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.
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E(YX)−E(Y)E(X)
where denotes the probability density function
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:
Correlation is invariant under linear transformations, but not under general transformations:
Example, two lognormal RVs have a different correlation than the underlying normal RVs
A small correlation does not imply a small degree of dependency.
Correlation clustering:
periods of high (low) correlation are likely to be followed by periods of high (low) correlation
Asymmetry and comovement 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 stockmarket declines.
Define xi=(Xi−Xi1)/Xi1 and yi=(Yi−Yi1)/Yi1
Also
varx,n: daily variance of X calculated on day n1
vary,n: daily variance of Y calculated on day n1
covn: covariance calculated on day n1
The correlation is
E(xnyn)−E(xn)E(yn)
A variancecovariance matrix, , is internally consistent if the positive semidefinite condition
holds for all vectors w
The variance covariance matrix
is not internally consistent. When w=[1,1,1] the condition for positive semidefinite is not satisfied.
Probability density function of a bivariate normal distribution:
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
independence
dependence
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 Uiand Ujis ai aj
OneFactor Model continuedA powerful concept to aggregate the risks — the copula function — has beenintroduced 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.
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].
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).
must be nonnegative
Let Si be the value of Stock i. Let Vpf be the value of a portfolio
5% ValueatRisk of a Portfolio is defined as follows:
Gaussian Copulas have been used to model dependence between (S1, S2, …..,Sn)
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The Correlation Structure Between the V’s is Defined by that Between the U’s0.2
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Use function NORMINV in Excel to get values in for U1
Use function NORMINV in Excel to get values in for U2
M( −0.84, −1.41, 0.5) = 0.043
where M is the cumulative distribution function for the bivariate normal distribution
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 .
In a factor copula model the correlation structure between the U’s is generated by assuming one or more factors.
where F and the Zi have independent standard normal distributions
where L is loan principal and R is recovery rate
=IF(NORMSDIST(C1)<0.5,TINV(2*NORMSDIST(C1),df),TINV(2*(1NORMSDIST(C1)),df))
where df stands for degrees of freedom parameter