Fat Tails, Expected Shortfall

No matter how you slice it, there is no escaping tail events in investing. It is the nature of the beast and every attempt you make eliminating the risk results in you giving up a significant portion of your returns. But given two investment opportunities, how do you go about figuring out which one is more susceptible to tail events?

Expected shortfall (ES) is a risk measure that can be used to estimate the loss during tail-events. The “expected shortfall at q% level” is the expected return on the portfolio in the worst q% of cases. ES estimates the risk of an investment in a conservative way, focusing on the less profitable outcomes. For high values of q it ignores the most profitable but unlikely possibilities, while for small values of q it focuses on the worst losses. Typically q is 5% and in formulae, p (= 100% – q) is often used as a substitute.

ES of Weekly Returns

Here’s a dilemma that most investors face: Mid-caps have given higher returns in the past compared to large-caps. But, how do their tail-risks compare?

Turns out, ES of the NIFTY MIDCAP 150 TR index is -6.73% vs. NIFTY 50 TR’s -5.65%. This is how much an investor would have lost in the worst 5% of weeks since 2011.


In our previous post, we showed how strata-sampling can be used to make sure that you don’t end up ignoring tail-risk in your simulations. By definition, tail-events are rare. So, the differences are subtle.

Tactical Allocation

Reducing tail-risk is one of the biggest draws of tactical allocation. Anything that reduces deep drawdowns has the effect of keeping investors faithful to their investment process.

One way to setup a tactical allocation strategy is to use a Simple Moving Average (SMA) to decide between equity and bond allocations. Different SMA look-back periods will result in different levels of risk and reward. From an ES point of view, here’s how things for NIFTY shakes out:

Since 1999
Since 2010

Using an SMA and re-balancing weekly significantly reduces tail-risk.

How far back should you go?

The problem with tail-events is that there aren’t enough of them to build an effective model. There’s always a temptation to use as much data as possible so that these events find sufficient representation. However, markets evolve, regulatory structures change and past data stop being representative.

For example, if you run a tactical allocation back-test with all the data that is available, you’ll conclude that shorter the SMA, the better:

However, if you remove 2008 and its aftermath and look only are the data from 2011 onward, you get a different picture:

While metrics like ES and strategies like SMA are useful, the data that they are presented will give different results based on the regime that they are drawn from.

Risk management is a continuous process and cannot be reduced to single number.

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