Correlation Coefficient

The correlation coefficient is a statistical measure of the linear relationship between two variables. In finance, the coefficient measures how strongly and in what direction two data points, such as a financial security and an economic indicator, move with one another. The correlation coefficient expresses the relationship of the two variables on a scale of -1 to 1.

For example, the US dollar and gold tend to have a negative correlation, meaning upward trajectory in the value of USD often occurs in tandem with downward pressure on the price of gold. The correlation coefficient value will be closest to -1 when linear correlation between the two variables expresses strong negative correlation.

Understanding correlation coefficients

Values used in correlation coefficients can express positive, negative or zero correlation. Correlation coefficients are often expressed as a line charted on a scatter plot or a correlation matrix. While there are more complex formulations of correlation coefficients, the common variants used in financial analysis measure the relationship between just two variables. The relationship will be expressed in one of three ways detailed below.

Positive correlation

Positive correlation indicates that the value of two data points currently tend to move in the same direction. In finance, positive correlation coefficients are most common between similar markets, like the forex pairs EUR/USD and GBP/USD. A sample correlation coefficient for these two pairs ranges primarily between 0.81 and 0.95.

In other fields that use correlation coefficients such as scientific fields like sociology and biology, there may be perfect positive correlation. Perfect correlation means the linear relationship between correlated variables is an exact one-to-one match.

Negative correlation

A negative correlation coefficient expresses linear relationships that move inverse to one another. In the case of a perfect negative correlation, if one variable increases the other variable decreases at exactly the same degree.

Like positive correlation, perfect negative correlation rarely if ever occurs in financial markets. However, there are some long-standing negative correlations such as the relationship between stocks and bonds. When stock prices fall, bond prices typically rise as investors seek safer investments.

No linear association

A correlation coefficient of zero means no linear relationship exists between the two variables. In these instances, the movement of one variable has no discernable statistical significance for the direction of another variable.

Formula for the correlation coefficient 

The formula for the correlation coefficient is depicted below. When you calculate the correlation coefficient, you take multiple data points of both two variables, ideally within a defined timeframe.

Once you have multiple iterations of paired data points, you calculate the sample means and the distance of each data point from its mean. This calculation is done for every pair of data points, before taking the summation of both data points as represented on the top half of the coefficient calculation.

The correlation coefficient can be visualized with the aid of a scatter plot. The two variables in question are plotted along the x and y axis, once for every iteration of paired data points. A trend line is drawn across the scatter plot that best fits the plotted points.

Correlations are inferred when the trend line moves upward from left to right (positive) or downward from left to right (negative). The closer all plotted points fall to landing on the trend line, the stronger the correlation. When the correlation coefficient is zero, a trend line is simply not helpful in depicting a relationship between the plotted points.

Correlation statistics and investing 

Correlation statistics like the correlation coefficient help investors and traders assess and manage investment risks. In modern portfolio theory, investment risks are heavily assuaged by increased diversification. Using the correlation coefficient between historical returns of different assets and your entire portfolio can suggest whether the new asset would increase or decrease a portfolio's diversification.

Financial professionals like portfolio managers and quantitative traders may use the correlation coefficient in more complex ways. For example, portfolio managers may use it when building a portfolio focused on factors associated with excess returns, a technique known as factor investing. Quant traders can use correlation statistical significance to anticipate near-term changes in the price of securities. These methods however require a strong understanding of mathematical statistics and what correlation analysis can and can't tell us.

What are the types of correlation coefficients in finance? 

Different types of correlation coefficients are used when assessing data in finance depending on the properties of the data compared.

Pearson correlation coefficient

The Pearson correlation coefficient is the most common type due to its broad application. Typically, it is used when both variables are quantitative and normally distributed, there are no outliers within the data and the relationship between two variables is linear.

The Pearson correlation coefficient measures the strength and direction of a linear relationship between two quantitative variables by charting the trend as a straight line.

Spearman correlation coefficient

If any of the requirements necessary to apply the Pearson coefficient are absent, i.e. the variables are ordinal instead of quantitative, the data includes outliers or the relationship between two variables is non-linear, then Spearman's correlation coefficient can be used instead.

If the Pearson correlation coefficient would be charted as a straight line moving across the graph, the Spearman correlation coefficient by comparison often depicts a curved line to account for abnormal distributions of variables.

Limitations of correlation coefficient 

Because the value of any financial instrument is influenced by numerous factors, you should not mistake a correlation coefficient of a strong positive or negative value to hold any indication that the movement of one variable exerts pressure on the other variable. It's more likely that both variables are experiencing upward or downward pressure from a confluence of similar forces than either variable affecting the other directly.

You may have heard the phrase "correlation does not imply causation." That holds true with the correlation coefficient. Misusing or misinterpreting the correlation coefficient can undermine its value in your financial strategy.

Beyond a lack of direct influence between the two variables, you should also note that correlation coefficients are not constant. Strong positive or negative correlation between two financial variables can shift as market influences on both variables change.

Correlation coefficient FAQs

How do you calculate the correlation coefficient? 

 You can calculate the correlation coefficient using statistics programs or software like Excel. The correlation coefficient formula is detailed earlier in the article, but the computing is too involved to do manually with any efficient range of data points.

In Excel, the easiest way to calculate the correlation coefficient is simply to arrange the two sets of data side by side and apply the correlation coefficient from the drop-down list of formulas.

How is the correlation coefficient used in investing? 

 The correlation coefficient is used in investing to identify viable financial instruments you can pick up to diversify your portfolio or hedge against risk. Assets with low to zero correlation are good for diversification, and hedging strategies work best between assets with strong negative correlation.

The correlation coefficient is also used in portfolio risk assessments, helping gauge how diverse in influence the assets in your portfolio are from one another.

Why is understanding correlation important for diversification? 

Understanding correlation is important for diversification because negative or zero correlation is a major principle to guiding your portfolio diversification choices. Modern portfolio theory recommends diversification by allocating different amounts of your portfolio to different levels of volatility, with less volatile assets making up the majority of your holdings.

However, understanding the correlation among different assets within your portfolio is also important managing investment risks. If every asset from the most volatile to the least share a high correlation, your entire portfolio is at danger of sinking if the right causes come into effect. The correlation coefficient therefore can be used when assessing investment risks by gauging how correlated your portfolio is and whether its diverse to risk, not only volatility.