VaR and subadditvity

Could somebody please explain why VaR is not subaddtive and why this matters?

I can't think of an example where the VaR from two risks together is greater than from the sum of the individual VaRs, providing those risks don't interact. Also why is TVaR subaddtive, but VaR not?

I'm not particularly interested in the mathematical argument. I'm more interested in the intuition and a realistic example.

If a risk measure is subaddtive, how does it actually help? As far as I can see to actually work out the total risk you'll always need some knowledge/make assumptions about the risk dependency structure to aggregate the risks, regardless of whether or not the risk metric is subaddtive.

Thanks.

 

A simple example:

Suppose that your VaR is at the 99.5th percentile. Consider two identical and independent loss distributions which take the value 100 0.3% of the time and 0 otherwise.

The VaR of each distribution is 0, as at the 99.5th percentile of the loss distribution we expect to have no loss.

The sum of the distributions takes the value 0 99.4009% of the time (97%*97%) and is non-zero the rest of the time. So in this case the sum has a VaR which is strictly greater than 0, but because the individual distributions have VaR of 0, the sum of the VaRs is still 0.

Hope this answers your question, basically when you have a very skew distribution the VaR fails subadditivity.

It is important to realise that VaR is not always subadditive because often people take two VaR and add them together and say "well this is an upper bound". Since this isn't true it can cause wrong decisions to be made.

 

Thanks Simon and Shillington that clears things up. I hadn't seen Simon's final comment on the linked thread when I originally posted my question.

 

Hello, going back to the VaR measure and its disadvantages, the ActEd notes state one of the disadvantages as:

"if used in regulation it may encourage "herding" thereby increasing systemic risk"

- can someone explain this concept and/ or give an example?

Thanks in advance

 

This is similar to the general Standard formulas in Solvency II and Basel;-

If too many companies are using VAR and the model handles a particular branch of business favourably (think of a situation where Risk is calculated at a product level) > capital for that business will be small > RAROC will seem high > other companies will be incentivised to write the business > overexposure of entire industry to the specific risk associated with that product (say 30% of products are MBS) > if the risk materialises everyone goes down!

 

Thanks, Edwin. It makes total sense.

Can I ask one more question on the advantages of VaR?

The ActEd notes mentions: "its inherent allowance for the way in which different risks interact to cause losses".....

Can you illustrate this with an example?

 

Hi RobyWi, you were looking for an example. See attached xls file, can someone please confirm my TVarX and TVar(X+Y).

 

Does this simply mean that VaR can be expressed for several risk altogether as one VaR amount and we would value this bandle of risks with one calculation? That would assume that they all follow the same distribution.

Am I thinking along the correct lines?

 

Hi Edwin,

For your example, I agree with the combined probability. However,

1. Isn't VaR_X@97% = 1, but VaR_X@99.5% should be 2?
2. Similarly VaR_Y@99.5% = 2.5.
3. VaR_X+Y @ 99.5% = 2.5.

So it does't seem to prove at 99.5%. But at say 95%, it works.

Happy to be corrected.

 

The spreadsheet actually refers to VAR at a 95% (thanks), as you go into the tails of the distribution, VAR should be subadditive since you are increasing the level of your confidence level.

For example for VAR to fail subadditivity at a 100% CI, then you have to lose more than you have.

See the spreadsheet attached, VAR_X@100% = 2, VAR_Y@100% = 2.5, VAR_(X+Y)@100% = 4.5, it cannot exceed 4.5 since you will have to lose more than your portfolio has!

 

Hi guys, I don't think that anyone responded. I would be very thankful for an example of one of the arguments on advantages of VaR:

"VaR also enables the aggregation of risks taking account of the ways in which risk factors are associated with each other."

I am struggling to see how you can use VaR for aggregation of risks, whereas Var is not a subadditive measure.....

 

Some time ago (I am not sure if ES had been approved for Basel yet) VAR was used in Basel for assessing Market risk collecting Values of Market factors and using Historical simulation.

ES was available in the early 1990's and it didn't win out over VaR...
The big reasons VaR triumphed as a risk measure is that it was backtestable without assumptions, it was faster to compute, it used only information from normal markets and it was instantly graspable by all constituencies. The people who liked ES couldn't demonstrate any practical difference, and the theoretic advantage of coherency was much less important than the advantages of VaR. However, the ES debate did force VaR people to incorporate precautions such as stress testing to ensure that tail was not ignored.

(P.S Someone finally made people open their ears after years of continuing to repeat, "but VaR is not subadditive.")