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tag : Chief Financial Officer (CFO), Chief Risk Officer (CRO), Conditional VaR, Expected Shortfall, Market Risk, P&L, Risk Management Policy, Volatility

by : Nitin Gupta

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First, the bad news – Value at Risk (VaR) of a position / portfolio just gives the maximum loss you can have, with a certain confidence. But it doesn’t tell you what your position / portfolio could lose beyond that confidence. If you are the one managing risks in your organization using the VaR framework, chances are you would already know this. So the real question is, is there and extension of Commodity Risk Management beyond VaR that solves this problem? Expected Shortfall (ES) is part of the answer to this question. Let us take a look at the Issues with using only VaR and how Expected Shortfall can help us overcome those issues to a certain extent.

Let’s start by putting some numbers around this problem. Consider a simple, 1 commodity portfolio below:

The portfolio is currently worth $ 250 mn with a M2M P&L of $ 4.1 mn. However, the VaR, which is calculated for 1-day Holding period and 95 percentile confidence level, is greater than M2M P&L, and is $ 4.5 mn. This means that on 19 out of 20 days the position is not expected to have a M2M Loss exceeding $ 4.5 mn (which is fairly bad by itself, since it can easily wipe out the entire M2M P&L !). But on 1 out of 20 days (on an average) this M2M loss will exceed that figure. But, to what extent can this loss be above $ 4.5 mn? This is a question that VaR does not answer. On that 1 very bad day, the loss could be anything above $4.5 mn.

Now, while it is comforting (and in compliance with regulations for a lot of industries) to know what portfolio’s worst loss could be on 19 out of 20 days (on an average), it is severely discomforting to NOT know how much could the portfolio lose on that 1 very bad day ! Could it be $ 5mn, or 10, or 50? It makes a huge difference, right?

And now for some good news, this is where Expected Shortfall (ES) comes to our rescue. ES, also sometimes known as Conditional VaR or Expected Tail loss, tells us how big the number could be on that 1 very bad day. To represent it mathematically,

where,

X is the random variable of loss and

α is the confidence percentile, 0<α<1, (α=0.05 in our case, since we are using 95^{th} percentile for VaR)

The above equation is just a complicated way of saying that ES for a position at any confidence level is an expected to be greater than or equal to the VaR for that position at that confidence level !

And here is how the ES is calculated:

This equation too, is just a complicated way of saying that ES is an average of VaRs, between 0 and α confidence levels !

With this knowledge, let’s look at the same table again:

Now, with this new knowledge it is easier for any Risk Manager to commit about the loss on that 1 very bad day, which is $ 7.8 mn. It might be less or more for the organization, depending upon several factors including risk appetite, but this knowledge surely *extends* our comfort to an area where VaR doesn’t reach.

There are still a lot of questions left unanswered though, like:

- How can I be sure that the worst case loss (under any circumstance), will not exceed ES?
- How do I determine whether the ES amount is less or more for my organization?
- How does knowledge of ES impact my Risk Policy?

These and many other such questions will be dealt with in our future posts. So keep coming back !

If you’d like to have a more detailed view of Expected Shortfall and its usage, you might want to read our whitepaper on “Back-testing Expected Shortfall”.

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