Portfolio Construction: Three Blind Mice
Accounting for fat tails of individual instruments is not the same as managing those tails at portfolio level. Svetlozar (Zari) Rachev and Georgi Mitov explore how advanced copulas might address the problem of fat tails, dependence models and portfolio risk
Today, risk and portfolio managers are finally looking beyond the bell curve and now accept that asset returns are fat-tailed. Indeed, practitioners are ramping up their processes to account for the phenomena that give rise to fat-tails. Most efforts however concentrate on improving the tails of financial instruments. Few have really gone beyond linear correlations and tackled the process of capturing the joint dependencies among risky assets. Continuing blindly down this path will surely lead to future catastrophic losses even with application of distributions that accurately describe the behavior of the individual assets.
This is clearly illustrated in scatter plots of the NASDAQ 100 and Russell 2000 historical returns (figure 1). In the empirical data shown in the first plot, we see concentrations of extremes, downside and upside, in the lower left and upper right quadrants. The observations in the lower quadrant are visually significant, being more dependent and concentrated. Applying a sophisticated fat-tailed model to each index but maintaining classical correlations to describe the joint dependence, the lower left quadrant of the simulated data in the second plot is suspiciously empty. Even with high correlation, this risk analysis dangerously misleads us to believe that there is no extreme dependence between these two indices.
As demonstrated, any practical dependency model has to be flexible enough to account for the phenomena observed empirically in real-world financial data. It is well known that the dependence between assets is non-linear, being greater during periods of market stress. Furthermore, the dependency is asymmetric: most asset prices become relatively more dependent during significant market downturns than upturns.
The industry standard dependence model implied by the multivariate normal distributions fails to incorporate both of the aforementioned phenomena. Under a normal model, the covariance matrix defining the dependence structure only determines linear dependencies, and is symmetric. Tail events are asymptotically independent under the multivariate normal (Gaussian) distribution: if a large bivariate sample is generated using bivariate normal distribution with a correlation very close but not equal to +1, the extreme values (losses and gains) will be approximately uncorrelated. Finally, the multivariate normal distributions describe only bivariate dependences.
To overcome the covariance matrix deficiencies, a multivariate fat-tailed distribution is often used to account for tail dependence. The multivariate Student's t-distribution is the easiest and most frequently used. However, this approach leads to making two separate assumptions simultaneously. First, there is an assumption about the symmetry with one and the same tail of one-dimensional distribution of the individual variables, and then a second assumption on the elliptical dependence between the variables. This significantly limits the flexibility of the model.
Given the simplicity, it is also tempting to use the historical approach for dependence modelling. This approach relies purely on historical returns and thus the implied dependence. But this method, including enhanced variations that adjust a long data series to current correlation levels, all suffer from the same deficiency - joint multivariate tail events not seen in the past will never be predicted. Only a model based on a parametric distribution (Normal, Student-T, Stable, Hyperbolic) will have the predictive power to produce scenarios not observed in the past.
Copulas provide the most general function for describing dependence structures. They can be based on any multivariate distribution with marginals (the distribution that describes the individual assets) transformed to values between 0 and 1. The values of the copula function show the joint probability of two or more assets moving together to extreme (or non-extreme) quantiles without specifying the values of the quantiles for each individual asset. The main concept is to first specify the statistically most preferable marginal distributions and then as a second step apply a copula to capture the dependency structure. For example, marginal distribution models could be fit to risk factors with varying tail-fatness parameters and GARCH to remove volatility clustering (as described in our article ‘The emergency room', IPE April 2010), before a copula is specified to capture the multivariate dependency structure. This separation greatly extends the flexibility of the models. The multivariate distribution functions that copulas are based on can be either empirical (historical) or parametric.
Using copulas, based on extreme value distributions, were an attractive starting point for modelling extremes in finance, but unfortunately their application in a large-scale setting is prohibitive, due to numerical complexities mainly in the scenario generation process. As an alternative, the Gaussian copula, based on a multivariate normal distribution, provides a seemingly easy approach for describing the dependence structure between assets. However, it has a major drawback in that the entire dependence is described only by correlations implying that extreme events are asymptotically independent. Thus, the probability of large negative returns for two stocks occurring jointly is significantly underestimated. Bond traders and rating agencies relied on the Gaussian copula to describe the joint dependence between mortgage tranches with disastrous results.
A better alternative to the Gaussian copula is the Student's t-copula. It more accurately models the probability of joint extreme events. The classical Student's t-copula has the disadvantage of having symmetric tail dependence, resulting in the probability for the joint occurrence of very large negative and positive returns being the same, and also equal across assets. Several extensions are possible to overcome these deficiencies. The t-copulas and their extensions are from a rich class of so-called sub-Gaussian or subordinated copulas. Simply speaking, all subordinated copulas are an extension of the Gaussian copula, but volatility and correlations are random, and the copula can be skewed in different directions. The effect of this ‘randomness' on the shape of the dependence is controlled by the additional parameters of the copula.
FinAnalytica compares the models using a portfolio of the top 10 best performing DJIA Stocks based on 14 years of historical returns (1996-2009). We fit four models: historical, normal (Gaussian), fat-tailed with a Gaussian Copula and fat-tailed with an asymmetric subordinated copula. FinAnalytica looks at the joint probability of each pair of assets to drop simultaneously, exceeding the 98% VaR threshold of each stock in one month.
The first chart in figure 2 represents these probabilities based on historically observed data from 30 September 1996 through 1 September 2008. We see that Cisco and Intel have the highest probability of a joint drop at 0.6%. DD and IBM have the lowest at 0.2%. Gaussian approaches (charts ii and iii) do not provide good models, either by using a factor model nor by directly modelling the stock returns with a 10-variate Gaussian distribution. The modelling is dramatically improved (chart iv) if we add fat-tailed marginal distributions. However, comparing chart iv to chart vi, we see that the model is still short of predictive power. Chart vi shows the empirical probabilities after adding data through 1 September 2009. The model becomes predictive when we combine a fat-tailed parametric copula to describe the dependence combined with the fat-tailed marginals (chart v).
Now applying these models to risk estimation in an equally weighted portfolio of these stocks, we find that normal-based methods underestimate the possible risks as of 15 September 2008 when we compared them with the actual losses observed soon after. The one month loss as of 9 October 2008 is 24% and the one month loss as of 24 October 2008 is 28%. Normal-based methods predicted 99% 1 month VaR at 16.5% and 99% expected tail loss of 18%. The fat-tailed method with the Gaussian copula is still short in predicting the risk with 99% expected tail loss (the average loss beyond 99% VaR) of 20%. The fat-tailed with subordinated copula method has good predictive power for the crisis which can be seen by the 99% VaR of 26% and, more importantly, the 99% expected tail loss (the average loss beyond 99% VaR) of 33%.
If there are any positives to be gleaned from the 2008 crisis, clearly the focus on the underlying risk estimation methods benefits everyone. Moving beyond the bell curve is significant. But most approaches continue to blindly assume that linear correlations can describe dependence structures. Even in accepting this challenge and moving beyond it, wide adoption of those copulas that capture real world dependencies will remain elusive until more large-scale commercial products are seen in the market.
Svetlozar (Zari) Rachev is chief scientist and Georgi Mitov is senior quantitative analyst with FinAnalytica.