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

Types of Dependence

7

E(Y)

X

E(Y)

E(Y)

X

(a) (b)

(c)

X

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

Markowitz Theory

10

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 transformation

13

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

23

24

25

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

28

THANKS

FOR

YOUR ATTENTION