Bivariate Copula

Also see: Modeling correlation , Help on ModelRisk, Copulas in ModelRisk, Vose Multivariate Copula, Common elements of ModelRisk windows

Introduction

The bivariate copula window from ModelRisk

Copulas are used for correlating two or more random variables without affecting the distributions themselves. Copulas provide greater flexibility than the older rank_order_correlation.

The following bivariate copulas are available for use in spreadsheet models in ModelRisk :

• Bivariate Clayton Copula

• Bivariate Frank Copula

• Bivariate Gumbel Copula

• Bivariate Normal Copula

• Bivariate T Copula

Associated with every bivariate copula is its copula density, which is much like the probability density of a (bivariate) distribution.

The output of a bivariate Copula in ModelRisk is an array function of two spreadsheet cells. These cells will contain correlated Uniform(0,1) random variables, with a pattern of correlation defined by the copula.

Next, these correlated Uniform(0,1) variables are used as the U-parameter in the two desired distributions. The result is that the two final spreadsheet cells will contain variables sampled from your chosen distribution(s), and at the same time generating the pattern of correlation defined by the chosen copula.

You can read more about the mathematical details of copulas here.

To see the output functions of this window, click here.

Window elements

You first select the type of copula to use in the Copulas section of the bivariate copula interface.

The copula's parameters can be set manually or linked to a spreadsheet cell. In the picture above, we see the Bivariate Clayton copula which takes one parameter named Alpha.

The copula direction can be set to 1, 2, 3 or 4. This is for rotating the copula 90, 180 and 270 degrees respectively. This allows for more flexibility in modeling correlations.

Also see: Direction of a copula

Correlated distributions

In the Correlated distributions area, the distributions to be correlated can be selected. These can be either typed directly, chosen from the Select Distribution window, or inserted from a spreadsheet cell.

Copula graph

Copula graph in the Bivariate Copula window

In the middle pane, a graph for the copula is shown. The points represent randomly generated (x,y) values generated by this copula: the X- and the Y- axis represent the correlated variables associated with the first and second selected distribution, respectively.

By default, the percentiles of these 2 correlated variables can be shown: these are values between 0 and 1. As explained above, certain pairs will have a higher probability of being generated, as determined by their correlation (i.e. the copula used).

Optionally, the actual values of sampled random variables can be shown, with both axes rescaling appropriately. This goes one step further: the (x,y) pairs represent sampled random variables from the chosen distributions, with the percentiles now being driven by the copula. Internally, this is the U-parameter in action: it takes the random value generated by the copula.

See also: Distribution functions and the U parameter

Copulas are directly connected to classical measures of correlation, like rank order correlation. The equivalent Rank Order Coefficient of the current copula is shown on the left.