Folio Management | Fat Tails And Kurtosis - A Graphical Explanation

24th Sep '17 by Nick Ray

3 comments 5042 reads

If you have read any of Taleb's books you will be aware that stock returns have "fat tails" and that there are "black swan" events which make investing much more dangerous than simple models would suggest. Talk of "black swans" seems to suggest that these unlikely events are very rare, but in fact lesser "dark grey" swans are occurring all the time. In this post I hope to use graphs to make this effect easier to appreciate.

What do fat tails look like in a real example?

Let's have a look at the price action of Rio Tinto (LON:RIO) from 2007 until now. (Right-click: view image to see a bigger version)

Underneath the price chart I have computed the "daily returns" - that is to say the (price today)/(price yesterday) for each day. For the human eye the normal price chart is much easier to understand than the returns chart which just looks like noise.

If you look at this chart of returns (the blue "noise") you can see that somewhere around November 2008 there was a very big fall (over -40%) in a single day, but mostly the daily moves are very much less.

It's impossible to really see any other patterns from the time-series plot, so let's look at these returns as a histogram instead.

We no longer care about the order in time that they occurred - just how often moves of a certain size happened. That one massive fall has skewed the histogram across to the right. But mostly the returns form the familiar shape of a bell curve similar to a Gaussian (Normal) distribution. It is natural to assume the distribution is basically close to Gaussian, with any deviation due to simple "experimental error" but is this true? (Hint: no - as we shall see!)

Superimposed on the histogram are two attempts to fit a Gaussian (Normal) distribution. You can see from the coloured labels in the top right the values of mean and standard deviation (sd) that were used to draw these two curves. The blue curve is based on the values for mean and standard deviation computed using the standard formulae. The red curve however was computed using a different technique which we will discuss in more…

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gus 1065

24th Sep '17

Morning Ray.

Very interesting discussion of the phenomena. I am no student of statistics (and what little studying I did was more than 30 years ago) but at risk of embarrassment through ignorance, here are a few "lay" observations.

For a simple investor who takes only a basic long or cash strategy, the distribution of possible returns is skewed in that the maximum loss on any position will be 100% (the original investment) but the possible gain is potentially infinite (or at least significantly larger than 100%) so as to make a high kurtosis (leptokurtic) investment more attractive. (Possibly a theoretical vindication of the story stock value investor buying a wide spread of penny shares).

Presumably the holding period is an important real world variable in the analysis. If one holds a stock for a very long period (say 30 years), presumably the negative effect of the occasional black swan event (a one day precipitous price drop a la Crawshaw (LON:CRAW) ) is diluted by the size of the sample suggesting kurtosis risk is mitigated by a buy and hold approach. Alternatively, if one took a very short term view, successful intra day trading in and out of a position (at an extreme having minimal open holdings at the time of the possible BS event) could reduce the exposure to incurring a negative tail. (I guess at an extreme this is tantamount to being an arbitrage trader which is impractical for most PIs due to transaction expenses and un-level playing field advantage of the professionals).

If we accept the risk of kurtosis (s**t happens), is one possible mitigant to have either a widely dispersed portfolio of small positions or ideally a portfolio of positions where the impact of potential black swan events are expected to have opposite effects. Given the unpredictability of such events, I accept it would be very difficult to construct the latter other than by maybe combining long and short positions in the same investment, at which point I guess your investment return simply becomes the dividend yield minus the transaction costs/cost of carry.

I think your idea of using options is theoretically attractive but as you infer given the significant cost component (both counter party profit and the wasting nature of the time value of an option) is most likely impractical. It might add value in the short term if one buys protection at a time of "bad fat tail" risk but by their very nature Black Swan events are unpredictable making such an approach most likely imperfect. (Also bear in mind that when there is a "knowable" high risk this usually translates into higher option/ insurance prices).

All in all, I guess it's one more thing to be aware of but not lose to much sleep over. Add it to the list of life's unavoidables, death, taxes and fat tail kurtosis risk.

Gus.

Nick Ray

Hi Gus,

Your comment about the downside being limited to -100% is absolutely right. The limited liability is a huge benefit. I reckon that any stop-loss you might set to avoid Black Swan style losses would need to be around the -50% point or lower anyway and frankly the "free" stop at -100% is probably good enough. So in truth, when individual stocks fall to zero this need not be a concern.

The real threat then is when all stocks fall together. You've carefully chosen stocks with low covariance (i.e. they tend to move in different directions at different times) and suddenly BANG! they all drop precipitously together.

As for the holding period argument (holding for along time dilutes the effect of Black Swan events) the problem is that this is exactly what the problem with kurtosis is. If we had a Gaussian distribution then the more samples (i.e. longer holding time) the closer the performance moves to what the theory predicts. (Slightly modified by the fact that the mean and sd of a stock are not going to stay the same for many years.) But fat tails can mean that the mean and sd do not converge. This is a rather subtle mathematical idea but the simple version is: things don't become more stable with time. Of course, very extreme events such as the Wall Street crash in the 1920s are still very rare - but just more common than they should be and common enough to prevent convergence.

On the other hand there does seem to be an invisible hand which makes sure that over long enough periods of time there is a positive trend. This of course is basically because the market is simply a reflection of the world or national economy and there is always going to be positive pressure over long enough periods. However, in the long run we are all dead. (as a wiser head once said.)

I threw in the thoughts about options because there seems to be very little interest in them in the UK but they are very popular in the US. On the face of it they look to be more appealing than spreadbets and a useful tool for portfolio protection but I really am not sure.

The other method, which you hinted at, is creating a "market neutral" portfolio by trying to balance out the beta component altogether. I think such ideas were the genesis for the original hedge funds which allowed themselves to go short and long specifically to try to return "absolute" returns". But it has proved very much more difficult to do than it sounds. (Which makes sense to me. I don't think you can reliably cancel out the overall behaviour of the economy you are embedded in. It would create a contradiction in the maths of the economy and some unexpected effect (such as counter-party risk) would pop up and destroy the singularity you've created.)

What I found interesting about these graphs though is that I used to think of "Black Swans" as just those very rare events you listed under "life's unavoidables". Something far away that will probably happen but maybe I'll get lucky, etc etc.

But the reality is that dark grey swans are permanently happening all the time. Rare events are far too common compared with the Gaussian prediction and that effect starts with mildly rare events being just a little bit more common. I'm knee-deep in smallish black swans; occasionally meet a medium black swan; and once in a long while a huge black swan comes along and everyone says "How on earth did that happen?" when black swans are everywhere hiding in plain sight!

Effortless Cool

Some years ago, an institution pitched an equity tracker fund to me that purported to resolve the fat tail issue. Basically, it bought downside protection at -2% on a monthly basis, and sold upside to pay for this on a cost neutral basis. The remaining potential upside gain was always above 2% because of the positive mean of the distribution of monthly returns. The cost-neutral methodology led to this being sold as "free" protection.

When I analysed it, however, of course this was anything but free - the cost was the favourable upside kurtosis. I found that a mix of, say, 75% equities, 25% gilts (I actually can't remember the exact percentages) produced a similar risk profile and almost guaranteed better returns over the long term.

What I took from this is that it is best to just accept equities as what they are - fat tails and all - rather than try and change their risk characteristics. If you are concerned about equity kurtosis, the best approach is to temper it through putting a proportion of your portfolio into lower risk asset classes (gilts and/or cash).

Fat tails and kurtosis - a graphical explanation

What do fat tails look like in a real example?

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Nick Ray

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