Correlation coefficients are indicators of the strength of the linear relationship between two different variables, x and y. A linear correlation coefficient that is greater than zero indicates a positive relationship. A value that is less than zero signifies a negative relationship. Finally, a value of zero indicates no relationship between the two variables x and y.

This article explains the significance of linear correlation coefficients for investors, how to calculate covariance for stocks, and how investors can use correlation to predict the market.

### Key Takeaways:

- Correlation coefficients are used to measure the strength of the linear relationship between two variables.
- A correlation coefficient greater than zero indicates a positive relationship while a value less than zero signifies a negative relationship.
- A value of zero indicates no relationship between the two variables being compared.
- A negative correlation, or inverse correlation, is a key concept in the creation of diversified portfolios that can better withstand portfolio volatility.
- Calculating the correlation coefficient is time-consuming, so data are often plugged into a calculator, computer, or statistics program to find the coefficient.

## Understanding Correlation

The correlation coefficient (*ρ*) is a measure that determines the degree to which the movement of two different variables is associated.The most common correlation coefficient, generated by the Pearson product-moment correlation, is used to measure the linear relationship between two variables.However, in a non-linear relationship, this correlation coefficient may not always be a suitable measure of dependence.

The possible range of values for the correlation coefficient is -1.0 to 1.0. In other words, the values cannot exceed 1.0 or be less than -1.0. Acorrelation of -1.0 indicates a perfect negative correlation, anda correlation of 1.0 indicates a perfectpositive correlation. Ifthe correlation coefficient is greater than zero, it isa positive relationship. Conversely, ifthe value is less than zero, it isa negative relationship. A value of zero indicates that there is no relationship between the two variables.

When interpreting correlation, it's important to remember that just because two variables are correlated, it does not mean that one causes the other.

### Correlation and the Financial Markets

In the financial markets, the correlation coefficient is used to measurethe correlation between two securities. For example, when two stocks move in the same direction, the correlation coefficient is positive. Conversely, when two stocks move in opposite directions, the correlation coefficient is negative.

If the correlation coefficient of two variables is zero, there is no linear relationship between the variables. However, this is only for a linear relationship. It is possible that the variables have a strong curvilinear relationship. When the value of ρ is close to zero,generally between -0.1 and +0.1, the variables are said to have no linear relationship (or a very weak linear relationship).

For example, suppose that the prices of coffee and computers are observed and found to have a correlation of +.0008. This means that there is no correlation, or relationship, between the two variables.

## Calculating ρ

Thecovarianceof the two variables in question must be calculated before the correlation can be determined. Next, each variable'sstandard deviation is required. The correlation coefficient is determined by dividing the covariance by the product of the two variables' standard deviations.

Standard deviation is a measure of thedispersionof data from its average. Covariance is a measure of how two variables change together. However, its magnitude is unbounded, so it is difficult to interpret. Thenormalized version of the statistic is calculated by dividing covariance by the product of the two standard deviations. This is the correlation coefficient.

$\text{Correlation}=\rho=\frac{\text{cov}(X,Y)}{\sigma_X\sigma_Y}$Correlation=ρ=σXσYcov(X,Y)

## Positive Correlation

A positive correlation—when the correlation coefficient is greater than 0—signifies that both variables move in the same direction. When ρ is +1, it signifies that the two variables being compared have a perfect positive relationship; when one variable moves higher or lower, the other variable moves in the same direction with the same magnitude.

The closer the value of ρ is to +1, the stronger the linear relationship. For example, suppose the value of oil prices is directly related to the prices of airplane tickets, with a correlation coefficient of +0.95. The relationship between oil prices and airfares has a very strong positive correlation since the value is close to +1. So, if the price of oil decreases, airfares also decrease, and if the price of oil increases, so do the prices of airplane tickets.

In the chart below, we compareone of the largest U.S. banks, JPMorgan Chase & Co. (JPM), with the Financial Select SPDR Exchange Traded Fund (ETF) (XLF). As you can imagine, JPMorgan Chase & Co. shouldhave a positive correlation to the banking industry as a whole. We can see the correlation coefficientis currently at 0.98, which is signaling a strong positive correlation. A readingabove 0.50 typically signalsa positive correlation.

Understanding the correlation between two stocks (or a single stock) and its industrycan help investors gauge how thestock is tradingrelative to its peers. All types of securities, including bonds, sectors, andETFs,can be compared with the correlation coefficient.

## Negative Correlation

A negative (inverse) correlationoccurswhen the correlation coefficientis less than 0. This is an indication that both variables move in the opposite direction. In short, any reading between 0 and -1 meansthat the two securities move in opposite directions. When *ρ* is -1, the relationship is said to be perfectly negatively correlated.

In short, if one variable increases, the other variable decreases with the same magnitude (and vice versa). However, the degree to which two securities are negatively correlated might vary over time (and they are almost never exactly correlated all the time).

### Examples of Negative Correlation

For example, suppose a study is conducted to assess the relationship between outside temperature and heating bills. The study concludes that there is a negative correlation between the prices of heating bills and the outdoor temperature. The correlation coefficient is calculated to be -0.96. This strong negative correlation signifies that as the temperature decreases outside, the prices of heating bills increase (and vice versa).

When it comes to investing, a negative correlation does not necessarily mean that the securities should be avoided. The correlation coefficientcan help investors diversifytheir portfolio by including a mix of investmentsthat have a negative, or low, correlation to the stock market. In short, when reducing volatility risk in a portfolio,sometimes opposites do attract.

For example, assume you have a $100,000 balanced portfolio that is invested 60% in stocks and 40% in bonds. In a year of strong economic performance, the stock component of your portfolio might generate a return of 12% while the bond component may return -2% because interest rates are rising (which means that bond prices are falling).

Thus, the overall return on your portfolio would be 6.4% ((12% x 0.6) + (-2% x 0.4). The following year, as the economy slows markedly and interest rates are lowered, your stock portfolio might generate -5% while your bond portfolio may return 8%, giving you an overall portfolio return of 0.2%.

What if, instead of a balanced portfolio, your portfolio were 100% equities? Using the same return assumptions, your all-equity portfolio would have a return of 12% in the first year and -5% in the second year. These figures are clearly more volatile than the balanced portfolio's returns of 6.4% and 0.2%.

## Linear Correlation Coefficient

The linear correlation coefficient is a number calculated from given data that measures the strength of the linear relationship between two variables:xandy. The sign of the linear correlation coefficient indicates the direction of the linear relationship betweenxandy. Whenr (the correlation coefficient)is near 1 or −1, the linear relationship is strong; when it is near 0, the linear relationship is weak.

Even for small datasets, the computations for the linear correlation coefficient can be too long to do manually. Thus, data are often plugged into a calculator or, more likely, a computer or statistics program to find the coefficient.

### The Pearson Coefficient

Both the Pearson coefficient calculation and basic linear regression are ways to determine how statistical variables are linearly related. However, the two methods do differ. The Pearson coefficient is a measure of the strength and direction of the linear association between two variables with no assumption of causality. The Pearson coefficient shows correlation, not causation. Pearson coefficients range from +1 to -1, with +1 representing a positive correlation, -1 representing a negative correlation, and 0 representing no relationship.

Simple linear regression describes the linear relationship between a response variable (denoted by y) and an explanatory variable (denoted by x) using a statistical model. Statistical models are used to make predictions.

Simplify linear regression by calculating correlation with software such as Excel.

In finance, for example, correlation is used in several analyses including the calculation of portfolio standard deviation. Because it is so time-consuming, correlation is best calculated using software like Excel. Correlation combines statistical concepts, namely, variance andstandard deviation.Variance is the dispersion of a variable around the mean, and standard deviation is the square root of variance.

### Finding Correlation Using Excel

There are several methods to calculate correlation in Excel. The simplest is to get two data sets side-by-side and use the built-in correlation formula:

If you want to create a correlation matrix across a range of data sets, Excel has a Data Analysis plugin that is found on the Data tab, under Analyze.

Select the table of returns. In this case, our columns are titled, so we want to check the box "Labels in first row," so Excel knows to treat these as titles.Then you can choose to output on the same sheet or on a new sheet.

Once you hit enter, the data is automatically created.You can add some text and conditional formatting to clean up the result.

## Linear Correlation Coefficient Frequently Asked Questions

### What Is the Linear Correlation Coefficient?

The linear correlation coefficient is a number calculated from given data that measures the strength of the linear relationship between two variables,xandy.

### How Do You Find the Linear Correlation Coefficient?

Correlation combines several important and related statistical concepts, namely, variance and standard deviation.Variance is the dispersion of a variable around the mean, and standard deviation is the square root of variance.

The formula is:

$\bold{r}=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2)]}}$r=[n∑x2−(∑x)2][n∑y2−(∑y)2)]n(∑xy)−(∑x)(∑y)

The computing is too long to do manually, and sofware, such as Excel, or a statistics program, are tools used to calculate the coefficient.

### What Is Meant By Linear Correlation?

Thecorrelationcoefficient is a value between -1 and +1. A correlation coefficient of +1 indicates a perfect positive correlation. As variable x increases, variable y increases. As variable x decreases, variable y decreases. A correlation coefficient of -1 indicates a perfect negative correlation. As variable x increases, variable z decreases. As variable x decreases, variable z increases.

### How Do You Find the Linear Correlation Coefficient on a Calculator?

A graphing calculator is required to calculate the correlation coefficient. The following instructions are provided by Statology.

**Step 1: Turn on Diagnostics**

You will only need to do this step once on your calculator. After that, you can always start at step 2 below. If you don’t do this, r (the correlation coefficient) will not show up when you run the linear regression function.

- Press [2nd] and then [0] to enter your calculator’s catalog. Scroll until you see “diagnosticsOn”.
- Press enter until the calculator screen says “Done”.

This is important to repeat:You never have to do this again unless you reset your calculator.

**Step 2: Enter Data**

Enter your data into the calculator by pressing [STAT] and then selecting 1:Edit. To make things easier, you should enter all of your “x data” into L1 and all of your “y data” into L2.

**Step 3: Calculate!**

Once you have your data in, you will now go to [STAT] and then the CALC menu up top. Finally, select 4:LinReg and press enter.

That’s it! You’re are done! Now you can simply read off the correlation coefficient right from the screen (its r). Remember, if r doesn’t show on your calculator, then diagnostics need to be turned on. This is also the same place on the calculator where you will find the linear regression equation and the coefficient of determination.

## The Bottom Line

The linear correlation coefficient can be helpful indetermining the relationship between an investment and the overall market or other securities. It is often used to predict stock market returns. This statistical measurement is useful in many ways, particularly in the finance industry.

For example, it can be helpful in determining how well a mutual fund is behaving compared to itsbenchmarkindex, or it can be used to determine how a mutual fund behaves in relation to another fund orasset class. By adding a low, or negatively correlated, mutual fund to an existing portfolio,diversificationbenefits are gained.

## FAQs

### What does a 0.00 correlation coefficient mean? ›

Thecorrelation coefficient (r) is a statistic that tells you the strengthand direction of that relationship. It is expressed as a positive ornegative number between -1 and 1. The value of the number indicates the strengthof the relationship: r = 0 means **there is no correlation**.

**What do you understand by positive negative and zero correlation write with example? ›**

An example of negative correlation would be height above sea level and temperature. As you climb the mountain (increase in height) it gets colder (decrease in temperature). A zero correlation exists when there is no relationship between two variables.

**What does a correlation coefficient of 1 0 and +1 indicate? ›**

A correlation coefficient of -1 describes a perfect negative, or inverse, correlation, with values in one series rising as those in the other decline, and vice versa. **A coefficient of 1 shows a perfect positive correlation, or a direct relationship.** **A correlation coefficient of 0 means there is no linear relationship**.

**What does positive and negative correlation mean? ›**

A positive correlation exists when two variables operate in unison so that when one variable rises or falls, the other does the same. A negative correlation is when two variables move opposite one another so that when one variable rises, the other falls.

**How do you know if a correlation is strong or weak? ›**

The sign of the linear correlation coefficient indicates the direction of the linear relationship between x and y. **When r (the correlation coefficient) is near 1 or −1, the linear relationship is strong; when it is near 0, the linear relationship is weak**.

**How do you know if a correlation is significant? ›**

Compare r to the appropriate critical value in the table. **If r is not between the positive and negative critical values, then the correlation coefficient is significant**.

**What does a negative correlation tell us? ›**

A negative correlation is a relationship between two variables such that **as the value of one variable increases, the other decreases**.

**What does a perfect positive and perfect negative correlation mean explain with example? ›**

In statistics, a perfect negative correlation is represented by the value -1.0, while a 0 indicates no correlation, and +1.0 indicates a perfect positive correlation. **A perfect negative correlation means the relationship that exists between two variables is exactly opposite all of the time**.

**Why is a correlation coefficient never greater than 1 or less than − 1 )? ›**

So **there's no way you can get the correlation to be bigger than 1**, and it's equal to 1 when the two variables are identical or when one is a positive multiple of the other, or (more generally) when one is a positive multiple of the other plus a constant difference -- ie, a straight line relationship.

**Is a correlation always between 0 and 1? ›**

**The correlation value always lies between -1 and 1** (going thru 0 – which means no correlation at all – perfectly not related). Correlation can be rightfully explalined for simple linear regression – because you only have one x and one y variable.

### What is considered a good correlation coefficient? ›

**Correlation Coefficient = +1**: A perfect positive relationship. Correlation Coefficient = 0.8: A fairly strong positive relationship. Correlation Coefficient = 0.6: A moderate positive relationship. Correlation Coefficient = 0: No relationship.

**What does a positive correlation tell us? ›**

Understanding positive correlation

In statistics, a positive correlation shows that changes in one variable will relate to the same type of changes in a second variable. The data is usually displayed in a scatterplot, which shows the linear relationship between variables in a positive correlation graph.

**How do you analyze correlation coefficient? ›**

The strength of relationship can be anywhere between −1 and +1. The stronger the correlation, the closer the correlation coefficient comes to ±1. If the coefficient is a positive number, the variables are directly related (i.e., as the value of one variable goes up, the value of the other also tends to do so).

**Is a negative correlation strong or weak? ›**

A negative correlation indicates two variables that tend to move in opposite directions. **A correlation coefficient of -0.8 or lower indicates a strong negative relationship**, while a coefficient of -0.3 or lower indicates a very weak one.

**What is a strong negative correlation example? ›**

For example, **the correlation between rainy days and sales per week is -0.9**. This means there is a strong negative correlation between rainy days and sales, or the more it rains, the less sales you make, or the less it rains, the more sales you make.

**What does it mean for two variables to be perfectly negatively correlated? ›**

A "perfect" negative correlation of -1.0, by contrast, would indicate that the two variables move in opposite directions with equal magnitude—if A increases by two, B decreases by two.

**Which correlation coefficient indicates the weakest relationship between two variables? ›**

The weakest linear relationship is indicated by a **correlation coefficient equal to 0**. A positive correlation means that if one variable gets bigger, the other variable tends to get bigger. A negative correlation means that if one variable gets bigger, the other variable tends to get smaller.

**When interpreting a correlation coefficient it is important to look at? ›**

When interpreting a correlation coefficient, it is important to look at: **The magnitude of the correlation coefficient**.

**Is correlation stronger at 1 or 0? ›**

Correlation coefficient

Furthermore: Positive values denote positive linear correlation; • Negative values denote negative linear correlation; • A value of 0 denotes no linear correlation; • **The closer the value is to 1 or –1, the stronger the linear correlation**.

**What does it mean if correlation is close to 0? ›**

A correlation close to zero suggests **no linear association between two continuous variables**. It is important to note that there may be a non-linear association between two continuous variables, but computation of a correlation coefficient does not detect this.

### Does 0 correlation mean no relationship? ›

If there is zero correlation (rxy=0), it means the two variables are uncorrelated and there is no linear relation between them. However, other types of relations may be there and they may not be independent.

**Is 0.1 A negative correlation? ›**

**Negative correlation is measured from -0.1 to -1.0**. Weak negative correlation being -0.1 to -0.3, moderate -0.3 to -0.5, and strong negative correlation from -0.5 to -1.0.

**How do you interpret a correlation coefficient in r for dummies? ›**

The value of r ranges between −1 and 1. **When r = zero, it means that there is no linear association between the variables**. However, there might be some nonlinear relationship but if r = zero then there is no consistent linear component to that relationship.

**Is positive correlation better than negative? ›**

The number portion of the correlation coefficient indicates the strength of the relationship. **The closer the number is to 1 (be it negative or positive), the more strongly related the variables are**, and the more predictable changes in one variable will be as the other variable changes.

**Why is 0.7 A strong correlation? ›**

Values between 0.7 and 1.0 (−0.7 and −1.0) **indicate a strong positive (negative) linear relationship through a firm linear rule**. It is the correlation coefficient between the observed and modelled (predicted) data values. It can increase as the number of predictor variables in the model increases; it does not decrease.

**How do you know if a correlation is direct or indirect? ›**

**If variables change in the same direction, the correlation is called a direct correlation or a positive correlation**. If variables change in opposite directions, the correlation is called an indirect correlation or a negative correlation.

**What does a zero 0 Pearson's coefficient mean? ›**

The Pearson correlation coefficient, r, can take a range of values from +1 to -1. A value of 0 indicates that **there is no association between the two variables**. A value greater than 0 indicates a positive association; that is, as the value of one variable increases, so does the value of the other variable.

**Does a correlation near 0 mean an association is weak? ›**

r is always a number between -1 and 1. r > 0 indicates a positive association. r < 0 indicates a negative association. **Values of r near 0 indicate a very weak linear relationship**.