Fat tails in FX: how obese are they?

B. Hathi 4^th July 2011.

The term ‘fat tail’, when talking about distributions, is [mathematically] reserved for those distributions with infinite sigma (i.e. variance does not exist). Such a distribution – for example, a Cauchy distribution - is said to have fat tails and its tails only approach zero when the parameter being modelled approaches +/- infinity.


However in many fields and modelling talk, often the term ‘fat tail’ comes in as soon as a normal distribution is breached – and there is a good reason. If a Risk Practitioner models for daily occurrences properly, then he or she will find that, on rare occasions, their models are being breached and sometimes spectacularly. This leads many to panic and resort to using some form of fat tail distributions to cover the outliers (or outlying scenarios) . In the case of being too risk averse, one might find that their model is invalid for ‘normal’ daily occurrences.

I was wondering if using the Gaussian Kernel will cover the tail risk adequately – if nothing else, as a compromise between extreme caution (fat tail modelling) and outright risky calculations that result from calculations using the normal distribution. My approach to the problem is demonstrated by using the Swiss Franc’s valuation versus the US Dollar (Usd/Chf).

Figure 1. Shows the Swiss Franc’s daily spot prices from 1971 to present. Looking at the plot, you might wonder if some sort of autoregressive (or Box Jenkins), non linear model might be able to track  [a smoothed] moving average – this might be a topic for some other day.

Since valuation of all currencies are time series dependent – i.e. their value today depends [to some extent] on their price yesterday – there is little point in fitting statistical curves to Fig 1. However using the data in Fig 1, we calculate the daily percentage changes in Usd/Chf: a histogram for which is shown in Fig 2.

Figure 2. Histogram of Usd/Chf daily percentage variations. The zoomed insets on the left and right side of the main plot show the tails outside the mean +/- 3.sigma. Common VaR models assign mean +/- 1.65sigma as 95% confidence interval and +/- 2.33sigma as covering 99% of the variations in the underlying data.

If we were to fit a normal distribution to Fig 2 (which, by eye, is a perfectly normal-looking distribution), then the fit covers most cases…until you zoom in on the tails.

Figure 3. Normal distribution fit to daily percentage changes in Usd/Chf shows that some outliers exist as shown by the zoomed insets showing the tails. On [rare] occasions such a normal fit will be breached.

If on other hand we fit a GKDE (or Gaussian Kernel Density Estimator) to the data in Fig 2, we find much better coverage of the tail data.

Figure 4. Gaussian kernel fit to daily percentage changes in Usd/Chf shows that most outliers get covered satisfactorily as shown by the zoomed insets showing the tail coverage. Note, also the kurtosis is less pronounced cf Fig 3 (i.e. central cases also get a more accurate profile).

In summary, our underlying data set covers the period full period from major currencies free floatation and we can see that applying a Gaussian kernel does not harm the effort of estimating risk of daily movements. I estimate through my back-of-an-envelope calculation that a +/-3sigma normal fit will be out by approximately 2%  in its tail coverage compared to a kernel fit, while the VaR values (i.e. 2.33sigma for 99% coverage) will be out by 5 to 7% for most freely traded currencies.

Source | The Currency Forecasting Blog (CFB) | http://currency-forecasting.blogspot.com/2011/07/fat-tails-in-fx-how-obese-are-they.html