Is VaR Subadditive in Practice?

[IAmEric]

Hi paul1,



You'll soon learn (as I did with similar questions) that there are multiple definitions of cvar. Which do you mean? To me, CVaR is "component VaR" in which case you have



VaR = sum_i CVaR_i



where you sum over "components" (could be risk factors, could be desks, etc).



Your question sounds like you are referring to what I would call "expected tail loss".



If that is the case, I would prefer to look at ETL than VaR. One nice thing about subadditivity is that it allows you to draw some nice pictures to illustrate "risk", i.e. ETL defines a "vector dot product" and you can view P&L and/or returns as vectors. That is nice because it provides a theory with absolutely no practical implications. A mathematician's dream.

[Graeme]

The lack of subadditivity IS a problem in VaR precisely because people use it as a default measure for reporting risk. (Thus the problem is in the usage of the measure, not necessarily the measure itself - so maybe I am agreeing with Aaron after all.)



As has been pointed out a couple of times, lack of subadditivity does occur in practice. This is a problem if you (logically) define a measure of diversification of a new portfolio being added to an existing portfolio to be f(Existing P) + f(New P) - f(Exisiting P + New P) where f is the risk measure and that risk measure is NOT subadditive.



The fact that VaR is the default measure is partially the fault of the regulators, although they do require stress testing as more than just a supplement to VaR to get the (VaR) model approved.



Trouble currently with something like the canonical coherent measure expected shortfall is that there is no obvious way to backtest it. A collaborator and I are working on some results in this area, although currently our results will allow a bank to backtest its expected shortfall but not for the regulator to backtest it (we require a truthful disclosure of some statistics of the simulated p&l distribution for every day in the sample).

[paul1]

thanks IAmEric, yes, i meant conditional value at risk, aka expected shortfall.

[DrTarr]

[i]expected shortfall...no obvious way to backtest it.......our results will allow a bank to backtest its expected shortfall but not for the regulator to backtest it......[/i]



It would be a wonderful start if a bank could do it!  Not sure why the regulator would not be able to follow up on the method used by the bank and if appropriate (you know documentation-assumptions- validate) call it good.



Edit: "[i]require a truthful disclosure[/i]" unless this has something to do with it?

[Graeme]

Yes, exactly. The data is quite 'heavy', it isn't just a draw from a distribution as the VaR backtest is.

[djanklod]

Is the question really whether a single number is sufficient to consistently summarize the risk of a portfolio and its components?



Unless your portfolio is really really simple [i.e. a single risk factor with a gaussian behavior], this sounds like a chimera doesn't it?



Why even wonder about inconsistent behavior when the concept is flawed from the start?

[aaron]

I missed the leap between "not sufficient" to "flawed from the start."



We want to monitor risk quantitatively. That means we need numbers. None of them will be sufficient for all monitoring, but we have to start somewhere. VaR does give valuable information. It won out over a lot of other contenders as the first-choice, general purpose, cross-product, short-term risk number.



Think of it like a speed limit. The highway department can put one number on a sign to manage the risk of drivers. No one number is enough. A sober, careful, skilled driver in a well-maintained sports car can drive at twice the speed with less risk than a bad driver in a clunker that hasn't been to a mechanic since it left the factory, who is fighting with three different girlfriends on three different cell phones while chugging vodka and trying to put the moves on potential girlfriend number four. But that's a lot of numbers to put on a sign:



Speed x exp(Blood Alcohol Level) x Number of Cell Phone Conversations^2 x Number of Girlfriends^3/(Driving Skill Index x Car Quality Index x Maintenance Index) < 55



It would make it hard to design car dashboards and radar guns. And it would be no more generally useful than the simple speed number.

[IAmEric]

And whether VaR is a "good number" or not, I still tell people I think it is a good idea to measure it because it forces you to look under the hood. The infrastructure required to compute VaR provides a window into the portfolios that might not otherwise be available if you weren't measuring VaR. It is that window that is of value even if the ultimate number coming out is bordering on meaningless. When you pay that much attention to your portfolios you might see things that the actual risk number misses.

[djanklod]

eric: You'd still get a decent "under the hood" view if you looked at something more meaningful like a distribution rather than at a single and arbitrary statistics.



aaron: I have to say though that my previous post was probably too provocative and I thank you for pointing out that not quite sufficient is not quite the same as flawed from the start. I'm just so passionate about this, but passion alone does not convince! Trying to be more measured and specific:

My biggest problem with VaR is that despite all the lip service to the need to complement it with other measures, I still see it as [img]/User%20Files/3310/Latex-Equation-6732.gif[/img] single number with which everyone is obsessed (maybe this is a sad reflection on my current shop, but for some reason I doubt it). Put another way, VaR seems to breed intellectual laziness. That's the point where I have to restrain myself from starting to rant about management managing through the blind application KPIs and thinking they know everything about their business because they get a report with three numbers every day...



Real world conversation:

- "Isn't this position a bit large given the depth of the market?"

- "oh, not to worry, we've got a VaR limit"



Now coming back to what I was trying to say (maybe not very clearly): if only limitation of VaR was that it turned out to be not subadditive from time to time!

[aaron]

That’s a fair point. Here’s the story from my point of view, overworking my speed limit analogy. 



At one time, there were no speedometers or speed limits. People drove as seemed prudent to them at the time. Police gave tickets for “reckless driving,” based on how they felt at the time. The system didn’t work, people drove too slowly on average, and there were too many accidents, and the wrong people got tickets. 



Lots of ideas were thrown around to solve the problems. The winning solution was to put speedometers in all cars, and charge highway departments with determining maximum safe speeds for all roads. This was never intended to be the only safety precaution or driving rule. 



One big advantage came directly from the implementation, as IAmEric pointed out. Automotive engineers and highway engineers had to think about risk, leading to a lot of useful new theory, and many outmoded designs and roads being upgraded.



After the limits were implemented and average speeds picked up while accidents fell, people started criticizing limits. Theoreticians pointed to Richard Feynman’s famous example from his physics textbook that speed is not well defined. Practical people pointed out that weather, car quality and maintenance, traffic conditions and other factors affected the maximum safe driving speed. Problems arose due to measurement difficulties, speedometers gave different readings than radar guns, tire inflation affected speedometer readings and some measuring devices were broken altogether. In some major accidents, drivers claimed their speedometers had shown they were under the limit at all times. 



Despite all the theoretical and practical issues, which were valid, some people insisted that if the speedometer needle is under the posted speed limit, everything is fine. Drivers need not look out the windshield or rear view mirror, they can drive staring at the speedometer. Also, speed limits began to be set for political reasons, like saving gas or getting cars to slow down in hopes some would stop at shopping areas, rather than by highway engineers considering safety. Drivers figured out ways to fool their speedometers by installing extra-large tires, police figured out ways to overstate speed. 



For all the problems, things were much better with speed limits than without. Sensible people worked to fix the problems, and to come up with additional risk measures. Other people either pointed to the problems and wanted to go back to no measurement at all, or advocated unworkably complex alternatives. Sensible people made slow but steady progress, mostly because the others either crashed or never got anywhere in the first place.