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Correlations and Copulas
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Measures of Dependence
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•The risk can be split into two parts:
• the individual risks and • the dependence structure between them
• Measures of dependence include:
• Correlation • Rank Correlation• Coefficient Tail Dependence• Association
Correlation and Covariance
• The coefficient of correlation between two variables X and Y is defined as
• The covariance is
E(YX)−E(Y)E(X)
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SD(X)SD(Y)
E(X)E(Y)E(YX)
Independence
• X and Y are independent if the knowledge of one does not affect the probability distribution for the other
where denotes the probability density function
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f(Y)x)XYf(
)f(
Correlation Pitfalls• A correlation of 0 is not equivalent to independence• If (X, Y ) are jointly normal, Corr(X,Y ) = 0 implies
independence of X and Y• In general this is not true: even perfectly related
RVs can have zero correlation:
0Y)Corr(X,
XY and N(0,1)~X 2
Types of Dependence
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E(Y)
X
E(Y)
E(Y)
X
(a) (b)
(c)
X
Correlation Pitfalls (cont.)
• Correlation is invariant under linear transformations, but not under general transformations:– Example, two log-normal RVs have a different
correlation than the underlying normal RVs
• A small correlation does not imply a small degree of dependency.
Stylized Facts of Correlations
• Correlation clustering:
– periods of high (low) correlation are likely to be followed by periods of high (low) correlation
• Asymmetry and co-movement with volatility:
– high volatility in falling markets goes hand in hand with a strong increase in correlation, but this is not the case for rising markets
• This reduces opportunities for diversification in stock-market declines.
Monitoring Correlation Between Two Variables X and Y
Define xi=(Xi−Xi-1)/Xi-1 and yi=(Yi−Yi-1)/Yi-1
Also
varx,n: daily variance of X calculated on day n-1
vary,n: daily variance of Y calculated on day n-1
covn: covariance calculated on day n-1
The correlation is nynx
n
,, varvar
cov
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Covariance
• The covariance on day n is
E(xnyn)−E(xn)E(yn)
• It is usually approximated as E(xnyn)
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Monitoring Correlation continued
EWMA:
GARCH(1,1)
111 )1(covcov nnnn yx
111 covcov nnnn yx
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Correlation for Multivariate Case
• If X is m-dimensional and Y n-dimensional then
• Cov(X,Y) is given by the m×n-matrix with entries Cov(Xi, Yj )
= Cov(X,Y) is called covariance matrix
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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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0ww T
Example
The variance covariance matrix
is not internally consistent. When w=[1,1,-1] the condition for positive semidefinite is not satisfied.
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1 0 0 9
0 1 0 9
0 9 0 9 1
.
.
. .
Correlation as a Measure of Dependence
• Correlation as a measure of dependence fully determines the dependence structure for normal distributions and, more generally, elliptical distributions while it fails to do so outside this class.
• Even within this class correlation has to be handled with care: while a correlation of zero for multivariate normally distributed RVs implies independence, a correlation of zero for, say, t-distributed rvs does not imply independence
Multivariate Normal Distribution
• Fairly easy to handle
• A variance-covariance matrix defines the variances of and correlations between variables
• To be internally consistent a variance-covariance matrix must be positive semidefinite
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Bivariate Normal PDF• Probability density function of a
bivariate normal distribution:
2121
2121
2
1
σ)X,Cov(X
)X,Cov(XσMatrix Covariance
μ
μVector Mean
Σ
μ
),(~X
X
2
1 ΣμX MVN
)]()'exp[(||2
1),( 1
21 μxΣμxΣ
xxf
X and Y Bivariate Normal
• Conditional on the value of X, Y is normal with mean
and standard deviation where X, Y, X, and Y are the unconditional means and SDs of X and Y and xy is the coefficient of correlation between X and Y
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X
XYXYY σ
μXσρμ
2XYY ρ1σ
Generating Random Samples for Monte Carlo Simulation
• =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 generate random samples
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Bivariate Normal PDF independence
0.50)X|0Pr(Y0,ρ,10
01,
0
0
Σμ
Bivariate Normal PDF dependence
570.0)X|0Pr(Y,4830.ρ,1ρ
ρ1,
0
0
Σμ
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
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One-Factor Model continued
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• If Ui have standard normal distributions we can set
where the common factor F and the idiosyncratic component Zi have independent standard normal distributions
• Correlation between Ui and Uj is ai aj
iiii ZaFaU 21
Copulas
• A powerful concept to aggregate the risks — the copula function — has been introduced in finance by Embrechts, McNeil, and Straumann [1999,2000]
• A copula is a function that links univariate marginal distributions to the full multivariate distribution
• This function is the joint distribution function of N
standard uniform random variables.
Copulas• The dependence relationship between two random variables X and Y is obscured by the marginal densities of X and Y
• One can think of the copula density as the density that filters or extracts the marginal information from the joint distribution of X and Y.
• To describe, study and measure statistical dependence between random variables X and Y one may study the copula densities.
• Vice versa, to build a joint distribution between two random variables X ~G() and Y~H(), one may construct first the copula on [0,1]2 and utilize the inverse transformation and G-1() and H-1().
Cumulative Density Function Theorem
• Let X be a continuous random variable with distribution function F()
• Let Y be a transformation of X such that
Y=F(X).
• The distribution of Y is uniform on [0,1].
Sklar’s (1959) Theorem- The Bivariate Case
• X, Y are continuous random variables such that X ~G(·), Y ~ H(·)
• G(·), H(·): Cumulative distribution functions – cdf’s
• Create the mapping of X into X such that X=G(X ) then X has a Uniform distribution on [0,1] This mapping is called the probability integral transformation e.g. Nelsen (1999).
• Any bivariate joint distribution of (X ,Y ) can be transformed to a bivariate copula (X,Y)={G(X ), H(Y )} –Sklar (1959).
• Thus, a bivariate copula is a bivariate distribution with uniform marginal disturbutions (marginals).
CopulaMathematical Definition
• A n-dimensional copula C is a function which is a cumulative distribution function with uniform marginals:
• The condition that C is a distribution function leads to the following properties– As cdfs are always increasing is increasing in each
component ui.
– The marginal component is obtained by setting uj = 1 for all j i and it must be uniformly distributed,
– For ai<bi the probability
must be non-negative
)u,.....,C(u)C( n1u
)u,.....,C(u)C( n1u
ii u)1...,1,u,1,...,1C()C( u
])b,[aU],...,b,[aPr(U nnn111
An Example
N
1ii
N
1iiipf 1w ,SwV
Let Si be the value of Stock i. Let Vpf be the value of a portfolio
5% Value-at-Risk of a Portfolio is defined as follows:
Gaussian Copulas have been used to model dependence between (S1, S2, …..,Sn)
0.05VaR)Pr(Vpf
Copulas Derived from Distributions
• Typical multivariate distributions describe important dependence structures. The copulas derived can be derived from distributions.
• The multivariate normal distribution will lead to the Gaussian copula.
• The multivariate Student t-distribution leads to the t-copula.
Gaussian Copula Models: • 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
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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
V1V2
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
U1U2
One-to-one mappings
Correlation Assumption
V1V2
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
U1U2
One-to-one mappings
Correlation Assumption
Example (page 211)
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V1 V2
V1 Mapping to U1
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V1 Percentile (probability)
U1
0.2 0.20 -0.84
0.4 0.55 0.13
0.6 0.80 0.84
0.8 0.95 1.64
Use function NORMINV in Excel to get values in for U1
V2 Mapping to U2
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V2 Percentile (probability)
U2
0.2 0.08 −1.41
0.4 0.32 −0.47
0.6 0.68 0.47
0.8 0.92 1.41
Use function NORMINV in Excel to get values in for U2
Example of Calculation of Joint Cumulative Distribution
• Probability that V1 and V2 are both less than 0.2 is the probability that U1 < −0.84 and U2 < −1.41
• When copula correlation is 0.5 this is
M( −0.84, −1.41, 0.5) = 0.043
where M is the cumulative distribution function for the bivariate normal distribution
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Gaussian Copula – algebraic relationship
• Let G1 and G2 be the cumulative marginal probability distributions of V1 and V2
• Map V1 = v1 to U1 = u1 so that
• Map V2 = v2 to U2 = u2 so that
is the cumulative normal distribution function
)(u)(vG 111
)(u)(vG 222
)](u[Gv and )](u[Gv
)](v[Gu and )](v[Gu
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2211
11
221
2111
1
Gaussian Copula – algebraic relationship
• U1 and U2 are assumed to be bivariate normal
• The two-dimensional Gaussian copula
where is the 22 matrix with 1 on the diagonal and correlation coefficient otherwise. denotes the cdf for a bivariate normal distribution with zero mean and covariance matrix .
• This representation is equivalent to
)])(v[G)],(v[G()u,(uC 221
111
21Gaρ
212
2221
21
u u
2dsds)
)ρ2(1
ssρs2sexp(
ρ1π2
11 2
Bivariate Normal Copulaindependence
( , )BVN X Y Independence ( , )BVNCopula X Y Independence
21 2 1 2~ [0,1], ~ [0,1], ( , ) ~ ( , ) 1, ( , ) [0,1]X U Y X Y c u u u u
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Bivariate Normal Copula dependence
( , )BVN X Y Dependence ( , )BVNCopula X Y Dependence
21 2 1 2~ [0,1], ~ [0,1], ( , ) ~ ( , ), ( , ) [0,1]X U Y X Y c u u u u
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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
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
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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, Ti, to a new
variable Ui and assume
where F and the Zi have independent standard normal distributions
• The copula correlation is =a2
• Define Qi as the cumulative probability distribution of Ti
• Prob(Ui<U) = Prob(Ti<T) when N(U) = Qi(T)
iiii ZaFaU 21
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Analysis• To analyze the model we
– Calculate the probability that, conditional on the value of F, Ui is less than some value U
– This is the same as the probability that Ti is less that T where T and U are the same percentiles of their distributions
– And
– This is also Prob(Ti<T|F)
1
2 FUZ i
11Pr)|(Pr
FUN
FUobFUUob i
Analysis (cont.)
This leads to
where PD is the probability of default in time T
1
PD)(Prob
1 FNNFTTi
)]([1 TQNU
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
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1
)()]([ 11 XNTQNNWCDR(T,X)
),()1(),( XTWCDRRLXTVaR
The Model continued
ncorrelatio copula the is where
Prob
companies all for same the are s' and s' the Assuming
Prob
Hence
Prob
1
)()(
1
)()(
1)(
1
2
1
2
FTQNNFTT
aQ
a
FaTQNNFTT
a
FaUNFUU
i
i
iii
i
ii
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Appendix 1: Sampling from Bivariate Normal Distribution
Appendix 2: Sampling from Bivariate t Distribution
Appendix 3: Gaussian Copula with Student t Distribution
• Sample U1 and U2 from a bivariate normal distribution with the given correlation .
• Convert each sample into a variable with a Student t-distribution on a percentile-to-percentile basis.
• Suppose that U1 is in cell C1. The Excel function TINV gives a “two-tail” inverse of the t-distribution. An Excel instruction for determining V1 is therefore
=IF(NORMSDIST(C1)<0.5,TINV(2*NORMSDIST(C1),df),TINV(2*(1-NORMSDIST(C1)),df))
where df stands for degrees of freedom parameter