Copula Types

Just as there are many types of univariate distributions, there are many different copula patterns. Each pattern represents a different relationship that can be modeled between two input distributions. Similar to a correlation matrix, the pattern of the copula will control the possible values for the correlated distributions, preventing unrealistic values from being generated during a simulation. It is the shape of the pattern that differentiates a copula from a correlation.

There are three broad classifications of copulas available in @RISK.

Archimedean

These simple copulas are used to correlate a potentially large number of similar variables, such as the returns for all the stocks in a particular industry. They are simple in that they require only a single parameter - theta - which controls the degree of "spread".

  • Clayton - Asymmetric, with greater spread in the positive tail. Coefficient Matrix: No
  • Gumbel - Asymmetric, with greater spread in the negative tail. Coefficient Matrix: No
  • Frank - Symmetric. Coefficient Matrix: No

In their standard form, the Archimedean copulas can model only positive correlations; however, "reflected" versions of these copulas also exist:

  • ClaytonR - Clayton copula, reversed (flipped on both X and Y-axis). Coefficient Matrix: No
  • ClaytonRX - Clayton copula, flipped on the Y-axis. Coefficient Matrix: No
  • ClaytonRY - Clayton copula, flipped on the X-axis. Coefficient Matrix: No
  • GumbelR - Gumbel copula, reversed (flipped on both X and Y-axis). Coefficient Matrix: No
  • GumbelRX - Gumbel copula, flipped on the Y-axis. Coefficient Matrix: No
  • GumbelRY - Gumbel copula, flipped on the X-axis. Coefficient Matrix: No
  • FrankRX - Frank copula, flippedo n the Y-axis. Coefficient Matrix: No

For three-dimensional or higher-dimensional copulas, only the "R" reflections are allowed, where all the axes are reflected - other reflections are mathematically impossible. The Frank copula does not have a reflected version because of its symmetry.

Elliptical

There are two elliptical copulas - Gaussian and t. A Gaussian copula is identical to a standard @RISK Correlation Matrix, so it requires the specification of a correlation matrix in the same manner as a Correlation.

  • Gaussian - Coefficient Matrix: Yes
  • t - Coefficient Matrix: Yes

Empirical

Copulas can also be based on a set of existing data; using an empirical copula will model the relationship between inputs in the same pattern that the existing data is related. When using an Empirical copula, the copula can be configured to use either Interpolated values or Non-Interpolated using the 'Allow Interpolation' checkbox beside the Copula Type pull-down menu.

Given a set of data, an empirical copula first removes the margins from the data, and then creates a copula that allows for the correlation of any distributions with that same pattern. When specifying an empirical copula, it is possible to choose whether or not to interpolate values. If the data is not interpolated, the copula will only display the demarginalized data. When interpolation is selected, @RISK uses Bayesian statistics to compute values between the actual values of the data set; it is more common to allow interpolation.