Correlations and Copulas

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Correlations and Copulas 1

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Correlations and Copulas. Measures of Dependence. 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. - PowerPoint PPT Presentation

Transcript of Correlations and Copulas

Page 1: Correlations and Copulas

Correlations and Copulas

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Page 2: Correlations and Copulas

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

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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)

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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(

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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

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Types of Dependence

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E(Y)

X

E(Y)

E(Y)

X

(a) (b)

(c)

X

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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.

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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.

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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

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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

.

.

. .

Page 15: Correlations and Copulas

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

Page 16: Correlations and Copulas

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

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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

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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

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Bivariate Normal PDF dependence

570.0)X|0Pr(Y,4830.ρ,1ρ

ρ1,

0

0

Σμ

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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

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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.

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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().

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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].

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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).

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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

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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

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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.

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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

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Example (page 211)

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V1 V2

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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

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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

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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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Page 37: Correlations and Copulas

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

21

2211

11

221

2111

1

Page 38: Correlations and Copulas

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

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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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Page 40: Correlations and Copulas

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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Page 41: Correlations and Copulas

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 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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Page 45: Correlations and Copulas

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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Page 46: Correlations and Copulas

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

Page 48: Correlations and Copulas

Analysis (cont.)

This leads to

where PD is the probability of default in time T

1

PD)(Prob

1 FNNFTTi

)]([1 TQNU

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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

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1

)()]([ 11 XNTQNNWCDR(T,X)

),()1(),( XTWCDRRLXTVaR

Page 50: Correlations and Copulas

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

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Appendix 2: Sampling from Bivariate t Distribution

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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