What is a copula?
A copula can be defined as a
multivariate distribution with marginals that are uniform over the unit
interval (0,1). Copula functions can be used to simulate a dependence
structure independently from the marginal distributions.
Based on Sklar's theorem the multivariate distribution F can be represented by copula C as follows:
Based on Sklar's theorem the multivariate distribution F can be represented by copula C as follows:
F(x1…xp)
= C{ F1(x1),…, Fp(xp); θ}
where each
Fi(xi) is a uniform marginal distribution and θ is a dependence parameter. ‘Copula’ means to
connect or join. Copula functions in essence connect or join the information in
the marginal to form a multivariate distribution.
Why not just use
measures of correlationn or multivariable regression to describe relationships
between varaibles?
As
described by Tang and Valdez
"While
being computationally convenient and straightforward to understand, linear
correlations fail to capture all the dependence structure that exist between
losses from multiple business lines."
Zimmer
and Trivedi (2005) also elaborate on how copula methods can be more useful than
traditional correlation approaches:
“econometricians often possess more information about marginal
distributions of related variables than their joint distribution. The copula
approach is a useful method for deriving joint distributions given the marginal
distributions, especially when the variables are nonnormal....When fairly
general and/or asymmetric modes of
dependence are relevant, such as those that go beyond correlation or linear
association, then copulas play a special role in developing additional concepts
and measures.”
Dorey and Joubert discuss the
importance of copulas in actuarial work:
"The increasing complexity of insurance and reinsurance
products has led to interest in the modelling of dependent risks, especially
with the emergence of intricate multi-line products.....Extreme events, such as
hurricanes, can also result in dependencies between classes of business which
are unrelated in normal conditions"
Example
Let’s take a subset of data (from the SAS
Online Documentation) for stock returns.
data
returns;
input ret_msft
ret_duk;
cards;
0.004182 0.003503
-0.027960 -0.000582
0.006732 0.025611
-0.033435 -0.002838
0.029560 0.010814
-0.003054 -0.001689
-0.012255 -0.012408
0.013958 0.003427
-0.011318 -0.017075
-0.022587 0.002316
;
run;
The next block of data uses
PROC COPULA to estimate the covariance structure of the returns of Microsoft
and Duke Energy.
/* Copula
estimation */
proc
copula data = returns;
var ret_msft ret_duk;
fit normal / outcopula=estimates;
run;
/* keep only
correlation estimates */
data
estimates;
set
estimates;
keep
ret_msft ret_duk;
run;
Next, we can use
PROC COPULA to simulate the multivariate distribution of returns based on the
correlation estimates derived before, using the Gaussian Copula.
/* Copula
simulation */
proc
copula;
var
ret_msft ret_duk;
define
cop normal (cor = estimates);
simulate
cop / ndraws = 500
outuniform =
simulated_uniforms
plots=(data=uniform scatter);
run;
The Financial Crisis:
A Case Study for Copula Specifications Using SAS
For an interesting 2 part video visualizing the construction of collateralized debt obligations and tranches see here and here.
From: In
defense of the Gaussian copula, The Economist
"The Gaussian copula provided a convenient way to describe a relationship that held under particular conditions. But it was fed data that reflected a period when housing prices were not correlated to the extent that they turned out to be when the housing bubble popped."
"The Gaussian copula provided a convenient way to describe a relationship that held under particular conditions. But it was fed data that reflected a period when housing prices were not correlated to the extent that they turned out to be when the housing bubble popped."
Various functional forms
for copula functions exist, each based on different assumptions about the
dependence structure of the underlying variables.
Using SAS PROC COPULA , and loosely following Zimmer
2012, I have simulated data based on 3 different copula formulations and
produced scatter plots for each. Using the table provided in Zimmer, I solved
for the dependence parameter θ in each copula formulation giving a dependence
structure consistent with Kendall’s tau = .60 (SAS code follows).
A mentioned in the Economist article at the beginning of this post, the Gaussian copula was used widely before the housing crisis to simulate the dependence between housing prices in various geographic areas of the country. Looking at the plot for the Gaussian copula above, it can be seen that extreme events (very high values of Y1 and Y2 or very low values of Y1 and Y2) seem very weakly correlated compared to the plots for the Clayton and Gumbel copulas. We can see that with the Gumbel copula, extreme events (very high values of Y1 and Y2) are more correlated, while with the Clayton copula, extreme events (very low Y1 and Y2) are more correlated.
Recall, as previously stated, linear correlations may provide a measure of dependence, but they fail to capture the complete structure of dependence. For instance, in each of plots above, we get the following measures of the Pearson Correlation Coefficient (.81, .78, & .79) for the Gaussian, Gumbel, and Clayton copula simulations respectively. This measure is in essence the same across all three simulations, ~ .80. However, this tells us nothing about the dependence relationship for extreme values of Y1 and Y2. Copulas, if properly specified, help us to better specify the structure of dependence.
How might this
relate to the financial crisis specifically? In The Role of Copulas in the Housing Crisis, Zimmer
states:
"Due to its
simplicity and familiarity, the Gaussian copula is popular in the calculation
of risk in collaterized debt obligations. But the Gaussian copula imposes
asymptotic independence such that extreme events appear to be unrelated. This
restriction might be innocuous in normal times, but during extreme events, such
as the housing crisis, the Gaussian copula might be inappropriate"
In the paper a
mixture of Clayton and Gumbel copulas is investigated as an alternative to
using the Gaussian copula.
SAS CODE
*-------------------------------------------------------------------------
| theata : kendal's tau = .60
*--------------------------------------------------------------------------;
* gaussian;
data inparm;
input COL1
COL2;
cards;
1 .81
.81 1
;
run;
proc copula ;
var Y1-Y2;
define cop normal (cor=inparm);
simulate cop /
ndraws = 500
seed = 1234
outuniform
= normal_unifdata
plots = (data = original scatter
distribution = cdf);
run;
* gumbel ;
proc copula ;
var Y1-Y2;
define cop gumbel (theta=2.5);
simulate cop /
ndraws = 500
seed = 1234
outuniform
= normal_unifdata
plots = (data = original scatter
distribution = cdf);
run;
* clayton copula;
proc copula ;
var Y1-Y2;
define cop clayton (theta=3);
simulate cop /
ndraws = 500
seed = 1234
outuniform
= normal_unifdata
plots = (data = original scatter
distribution = cdf);
run;
For a related visualization of copulas using excel, see below:
References:
SAS Online Documentation:
PROC COPULA
The Role of Copulas in the Housing Crisis. David M. Zimmer.
The Review of Economics
and Statistics, May 2012, 94(2): 607–620
Copula Modeling: An Introduction for Practitioners
Pravin K. Trivedi1 and
David M. Zimmer2
Foundations and Trends in Econometrics
Vol. 1, No 1 (2005) 1–111
c 2007 P. K. Trivedi and
D. M. Zimmer
DOI: 10.1561/0800000005
Economic Capital and the
Aggregation of Risks using Copulas
Andrew Tang and Emiliano
A. Valdez†
School of Actuarial
Studies
Faculty of Commerce &
Economics
University of New South
Wales
Sydney, AUSTRALIA
Understanding Relationships Using Copulas
Edward W. Frees† and
Emiliano A. Valdez‡
North American Actuarial Journal, Vol. 2, No. 1.
Modelling Copulas: An
Overview
Martyn Dorey Phil Joubert
The Staple Inn Actuarial
Society
