Fat-tailed models need to adapt well in normal market conditions and differentiate between asset classes. Boryana Racheva-Iotova compares fat-tailed models with GARCH based on stable Paretian distributions, t-distributions and extreme value theory

To better anticipate and prepare for large downward market moves, we must first understand the anatomy of a market crash. Viewed through the high-powered lens of a fat-tailed model, we are able to isolate the two key components - tail-thickness parameter and volatility. The tail-thickness parameter controls the probability of an isolated extreme event, such as a very large loss in one day. The volatility concerns clusters of large returns - both positive and negative.

Anatomy of crash
A crisis evolves with the onset of some isolated, larger-than-usual movements, similar to the foreshocks of an earthquake. At this time, the overall volatility does not increase because it is a period of unawareness. But foreshocks, even small ones, cause the tail-thickness parameter to go up, and this is the first indication of a possible market crash. The first few foreshocks increase the nervousness of the market and bring the second component into place - increased volatility. This is when market participants start asking questions. They don’t know the answers, so they push volatility up by starting to price this uncertainty into their trading activity. (For markets that are not the primary location of the crisis, it is typical for the tail-thickness increase to be minimal, or possibly lagging, with the transition to higher volatility first being observed.)

As the crash picks up steam, we observe clusters of extreme events, which are a joint effect of increased volatility and high tail thickness. When the crash starts to diminish, the tail thickness usually diminishes faster than the volatility. The gradual increase in tail thickness started about a year before September 2008, and is clearly visible on the generalised autoregressive conditional heteroskedasticity (GARCH) indicator for the Dow Jones Industrial Average (DJIA).

However, the fact that the crash originated in the US and that the US stock market is the most liquid and forward looking contributes to this result. In Europe, the crash is best captured by volatility, and there is not a similar gradual increase of the tail thickness before the crash, perhaps because the market nervousness increased due to US events before the Europe-intrinsic events occurred, and European markets individually are not as large or liquid as the US market.

So, how is the patient doing today? On 19 January 2010, the DJIA hit a recent high of 10,725 and then traded to an intraday low of 9,908 on 8 February. The FTSE 100 started going down on 11 January, with its recent low on 5 February. These drops were both approximately 7.5%. Using our anatomy lesson, let us compare the VaR, expected tail loss (ETL), volatility and tail thickness of the current situation with what we were seeing before the crisis.

Compared with previous regimes, particularly the crisis, tail thickness is not significant for either the DJIA or FTSE 100. But volatility has increased, probably reflecting the increased cautiousness of the markets under which they price in the risk premium of several unknowns (that is, Greece’s difficulties, questions about the strength of the recovery in various economies and the potential for higher interest rates). The fat tail indicator (FTI) on the DJIA is less than 50% of its (average) reading as compared with the pre-crisis period. Unless we see an increase in the FTI, the potential for a series of continued large losses in a timeframe of a month is less probable. The FTI on the FTSE 100, while relatively low, does warrant close monitoring to see if it or any other European market is exhibiting key symptoms, indicating that problems are building.

Second opinions
The above analysis highlights a number of conditions that fat-tailed models must account for: clustering of volatility (large price changes tend to be followed by large price changes and small price changes tend to be followed by small price changes); temporal behaviour of tail thickness (the probability of extreme price movements is smaller in regular markets and much larger in turbulent markets); and cross-asset tail thickness (tail thickness varies from asset to asset, as seen between the FTSE and DJIA). The FTI demonstrated here and the associated fat-tailed VaR and fat-tailed ETL risk estimates are only possible when the assumption of normality is discarded and replaced with a Stable Paretian distribution model that accounts for fat tails and skewness in asset returns, as well as the three conditions mentioned above. Stable Paretian distributions are attractive because their tails decay more slowly than the tails of the normal distribution and, therefore, they better describe the extreme events present in the data.

A good fat-tailed model collapses to a normal model during normal market periods, but quickly reacts and starts showing increasing risk with the first signs of upcoming market turmoil. FinAnalytica’s commercial deployment is based on stable Paretian distributions, but two other widely discussed fat-tailed approaches are the classical Student’s t-model and extreme value theory (EVT).

The Student’s t-distribution is the most commonly used alternative to the normal distribution for modelling asset returns. Like the normal distribution, Student’s t-densities are symmetric with a single peak density function, but they are more peaked around the centre and have fatter tails. While these properties make them acceptable for modelling asset returns, the primary reasons for their widespread use are easy implementation and their wide availability in popular statistical software packages. The most significant limitation of the t-model is that the residuals, after cleaning the volatility clustering and the auto-regression, are assumed to have a multivariable t-distribution with the degrees of freedom parameter usually fixed to four or five for all assets, irrespective of their type and of the time period under consideration, and this does not allow for a smooth transition between Gaussian and fat-tailed data. Risk is thereby significantly overestimated for assets with returns that are close to being normally distributed, making the probability of extreme events over time a constant. In a large universe of risk drivers, the ‘on average’-type performance of these models eliminates all potential as an early warning indicator.

EVT and the related peaks-over-threshold (POT) method have for a long time been used to model the occurrence of severe weather, earthquakes and other extreme natural phenomena.

The POT method is based on the generalised Pareto distribution (GPD) that represents a limiting distribution of the exceedances of a given return distribution over a given threshold when the threshold, which can be viewed as the left or right tail of the original distribution, goes to infinity. It is a model for the tails only, and does not model the body of the distribution. This creates two major challenges - sourcing a large enough sample to get a sufficient number of tail observations, and selecting the point (threshold) where the body of the distribution ends and the tail begins.

Overcoming these challenges leads to subjective choices on the tail behaviour and large instability of the final risk estimates. Automated EVT models for commercial large-scale, day-to-day risk measurement have yet to be demonstrated - and even if they could be, the model cannot warn on changing market regimes or differentiate between assets that are current risk contributors or diversifiers, due to the enormous amount of data needed and the large estimation error.

Proactive and pre-emptive attention without overly defensive examination and procedures is required to survive extreme events. Stable Paretian distributions and certain generalisations (operator Stable) represent a model for the entire distribution, not just the tails and, most significantly, they have a tail-thickness parameter that varies from asset to asset and through time (indeed, during normal markets stable VaR is often lower than normal VaR). As such, they can serve as an early warning indicator of extreme events.

Boryana Racheva-Iotova is president and head of research and development at Finanalytica, a provider of portfolio risk management solutions