Question

A bank offers loan to its customers on different types of interest 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 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?

Answer 1

E₁ : Loan at fixed rate

E₂ : Loan at floating rate

E₃ : Loan at variable rate A :

A person defaults on loan

\(P(E_1) = \frac{10}{100}; \ P (E_2) = \frac{20}{100}; \ P(E_3)=\frac{70}{100}\)

\(P(A/E_1) = \frac{5}{100}; \ P(A/E_2) = \frac{3}{100}; \ P(A/E_3) = \frac{1}{100}\)

(i) \(P(A) = P(E_1)P(A/E_1)+P(E_2)P(A/E_2)+P(E_3)P(A/E_3))\)

\(= \frac{10}{100}\times \frac{5}{100}+\frac{20}{100}\times \frac{3}{100}+\frac{70}{100} \times \frac{1}{100}\)

\(= \frac {50+60+70}{10000}\)

\(\frac{180}{10000} = 0.018\) 

(ii) \(P(E_3/A)= \frac{P(E_3)(P \ A/E_3)}{P(A)}\)

\(= \frac{\frac{70}{100}\times\frac{1}{100}}{\frac{180}{10000}} = \frac{70}{10000\times\frac{180}{10000}} = \frac{7}{18}\)