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I found on pages 2 and 3 of Martin Haugh's "Risk Measures, Risk Aggregation and Capital Allocation" from 2010 an example showing non sub-additivity of VaR (excerpts given at the end).

I understood the example on the whole, but I would like to clear out this doubt: when we compute the quantile, why for portfolio A we use -5 and l<-5, whereas for portfolio B we use +5 and +4?

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  • $\begingroup$ It would be best if you could reproduce the example in summary at least up until the point you are asking about in your post, preferably using MathJax/LaTeX. If you are unable to do that, pls at least include in your post a picture of the part of the paper you are specifically enquiring about in addition to the link to the original paper so that future readers of this discussion can figure out the details in case the link gets broken. $\endgroup$
    – Alper
    Aug 12, 2022 at 15:39

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What Alper says but from a quick look, I'll give you hint:

They figured out that (by trail and error for B) these bounds result in the various $P(\cdot)$ to have value close to the desired VaR levels (0.98 and 0.95).

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