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covariance

(kō-vâr'ē-əns) pronunciation

A statistical measure of the variance of two random variables that are observed or measured in the same mean time period. This measure is equal to the product of the deviations of corresponding values of the two variables from their respective means.

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Investment Dictionary: Covariance

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.

Related Links:
See why investors today still follow this set of principles to reduce risk and increase returns through diversification. Modern Portfolio Theory: An Overview
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Volatility is not the only way to measure risk. Learn about the "new science of risk management". Introduction to Value at Risk (VAR) - Part 2

Business Dictionary: Covariance

Statistical term for the correlation between two variables multiplied by the standard deviation for each of the variables. A positive covariance indicates that the two variables tend to move up and down together; a negative covariance indicates that when one moves higher, the other tends to go lower.

Geography Dictionary: covariance

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(x_i - x)(y_i - 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.

Veterinary Dictionary: covariance

The expected value of the product of the deviations of corresponding values of two random variables from their respective means.

c. method — used for the calculation of relationship and inbreeding in large populations.

Wikipedia: covariance

For the physics topics, see covariant transformation; about the mathematics example for groupoids, see covariance in special relativity; for the computer science topic see Covariance and contravariance (computer science).

In probability theory and statistics, covariance is the measure of how much two random variables vary together (as distinct from variance, which measures how much a single variable varies). If two variables tend to vary together (that is, when one of them is above its expected value, then the other variable tends to be above its expected value too), then the covariance between the two variables will be positive.

On the other hand, if one of them is above its expected value and the other variable tends to be below its expected value, then the covariance between the two variables will be negative.

The covariance between two real-valued random variables X and Y, with expected values Failed to parse (unknown function\scriptstyle): \scriptstyle E(X)\,=\,\mu

and Failed to parse (unknown function\scriptstyle): \scriptstyle E(Y)\,=\,\nu
is defined as

$\operatorname{Cov}(X, Y) = \operatorname{E}((X - \mu) (Y - \nu)), \,$

where E is the expected value operator. This can also be written:

$\operatorname{Cov}(X, Y) = \operatorname{E}(X \cdot Y) - \mu \nu. \,$

If X and Y are independent, then their covariance is zero. This follows because under independence,

$E(X \cdot Y)=E(X) \cdot E(Y)=\mu\nu.$

Recalling the second form of the covariance given above, and substituting, we get

$\operatorname{Cov}(X, Y) = \mu \nu - \mu \nu = 0.$

The converse, however, is not true: if X and Y have covariance zero, they need not be independent.

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.

Random variables whose covariance is zero are called uncorrelated.

Properties

If X, Y are real-valued random variables and a, b are constant ("constant" in this context means non-random), then the following facts are a consequence of the definition of covariance:

$\operatorname{Cov}(X, X) = \operatorname{Var}(X)\,$

$\operatorname{Cov}(X, Y) = \operatorname{Cov}(Y, X)\,$

$\operatorname{Cov}(aX, bY) = ab\, \operatorname{Cov}(X, Y)\,$

$\operatorname{Cov}(X+a, Y+b) = \operatorname{Cov}(X, Y)\,$

$\operatorname{Cov}(aX+bY, cW+dV) = ac\,\operatorname{Cov}(X,W)+ad\,\operatorname{Cov}(X,V)+bc\,\operatorname{Cov}(Y,W)+bd\,\operatorname{Cov}(Y,V)\,$

For sequences X₁, ..., X_n and Y₁, ..., Y_m of random variables, we have

$\operatorname{Cov}\left(\sum_{i=1}^n {X_i}, \sum_{j=1}^m{Y_j}\right) = \sum_{i=1}^n{\sum_{j=1}^m{\operatorname{Cov}\left(X_i, Y_j\right)}}.\,$

For a sequence X₁, ..., X_n of random variables, we have

$\operatorname{Var}\left(\sum_{i=1}^n X_i \right) = \sum_{i=1}^n \operatorname{Var}(X_i) + 2\sum_{i,j\,:\,i<j} \operatorname{Cov}(X_i,X_j).$

Relationship to inner products

Many of the properties of covariance can be extracted elegantly by observing that it satisfies the abstract properties of an inner product:

(1) bilinear: for constants a and b and random variables X, Y, and U, Cov(aX + bY, U) = a Cov(X, U) + bCov(Y, U)

(2) symmetric: Cov(X, Y) = Cov(Y, X)

(3) positive definite: Var(X) = Cov(X, X) ≥ 0, and Cov(X, X) = 0 implies that X is a constant random variable (K).

It follows that covariance is an inner product over a vector space whose elements are random variables, with the vector addition operation defined as the sum of random variables X + Y and the scalar multiplication operation defined as multiplication of a random variable by a scalar, aX. Here, the "random variable" equated with a single vector is actually a probability distribution, not just a particular value that the random variable might take.

Covariance matrices and operators

For column-vector valued random variables X and Y with respective expected values μ and ν, and respective scalar components m and n, the covariance is defined to be the m×n matrix called the covariance matrix:

$\operatorname{Cov}(X, Y) = \operatorname{E}((X-\mu)(Y-\nu)^\top).\,$

For vector-valued random variables, Cov(X, Y) and Cov(Y, X) are each other's transposes.

Even more generally, for a probability measure P on a Hilbert space H with inner product 〈 , 〉, the covariance operator of P is the operator Cov : H → H given by

$\langle \mathrm{Cov} x, y \rangle = \int_{H} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \mathbf{P} (z)$

for all x and y in H. Cov is a self-adjoint operator (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case); when P is a centred Gaussian measure, Cov is also a nuclear operator.

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.

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		Investment Dictionary. Copyright ©2000, Investopedia.com - Owned and Operated by Investopedia Inc. All rights reserved. Read more
		Business Dictionary. Dictionary of Business Terms. Copyright © 2000 by Barron's Educational Series, Inc. All rights reserved. Read more
		Geography Dictionary. A Dictionary of Geography. Copyright © Susan Mayhew 1992, 1997, 2004. All rights reserved. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Covariance". Read more

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Covariance matrices and operators

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