It is now a well known fact that financial returns have high extreme and asymmetric dependence. Simple correlation is a measure for linear dependence only. It cannot model the phenomena of increased correlations of assets and poor diversification effects experienced in down-market conditions. Linear correlation only measures the degree of dependence but does not clearly discover the structure of dependence. The last caveat has an especially important implication in light of the current crisis. It has been widely observed that market crashes or financial crises often occur in different markets and countries at about the same time period even when the correlation among those markets is fairly low.

A more accurate approach that overcomes the disadvantages of linear correlation is to model dependency by using copulas. With the copula method, the nature of dependence that can be modeled is more general and the dependence of extreme events can be considered. Generally, a copula is used to separate the pure randomness of one variable (for example, a financial asset) from the interdependencies between it and other variables. By doing so, one can model each variable separately and also have a measure of the relations between those variables..

Along with the flexibilities of the copula model come certain challenges. First, there is the choice or estimation of an adequate univariate model to explain the randomness of a risk driver on a standalone basis. Then there is the choice and estimation of the copula model. The former challenge has been well discussed in recent decades, from the beginning of portfolio theory and Markowitz’ work using Gaussian distribution functions to the most recent developments such as using fat-tailed distributions. The latter challenge, however, is much more recent.

The right plot above shows scenarios generated from the skewed student’s-t copula model in Cognity with fat-tailed models for the marginals fitted on the daily returns of Russell 2000 and NASDAQ during the 1987 crash.

Illustrating the importance of choosing an appropriate copula, consider the frequent use of multivariate Gaussian copulas to price risky assets. This is especially true in the credit markets based on David X. Li’s contribution in the Journal of Fixed Income in 2000. While it is widely accepted that use of one-dimensional Gaussian distributions is flawed due to the fact that this distribution type attributes too low probabilities to extreme observations, the same is also true for the multivariate Gaussian copula. This copula is not appropriate to model financial data because it cannot explain the tail dependence phenomenon..

Using the same data in the previous example, the left plot above shows scenarios generated from a multivariate normal distribution fitted on the daily returns of Russell 2000 and NASDAQ during the 1987 crash. The right plot shows scenarios generated from the normal copula with fat-tailed models for the marginals fitted on the same data. Notice the extreme scenarios on the plot tend to cluster along the horizontal and the vertical axes, showing the extreme events not occurring jointly. This confirms that the Gaussian copula cannot explain tail dependence.

Cognity's copula model is constructed with the goals of:

• Accounting for tail dependence
• Capturing dependence asymmetry in the sense that the dependence between stocks returns in a bearish market should differ from that in a bullish market;
• Providing enough flexibility to be used in high dimensions.