Enterprise Risk Management Formula Book
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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.
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
3. for all and
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
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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