Question

A bank offers loans to its customers on different types of interest rates namely, fixed rate, floating rate, and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate, or variable rate with probabilities 10%, 20%, and 70% respectively. A customer after availing a loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate, and variable rate is 5%, 3%, and 1% respectively. Based on the above information, answer the following:
(i) What is the probability that a customer after availing the loan will default on the loan repayment?
(ii) A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest?

Hint:

To solve probability problems involving conditional probabilities, use the law of total probability for finding the total probability and Bayes' theorem for finding conditional probabilities.

Answer 1

We are given the following probabilities:
- Probability of availing loan at fixed rate = \( P(F) = 0.1 \)
- Probability of availing loan at floating rate = \( P(Fl) = 0.2 \)
- Probability of availing loan at variable rate = \( P(V) = 0.7 \)
- Probability of defaulting on loan after availing at fixed rate = \( P(D|F) = 0.05 \)
- Probability of defaulting on loan after availing at floating rate = \( P(D|Fl) = 0.03 \)
- Probability of defaulting on loan after availing at variable rate = \( P(D|V) = 0.01 \)
(i) What is the probability that a customer after availing the loan will default on the loan repayment?
To find the total probability of defaulting on loan repayment, we use the law of total probability: \[ P(D) = P(D|F) \cdot P(F) + P(D|Fl) \cdot P(Fl) + P(D|V) \cdot P(V). \] Substitute the given values: \[ P(D) = (0.05 \times 0.1) + (0.03 \times 0.2) + (0.01 \times 0.7) \] \[ P(D) = 0.005 + 0.006 + 0.007 = 0.01. \] Thus, the probability that a customer after availing the loan will default on the loan repayment is \( 0.01 \) or 1.%. (ii) A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest?
We need to find \( P(V|D) \), the probability that the customer availed the loan at a variable rate given that they defaulted. We use Bayes' theorem for this: \[ P(V|D) = \frac{P(D|V) \cdot P(V)}{P(D)}. \] Substitute the values: \[ P(V|D) = \frac{0.01 \times 0.7}{0.01} = \frac{0.007}{0.01} \approx 0.39. \] Thus, the probability that the customer availed the loan at a variable rate of interest, given that they defaulted, is approximately \( 0.39 \) or 3.9%.