While sitting through your Credit C part 2 screencast—I got bit by this wisdom that VaR is not sub-additive while ES is....(I find myself an unwitting victim of having forgotten some concept which I first came across say 6 weeks back...)

1. Can you pithily describe what sub-additive means? I tried to look up Wilmott, but couldn’t zero in…

2. Why is it a big deal that VAR is not a sub-additive risk measure? I thought there is value in reserving capital to cushion against unexpected losses, i.e. the VaR number.

--sridhar

Hi sridhar,

Ah, then may i suggest allocating a meaningful chunk of your time in the six weeks before the exam?

1. Sub-additive is one of the 4 things needed for a COHERENT RISK MEASURE (a new FRM 2008 AIM, testable). K Dowd says it’s the most important violation of VaR and renders VaR “not a proper risk measure.” My pithy try: a sub-additive measure does not ever say “diversification is bad.” VaR can sometimes under circumstances draw this conclusion. If not sub-additive, then the VaR of a portfolio can be larger than the sum of the individual VaRs, which makes no sense. At worst, portfolio VaR = sum of [individaul VaRs]. Sub-additive will never reach the ridiculous conclusion that portfolio VaR > sum of [individaul VaRs]

Here is a recent blog article with the best concrete example i could find, stolen of course, from Jorion. It shows has VaR can make a portfolio of bonds riskier than the sum of their individual VaRs.

2. Theoretically the consequences are significant. You can see from above, lack of S-A can tell a firm that it should set aside more economic/regulatory capital than it really needs. Or, conversely, a firm yoked under VaR might be encouraged to split up (as the “whole is riskier than the parts"). In practice, I don’t *really* know if sub-additivity is a huge deal, they don’t let me out much...the academics seem to get quite upset about it...but i don’t know about actual practice so much

David

sridhar,

I just blogged a summary on BIS’ recent economic capital survey (link here). They make an interesting point: that banks do find lack of sub-additive VaR to be a problem for credit and operational risks where the distributions are especially non-normal:

“VaR is subadditive for elliptical distributions, such as the Gaussian (or normal) distribution, whereas it is not
subadditive for non-elliptical distributions. The non-subadditivity of VaR can occur when assets in portfolios
have very skewed loss distributions; when the loss distributions of assets are smooth and symmetric, but their
dependency structure or copula is highly asymmetric; and when underlying risk factors are independent but
very heavy-tailed. The lack of subadditivity for VaR is probably more of a concern for credit risk and
operational risk than for market risk, where an elliptical model may be a reasonable approximate model for
various kinds of risk-factor data. “