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Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.1
Correlations and
Copulas
Chapter 9
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.2
Coefficient of Correlation
The coefficient of correlation between two
variables V1 and V2 is defined as
)()(
)()()(
21
2121
VSDVSD
VEVEVVE
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.3
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.4
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.5
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.6
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.7
)(
)()()(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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.8
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.9
Monitoring Correlation continued
EWMA:
GARCH(1,1)
111 )1(covcov nnnn yx
111 cov cov nnnn yx
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.10
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.11
Example
The variance covariance matrix
is not internally consistent
1 0 0 9
0 1 0 9
0 9 0 9 1
.
.
. .
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.12
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.13
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.14
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.17
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.18
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.19
M-factor Model
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.20
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.21
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.26
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.27
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.28
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.29
5000 Random Samples from the
Bivariate Student t
-10
-5
0
5
10
-10 -5 0 5 10
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.30
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.31
Factor Copula Model
In a factor copula model the correlation
structure between the U’s is generated by
assuming one or more factors.
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.32
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.33
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.34
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.35
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
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.36
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 $100
million of retail exposures. The one-year
probability of default averages 2% and the
recovery rate averages 60%. The copula
correlation parameter is estimated as 0.1.
Risk Management and Financial Institutions, Chapter 6, Copyright © John C. Hull 2006 6.37
million 13.5$)6.01(128.0100
are case in this Losses
128.0
1.01
)999.0(1.0)02.0()999.0,1(
11
NNNWCDR