1. ## probability question help

Can someone help me with this question: I have 5 buckets of names with different default probabilities. BKT1 has 30 names with 0.61% default probability each, BKT2 has 56 names with 1.22% default probability each, BKT3 has 15 names with 7.54% default probability each, BKT4 has 14 names with 18.19% default probability each, and BKT5 has 3 names with 38.34% default probability each. What is the probability that at most X names default? (say, what is the probability that at most 2 names default)

I am using the cumulative binomial distribution to get the probability of X defaults PER BUCKET but I am not sure how to combine this to get the result of a total of X defaults in the whole portfolio.

2. ## Re: probability question help

I'm not at all clear about your setup -- are you drawing one name from each bucket?

3. ## Re: probability question help

I am looking for the probability that at most 2 (i.e. 0, 1, or 2) of all the names in the whole portfolio, comprised of all 5 buckets or 118 names, will default. I start by calculating the probability that at most 0 of the names in each bucket will default (i.e. in excel: for BKT1: BINOMDIST(0,30,0.0061,TRUE), for BKT2: BINOMDIST(0,56,0.0122,TRUE), for BKT3: BINOMDIST(0,15,0.074,TRUE) etc.); then I calculate that at most 1 name will default in each bucket (BKT1: BINOMDIST(1,30,0.0061,TRUE), BKT2: BINOMDIST(1,56,0.0122,TRUE), etc.); then I calculate that at most 2 names will default in each bucket (BKT1: BINOMDIST(2,30,0.0061,TRUE), BKT2: BINOMDIST(2,56,0.0122,TRUE), etc.).

This gives me the probability that at most 2 names will default in each bucket. I don't know how to combine this to get the result that at most 2 names will default out of all the 5 buckets (118 names). I am attaching an excel spread sheet with the setup.

4. ## Re: probability question help

From the setup it seems that you assume the 5 buckets are mutually independent.

Let be the number of defaults in the -th bucket, as suggested.

As a whole portfolio, you want to calculate

Note that not all are the same so you do not have any shortcut to calculate the distribution of the sum.
i.e. you have to calculate by counting all the relevant scenarios.

Since is not too big, listing/counting them is not too difficult:

- 1 case of {0, 0, 0, 0, 0}
- cases of {1, 0, 0, 0, 0} and {2, 0, 0, 0, 0}
- cases of {1, 1, 0, 0, 0}

and you just sum up the probability for the above scenarios.

5. ## Re: probability question help

BGM, thank you very much for your help!

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