What you will learn
- Explain how correlation standardizes covariance
- Understand why correlation always falls between negative one and positive one
- Interpret correlation values from negative one to positive one
- Explain why correlation does not prove causation
Covariance told you the direction two assets move together, but its size was uninterpretable. Correlation fixes that. It takes the covariance and rescales it into a clean number that always lands between negative one and positive one, so you can finally say not just whether two things move together but how strongly. Correlation is one of the most used numbers in finance, and this lesson explains why it is built the way it is.
Pearson's correlation, clearly explained (StatQuest)
Shows correlation as a standardized covariance bounded to between negative one and positive one, and what the value means. A bit long, so focus on the standardizing step.
Correlation is standardized covariance
Correlation is computed by dividing the covariance by the product of the two variables' standard deviations. That division is doing something specific. The covariance's messy size came from the units and scale of the two variables, and each variable's standard deviation carries exactly that scale. Dividing by both standard deviations cancels out the units entirely, leaving a pure, unitless number. Because the standard deviations are always positive, the division never changes the sign, so correlation keeps the same positive or negative direction as the covariance. It just rescales the magnitude into something comparable.
Key terms
- Correlation
- The covariance divided by the product of the two standard deviations. A unitless measure of linear relationship from negative one to positive one.
- Perfect positive correlation (+1)
- The two variables move in exact lockstep in the same direction.
- Perfect negative correlation (-1)
- The two variables move in exact lockstep in opposite directions.
- Zero correlation
- No linear relationship between the two variables.
Why it is bounded between negative one and positive one
The reason correlation cannot exceed one in either direction comes from that division by both standard deviations. The covariance can never be larger, in size, than the product of the two standard deviations. It reaches that maximum only when the two variables move in perfect lockstep. So when you divide covariance by that product, the result can never be bigger than one or smaller than negative one. This bounding is the whole point. It turns an unbounded, uninterpretable covariance into a fixed scale where negative one, zero, and positive one always mean the same thing no matter what the variables are. That is what makes correlation comparable across any pair of assets.
The formula in symbols
- Cov(X, Y) = covariance of the two variables
- SDx, SDy = the two standard deviations
You divide the covariance by the product of the two standard deviations. Because dividing by both standard deviations cancels the units, the result is always a pure number between negative one and positive one. The worked example below turns a covariance of 6 into a correlation using the two standard deviations.
From covariance to correlation
Two assets have a covariance of 6. Asset X has a standard deviation of 3, and Asset Y has a standard deviation of 4. What is their correlation?
- Multiply the two standard deviations. 3 times 4 is 12.
- Divide the covariance by that product. 6 divided by 12.
- Read the result. That is 0.5, safely between negative one and positive one.
Why it matters: A correlation of 0.5 means a moderate positive relationship. Dividing the covariance by the two standard deviations stripped away the units and put the answer on the fixed negative-one-to-positive-one scale.
Calculate a correlation
Two assets have a covariance of negative 8. One has a standard deviation of 4 and the other 5. What is their correlation?
| Correlation | Meaning |
|---|---|
| +1 | Perfect positive: move in exact lockstep, same direction |
| +0.5 | Moderate positive: tend to move together |
| 0 | No linear relationship |
| -0.5 | Moderate negative: tend to move oppositely |
| -1 | Perfect negative: move in exact lockstep, opposite directions |
Correlation is not causation
Here is one of the most important warnings in statistics, and it applies far beyond finance. That two variables are correlated does not mean one causes the other. There are always other explanations. A third factor might drive both, the way hot weather drives up both ice cream sales and drowning deaths without either causing the other. The link might run the opposite way from what you assume. Or it might be pure coincidence, since if you check enough pairs of things, some will line up by chance alone. Concluding that A causes B just because they move together is one of the most common and costly errors in reasoning. Correlation can hint at a relationship worth investigating, but establishing causation requires far more, usually a controlled experiment or a solid theory for why the link should exist.
Correlation earns you the right to ask why two things move together. It never gives you the answer that one caused the other.
Correlation does not equal causation (CrashCourse Statistics)
A memorable, example-driven treatment of why correlation never proves causation. Watch for the third-factor and coincidence explanations.
Spot the causation trap
An analyst notices that over the past few years, a certain small country's butter production has been highly correlated with the US stock market. He suggests trading stocks based on butter reports. What is the flaw?
This is a spurious correlation. With countless variables in the world, some will line up by pure chance, and there is no mechanism for butter production to move the stock market. Correlation never proves causation.Why correlation matters in finance
Correlation is the number at the heart of diversification, which is the closest thing to a free lunch in investing. Combining assets that are not perfectly correlated reduces a portfolio's overall risk, because when some assets zig, others zag, smoothing the combined result. Two volatile stocks can form a much calmer portfolio if their correlation is low or negative. You will build this idea into portfolio variance later in this unit, and it is why professionals watch correlations so closely. One caution to carry forward: correlations are not stable and tend to spike toward positive one during market crises, so diversification can weaken exactly when it is needed most.
Why standardize?
Explain in a sentence or two why correlation is more useful than raw covariance for comparing the relationships between different pairs of assets.
Write an answer before comparing it with the model response.
Model answer
Covariance shows the direction two assets move together, but its size depends on the units and scale of the assets, so a covariance of 6 for one pair and 600 for another cannot be compared. Correlation divides by both standard deviations, which cancels the units and forces the result onto a fixed scale from negative one to positive one. That means a correlation of 0.5 means the same strength of relationship for any pair, so I can compare relationships across different assets directly.
You now understand covariance and correlation, the tools for how two variables move together. The next lesson tackles a subtle but crucial point the user of these tools must know: that zero correlation does not always mean the two variables are independent.