Fat-tailed Distributions
All standard risk management and portfolio optimization solutions make one of two assumptions about the probability distribution of asset returns: (a) explicitly or implicitly the asset returns in the portfolio have a multivariate normal (Gaussian) distribution or (b) the empirical (sample) multivariate distribution of the returns (the historical distribution) is an adequate estimate of the true distribution.
Unfortunately the normal distribution is inadequate for modeling the probabilities of extreme movements in asset returns. While historical return distributions capture non-normality to some degree, they assign zero probability to values outside the range of the historical returns. As a consequence, both approaches underestimate downside risk. There are many potential reasons for extreme events:
- Time-varying volatility is sometimes said to be the main cause for extreme events.
This phenomenon, as noted by Mandelbrot, concerns the fact that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes." To account for this volatility clustering effect, the class of Robert Engle’s Nobel Prize winning “autoregressive conditional heteroskedasticity” (ARCH) models, their generalized (GARCH) extensions and the simpler exponentially weighted moving average (EWMA) volatility models are widely used. GARCH models assume that the volatility on a given day depends on the volatilities and also on the squared residuals of the one or two previous days. Thus, when volatility starts increasing, this is captured by the model and the next-day volatility can be forecast.
Many studies, however, demonstrate that even after removing this time-varying effect, fat-tails, though smaller in magnitude, continue to be present. Therefore, a more realistic model should allow for a fat-tailed residual. The time series models of FinAnalytica implemented in Cognity were designed with this goal in mind. The methods for parameter estimation are completely consistent with the assumption of a fat-tailed residual.
- Changes in regulation, announcements of events (e.g. mergers, acquisitions, etc.), resulting in random and unpredictable jumps in the value of a given factor or variable.
A possible way to capture this phenomenon is through Poissonian-type jump models. Note that the nature of these events is such that their exact occurrence cannot be predicted. However, a non-zero probability for such an event can be included in the model. This makes the risk numbers more conservative and adequate should it occur.
FinAnalytica developed such a jump model to take into account this behavior. It can identify isolated outliers and calculate the frequency of occurrence. Then, in the scenario generation phase, similar outliers are inserted into the scenarios.
- Market structure factors such as block trades, liquidity, market depth, concentration, etc.
All these characteristics are related to the idea of market timing – the fact that the market exhibits its own, non-physical time which depends on the intensity of the information arriving on the market. The information can be expressed in terms of trade order or news announcements. When there is no new information, trading is slow and the market time almost stops. Thus, the corresponding price process almost does not evolve. However, when new information arrives, market trading becomes more or less hectic depending on the intensity of the new information and the price process starts to progress quicker which leads to observing larger changes in magnitude (positive or negative) for the same period of physical time.
A simple approach to capture this idea is to consider a model with two states – one with low trading volume and one with high trading volume, which can be well distinguished especially for low liquidity assets. This is a simple form of an Embedded Markov Chain (EMC) model. This model can identify the two states and can calculate the transition probability matrix.
Another approach is to employ the idea of subordinated processes in which the returns of risk drivers in physical time are modeled as location-scale normal mixtures in which the mixing distribution is unobservable and describes the rate of flow of market time. This construction naturally leads to many of the fat-tailed models available in the academic literature, the most natural class of which is represented by the so called Stable Paretian models.
Building realistic fat-tailed models is a difficult task.
Based on Chief Scientist Svetlozar Rachev’s 20 years of extensive work on stable distributions in finance, Cognity uses proprietary multivariate stable distributions as a central foundation for its risk management and portfolio allocation solutions. These distributions account for the different marginal stable distributions in the returns of each individual asset in a portfolio, thereby accurately representing the differential tail-fatness and skewness across assets.
While stable distributions have certain advantages, in some circumstances, it is preferable to use a generalized multivariate t-distribution with different degrees of freedom and possible skewness features for each asset. For example, with smaller sample sizes typical of monthly return data, a t-distribution may be more reliably fit than a stable distribution. Thus, Cognity provides the capability of fitting such alternatives.
It should also be noted that the fat-tailed models constructed using stable distributions include the normal distribution as a special case and, therefore, nothing is lost by using a fat-tailed model. If financial returns do not exhibit fat-tails, then this will be recognized by the model and be properly taken care of by using a normal model. FinAnalytica has deployed the only commercial implementation of Stable Paretian models in a risk management platform.
The extreme events observed in returns may have different sources and may require the combining of different models in order to explain them. All models mentioned above can be combined into one model that best explains the data. For example, in order to explain the daily returns of an illiquid stock, the EMC model is needed to identify the corresponding two states and to calculate the transition probabilities. Then, in the state in which the stock is actively traded, clustering of volatility is observed and can be explained by a GARCH model. In the residuals of the GARCH model, typical outliers may be seen which can be filtered by the jump model. Finally, fat tails due to the market timing phenomenon can be captured by the class of stable distribution models.