Is VaR Subadditive in Practice?

doobs · Post by **doobs** » Sat Jun 02, 2007 7:38 pm

I would be interested to hear what others on NP think about VaR and if it's Subadditive in Practice?

I just came across this interesting paper by the Heavy Tailed crew

http://www.orie.cornell.edu/orie/people/faculty/profile.cfm?netid=gs18

nsande · Post by **nsande** » Mon Jun 04, 2007 2:15 pm

No.

We have had periods where sub-additivity has been violated in our portfolio.

That was because we were using historical VaR.

Graeme · Post by **Graeme** » Mon Jun 04, 2007 6:06 pm

Ditto. In fact, as I recall, there were 2 portfolios that we reported on (of several) which were very infrequently subadditive.

doobs · Post by **doobs** » Mon Jun 04, 2007 6:38 pm

Is this strictly a function of the asset class?

Graeme · Post by **Graeme** » Mon Jun 04, 2007 8:37 pm

No. Why should it be?

doobs · Post by **doobs** » Mon Jun 04, 2007 8:47 pm

Certain asset class returns are more normal than others, and according to this

paper certain fat-tail properties produce Sub additivity, and apparently most asset classes have these properties.

Graeme · Post by **Graeme** » Tue Jun 05, 2007 2:17 pm

Notice how in the paper there is no reference to real portfolios. Yet the abstract refers to a real world problem. The end.

doobs · Post by **doobs** » Tue Jun 05, 2007 5:48 pm

No Kidding! I agree 100%

aaron · Post by **aaron** » Sun Jul 01, 2007 1:59 am

I would reverse the question.

If VaR is not subadditive, it means that the combined portfolio VaR is dominated by at least two different outlier events with individual probabilities less than the VaR point. If this is the case then the individual VaRs are not meaningful.

It's not hard to come up with a case in which this is true. Suppose you have a large portfolio of subinvestment grade bonds, say each with a 1 basis point probability of defaulting over a one-day horizon. These short-term default events will not be highly correlated (probability of default will move up or down in a correlated way, but actual default events probably won't be highly correlated on a one-day scale). So the 99% 1-day VaR will jump as you go from 99 to 100 bonds in the portfolio. But no one uses VaR for a single subinvestment grade bond.

Or, as nsande implied, if you use historical VaR, it's easy to come up with examples that violate subadditivity. But these will be mostly examples where no one would have used VaR on the individual portfolios in the first place.

I don't know of any examples of VaR not being subadditive for reasonable individual portfolios, ones in which people would use VaR. If it did happen, you should catch it by stress testing or some other technique.

Also, we almost never know enough about the tails to know if VaR really is subadditive or not. Even if we suspected it wasn't, there's rarely much we can do about it. Only people who put too much reliance on models think of subadditivity as a well-defined concept.

Therefore, I don't consider lack of subadditivity to be a practical problem for VaR.

paul1 · Post by **paul1** » Tue Jul 03, 2007 3:05 pm

sorry to interrupt...i'm curious: what do you think of cvar vs var? (academic) proponents of cvar state that unlike var, cvar is subadditive, convex, a coherent measure of risk and other nice things they say makes it preferrable to var.