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Correlations andCopulas
Chapter 10
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 1
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Correlation and Covariance
The coefficient of correlation between two
variables V 1 and V 2 is defined as
The covariance is E (V 1V 2)− E (V 1 ) E (V 2)
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)()(
)()()(
21
2121
V SDV SD
V E V E V V E −
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Independence
V 1 and V 2 are independent if the
knowledge of one does not affect the
probability distribution for the other
where f (.) denotes the probability densityfunction
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)()( 212 V f xV V f ==
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Independence is Not the Same as
Zero Correlation
Suppose V 1 = –1, 0, or +1 (equally
likely)
If V 1 = -1 or V 1 = +1 then V 2 = 1 If V 1 = 0 then V 2 = 0
V 2 is clearly dependent on V 1 (and vice
versa) but the coefficient of correlationis zero
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Types of Dependence (Figure 10.1, page 204)
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 5
E (Y )
X
E (Y )
E (Y )
X
(a) (b)
(c)
X
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Monitoring Correlation Between
Two Variables X and Y
Define xi=( X i− X i-1) /X i-1 and yi=(Y i−Y i-1) /Y i-1
Also
var x,n: daily variance of X calculated on day n-1var y,n: daily variance of Y calculated on day n-1
covn: covariance calculated on day n-1
The correlation is
n yn x
n
,, var var
cov
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Covariance
The covariance on day n is
E ( xn yn)− E ( xn) E ( yn)
It is usually approximated as E ( xn yn)
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Monitoring Correlation continued
EWMA:
GARCH(1,1)
111 )1(covcov −−− λ−+λ= nnnn y x
111 covcov −−− β+α+ω= nnnn y x
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Positive Finite Definite Condition
A variance-covariance matrix, , is
internally consistent if the positive semi-definite condition
holds for all vectors w
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0≥Ωww
T
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Example
The variance covariance matrix
is not internally consistent
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 10
1 0 0 9
0 1 0 90 9 0 9 1
.
.. .
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V 1 and V 2 Bivariate Normal
Conditional on the value of V 1, V 2 is normal with
mean
and standard deviation where µ1,, µ2, σ1,
and σ2 are the unconditional means and SDs ofV 1 and V 2 and ρ is the coefficient of correlation
between V 1 and V 2
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1
1122
σµ−ρσ+µ V
2
2 1 ρ−σ
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Multivariate Normal Distribution
Fairly easy to handle
A variance-covariance matrix defines
the variances of and correlationsbetween variables
To be internally consistent a variance-
covariance matrix must be positivesemidefinite
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Generating Random Samples for
Monte Carlo Simulation (pages 207-208)
=NORMSINV(RAND()) gives a random
sample from a normal distribution in
Excel
For a multivariate normal distribution a
method known as Cholesky’s
decomposition can be used to generaterandom samples
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One-Factor Model continued
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 15
If U i have standard normal distributions
we can set
where the common factor F and the
idiosyncratic component Z i have
independent standard normal
distributions Correlation between U i and U j is ai a j
iiii Z aF aU 21−+=
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The Correlation Structure Between the V’s is
Defined by that Between the U’s
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-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 1V 2
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
U 1U 2
One-to-one
mappings
Correlation
Assumption
V 1V 2
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
U 1U 2
One-to-one
mappings
Correlation
Assumption
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Example (page 211)
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V 1 V 2
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V 1 Mapping to U 1
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 19
V 1 Percentile U 1
0.2 20 -0.84
0.4 55 0.13
0.6 80 0.84
0.8 95 1.64
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V 2 Mapping to U 2
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 20
V 2 Percentile U 2
0.2 8 −1.41
0.4 32 −0.47
0.6 68 0.47
0.8 92 1.41
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Example of Calculation of Joint
Cumulative Distribution
Probability that V 1 and V 2 are both less
than 0.2 is the probability that U 1 < −0.84
and U 2 < −1.41 When copula correlation is 0.5 this is
M ( −0.84, −1.41, 0.5) = 0.043
where M
is the cumulative distributionfunction for the bivariate normal
distribution
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Other Copulas
Instead of a bivariate normal distribution
for U 1 and U 2 we can assume any other
joint distribution
One possibility is the bivariate Student t
distribution
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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
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5000 Random Samples from the
Bivariate Student t
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-10
-5
0
5
10
-10 -5 0 5 10
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Multivariate Gaussian Copula
We can similarly define a correlation
structure between V 1, V 2,…V n
We transform each variable V i to a newvariable U i that has a standard normal
distribution on a “percentile-to-percentile”
basis.
The U’s are assumed to have a
multivariate normal distribution
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Factor Copula Model
In a factor copula model the correlation
structure between the U ’s is generated by
assuming one or more factors.
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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
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Model for Loan Portfolio
We map the time to default for company i, T i, to a
new variable U i and assume
where F and the Z i have independent standard
normal distributions
Define Qi as the cumulative probability distributionof T i
Prob(U i
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The Model continued
[ ]
[ ]
ncorrelatiocopulatheiswhere
Prob
companiesallfor sametheares'ands'the Assuming
Prob
Hence
Prob
ρ
ρ−
ρ−
=<
−
−=<
−
−=<
−
−
1
)(
)(
1
)()(
1)(
1
2
1
2
F T Q N
N F T T
aQ
a
F aT Q N N F T T
a
F aU N F U U
i
i
iii
i
ii
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The Model continued
The worst case default rate for portfolio for a
time horizon of T and a confidence limit of X is
The VaR for this time horizon and confidence
limit is
where L is loan principal and R is recovery rate
Risk Management and Financial Institutions 2e, Chapter 10, Copyright © John C. Hull 2009 30
ρ−
ρ+
=
−−
1
)()]([ 11 X N T Q N
N WCDR(T,X)
),()1(),( X T WCDR R L X T VaR ×−×=