Which is better, ES or VaR?

November 30, 2024

Almost everyone insists that Expected Shortfall (ES) is superior to Value-at-Risk (VaR). But is that truly the case?

I've been involved in a somewhat chaotic debate about which is better: ES or VaR.

The argument seems to boil down to two main points. The first is whether subadditivity is desirable, which would favour ES, as argued by Artzner et al. (1999).

The second, more curious claim is that ES is estimated more accurately than VaR.

It is easy to show that at the 99% probability level, ES estimates are less accurate than those of VaR. My paper, Why Risk Is So Hard to Measure, demonstrates this, as do others. I'll soon post more analysis and code to illustrate that point. But I don't think this result is controversial.

However, the argument is usually presented differently.

"While it's correct that 99% VaR is estimated more accurately than 99% ES, that's not relevant. What matters is that 97.5% ES is more accurate than 99% VaR."

For context, 99% VaR was used in Basel II, and 97.5% ES is used in Basel III.

Some authors correctly find that 97.5% ES is approximately the same as 99% VaR. Our paper shows this theoretically, showing that approximately: $$ ES\approx \frac{ \alpha }{ \alpha-1 }VaR $$ where \(alpha\) is the tail index.

In other words, because 97.5% ES is basically the same as 99% VaR and is more accurately estimated — both of which are true — therefore the 97.5% ES is preferred.

I find this argument curious and wrong.

These measures are measuring different risk at different frequencies. A 99% measure corresponds to an event that happens once every five months (assuming a daily holding period, which is usual in these discussions). The 97.5% event happens less than once every two months.

Arguing that we should prefer ES because the 97.5% ES is basically the same as the 99% VaR while being more accurately estimated is akin to claiming that an event occurring once every two months is the same as one occurring once every five months. That's just wrong.

The only way to maintain the argument that 97.5% ES should be chosen over 99% VaR is to ignore the underlying mathematics and reality and say:

"here are two numbers that are approximately the same, but one is more accurately estimated than the other. I prefer the one more accurately estimated. I don't care what the numbers represent."

I suspect that's exactly what's happening. But I wouldn't want any financial institution I do business with to take such a cavalier attitude to risk management, and I'm not comfortable with important regulations like Basel III being based on this.

En fin

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