Pearson correlation coefficient

Pearson correlation coefficient is a statistical measure of the strength and direction of the linear relationship between two variables. It is commonly used in statistics, economics, finance, and other quantitative fields.

The Pearson correlation coefficient is commonly denoted by $r$ for a sample and $\rho$ for a population.

Its value ranges from −1 to +1:

$-1\le r\le 1$

A value close to +1 indicates a strong positive linear relationship, a value close to −1 indicates a strong negative linear relationship, and a value close to 0 indicates a weak or absent linear relationship.

Formula

For a sample of $n$ paired observations, the Pearson correlation coefficient is:

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

where:

• $x_i$ = observation $i$ of variable $X$;
• $y_i$ = observation $i$ of variable $Y$;
• $\overline{x}$ = sample mean of $X$;
• $\overline{y}$ = sample mean of $Y$;
• $n$ = number of paired observations.

Covariance formulation

The Pearson correlation coefficient can also be expressed using covariance and standard deviations:

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

For sample data:

$r_{XY}=\frac{s_{XY}}{s_X{s}_Y}$

where:

• $s_{XY}$ = sample covariance;
• $s_X$ = sample standard deviation of $X$;
• $s_Y$ = sample standard deviation of $Y$.

This standardization makes correlation dimensionless and restricts its value to the interval from −1 to +1.

Interpretation

A positive Pearson correlation indicates that the two variables tend to increase or decrease together.

$r_{XY}>0$

A negative Pearson correlation indicates that one variable tends to increase when the other decreases.

$r_{XY}<0$

A value near zero indicates little or no linear association:

$r_{XY}\approx 0$

The sign indicates the direction of the linear relationship, while the absolute value indicates its strength.

Perfect correlation

A Pearson correlation of +1 represents a perfect positive linear relationship:

$r_{XY}=1$

A Pearson correlation of −1 represents a perfect negative linear relationship:

$r_{XY}=-1$

A perfect positive relationship can be represented as:

$Y=a+bX,b>0$

A perfect negative relationship can be represented as:

$Y=a+bX,b<0$

Zero correlation

A Pearson correlation of zero indicates that there is no linear correlation between the variables:

$r_{XY}=0$

However, zero Pearson correlation does not necessarily imply that the variables are independent. A nonlinear relationship may exist even when the Pearson correlation is zero.

Pearson correlation in finance

In finance, the Pearson correlation coefficient is commonly used to measure the relationship between the returns of two assets.

For two asset returns, it can be written as:

${\rho }_{ij}=\frac{\mathrm{Cov}(R_i,R_j)}{{\sigma }_i{\sigma }_j}$

where:

• $R_i$ = return of asset $i$;
• $R_j$ = return of asset $j$;
• ${\sigma }_i$ = standard deviation of asset $i$'s returns;
• ${\sigma }_j$ = standard deviation of asset $j$'s returns.

Correlation is an important input in portfolio risk analysis because the correlation between assets affects the volatility of a combined portfolio.

Portfolio application

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:

• $w_1,w_2$ = portfolio weights;
• ${\sigma }_1,{\sigma }_2$ = standard deviations of the assets;
• ${\rho }_{12}$ = Pearson correlation between the asset returns;
• ${\sigma }_p^2$ = portfolio variance.

A lower correlation between assets can provide greater potential diversification benefits, subject to the assets' weights and individual volatilities.

Correlation and regression

The Pearson correlation coefficient is closely related to linear regression.

A simple linear regression can be represented as:

$Y=\alpha +\beta X+ϵ$

The correlation coefficient measures the strength and direction of the linear association between $X$ and $Y$, whereas regression additionally estimates the relationship between the variables through parameters such as the slope and intercept.

For a simple linear regression, the coefficient of determination is related to Pearson correlation by:

$R^2=r^2$

where $R^2$ represents the proportion of variation in the dependent variable explained by the linear model in the simple-regression setting.

Correlation does not imply causation

A Pearson correlation measures statistical association and does not establish a causal relationship.

A high correlation between two variables may arise because:

• one variable influences the other;
• both variables are influenced by another factor;
• the relationship is coincidental;
• the variables share a common trend.

Therefore:

$\text{Correlation}\ne \text{Causation}$

Limitations

The Pearson correlation coefficient has several limitations:

• it measures linear association;
• it can be sensitive to outliers;
• it does not establish causation;
• historical correlations may change over time;
• it may be misleading when variables have strong nonlinear relationships;
• correlation does not measure the magnitude of individual price or return movements.

In financial markets, correlations can also change significantly during periods of market stress.

See also

• Correlation
• Covariance
• Standard deviation
• Variance
• Linear regression
• Coefficient of determination
• Beta (finance)
• Portfolio (finance)
• Diversification (finance)
• Volatility (finance)
• Risk management