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  1. #1
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    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.

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    Re: probability question help

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

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    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.
    Attached Files

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    Re: probability question help

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

    Let D_i \sim \text{Binomial}(n_i, p_i) be the number of defaults in the i-th bucket, i = 1, 2, \ldots 5 as suggested.

    As a whole portfolio, you want to calculate \Pr\left\{\sum_{i=1}^5 D_i \leq 2 \right\}

    Note that not all p_i 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 2 is not too big, listing/counting them is not too difficult:

    - 1 case of {0, 0, 0, 0, 0}
    - \binom {5} {1} = 5 cases of {1, 0, 0, 0, 0} and {2, 0, 0, 0, 0}
    - \binom {5} {2} = 10 cases of {1, 1, 0, 0, 0}

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

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    Re: probability question help


    BGM, thank you very much for your help!

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