default given that the customer missed one or more monthly payments is 0.2083.
If the customer misses a monthly payment, the probability that a customer will default is 0.2083.
Here, $2083>0.2$.
Since, the probability is greater than 0.2, the bank should recall its card.
Therefore, the bank should recall its card if the customer misses a monthly payment.

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