Enterprise Risk Management Formula Book
12. Copulas
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12.1 Definition
A copula
is a multivariate cumulative distribution function for an 
dimensional
random vector 
in the unit hypercube (
) that has uniform
marginals, 
, each distributed
according to 
but not in general
independent of each other. Let 
also be restricted to
the unit hypercube . Then a copula is
defined as a function of the form:
Equivalently 
is the joint cumulative
distribution function for the random vector 
.
The copula density (for a continuous copula) is the
pdf for which the cdf is the copula.
12.2 Properties
In the bivariate case (
) for a general function 
to be a copula it must
satisfy the following properties:
1. 
for all 
2. must be increasing in
both 
and 
3. 
for all 
and 
4. 
5. 
12.3 Sklar’s theorem
If 
is a joint (cumulative)
distribution with marginal cdf’s 
then there exists a
copula 
which maps the unit
hypercube onto the interval 
such that for all 
we have:
Moreover, if the 
are continuous functions
then the copula is unique and
Conversely, suppose is a copula and 
are univariate cdf’s.
Then the function is a joint distribution
function with marginal cdf’s .
12.4 Example copulas
The Archimedean family
involves copulas of the following form, where 
, 
, 
, 
is
continuous and strictly decreasing and 
Special cases include the Clayton copula which has
(for some suitable value
of 
) and the independence or
product copula which has 
.
12.5 Tail dependence
If 
and 
are continuous random
variables with copula then their coefficient
of (joint lower) tail dependence (if it exists) is:
For continuous random variables 
and 
each
with lower limit of 
the coefficient of
(lower) tail dependence is also:
12.6 Simulating copulas
Correlated Gaussian (i.e. multivariate normal) random
variables (i.e. random variables with a Gaussian copula and Gaussian marginals)
can be generated using Cholesky
decomposition.
For random variables that have a Gaussian copula but
non-normal marginal (with cdfs 
) we can generate a
vector 
of correlated Gaussian
random variables as above and then transform as per 
.
In general, for non-Gaussian copulas we may need to generate
a vector of unit uniform random variables 
and then transform them
using 
, 
etc.
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