Correlation

Correlation is a statistical measure of the degree to which two variables move together. In finance, correlation is commonly used to measure the strength and direction of the relationship between the returns of two assets.

The Pearson correlation coefficient is commonly used to measure linear correlation.

A correlation coefficient ranges from −1 to +1:

$-1\le {\rho }_{XY}\le 1$

where:

• ${\rho }_{XY}=+1$ indicates perfect positive linear correlation;
• ${\rho }_{XY}=0$ indicates no linear correlation;
• ${\rho }_{XY}=-1$ indicates perfect negative linear correlation.

Pearson correlation coefficient

For two variables $X$ and $Y$, the Pearson correlation coefficient can be calculated as:

${\rho }_{XY}=\frac{\mathrm{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$

where:

• ${\rho }_{XY}$ = correlation coefficient;
• $\mathrm{Cov}(X,Y)$ = covariance between $X$ and $Y$;
• ${\sigma }_X$ = standard deviation of $X$;
• ${\sigma }_Y$ = standard deviation of $Y$.

For a sample of observations, the correlation coefficient can also be written as:

$r_{XY}=\frac{\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{\sum _{i=1}^n(x_i-\overline{x}{)}^2\sum _{i=1}^n(y_i-\overline{y}{)}^2}}$

where:

• $x_i$ and $y_i$ = individual observations;
• $\overline{x}$ and $\overline{y}$ = sample means;
• $n$ = number of observations.

Positive correlation

Two variables have positive correlation when they tend to move in the same direction.

For example, if the returns of two assets generally increase and decrease together, their correlation may be positive.

A simplified relationship is:

${\rho }_{XY}>0\Rightarrow X\uparrow \text{is generally associated with}Y\uparrow$

Positive correlation does not necessarily mean that one variable causes the other to change.

Negative correlation

Two variables have negative correlation when they tend to move in opposite directions.

A simplified relationship is:

${\rho }_{XY}<0\Rightarrow X\uparrow \text{is generally associated with}Y\downarrow$

A correlation of −1 represents perfect negative linear correlation.

Zero correlation

A correlation coefficient of zero indicates the absence of a linear relationship between two variables:

${\rho }_{XY}=0$

Zero correlation does not necessarily mean that the variables are completely unrelated. They may have a nonlinear relationship that is not captured by the Pearson correlation coefficient.

Correlation in finance

Correlation is important in portfolio management because the relationship between asset returns affects overall portfolio risk.

For a two-asset portfolio, portfolio variance can be expressed as:

${\sigma }_p^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\sigma }_1{\sigma }_2{\rho }_{12}$

where:

• ${\sigma }_p$ = portfolio standard deviation;
• $w_1,w_2$ = portfolio weights;
• ${\sigma }_1,{\sigma }_2$ = standard deviations of the two assets;
• ${\rho }_{12}$ = correlation between the assets.

The correlation term affects the amount of diversification achieved by combining the assets.

Correlation and diversification

When two assets have a correlation below +1, combining them can potentially reduce portfolio volatility relative to holding the assets independently in the same proportions.

The strongest diversification benefit from correlation alone occurs when the correlation approaches −1.

A simplified relationship is:

${\rho }_{12}\downarrow \Rightarrow \text{Potential Diversification Benefit}\uparrow$

However, the actual effect on portfolio risk also depends on asset weights and individual asset volatilities.

Correlation matrix

A correlation matrix displays the pairwise correlations among multiple assets or variables.

For three assets, a simplified correlation matrix is:

$𝐑=\left[\begin{array}{ccc}1& {\rho }_{12}& {\rho }_{13}\\ {\rho }_{21}& 1& {\rho }_{23}\\ {\rho }_{31}& {\rho }_{32}& 1\end{array}\right]$

The diagonal elements are equal to 1 because each variable is perfectly correlated with itself.

Correlation is symmetric:

${\rho }_{ij}={\rho }_{ji}$

Correlation and covariance

Correlation is closely related to covariance.

Covariance measures the direction in which two variables tend to move together but is dependent on the scale of the variables.

Correlation standardizes covariance by the standard deviations of the two variables:

${\rho }_{XY}=\frac{\mathrm{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$

Consequently, correlation is dimensionless and always lies between −1 and +1.

Correlation and regression

Correlation and regression are related but serve different purposes.

Correlation measures the strength and direction of a linear relationship, whereas regression can be used to estimate a relationship in which one variable is modeled as a dependent variable and another as an explanatory variable.

A simplified regression model is:

$Y=\alpha +\beta X+ϵ$

Correlation alone does not establish causation.

Correlation and beta

In finance, an asset's beta relative to a market can be expressed in terms of correlation:

${\beta }_i={\rho }_{i,m}\frac{{\sigma }_i}{{\sigma }_m}$

where:

• ${\beta }_i$ = beta of asset $i$;
• ${\rho }_{i,m}$ = correlation between asset $i$ and the market;
• ${\sigma }_i$ = standard deviation of the asset's returns;
• ${\sigma }_m$ = standard deviation of market returns.

Thus, correlation is one component of the relationship between an asset's volatility and its sensitivity to market movements.

Limitations

Correlation has several limitations:

• it measures linear association when using Pearson correlation;
• it does not establish causation;
• historical correlation may change over time;
• correlations can increase during periods of market stress;
• correlation does not measure the magnitude of price movements;
• a low correlation does not guarantee low portfolio risk.

Financial correlations are therefore commonly analyzed over different time periods and under different market conditions.

See also

• Covariance
• Pearson correlation coefficient
• Standard deviation
• Variance
• Beta (finance)
• Regression analysis
• Portfolio (finance)
• Diversification (finance)
• Volatility (finance)
• Risk management
• Correlation matrix