When the ratio of variations in the related variables is constant it is called positive correlation?

2.3.1 Covariance

In Section 1.5.1 the concept of the standard deviation was introduced as a measure of variation for a single continuous or discrete variable. This concept can be extended to the case where we have multiple variables. The corresponding measure of variation for multiple variables (i.e. multivariate data) is known as covariance. Covariance is a measure of how much the variations of two variables are related. A positive covariance between two variables reveals that the paired values of both variables tend to increase together. A negative covariance reveals that there is an inverse relationship between the variables, that is, as one increases, the other tends to decrease. A zero covariance indicates that there is no link between the values of the two variables.

For two paired variables xi and yi, i=1,…,n (where n is the sample size), the covariance is defined by

(2.1)cov(x,y)=1n−1∑i=1n(x i−x¯)(yi−y ¯).

Interpretation of the magnitude of the covariance is not straightforward. A quick examination of Eq. (2.1) reveals that the covariance will be larger if the values of the variables themselves are larger. So without some form of normalization it is not possible to use covariance to determine how strong the relationship between the variables is. Pearson's correlation coefficient (see Section 2.4.1) does exactly this by normalizing the covariance by the product of the standard deviations of the variables.

5.1.1 Compound symmetry (CS)

In Chapter 3, I described the covariance structure for the random intercept model. Specifically, if b0i is considered to be the only random effect in a linear mixed model, variance of the between-subjects random effects is constant throughout all time points and so is covariance between two subject-specific observations. Such a covariance structure is referred to as compound symmetry (CS). In longitudinal data analysis, the CS covariance structure can also be specified for the R-sided matrix.

Let σ2 be the constant variance in mean μ across repeated measurements of the response and σ1 be the constant and positive covariance between any two successive observations. Given four equally spaced time points, the residual CS covariance structure can be written as

(5.1)R=σ 2σ1σ1σ1σ1σ2σ1σ1σ1σ1σ2σ 1σ1σ1σ1σ2,

where R, as defined previously, is the residual variance–covariance matrix for within-subject repeated measurements.

Given the intimate association between covariance and correlation, Equation (5.1) can be expressed in terms of a constant correlation coefficient by factoring σ2 out, written as

(5.2)R=σ21ρρρρ1ρρ ρρ1ρρρρ1.

The variance–covariance structure of CS is the simplest and the most parsimonious pattern model designed for longitudinal analysis. This model includes only two variance parameters, σ2 and σ1. If longitudinal data follow the pattern of constant variance and constant covariance across all time points, intraindividual correlation is addressed by the specification of σ1. Given this type of data, a linear regression model including a residual variance–covariance matrix of CS can yield statistically efficient and consistent parameter estimates. In some clinical experimental designs, where repeated measurements are narrowly spaced with equal distance, the use of CS in longitudinal analysis is effective. In most occasions, however, the hypothesis of CS on residuals is too strong to specify the R-sided covariance matrix in linear regression models. In the vast majority of longitudinal data, particularly those from observational surveys, intraindividual correlation is not constant over time without the specification of the between-subjects random effects. As a result, the CS covariance structure is empirically much less applied than the other more complex pattern models.

The case of two assets

We will denote the portfolio return with rp as follows (sometimes it is also referred to as the expected return of the portfolio):

rp=w1r1+w2r2.

The return is always associated to the risk of a portfolio, and finance takes from statistics its measure of risk, which is a measure of dispersion around a central tendency of the given random variable. The standard deviation or equivalently its square, the variance, is taken therefore as one of the principal risk indicators of a portfolio. See also Section 13.3.

Theory of finance introduced as well some other “combined” measures, like the portfolio returns adjusted by the risk, and within this class of risk-adjusted performance indicators, the Sharpe Ratio plays a key role.

The variance of a portfolio of two assets is calculated as follows (from the property of the variance of two variables in statistics and probability theory):

σp2=w12σ12+w22σ22+2w1w2ρ12σ1σ2

where

σ1=standarddeviation ofasset1

σ1=sta ndarddeviationofasset2

ρ12=Cov(r1,r2)σ1σ2correlationcoefficientbetwee nthetwofinancialassets.

The index of covariance Cov(r1, r2) is an index to study the relationship between two assets returns (normally the fund's returns and the benchmark's returns) which represent the statistical (or random) variables we are studying. The covariance ranges from negative values to positive values. A positive covariance indicates that the two variables tend to move together and with the same sign, a negative covariance indicates that the two variables tend to move in the opposite direction. A covariance close to zero indicates that the relation between the fund's returns and the benchmark's returns is neutral and therefore there is little relationship between the two variables.

The formula of the covariance considers the two variables (r1, r2) at the same time, and it is as follows:

Cov(r1,r2) =1N∑i=1N(r1i−μ1)⋅(r2i−μ2)

with μ1 and μ2 the expected (average) returns of the first and the second assets, respectively.

To measure the correlation between two variables, we calculate the correlation coefficient ρ12 with:

−1≤ρ12≤ +1.

Since via the covariance we can only recognize the magnitude of the relationship in absolute terms, this index helps us to investigate more about the magnitude of the relationship in relative terms. This means that ρ12 ranges from −1 to +1, and therefore we can recognize immediately the strength and the quality of the relationship between the two variables that we are studying.

When we find values of ρ12 = 1 we have got perfect positive correlation, when we find values of ρ12 = −1 we have got perfect negative correlation. Values close to zero indicate neutral correlation between the two variables, in other words they indicate independence between the two variables.

Let us represent the variances and covariances of the portfolio in the following symmetric matrix, called the matrix of variances and covariances associated to the portfolio p:

Σ=[Cov(r1,r1)Cov(r1, r2)Cov(r2,r1)Cov(r2,r2)]

then σp 2 can be represented as a quadratic form notation (see Eq. 3.4-2):

σp2=wTΣ w

which is, recalling that Cov(ri,ri)=σi2:

σp2=[w1w2] [σ12Cov(r1,r 2)Cov(r2,r1)σ22][w1w2]

σp2=[w1σ12+w2Cov(r2,r1)w1 Cov(r1,r2)+w2σ 22][w1w2]

which corresponds to a scalar, namely the sought portfolio variance:

σp2=w12σ12+w1 w2Cov(r2,r1)+w1 w2Cov(r1,r2)+w2 2σ22

=w12σ12+w22 σ22+2w1w2Cov(r1,r2)

=w12σ12+w22 σ22+2w1w2ρ12σ1σ2 .

5.5.1 Covariance and Correlation

Let X and Y be random variables with, respective, expectations μx=E[X] and μy=E[Y]. The covariance of X and Y is defined by

Cov(X,Y) =E[(X−μx)(Y−μy)].

That is, the covariance of X and Y is the expected value of the product of X minus its expected value and Y minus its expected value. By expanding the product (X−μx )(Y−μy) and then using that the expected value of a sum is equal to the sum of the expected values, we can show that the covariance of X and Y can, alternatively, be expressed as the expected value of the product of X and Y minus the product of their expectations. We now give that derivation.

(5.5)Cov(X,Y)=E[(X−μx)(Y−μy)]

(5.6)=E[XY−Xμy−μ xY+μxμy]=E[XY]−E[Xμy]−E[ μxY]+E[μxμy]=E[XY]−μyE[X]−μxE[Y]+μxμy =E[XY]−μyμ x−μxμy+μxμy=E[XY]−μxμy

where the preceding used that the expected value of a constant times a random variable is just the constant times the expected value of the random variable, and that the expected value of a constant is just the constant.

To understand the significance of the covariance, consider a situation where X and Y have the property that when X is large then Y also tends to be large, and when X is small then Y also tends to be small. However, to understand the preceding, we have to first decide what it means to say that “X is large” or that “X is small.” Now, because E[X] is the average value of X, one reasonable interpretation of the preceding is that X is large when X is larger than E[X], and that X is small when X is smaller than E[X]. Thus, to say that “when X is large then Y also tends to be large” is to say that when X>E[X] then Y tends to be larger than E[Y]; and to say that when X is small then Y also tends to be small, is to say that when X<E[X] then Y tends to be smaller than E[Y]. But this is equivalent to saying that X−E[X] and Y−E[Y] typically have the same sign (that is, typically either both are positive or both are negative). But if X−E [X] and Y−E[Y] have the same sign, then their product (X−E[X ])(Y−E[Y]) will be positive, and thus its expected value E[(X−E[X])(Y−E[Y])] will be positive. That is, Cov(X,Y)>0 when X and Y tend to either both be large or both be small.

Now suppose X and Y have the property that when one is large the other tends to be small. Thus, when X> E[X] then, typically, Y<E[Y]. But then X−E[X] and Y−E[Y] typically have opposite signs (when one is positive the other is negative) and so their product will tend to be negative, and thus, E[(X−E [X])(Y−E[Y])] will be negative. Consequently, Cov(X,Y)<0 if when one is large the other tends to be small.

Whereas a positive value of the covariance between X and Y indicates that when one is larger than its mean then the other also tends to be larger than its mean, and a negative value indicates that when one is larger than its mean then the other tends to be smaller than its mean, the correlation measures the strength of the relationship. Its value is always between −1 and 1, with a large positive value of the correlation indicating that large values of one of the variables strongly go along with large values of the other, and a large negative value indicating that large values of one are almost always associated with small values of the other. A correlation near 0 indicates that the relationship is weak. The correlation is obtained by dividing the covariance by the product of the standard deviations of X and of Y. That is,