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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 a. Given that a customer missed one or more monthly payments, compute the posterior probability that the customer will default. b. 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?

Step by step solution

01

Define Events and Probabilities

Let's define the events: \( D \) as the event that a customer defaults, and \( M \) as the event that a customer misses a monthly payment. We know from the problem that: \( P(D) = 0.05 \), \( P(M|eg D) = 0.20 \), and \( P(M|D) = 1 \).

02

Use Total Probability Theorem

The probability of a customer missing a payment, \( P(M) \), can be found using the Total Probability Theorem: \[P(M) = P(M|D)P(D) + P(M|eg D)P(eg D)\]\[P(M) = (1)(0.05) + (0.20)(0.95)\]\[P(M) = 0.05 + 0.19 = 0.24\]

03

Use Bayes' Theorem

To find the posterior probability that a customer will default given that they missed a payment, \( P(D|M) \), use Bayes' Theorem:\[P(D|M) = \frac{P(M|D)P(D)}{P(M)}\]Replace the probabilities:\[P(D|M) = \frac{(1)(0.05)}{0.24} = \frac{0.05}{0.24} \approx 0.2083\]

04

Compare the Posterior Probability with the Bank's Threshold

The posterior probability that a customer will default given they missed a payment, \( P(D|M) \approx 0.2083 \), is now compared to the bank's threshold of 0.20. Since \( P(D|M) \) is greater than 0.20, the bank should recall the card if the customer misses a monthly payment.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Probability Theory

Probability theory is the mathematical framework that allows us to quantify the uncertainty of real-world events. It's like having a toolset for predicting how likely something is to happen based on known factors. In this exercise, probability theory helps the bank evaluate credit card defaults.
For instance, when the bank identifies that 5% of customers default, they calculate a probability of 0.05 for a default. This is a crucial estimation step. Probabilities range from 0 to 1, where 0 means an event will not happen, and 1 means it will definitely happen.

• Probabilities allow organizations to make informed decisions, like deciding whether to recall a credit card.
• Combining probabilities gives us a clearer picture of what to expect, as in the case of default probabilities and missed payments.

This theoretical framework underpins many real-world applications, assisting in everything from simple predictions to complex analyses, like evaluating credit risk.

The Role of Credit Risk Management

Credit risk management is the process banks use to manage the risk of default on credit. It's all about weighing potential losses against the benefits of lending credit. Here, the bank aims to minimize losses from customers failing to pay their credit card bills.
This process involves:

• Identifying Risks: Predicting which customers might default.
• Monitoring Accounts: Tracking payment behaviors.
• Taking Actions: Deciding whether to recall cards.

By setting a default probability threshold, like 0.20 in our example, the bank delineates acceptable risk levels. They use probability to guide decisions, such as recalling a card. Effective credit risk management balances aggressive collection with customer retention and satisfaction.

Calculating Posterior Probability

Posterior probability is the revised probability of an event occurring after taking into account new information. Bayesian methods update the likelihood of a hypothesis as new evidence becomes available. In the scenario, calculating the posterior probability involves using Bayes' Theorem to find out how likely a customer is to default if they've missed one or more payments.
Bayes' Theorem is given by:\[ P(D|M) = \frac{P(M|D)P(D)}{P(M)} \]This formula integrates prior probability with new evidence (i.e., missed payments). The outcome here was roughly 0.2083, indicating there's a 20.83% chance of default once a payment is missed, aligning above the bank's acceptable threshold.

• Posterior probability changes your decision-making from mere guesswork to evidence-based precision.
• It allows the bank to make more informed, data-driven decisions regarding their credit policies.

Thus, bayesian updating is pivotal in the constant assessment of risk within credit management systems.