correlations and copulas.pdf

Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.1

Correlations and

Copulas

Chapter 9


Coefficient of Correlation

The coefficient of correlation between two

variables V1 and V2 is defined as

)()(

)()()(

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

)()( 212 VfxVVf


Independence is Not the Same as

Zero Correlation

Suppose V1 = –1, 0, or +1 (equally likely)

If V1 = -1 or V1 = +1 then V2 = 1

If V1 = 0 then V2 = 0

V2 is clearly dependent on V1 (and vice

versa) but the coefficient of correlation is

zero

Correlation measures linear dependence


Types of Dependence

E(Y\X )

X

E(Y\X )

E(Y\X )

X

(a) (b)

(c)

X

Monitoring Correlation

Variance rate per day of a variable:

variance of daily returns

Covariance rate per day between two

variables: covariance between the daily

returns of the variables



)(

)()()(cov

:nday on rate Covariance

, ,

:are day on

returns the,day of end at the and

variables twoof values and

1

1

1

1

nn

nnnnn

i

iii

i

iii

ii

yxE

yExEyxE

Y

YYy

X

XXx

i

iY

XYX


Monitoring Correlation Between

Two Variables X and Y

Define

varx,n: daily variance rate of X estimated on day n-1

vary,n: daily variance rate of Y estimated on day n-1

covn: covariance rate estimated on day n-1

The correlation is

nynx

n

,, varvar

cov


Monitoring Correlation continued

EWMA:

GARCH(1,1)

111 )1(covcov nnnn yx

111 cov cov nnnn yx


Positive Finite Definite Condition

A variance-covariance matrix, W, is

internally consistent if the positive semi-

definite condition

holds for all Nх1 vectors w, W positive-

semidefinite

w wTW 0


Example

The variance covariance matrix

is not internally consistent

1 0 0 9

0 1 0 9

0 9 0 9 1

.

.

. .


V1 and V2 Bivariate Normal

Conditional on the value of V1, V2 is normal with

mean

and standard deviation where m1,, m2,

s1, and s2 are the unconditional means and SDs

of V1 and V2 and r is the coefficient of

correlation between V1 and V2

1

112212 )|(

s

mrsm

VVVE

2

2 1 rs


Multivariate Normal

Many variables can be handled

A variance-covariance matrix defines the

variances of and correlations between

variables

To be internally consistent a variance-

covariance matrix must be positive

semidefinite

Generating Random Samples for

Monte Carlo Simulation



Factor Models

When there are N variables, Vi (i=1,2,..N), in a multivariate normal distribution there are N(N-1)/2 correlations

We can reduce the number of correlation parameters that have to be estimated with a factor model


Factor Models continued If Ui have standard normal distributions we

can set

F and Zi have independent standard

normal distributions, is a constant

between -1 and +1, Zi uncorrelated with

each other and F

All the correlation between Ui and Uj

arises from F

iiii ZaFaU 21

ia

Factor Models continued


M-factor Model


M

m

jmimij

i

iM

iiMiiMiMiii

aa

sF

ZFF

ZaaaFaFaFaU

1

1

22

2

2

12211

' andother each with eduncorrelat

, variablesnormal eduncorrelat edstandardiz ,...,

1...

r


Gaussian Copula Models: Creating a correlation structure for variables that are not

normally distributed

Suppose we wish to define a correlation structure

between two variable V1 and V2 that do not have normal

distributions

We transform the variable V1 to a new variable U1 that

has a standard normal distribution on a “percentile-to-

percentile” basis.

We transform the variable V2 to a new variable U2 that

has a standard normal distribution on a “percentile-to-

percentile” basis.

U1 and U2 are assumed to have a bivariate normal

distribution


The Correlation structure between the

V’s is define by that between the U’s

-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

V1V2

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6

U1

U2

One-to-one

mappings

Correlation

Assumption

-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

V1V2

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6

U1

U2

One-to-one

mappings

Correlation

Assumption

V1V2

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6

U1

U2

One-to-one

mappings

Correlation

Assumption


Other Copulas

Instead of a bivariate normal distribution

for U1 and U2 we can assume any other

joint distribution

One possibility is the bivariate Student t

distribution


5000 Random Samples from the

Bivariate Normal

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5


5000 Random Samples from the

Bivariate Student t

-10

-5

0

5

10

-10 -5 0 5 10


Multivariate Gaussian Copula

We can similarly define a correlation

structure between V1, V2,…Vn

We transform each variable Vi to a new

variable Ui that has a standard normal

distribution on a “percentile-to-percentile”

basis.

The U’s are assumed to have a

multivariate normal distribution


Factor Copula Model

In a factor copula model the correlation

structure between the U’s is generated by

assuming one or more factors.


Credit Default Correlation

The credit default correlation between two

companies is a measure of their tendency to

default at about the same time

Default correlation is important in risk

management when analyzing the benefits of

credit risk diversification

It is also important in the valuation of some

credit derivatives


The Model

We map the time to default for company i, Ti, to a

new variable Ui and assume

where F and the Zi have independent standard

normal distributions

Define Qi as the cumulative probability distribution

of Ti

Prob(Ui<U) = Prob(Ti<T) when N(U) = Qi(T)

iiii ZaFaU 21


The Model continued

(*)

1

)()(Prob

companies allfor same theare s' and s' theAssuming

1

)()(Prob

Hence

1)(Prob

1

2

1

2

r

r FTQNNFTT

aQ

a

FaTQNNFTT

a

FaUNFUU

i

i

iii

i

ii

The Model continued


r

r

-1

)()]([

angreater th be willratedefault that the

Yofy probabilita is thereso ,))(

rate.default the

F,on lconditiona T by time defaulting loans of

eprercentag theof estimate gooda provides

(*) equation loans, of portfolio largea For

11

1

YNTQNN

YYNP(F

The Model continued


r

r

1

)()]([),(WCDR

then,expression previous theinto X-1Y

Substitute rate".default case-worst" the

T, in time exceeded benot llcertain wi

X% are that weratedefault thecall We

11 XNTQNNXT

Example

Suppose that a bank has a total of $100

million of retail exposures. The one-year

probability of default averages 2% and the

recovery rate averages 60%. The copula

correlation parameter is estimated as 0.1.


million 13.5$)6.01(128.0100

are case in this Losses

128.0

1.01

)999.0(1.0)02.0()999.0,1(

11

NNNWCDR

correlations and copulas.pdf

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