VaR vs ES – A pictorial representation

In this article, I try to pictorially represent two methods used to “Quantify” Market Risk: One that has served the finance industry arguably well till now (VaR) and the other one that has been appointed its successor as per the FRTB “Commandments” laid down by the Basel Committee (ES).

Market Risk Explanation is already covered in my previous article.

Let’s take them one by one:

VaR

Value at Risk – As the name suggests, sole purpose to use this method is to answer the question: What Value of a given portfolio is at risk? Yes that’s it!

But as we know that future cannot be predicted with 100% surety (Unless you are Sachin Tendulkar), and the question raised above asks for predicting the future, we need to tweak VaR a little bit.

Incorporating the uncertainty of the future, question becomes: For a given confidence level, what value of a given portfolio is at risk?

However, this question still doesn’t seem present the complete picture. What about the timeline we are taking into consideration? Isn’t that important?

Yeah well… the final question we want answer for is:

Over a defined time period, for a given confidence level, what value of a given portfolio is at risk?

Phew...The question just mentioned is indeed the actual definition of VaR.

So if I say that for a given portfolio, one-week VaR at 99% confidence interval is $10 million, it means that there is only 1% probability that the Value of portfolio will drop by more than $10 million.

There are various method that we use to calculate VaR, namely:

1. Variance-CoVariance method
2. Historical Simulation
3. Monte Carlo Simulation

(Won’t explain these methods here or else this article would become too technical to digest)

Main Drawback:

There’s one huge drawback of using VaR. It says that one can be 99% sure that value won’t fall below the VaR calculated at 99% Confidence level. It, however, does not say anything what could that loss be if things go bad (the 1% probability of loss). In our example, that could vary from $10 million to $100 million or even $1 billion. One just doesn’t know. *Face-palm situation*

This drawback lays a foundation for Expected Shortfall

ES

ES or Expected Shortfall or Conditional VaR method aims to mitigate the main drawback of VaR methodology.

ES is like a nerdy student who just doesn’t ignore the worst this world has to offer. It basically takes VaR one step further by answering the question: What could the loss be if things go bad?

So if I say that ES at 97.5% confidence interval is $20 million, then it means that in worst 2.5% cases, average loss in value of a given portfolio would be $20 million.

Bottom-line: It considers the worst and hence it is the best ;)

Pictorial Representation:

Consider a man standing on a cliff. VaR answers that with a given confidence, how many steps the man can take before falling off the edge.

However, it does not tell that if the man falls, how far the ground is!

Expected Shortfall on the other hand answers that if the man is to fall, how far the ground is i.e. the height of the cliff.

PS: Examples are lame but I do believe they perform the task of explaining the major difference efficiently. In explaining these methodologies, I have omitted certain facts/assumptions which I thought would be too technical. Sole intent is to contrast both the methodologies while providing explanation in a simplified way.

Reference:

1. http://people.stern.nyu.edu/adamodar/pdfiles/papers/VAR.pdf
2. http://stat.wharton.upenn.edu/~steele/Courses/434/434Context/RiskManagement/VaRHistlory.pdf
3. https://en.wikipedia.org/wiki/Expected_shortfall
4. https://en.wikipedia.org/wiki/Value_at_risk
5. http://www.risk.net/risk-magazine/technical-paper/1506669/var-versus-expected-shortfall