Answer to Question #152019 in Statistics and Probability for Gcebile

Question

Question 1. Historically a bank expects about 5% of its borrowers to default (not repay). The bank currently has 250 loans outstanding.

(a) In order to use a binomial model to compute the probabilities associated with defaults, what must the bank assume about the behavior of these borrowers?

(b) Do the necessary assumptions listed in part (a) appear reasonable in the context of this problem?

(c) The bank has reserves on hand to cover losses if 25 of these loans were to default. Will these reserves will be enough?

Answer 1

It is given that a bank expects about 5% of its borrowers to default.

(a) In order to use a Binomial model to compute the probabilities associated with defaults, the bank must assume that the behaviors of its 250 borrowers are independent Bernoulli trials where a borrower default is called a success. The probability of success must remain the same for all the trials.

(b) In the given situation, the behaviors of the borrowers are Bernoulli trials. The trials are independent. If the first borrower defaults, it will not affect the behavior of the remaining borrowers. Here the probability of success is the same for all trials. The probability that a borrower defaults is the same and is equal to 0.05 for all the borrowers. Therefore, the necessary assumptions listed above are reasonable in the context of this problem.

(c) Here the number of borrowers who default is a random variable that follows the Binomial model with parameters n = 250 and p = 0.05.

The mean of the Binomial model is given by np.

Therefore, the expected number of borrowers who default is given by

$250 \times 0.05 = 12.5$

So, the expected number of borrowers who default is approximately equal to 13.

It is given that the bank has reserves on hand to cover losses if 25 of these loans were to default. So from the above calculations, it can be seen that the reserves will be enough.