The Mathematics of Financial Modelingand Investment Management

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