In order to manage the credit risk a bank regularly rates...

In order to manage the Credit Risk, a bank regularly rates each of its borrowers as A1 or A2 or A3, base on their credit history. Aj implies lowest risk and Az implies highest risk. Risk means the chance that a borrower might fail to payback the loan amount. On an average, 30% customers are rated A1, 60% customers are rated A2, and 10% customers are rated A3. It was found that 1% of customers who were rated A1, 10% of customers who were rated A2, 18% of customers who were rated A3, eventually became defaulters (failed to payback). (Hint: drawing a tree diagram may help you to solve it) a). If you randomly select a borrower from the bank, find the probability of he/she is a defaulter. b). If you randomly pickup a borrower from the defaulter's pool, what is the probability that he/she had received an Az rating?

Transcript

00:01 In this question, customers at a bank are rated with a1 to a3 rating, a1 being the lowest risk and a3 being the highest risk.

00:11 There are 30 % customers with lowest risk, a1, 60 % with a2 rating and 10 % with a3 rating.

00:19 And the defaulters with a1 rating are 1%.

00:23 With a2 rating are 10%, and with a3 rating are 18%.

00:29 Now, before we answer all this, let's define the event.

00:38 So i'm going to let a1 event be the event of person with a1 risk rating.

00:47 A2 be a person with a2 risk rating, a3 person with a3 risk rating, and d be the event of a defaulter.

00:55 So let's write out all the probabilities we are given.

01:00 So we have pa1 is 30%.

01:04 So that is 0 .3.

01:06 P -a -2 is 60%, so that's 0 .6.

01:12 Probability of a3, that's 10%, so that's 0 .1.

01:17 Now, we are also given that probability of defaulters to d given a1 rating is 0 .01, because 1%.

01:34 So, conditional probability means d intersect a1 over probability of a1 will be equals to 0 .01 so this means now p a1 is 0 .3 over here so this means it's 0 .01 this p a1 i move up okay times 0 .3 i'm going to get 0 .003 now similarly we can find the we are given probability of the default given a2 rating is 10 % over here so that is 0 .1 so again conditional probability okay now p a2 is 0 .6 so i'm going to bring this up which is 0 .6 times with my 0 .1 i will get 0 .06 now similarly you can find your pd intersect a 3 and that will be taking 18 % of 0 .18 times your p, a3, which is 0 .1, and you will get 0 .018.

03:05 Now, what does this mean all? now, in a 3, you have a1, a2, a3, right? so the probability of a1 is 0 .3, a2 is 0 .6, and a1 is 0 .1.

03:22 Now, you'll split into defaulters and non -defaulter.

03:28 Now, d prime means the complement of these.

03:30 So these are people who don't default.

03:36 Now, you can see that i wouldn't have this part yet, okay, but you can see that d intersect a1 means the probability of d and a1 will be 0 .03, so i'm just going to put the 0 .03 here.

03:54 And d intersect a2 is 0 .06.

03:57 So i'm going to put 0 .06 here.

03:59 And d intersect a3 is the same as a3 in a set d, so this is 0 .018.

04:07 Okay, let's answer the question in a...

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