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Thread: 3 Norm Dists (correlated), prob co-occurence of 1% right tail

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    3 Norm Dists (correlated), prob co-occurence of 1% right tail




    Hi,
    hoping someone can lead me in the right direction..

    I'm looking at three normal distributions, specifically looking at the 1% extreme in each of these (say in one of the tails). So I have 3 extreme events (in 3 data sets), each defined as being the 1% right tail.

    Now, if I assume the correlation between each of the data sets is 80%, can I work out the probability of all 3 of the 1% extreme events occuring?

    So its obviously less than 1% (which would be the joint probability if the correlations were 100%). And would be 1%^3 if 0 correlation I believe.

    I'd considered application of Bayes probability theorem, eg:
    Prob(A and B) = Prob(B given A) * Prob(A)
    [A being the first extreme, B the second]..

    But I think I can only calculate Prob(B given A) based on historical co-occurence, of which the data series doesnt have enough data points to be accurate (eg have less than 200 data points).

    Also considered the portfolio variance calculation, substituting in Probability (eg 1%, or Z(1%)=2.33) for standard deviation... but I don't know if thats a valid application of the formula (I know its normal application, just trying to find something that fits my purpose)..

    Also looked at a probability tree, but goes back to not knowing the Prob(B given A), so can't progress down the branch.

    Please point me in the right direction! Thanks

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    For the estimation, it should not be not hard to get the estimate of the mean vector and
    the variance-covariance matrix. Once you get all these estimates, you can compute the
    probability of the tail regions by (numerical) integration.

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    Quote Originally Posted by BGM View Post
    For the estimation, it should not be not hard to get the estimate of the mean vector and
    the variance-covariance matrix. Once you get all these estimates, you can compute the
    probability of the tail regions by (numerical) integration.
    Thanks for posting.

    Ok, with an actual 3x3 variance-covariance matrix, and a mean vector containing the 3 means..
    What do you mean by numerical integration? Can you explain that process or refer me to a link?

    Thanks

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