Correlation Between Two Variables

Updated for the 2026-2027 CFA® Level I curriculum.

Correlation helps you describe how two variables have moved in relation to one another. It shows whether their linear relationship is positive or negative and how closely their observations follow a straight-line pattern.

For CFA Level I, the main skill is interpretation. You should be able to read a correlation coefficient in context, explain what its sign and magnitude suggest, and recognize the conclusions that correlation alone cannot support.

Key Takeaways

• Correlation measures linear association between two variables.

• The correlation coefficient ranges from -1 to +1.

• The sign shows the direction of the relationship.

• The absolute value indicates the strength of the linear relationship.

• A value near zero does not rule out a nonlinear relationship.

• Correlation is unit-free because it standardizes covariance.

• Correlation does not establish that one variable causes changes in another.

• The estimate may change across samples, time periods, and market conditions.

What You Need to Know for CFA Level I

For CFA Level I, focus on:

• Interpreting positive, negative, and near-zero correlation.

• Recognizing perfect positive and perfect negative linear relationships.

• Using the absolute value of the coefficient to assess strength.

• Relating correlation to covariance and the standard deviations of both variables.

• Explaining why correlation has no units.

• Distinguishing correlation from causation.

• Identifying the effects of nonlinear patterns, outliers, and unstable samples.

• Interpreting the coefficient within the specific investment problem.

What Is Correlation?

Correlation is a standardized measure of covariance. Covariance identifies whether two variables tend to move together or in opposite directions, while correlation places that relationship on a fixed scale from -1 to +1.

The population correlation formula is:

Where:

• Inline equation code = population correlation coefficient between variables X and Y

• Inline equation code = population covariance between X and Y

• Inline equation code = population standard deviation of X

• Inline equation code = population standard deviation of Y

For sample data, the correlation formula is:

Where:

• Inline equation code = sample correlation coefficient between variables X and Y

• Inline equation code = sample covariance between X and Y

• Inline equation code = sample standard deviation of X

• Inline equation code = sample standard deviation of Y

The correlation coefficient must fall within the following range:

Covariance has units based on the two variables being measured. Dividing it by both standard deviations removes those units, which makes correlation easier to compare across different variable pairs.

Correlation is also symmetric:

The correlation between X and Y is therefore the same as the correlation between Y and X.

How Is Correlation Related to Covariance?

Covariance and correlation describe the direction of the same relationship.

• Positive covariance produces positive correlation.

• Negative covariance produces negative correlation.

• Covariance close to zero generally produces correlation close to zero.

Correlation adds a standardized measure of strength. A covariance of 20 may be large or small depending on the units and volatility of the variables. A correlation of 0.80 has a consistent interpretation because it always falls on the same -1 to +1 scale.

How Do You Interpret a Correlation Coefficient?

The sign and magnitude answer different questions.

• The sign shows the direction of the linear relationship.

• The absolute value shows how closely the observations follow a linear pattern.

A correlation of -0.85 is stronger than a correlation of +0.30 because 0.85 has the larger absolute value. The negative sign describes direction, not weakness.

Terms such as weak, moderate, and strong depend on context. A correlation that appears modest in one investment problem may still be economically useful in another. Review the coefficient alongside the variables, sample period, and purpose of the analysis.

What Does Positive Linear Correlation Mean?

Positive linear correlation means that higher values of one variable tend to occur with higher values of the other. Lower values also tend to occur together.

For example, a correlation of +0.80 between company revenue growth and earnings growth suggests a strong positive correlation in the sample. Companies or periods with higher revenue growth have generally also shown higher earnings growth.

The coefficient does not mean that earnings rise by 0.80% whenever revenue rises by 1%. Correlation describes how closely the two variables move together, not the size of the change in one variable caused by the other.

A perfect positive correlation of +1 means every observation lies on an upward-sloping straight line. The two variables may still change by different amounts.

What Does Negative Linear Correlation Mean?

Negative linear correlation means that higher values of one variable tend to occur with lower values of the other.

A correlation of -0.70 between interest-rate changes and long-duration bond returns suggests a meaningful negative linear correlation in the sample. Periods with rising interest rates have generally been associated with lower bond returns.

The coefficient summarizes the overall pattern. It does not mean the variables moved in opposite directions during every observation.

A perfect negative correlation of -1 means every observation lies on a downward-sloping straight line.

What Does a Correlation Near Zero Mean?

A correlation near zero indicates little linear association between the two variables.

The variables may still be related through a curved or more complex pattern. For example, one variable may rise whenever another variable moves far above or below its average. This U-shaped relationship can produce a correlation close to zero even though the variables are clearly connected.

A scatterplot helps reveal:

• Curved relationships

• Clusters of observations

• Extreme outliers

• Changes in the relationship across different ranges

The coefficient and the scatterplot should therefore be interpreted together whenever the underlying data are available.

Worked Example: Calculating and Interpreting Correlation

An analyst examines the monthly returns of asset X and asset Y. The sample statistics are:

• Sample covariance: -0.012

• Standard deviation of asset X: 0.10

• Standard deviation of asset Y: 0.20

Step 1: Apply the Sample Correlation Formula

Substitute the sample values:

Step 2: Calculate the Correlation

Step 3: Interpret the Result

The correlation of -0.60 indicates a moderately strong negative linear relationship in the sample.

Higher returns for asset X have generally been associated with lower returns for asset Y, and vice versa. The result does not imply that movements in one asset caused movements in the other.

Worked Example: Interpreting Correlation in an Investment Problem

An analyst calculates a correlation of -0.65 between monthly commodity-price changes and the monthly returns of a manufacturing company.

A careful interpretation would state that:

• The relationship is negative.

• The absolute value of 0.65 suggests a meaningful linear association.

• Larger commodity-price increases have generally occurred alongside lower company returns.

• The coefficient does not prove that commodity prices alone caused the returns.

• The relationship may differ during another sample period or market environment.

The analyst should also consider whether the relationship has an economic explanation. Higher input costs may reduce profit margins, but other factors such as product pricing, hedging, and demand could also affect company returns.

How Is Correlation Used in Investment Analysis?

Correlation can help analysts evaluate how financial and economic variables have moved together.

Common applications include:

• Comparing the returns of two assets.

• Studying how company returns respond to interest rates, commodity prices, or exchange rates.

• Assessing whether assets may provide diversification benefits.

• Identifying variables that may be useful in forecasting or risk analysis.

• Examining relationships before building a regression model.

Correlation provides evidence about historical co-movement. A sound investment conclusion also considers the economic relationship, data quality, and whether the observed pattern is likely to continue.

Detailed portfolio calculations involving covariance, correlation, and portfolio risk belong in the related portfolio mathematics notes.

Why Can Correlation Be Misleading?

Correlation summarizes a relationship in one number, which makes it useful but incomplete.

Correlation Does Not Establish Causation

Two variables may move together because one affects the other, because a third factor affects both, or because the observed relationship occurred by chance.

For example, company sales and share prices may both rise during a strong economy. Their correlation alone does not show that sales were the only cause of the price movement.

Correlation Measures Linear Relationships

The coefficient captures how closely the observations follow a straight line.

A curved relationship may produce a low correlation even when the variables are closely connected. Reviewing a scatterplot can expose patterns that the coefficient misses.

Outliers Can Distort the Estimate

One or two unusual observations may increase, reduce, or reverse the sample correlation.

An analyst should check whether the result reflects the general data pattern or is driven mainly by a small number of extreme values.

Correlation Can Change Over Time

Relationships between financial variables often shift as market conditions change.

Assets that showed low correlation during normal markets may move more closely together during a crisis. Estimates based on one market regime may therefore provide a poor description of another.

The Sample Period and Frequency Matter

A correlation calculated from daily observations may differ from one calculated using monthly or annual data.

A short sample may also produce an unstable estimate. Longer samples provide more observations but may combine market environments with very different relationships.

Spurious Correlation Can Occur

Two variables may appear strongly correlated without a meaningful economic connection.

A plausible economic explanation and additional analysis help determine whether the relationship is useful or coincidental.

Does Correlation Show How Much One Variable Explains?

A correlation of 0.60 does not mean that 60% of one variable is explained by the other.

In a simple linear regression with an intercept, the coefficient of determination is the square of the correlation coefficient.

Where:

• Inline equation code = coefficient of determination in a simple linear regression

• Inline equation code = sample correlation between X and Y

For a correlation of 0.60:

The simple regression would associate 36% of the sample variation in the dependent variable with variation in the independent variable. This interpretation belongs to regression analysis and still does not establish causation.

How Should You Interpret Correlation in an Exam Question?

Use the following sequence:

1. Check the Sign

A positive sign indicates movement in the same direction. A negative sign indicates movement in opposite directions.

2. Check the Absolute Value

A coefficient closer to 1 in absolute value indicates a stronger linear relationship.

3. Name the Relationship as Linear

Correlation describes linear association. Include the word “linear” when interpreting the result.

4. Connect the Result to the Variables

Explain what higher or lower values of one variable have tended to accompany in the other variable.

5. Avoid Causal Language

Use wording such as “associated with” or “tended to occur with” unless separate evidence supports causation.

6. Consider the Sample

Remember that the estimate depends on the observations, time period, and market conditions used.

Common Exam Traps

Common mistakes include:

• Interpreting a correlation of -0.80 as weak because the coefficient is negative.

• Comparing signed values instead of their absolute values when assessing strength.

• Saying that a correlation of zero means the variables have no relationship of any kind.

• Treating correlation as proof that one variable causes changes in another.

• Interpreting a correlation of 0.60 as meaning that 60% of one variable is explained.

• Assuming that perfectly correlated variables must change by the same amount.

• Ignoring a nonlinear pattern that the correlation coefficient does not capture.

• Overlooking the effect of one or two extreme observations.

• Assuming a historical correlation will stay constant.

• Applying informal labels such as strong or weak without considering the investment context.

Practice Question

An analyst calculates a correlation of -0.80 between two variables.

Which interpretation is most accurate?

1. The variables have a strong negative linear relationship

2. The variables have a weak relationship because the coefficient is negative

3. Changes in one variable cause 80% of the changes in the other variable

FAQs About Correlation Between Two Variables