The Clayton copula and
its impact on equity risk
Olivier Habinshuti
March 27, 2014
University of Oslo
1
Outline
Introduction
Dependence vs. correlation
Multivariate distributions
Tail dependence
Copulas
Clayton copula
Case study for equity risk
Others applications
2
The situation
You have a basket of stocks which, during
normal days, exhibit little relationship with
each other. But on days when the market
moves dramatically they all move together.
Such behaviour can be modelled by
copulas.
3
Data Days where more than 90 percent of stocks are
moving in the same direction
Up to 2006 this is about 3-5 days per year
2008-2011 this is above 30 (or about 1/7 trading days)
4
Correlation and Covariance
The coefficient of correlation between two
variables V1 and V2 is defined as
This coefficient measures the linear
correlation between variables
5
)()(
)()()(
21
2121
VSDVSD
VEVEVVE
Independence
V1 and V2 are independent if the
knowledge of one does not affect the
probability distribution for the other
where f(.) denotes the probability density
function.
Independence is not the same as Zero
Correlation
6
)()( 212 VfxVVf
Multivariate Normal Distribution
The univariate normal distribution function
is:
The multivariate normal distribution
function is:
The mean vector is μ
The covariance matrix is Σ
8
Multivariate Normal Distribution
Fairly easy to handle
A variance-covariance matrix defines
the variances of and correlations
between variables
If the returns on a set of assets have a
multivariate normal distribution, then the
return on any portfolio formed from
these assets will be normally distributed
9
The ugly truth
However, it is widely acknowledged that
prices, returns, and other financial
variables are not Normally distributed
They have fat tails, and exhibit ”tail
dependence”, in which correlations are
observed to rise during extreme events.
Their distributions change depending on
the time horizon.
11
Tail dependence
5000 Random Samples from the
Bivariate Normal
5000 Random Samples from the
Student t
12
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-10
-5
0
5
10
-10 -5 0 5 10
The Copula idea
14
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
V 1 V 2
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
U 1
U 2
One - to - one
mappings
Correlation
Assumption
V 1 V 2
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
U 1
U 2
One - to - one
mappings
Correlation
Assumption
Why copulas
Non-linear dependence Be able to measure dependence for heavy tail distributions Can be combined with any set of univariate distribution for marginal distributions
15
Definition
16
If (X, Y ) is a pair of continuous random variables with distribution function H(x, y) and marginal distributions Fx(x) and FY
respectively, then U = FX(x) ~ U(0, FY(y) ~ U(0, 1) and the distribution (U, V ) is a copula.
(y) 1) and V = function of
1.
2.
Properties
Sklar’s Theorem
17
Let H Then
be a joint df with marginal dfs F and G, there exists a copula C such that
If F and G are continuous,then the copula unique is
Copula models
Elliptical Copula
Student- copula
Gaussian copula
Archimedian Copula
Explicit copula function
Clayton, Gumbel, Frank copula
Gaussian Clayton Gumbel Frank
18
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Archimedian Copulas
19
continuous, strictly decreasing convex function. It is the generator of the copula
pseudo-inverse of g
Clayton’s copula
20
)0,]1max([),(
1
vuvuC
)1(1
)(
ttg
1
0
• With generator
0,1
• Special cases
Independent case
Perfect negative dependence
Perfect positive dependence
Monte Carlo Simulations
21
Z
VMU
j
j
)log(Jj 1
• Z is a positive r.v with moment generating function M
• V j sequence of independent uniform distributed r.v
• Then U j is also a sequence of independent and
uniform distributed r.v with
)()( 1 uMu
• If Z is Gamma distributed with some density
function, we get Clayton copula
22
SPECIFY MARGINAL DISTRIBUTIONS
Gaussian with mean 7% and volatility
25% for 2 assets
SPECIFY COPULA
Clayton copulas
and
SIMULATE DATA AND
CALIBRATE TO
GAUSSIAN MODEL
CONSTRUCT
PORTFOLIO weights
0:5 for both assets
COMPARE THE
MODELS
0 5
Observation
Gaussian models underestimate the
risk(equity) for low extreme returns.
Clayton copula captures the tail
dependence hence a higher ‘return at risk’
26
Copulas in Credit Risk
The credit default correlation between two
companies is a measure of their tendency to
default at about the same time
Portfolio of loans
Basel II framework for credit risk
Credit derivatives
27
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