# LO 9.1: Explain the purpose o f copula functions and the translation o f the copula

LO 9.1: Explain the purpose o f copula functions and the translation o f the copula equation.
A correlation copula is created by converting two or more unknown distributions that may have unique shapes and mapping them to a known distribution with well-defined properties, such as the normal distribution. A copula creates a joint probability distribution between two or more variables while maintaining their individual marginal distributions. This is accomplished by mapping multiple distributions to a single multivariate distribution. For example, the following expression defines a copula function, C, that transforms an ^-dimensional function on the interval [0,1] to a one-dimensional function.
C : [0,l]n -[0,1] Suppose (i.e., i is an element of set N). A copula function, C, can then be defined as follows:
G [0,1] is a univariate, uniform distribution with zq = zq,…, zzn, and z G N
C[G1(u1),…,Gn(un)] = Fn Fl l(Gl (ul )),…,Fn 1(Gn(un));p
-1
In this equation, G-(uJ are the marginal distributions, Fn is the joint cumulative distribution function, iq 1 is the inverse function of F , and structure of the joint cumulative function F .
is the correlation matrix
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Topic 9 Cross Reference to GARP Assigned Reading – Meissner, Chapter 4
This copula function is translated as follows. Suppose there are n marginal distributions,
to Gn(n). A copula function exists that maps the marginal distributions of G^(u^) to
Gn(n) yia F j^G^Uj) and allows for the joining of the separate values F -1Gi(ui) to a single w-variate function Fn Fl_1(G1(ui)),…,Fn_1(Gn(un)) that has a correlation matrix of pF. Thus, this equation defines the process where unknown marginal distributions are mapped to a well-known distribution, such as the standard multivariate normal distribution.
G a u s s i a n C o p u l a