Volatility and Rare Events
Volatility is undoubtedly a highly important indicator for the riskiness of an investment position. But could it be that the way volatility is measured might overestimate the impact of volatility respectively underestimate the probability of high risk rare events?
Rare events are located in the tails of data distributions. The frequency and the extent of rare events, hence the degree of risk, defines the scale of the tails. The fatter the tail, the higher the risk.
In order to understand this, we focus in this article on data which follow the location-scale family of distributions. This is -to some extent- a milder form, when it comes to the formation of tail risk compared to more extreme distribution families, such as the subexponential class of distributions or even power law classes.
It is to be noted — I am just a messenger here, i.e. a messenger with some own thoughts on the topic. Main input comes from N.N. Taleb’s latest book: Statistical Consequences of Fat Tails, which I would highly recommend to read.
Volatility Measures
First, let’s put the attention to two representatives of the location-scale family, i.e. the normal distribution and the Student’s t distribution. Here is a graphical comparison between those two distributions:
As can be seen, the Student’s t distribution has a more distinguished tail zone. And from the tail zone the heavier risk is coming from.
A familiar metrics for measuring volatility is the standard deviation. Formally, the standard deviation calculates the average distance (i.e. the scale) from the centre of the data sample (i.e. the location or the mean of the data sample, but could also be the median of the data sample). So:
variance = sum((x_i — mean(x))^2) / n
standard deviation = sqrt(variance)
with
- x_i … data point in the data sample
- mean(x) … average value of the data sample
- n … size of the data sample
Accordingly, compared to a (standardised) normal distribution the standard deviation (sd) of a Student t distribution is higher at some point and increases with the lowering of the degrees of freedom (df), i.e. the lower the df the fatter the tails:
Summarising, the closer a data sample is following a distribution with more accentuated tail areas, the higher the risk which is shown in higher volatility indicators. And with the growing of the tail, the volatility indicator in terms of standard deviation is starting to rise significantly. Here, the standard deviation for the Student’s t distribution with decreasing degrees of freedom as an example:
So far, so good…
Fat Tail Ratio
Let’s move to a different risk metrics: mean absolute deviation (mad).
Formally, mean absolute deviation is calculated as follows:
mad = abs(x_i — mean(x)) / n
- x_i … data point in the data sample
- mean(x) … average value of the data sample
- n … size of the data sample
The same volatility indication based on mean absolute deviation for Student’s t distribution with different degrees of freedom would look as follows:
It shows the same behaviour but on a much lower scale. The reason seems technical as the standard deviation is calculated as the sum of the squared differences between the value of each data point and the average value of its data sample while the mean absolute deviation represents the sum of these differences just in absolute values.
But what it means in practical terms is that within the framework of the standard deviation larger differences get much more weight than smaller differences. Hence, the standard deviation shows much faster a more significant volatility than this would be the case with the mean absolute deviation.
The ratio between standard deviation and mean absolute deviation can therefore be seen as some kind of fat tail indicator in a data distribution. For the normal distribution this ratio amounts to a mere 1.25 (as there is hardly a fat tail). But staying with the Student’s t distribution this ratio moves up steadily with decreasing degrees of freedom:
Volatility versus Rare Events
Let’s have a look at the Student’s t distribution with 3 different degrees of freedom.
Data observations which follow this kind of distribution would have an increasing — in fact, a significantly increasing — fat tail zone. This is nicely revealed by the rising scale of the x-axis from left to right in the plot above. And based on our understanding represented by the table below, the volatility of those data sets is therefore increasing as well.
Though, at a closer look the volatility in the data set did not change so much. In each of the three data distributions, the majority of the observations is still close to the average value of the data sample (= central location of the data distribution).
This gets even more obvious when setting the 2.5th and the 97.5th quantile, i.e. 95% of the data observations are within the two dashed red lines in each of the data observations in the graph below.
What did change though — and this is very important — is the extended impact rare events can have in each of these situations. In any of these scenarios, rare events represent 5 % of the overall observations. But the heavier the tail, the heavier the impact a rare event may have. This is represented in the scaling up of the x-axis (from left to right) in the graph above.
In other words, founding the volatility metrics on the standard deviation alone leads to an overestimation of the volatility and an underestimation of the extent of rare events.
On the other hand, mean absolute deviation seems to better cover the volatility question but definitely underestimates the issue of rare events.
In case those rare events have a disastrous impact on the position of an investor… well, we face the potential looming of Black Swan events which is the practical implication of all this. And this is in no case covered by that risk metrics!
And it gets worse
The whole situation is further aggravated when the data observations are forming out fat tails while at the same time they are “peaking out” in the centre. What this means in practical terms can be shown with the frequency of daily price changes of a certain real estate share:
Almost 50 % of daily price changes are within a range of +/- 0.5 percent (= this dominant peak of the daily price change distribution in the graph above) and 75 % of all daily price changes are within +/- 1.5 %. So, in the majority of cases, almost nothing happens. In fact, there is little volatility most of the time.
Nevertheless, this stock is exposed to fat tail behaviour as the empiric time line shows that the range of potential daily price changes is between -31 % and +22 %! Again, the x-axis is scaling up.
How the data points (= daily price changes) are much more pooled around the central location of the data sample (= establishing a peak) and at which extent fat tails are established gets clear when this empiric distribution is compared to the shape of a normal distribution or the shape of a Student’s t distribution (with e.g. 4 degrees of freedom):
From a statistical point of view, these daily price changes could not be covered by a normal distribution or a t distribution as they have too low a peak (therefore a broader “distribution body”) and too small tails (which is especially true for the normal distribution). Therefore, the risk behaviour of this stock could not be approximated close enough for further (predictive) analytics.
How to deal with this kind of empiric data distribution, read here.
The standard deviation of this stock price sample amounts to 2 %. The mean absolute deviation equals 1.2 %. Hence, the fat tail ratio is already at around 1.7.
Given the empiric volatility, the standard deviation seems to be overriding for most of the circumstances but in no case covers — from a statistical point of view — what is to be expected when one of those rare events is occurring.
To underpin the dramatics here, I’d like to quote from an article found on the Nasdaq homepage:
“According to general statistical principles, a 4-sigma event is to be expected about every 31,560 days, or about 1 trading day in 126 years. And a 5-sigma event is to be expected every 3,483,046 days, or about 1 day every 13,932 years.”
What does 4-sigma or 5-sigma mean? 4-sigma (5-sigma) equals 4 times (5 times) the distance of a standard deviation from the centre (i.e. sample average respectively sample median).
Well, based on the empiric standard deviation of this stock price sample a 4-sigma event — which would equal a daily loss higher than 8 % — would occur every 229 days. A 5-sigma event, i.e. in our case making a daily loss higher than 10 %, would occur every 321 days. This is much, much more frequent than those 31,560 days respectively those 3,483,046 days as quoted in the Nasdaq article.
And it bluntly reveals that the calculated standard deviation of the data set hints too high a volatility on the one hand but is no measure at all for rare events with quite an impact.
More on 3-, 4- and 5-sigma events, read here.
Conclusion
Remember, here we are just talking about location-scale family distributions with some kind of a milder formation of fat tails compared to other, more extreme distribution families.
But as it goes and the more data are prone to showing extreme behaviour (expressed in data distributions with fatter tails), the less volatility metrics — especially in terms of standard deviation — is a reliable indicator for the riskiness of an (investment) position.
To put it in N.N. Taleb’s words:
The fatter the tails, the less the “body” matters for the moments (which become infinite, eventually).
And with moments, things like mean, standard deviation, skewness or kurtosis are meant. The previous chapter regarding daily share price changes gave a good notion on the troubles that come along with measuring the riskiness of a position when fat tails are forming out and the remaining body of the data distribution gets quite slim.
Nevertheless, already in this area of distribution classes as discussed here it seems that standard risk metrics is overestimating the volatility in a data sample but is heavily underestimating the extent of rare events.
Still not convinced?
Then have a look at the empiric distribution of 95 % Value-at-Risk positions of SVB Financial Group over time as of March 10, 2023:
SVB Financial Group is the holding company of Silicon Valley Bank which got closed by Californian regulators the same day due to significant losses in their bonds positions. As the chart shows, there was relatively low volatility in the Value-at-Risk positions (= grey part) and nothing prepared for the risk of shutting down the bank (= rare event; represented by the dashed red line).
References
Statistical Consequences of Fat Tails by Nassim Nicholas Taleb/ Stem Academic Press, 2020
How Many Sigmas Was the Flash Correction Plunge? by Kevin Cook published on nasdaq.com/articles / March 4, 2020
Quantifying Risk in Turbulent Times by Christian Schitton published on LinkedIn/ April 4, 2020
Risk Management — Keeping Up Appearances by Christian Schitton published in Analytics Vidhya/ September 30, 2021
