6-ConceptsProbability Page 188 Wednesday, February 4, 2004 3:00 PM
188 The Mathematics of Financial Modeling and Investment Management
The covariance divided by the product of the standard deviations is a
dimensionless number called the correlation coefficient.
Given two random variables X,Ywith finite expected values and
finite variances, we can write the following definitions:
■ cov(XY, )= σXY, = EX X [( – )(YY– )]is the covariance of X,Y.
σXY,
■ ρXY, = -------------- is the correlation coefficient of X,Y.
σXσY
The correlation coefficient can assume values in the interval [–1,1].
If two variables X,Yare independent, their correlation coefficient van-
ishes. However, uncorrelated variables, that is, variables whose correla-
tion coefficient is zero, are not necessarily independent.
It can be demonstrated that the following property of variances holds:
i
var∑X = ∑var ()Xi + ∑cov(Xi,Xj)
i i ij≠
Further, it can be demonstrated that the following properties hold:
σXY, = EXY[ ]– EX[]EY[]
σXY, = σYX,
σaX bY , = abσYX,
σXY+ ,Z = σXZ, + σYZ,
cov∑a Xi,∑bjYj = ∑∑aibjcov(Xi,Yj)
i
i i i j
COPULA FUNCTIONS
Understanding dependences or functional links between variables is a
key theme of modern econometrics. In general terms, functional depen-
dences are represented by dynamic models. As we will see in Chapter
11, many important models are linear models whose coefficients are