
The covariance of two random variables is the difference between the expected value of their product and the product of their separate expected values. For random variables X and Y,Cov(X, Y)=E(XY)-E(X)×E(Y). If X and Y are independent then Cov(X, Y)=0. However, if Cov(X, Y)=0 then X and Y may not be independent. A useful result isVar(aX+bY)=a2Var(X)+2abCov(X, Y)+b2Var(Y),where Var denotes variance, and a and b are constants. The term 'covariance' was used by Sir Ronald Fisher in 1930. See also correlation.
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An index of the degree of association between two groups of variables. It is calculated in samples by transforming the scores in both groups to deviations from their respective means, multiplying pairs of scores from each group together, and calculating the mean. If the variables are x and y then the covariance of x and y is S(xi - x)(yi - y).
In the World Bank Poverty Profile, the degree to which values co-vary with a poverty indicator determines how much the total variation has been ‘explained’, and which variables contribute to that explanation.
A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns move inversely.
One method of calculating covariance is by looking at return surprises (deviations from expected return) in each scenario. Another method is to multiply the correlation between the two variables by the standard deviation of each variable.
Investopedia Says:
Possessing financial assets that provide returns and have a high covariance with each other will not provide very much diversification.
For example, if stock A's return is high whenever stock B's return is high and the same can be said for low returns, then these stocks are said to have a positive covariance. If an investor wants a portfolio whose assets have diversified earnings, he or she should pick financial assets that have low covariance to each other.
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See why investors today still follow this old set of principles that reduce risk and increase returns through diversification. Modern Portfolio Theory: Why It's Still Hip
Volatility is not the only way to measure risk. Learn about the "new science of risk management". An Introduction To Value at Risk (VAR)
Volatility is not the only way to measure risk. Learn about the "new science of risk management".
How To Convert Value At Risk To Different Time Periods
What to know about stationary and non-stationary processes before you try to model or forecast. Introduction To Stationary And Non-Stationary Processes
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The expected value of the product of the deviations of corresponding values of two random variables from their respective means.
In probability theory and statistics, covariance is a measure of how much two random variables change together. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the smaller values, i.e. the variables tend to show similar behavior, the covariance is a positive number. In the opposite case, when the greater values of one variable mainly correspond to the smaller values of the other, i.e. the variables tend to show opposite behavior, the covariance is negative. The sign of the covariance therefore shows the tendency in the linear relationship between the variables. The magnitude of the covariance is not that easy to interpret. The normalized version of the covariance, the correlation coefficient, however shows by its magnitude the strength of the linear relation.
A distinction must be made between (1) the covariance of two random variables, which is a population parameter that can be seen as a property of the joint probability distribution, and (2) the sample covariance, which serves as an estimated value of the parameter.
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The covariance between two jointly distributed real-valued random variables X and Y with finite second moments is
![\operatorname{Cov}(X,Y) = \operatorname{E}{\big[(X - \operatorname{E}[X])(Y - \operatorname{E}[Y])\big]}](http://wpcontent.answcdn.com/wikipedia/en/math/6/5/b/65b982bb5a97044743220aaf4f893a59.png)
where E[X] is the expected value of X. By using the linearity property of expectations, this can be simplified to
![\operatorname{Cov}(X,Y) = \operatorname{E}\big[X Y\big] - \operatorname{E}[X]\operatorname{E}[Y]](http://wpcontent.answcdn.com/wikipedia/en/math/5/0/4/504130ab84df630da90154c4e103f2dc.png)
For random vectors X and Y (of dimension m and n respectively) the m×n covariance matrix is equal to
![\begin{align}
\operatorname{Cov}(X,Y)
& = \operatorname{E}\left[(X - \operatorname{E}[X])(Y - \operatorname{E}[Y])^T\right]\\
& = \operatorname{E}\left[X Y^T\right] - \operatorname{E}[X]\operatorname{E}[Y]^T
\end{align}](http://wpcontent.answcdn.com/wikipedia/en/math/e/a/9/ea9eba835c4c27833e59ba6a2959d7f7.png)
where MT is the transpose of a matrix (or vector) M.
The (i,j)-th element of this matrix is equal to the covariance Cov(Xi, Yj) between the i-th scalar component of X and the j-th scalar component of Y. In particular, Cov(Y, X) is the transpose of Cov(X, Y).
For a vector
of n jointly distributed random variables with finite second moments, its covariance matrix is defined as:

Random variables whose covariance is zero are called uncorrelated.
The units of measurement of the covariance Cov(X, Y) are those of X times those of Y. By contrast, correlation, which depends on the covariance, is a dimensionless measure of linear dependence. (In fact, correlation can simply be understood as a normalized version of covariance.)


For sequences X1, ..., Xn and Y1, ..., Ym of random variables, we have

For a sequence X1, ..., Xn of random variables, and constants a1, ..., an, we have

Let
be a random vector, and let
denote its covariance matrix, and let
be a matrix that can act on
. Then 
If X and Y are independent, then their covariance is zero. This follows because under independence,
![\operatorname{E}\left[X \cdot Y\right] = E[X] \cdot E[Y].](http://wpcontent.answcdn.com/wikipedia/en/math/8/2/c/82ca14e29211e579d7228d895a597c92.png)
The converse, however, is not generally true. For example, let X be uniformly distributed in [-1, 1] and let Y = X2. Clearly, X and Y are dependent, but
![\begin{align}
\operatorname{Cov}(X, Y) &= \operatorname{Cov}(X, X^2) \\
&= \operatorname{E}\!\left[X \cdot X^2\right] - \operatorname{E}[X] \cdot \operatorname{E}\!\left[X^2\right] \\
&= \operatorname{E}\!\left[X^3\right] - \operatorname{E}[X]\operatorname{E}\!\left[X^2\right] \\
&= 0 - 0 \cdot \operatorname{E}\!\left[X^2\right] \\
&= 0.
\end{align}](http://wpcontent.answcdn.com/wikipedia/en/math/5/8/d/58d498c85bdd60b384d48d299f848df5.png)
Many of the properties of covariance can be extracted elegantly by observing that it satisfies similar properties to those of an inner product:
In fact these properties imply that the covariance defines an inner product over the quotient vector space obtained by taking the subspace of random variables with finite second moment and identifying any two that differ by a constant. (This identification turns the positive semi-definiteness above into positive definiteness.) That quotient vector space is isomorphic to the subspace of random variables with finite second moment and mean zero; on that subspace, the covariance is exactly the L2 inner product of real-valued functions on the sample space.
As a result for random variables with finite variance the following inequality holds via the Cauchy–Schwarz inequality:

Proof: If Var(Y) = 0, then it holds trivially. Otherwise, let random variable

Then we have:
![\begin{align}
0 \le \operatorname{Var}(Z) & = \operatorname{Cov}\left(X - \frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(Y)} Y,X - \frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(Y)} Y \right) \\[12pt]
& = \operatorname{Var}(X) - \frac{ (\operatorname{Cov}(X,Y))^2 }{\operatorname{Var}(Y)}
\end{align}](http://wpcontent.answcdn.com/wikipedia/en/math/d/8/e/d8e975aa5b1d029d942b6ddeb235b6ed.png)
QED.
The sample covariance of N observations of K variables is the K-by-K matrix
with the entries given by

The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector
, a row vector whose jth element (j = 1, ..., K) is one of the random variables. The reason the sample covariance matrix has
in the denominator rather than
is essentially that the population mean
is not known and is replaced by the sample mean
. If the population mean
is known, the analogous unbiased estimate is given by

The covariance is sometimes called a measure of "linear dependence" between the two random variables. That does not mean the same thing as in the context of linear algebra (see linear dependence). When the covariance is normalized, one obtains the correlation matrix. From it, one can obtain the Pearson coefficient, which gives us the goodness of the fit for the best possible linear function describing the relation between the variables. In this sense covariance is a linear gauge of dependence.
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