A practical look at VaR and TVaR

[Alastair_in_SA]

Chapter 10 - page 17 & 18.

1. "[VaR] can provide a consistent and comparable measure of risk across all instruments, products, trading desks and business lines."

I am just wondering how this interacts with VaR not being sub-additive? So we can compare that VaR between trading desks, for example, but really that does not meant much on an enterprise level?

2. "The ratio of TVaR to VaR can be used as an indication of the skewness of a distribution."

OK, this makes sense in theory, but unless you are using an empirical approach does this statement have any value? I say this because in both a parametric and stochastic approach you are defining the distribution of losses / returns. And so this ratio will be artificially implied by your choice of distributions. Am I missing something deeper here?

Thanks a lot
Alastair

 

[Shillington]

Chapter 10 - page 17 & 18.

1. "[VaR] can provide a consistent and comparable measure of risk across all instruments, products, trading desks and business lines."

I am just wondering how this interacts with VaR not being sub-additive? So we can compare that VaR between trading desks, for example, but really that does not meant much on an enterprise level?

2. "The ratio of TVaR to VaR can be used as an indication of the skewness of a distribution."

OK, this makes sense in theory, but unless you are using an empirical approach does this statement have any value? I say this because in both a parametric and stochastic approach you are defining the distribution of losses / returns. And so this ratio will be artificially implied by your choice of distributions. Am I missing something deeper here?

Thanks a lot
Alastair

I haven't studied ST9 but work in Capital Modelling.

a.) You're right. Each trading desk will have a distribution of losses/returns. We can then take the VaR for each desk and compare the results. The statement doesn't mention anything along the lines of adding up the VaR just simply that you can directly compare the VaR for different distributions.

b.) I think what you're saying is that if we define the distribution in the first place, what benefit is there in calculating this ratio if we already know it/decided what it was in our definition in the first place?

A lot of the time when you are defining these distributions you will apply dependencies between distributions to build up your eventual loss distribution. Analytic calculation of percentiles of say Gumbel dependent distributions is not an easy task. So although it is "defined" it's not always obvious what it will be. Even if no dependencies are applied, summing up 50 Pareto random variables and then calculating the TVaR is not easy (disclaimed: I haven't tried but assume that it's non-trivial).

 

[Alastair_in_SA]

Cool, thanks.

So even though VaR does not display subadditivity it is still used to compare different products, instruments, business lines etc.

And even if we have "defined" the probability distributions for the various risks affecting the loss distribution as well as their copulas, it is actually not obvious what the end result will be (especially as you ramp up the number of risk factors influencing the loss distribution). I guess in addition to this, if you were using stochastic methods, there will always be sampling variation which would make the ratio unpredictable to some extent as well.

Thanks again!

 

[Simon James]

VaR is not coherent as it can fail the subadditivity criteria.

However, for normally distributed variables, VaR is subadditive. Hence for many practical purposes we can treat VaR as subadditive and hence use it to "add across" risks/portfolios etc.

We do need to be careful however with non-normal distributions, especially very low likelihood/high impact events. For example if a single McDonalds has a 1 in a 10,000 chance of burning down, the risk would be efectively ignored for a single restaurant under even a 99.9% VaR, and would not figure if we simply summed across the VaRs for the restaurants. However with over 30,000 restaurants the whole portfolio VaR would include this risk, so the portfolio VaR is greater than the sum of the individual VaRs.

 

[Viki2010]

VaR is not coherent as it can fail the subadditivity criteria.

However, for normally distributed variables, VaR is subadditive. Hence for many practical purposes we can treat VaR as subadditive and hence use it to "add across" risks/portfolios etc.

We do need to be careful however with non-normal distributions, especially very low likelihood/high impact events. For example if a single McDonalds has a 1 in a 10,000 chance of burning down, the risk would be efectively ignored for a single restaurant under even a 99.9% VaR, and would not figure if we simply summed across the VaRs for the restaurants. However with over 30,000 restaurants the whole portfolio VaR would include this risk, so the portfolio VaR is greater than the sum of the individual VaRs.

Hello Simon, looking at the solution of last question on the last exam (Sept 2014) I can infer that VaR is subadditive when the risks are correlated, so not only when we are dealing with Normally distributed variables, but for any risks being correlated....can you confirm?

 

[Shillington]

Hello Simon, looking at the solution of last question on the last exam (Sept 2014) I can infer that VaR is subadditive when the risks are correlated, so not only when we are dealing with Normally distributed variables, but for any risks being correlated....can you confirm?

VaR is not subadditive.

My example (in the other thread on VaR and subadditivity) shows that even when the two risks are 100% correlated it still fails the subadditivity condition.

 

[Viki2010]

VaR is not subadditive.

My example (in the other thread on VaR and subadditivity) shows that even when the two risks are 100% correlated it still fails the subadditivity condition.

I know that VaR is not subadditive, but when you have two risks which are correlated, you can sum their VaRs.

 

[Shillington]

I know that VaR is not subadditive, but when you have two risks which are correlated, you can sum their VaRs.

No, you cannot.

Look at the example I gave in the other Thread.

In that example the two risks are 100% correlated and Var(X+Y)>VaR(X)+VaR(Y)

Also, be careful when saying "correlated". For two risks to be "correlated" all people usually mean is that Corr(X, Y)>0. This is just a statistic of two variables though and it doesn't mean much. There are lots of examples you can find of variables which have Corr(X, Y)>0 but they have very whacky behaviour which doesn't look anything like normal correlations.

 

[Viki2010]

No, you cannot.

Look at the example I gave in the other Thread.

In that example the two risks are 100% correlated and Var(X+Y)>VaR(X)+VaR(Y)

Also, be careful when saying "correlated". For two risks to be "correlated" all people usually mean is that Corr(X, Y)>0. This is just a statistic of two variables though and it doesn't mean much. There are lots of examples you can find of variables which have Corr(X, Y)>0 but they have very whacky behaviour which doesn't look anything like normal correlations.

I take your point on the correlated vs. 100% correlated.

Let me re-phrase, when two risks are 100% correlated, the overall VaR for both of these risks is the sum of indivudual VaRs for each of the risks.

 

[Shillington]

I take your point on the correlated vs. 100% correlated.

Let me re-phrase, when two risks are 100% correlated, the overall VaR for both of these risks is the sum of indivudual VaRs for each of the risks.

When you say "for both of these risks" do you mean the sum?

If so, then it's still not correct.

Have you read the example I gave in the other threat?

 

[td290]

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.

Shillington, are you talking about this example? If so, then in your example the risks are independent and therefore 0% correlated. Viki is talking about 100% correlated risks and I agree that in this case the VaR of the sum is the sum of the VaRs.

 

[Shillington]

Shillington, are you talking about this example? If so, then in your example the risks are independent and therefore 0% correlated. Viki is talking about 100% correlated risks and I agree that in this case the VaR of the sum is the sum of the VaRs.

Yep. You're right. I don't know what I was thinking last night.

Sorry Viki.

 

[Viki2010]

Hello, I would like to ask a question on the calculation methods. For VaR we can drop miu (average parameter) for calculations where VaR is considered over short time durations.

Does the same apply to TVaR and ES?

I did not find an answer in ActEd or Sweeting.

 

[Edwin]

Hello, I would like to ask a question on the calculation methods. For VaR we can drop miu (average parameter) for calculations where VaR is considered over short time durations.

Does the same apply to TVaR and ES?

I did not find an answer in ActEd or Sweeting.

In particular if you are looking at stock returns and the 1 day VaR, the same applies to ES and TVaR (I think), remember these are just linear combinations of VaR at varying confidence levels.