Hi David,

In one of your posts, you explain very nicely why VaR is NOT sub-additive. 

http://www.bionicturtle.com/learn/article/illustration_of_vars_failure_to_meet_coherence/

Although I understand why VaR is not sub-additive for non-elliptical loss distributions, I don’t quite understand why Expected Shortfall (ES) / Conditional VaR is sub-additive in all cases, even when the loss distribution is highly skewed. 

Can you explain why ES is sub-additive?  Can you show an spreadshet example where ES is sub-additive when VaR is NOT?

Thanks,

You might be interested in Acerbi, who gives proof in Appendix. I struggle with an *intuition* of it.

But to help, I added some rows to the bottom of the same spreadsheet at:
http://www.bionicturtle.com/learn/article/illustration_of_vars_failure_to_meet_coherence/
(see Expected shortfall at rows 20+)

Starting at row 20, I computed expected shortfall for 1, 2, and 3 bond portfolio. Given a 95%/5% threshold, then, ES (@ 5%) = the average loss for only the worst 5% losses (here one bond default = loss of $1). You can see ES(1 bond) = 0.4 and ES(3 bonds) = 1.02, as 3*ES(1 bond) > ES(3 bonds) sub-additivity is satisfied. Hope that’s helpful…

David