Correlation of Stocks Explained (Interpretation, Formula, Example)

In this article, we’re going to explore the correlation of stocks, including what it is, and how we calculate stock correlation. So let’s get into it.

Table of Contents hide

1 What is the Correlation of Stocks?

2 The precursor to the Correlation: The Covariance

2.1 Core Limitation of the Covariance

3 Correlation Bounds

4 Interpreting the Correlation of Stocks

4.1 Correlation = +1

4.2 Correlation = -1

4.3 Correlation = 0

4.4 Correlation is NOT Causality

4.4.1 Spurious Correlations

5 How to Calculate Stock Correlation

5.1 Correlation of Stock with Itself

6 Example on How to Calculate Stock Correlation

7 Strong vs Weak Correlation of Stocks

8 Wrapping Up – Correlation of Stocks

What is the Correlation of Stocks?

Firstly, what is the Correlation of stocks?

Fundamentally, the correlation (aka correlation coefficient, Pearson Correlation Coefficient) is just an alternative measure of the relationship between securities.

We tend to use the Greek letter $\rho_{j,k}$ (pronounced Rho with a silent-ish “h”) to denote the correlation of stocks.

The subscripts $j$ and $k$ in $\rho_{j,k}$ denotes the fact that this is the correlation coefficient between securities $j$ and $k$ .

You can substitute $j$ and $k$ with whatever you like – Apple and Alphabet; Facebook and Netflix; whatever you fancy.

Note that for the most part, we can only really measure the correlation of stocks listed on the stock market.

It’s difficult to do so for private companies (those that are not listed on the stock market) because of the lack of data.

And on that note, while everyone in Finance refers to it as the correlation of stocks, strictly speaking, it’s the correlation of stock returns.

Because that – stock returns – is the underlying data used for the correlation calculation.

Now, you can think of the correlation as an extension of the covariance.

The precursor to the Correlation: The Covariance

The covariance is a fundamental measure of the relationship – or strictly, the co-variability – of 2 variables.

The covariance of the returns of stocks $j$ and $k$ is estimated like this…

$\sigma_{j,k} = \sum_{t=1}^n (r_j - E[r_j])(r_k - E[r_k])$

Essentially, it takes the variability of asset $j$ , and it combines it with the variability of asset $k$ to get the co-variability between assets $j$ and $k$ .

The covariance is a pretty powerful variable in and of itself.

It helps us understand the relationships between securities.

So it helps us see where two individual stocks move:

with each other, or
against each other, or
if they move quite randomly

And this insight can help an investor in her asset allocation decisions.

The covariance is also a key variable used in the systematic risk calculation.

But the covariance does have its own limitations.

Core Limitation of the Covariance

Perhaps the biggest limitation of the covariance is the fact that it’s really hard to interpret.

That’s because the covariance tends to be a really small number, at least in the context of financial securities. Or specifically, the returns of financial securities.

Just by their nature, the returns of securities tend to be fairly small numbers – at least in the scheme of things.

So, if you were to calculate daily stock returns, for instance, you can be sure to get a number that’s very close to 0.

And when you then try to estimate the covariance of returns, you almost always end up with this really tiny number.

A number that’s difficult to read, and difficult to interpret.

Outside of finance, the covariance can theoretically be any value.

Because it’s not bounded by anything.

It could literally be any number. And the fact that it can be any number, at least theoretically, doesn’t really help with interpretation either.

For instance, you may think you’re getting a feel for the magnitude of the covariance if, say, it’s a really massive number.

But it could just be a massive number because the variables that you’re trying to measure the covariance for, just happened to be these really massive numbers as well!

The point is…

It can be really difficult to drive meaningful insights from the covariance.

It can be difficult to interpret what the covariance actually means.

And it’s this specific limitation of the covariance that the correlation addresses and overcomes.

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If you’re interested in exploring how correlation affects the risk of your own portfolio while working with real-world data, then you should definitely check out the course.

Correlation Bounds

The correlation of stocks is bounded between $-1$ and $+1$ , meaning we have a definitive range for the relationship between any two securities.

In other words, the correlation coefficient or $\rho_{j,k}$ for any security must be between minus one and plus one.

$-1 \leq \rho_{j,k} \leq +1$

This bounded nature of the correlation allows us to understand and comment on the strength of relationships between securities.

We can, for example, explicitly evaluate how strong or how weak the relationship between any two securities is.

And this brings us to the interpretation of the correlation of stocks.

Interpreting the Correlation of Stocks

The interpretation of the correlation is pretty straightforward. And incredibly powerful. Let’s get into it.

Correlation = +1

If the correlation between two securities is equal to $+1$ , or positive one, it means that the two securities are perfectly positively correlated.

Put differently, we say the stocks have a perfect positive correlation.

And that’s as strong as the relationship can get.

What this means is if, for instance, stock $j$ increases by 5%, then the stock $k$ will also increase by 5%.

In other words, if $\rho_{j,k} = +1$ , i.e. they are perfectly positively correlated, then both securities will move in exactly the same way.

So if $j$ increases by 5%, then so will $k$ .

And equally, if stock $k$ were to decrease by 10%, then stock $j$ will also decrease by 10%.

Correlation = -1

If on the other hand, the correlation of stocks between the two securities is equal to $-1$ , then we say that the stocks are perfectly negatively correlated.

Put differently, we say that the stocks have a perfect negative correlation.

And again, this is as strong as the relationship can get.

Only this time, the relationship is one where each security does the exact opposite of the other security given the negative correlation.

So if, for instance, stock $j$ were to increase by 5%, then stock $k$ will decrease by 5%.

And equally, if $k$ were to say, decrease by 10%, then $j$ will increase by 10%.

Thus, if $\rho_{j,k} = -1$ , i.e., if they’re perfectly negatively correlated, then the securities move in exactly opposite directions.

So when one increases by $x$ , the other must and will decrease by $x$ .

Note that a negative correlation between -1 and 0 (not inclusive) also means the stocks will move in opposite directions, but they won’t be at exactly the same level.

Correlation = 0

Finally, if the correlation is equal to zero, then we can say that the securities are completely uncorrelated; or perfectly uncorrelated.

It means that there’s no relationship between the two securities whatsoever.

So if $\rho_{j,k} = 0$ , then if $j$ were to increase by 5%, then $k$ could increase by 5% or increase by some other value.

Or in fact, it could decrease by some value or indeed remain entirely unchanged.

But the point is that it does not move in the same direction or the same value as the other security. In other words, the two stocks just move in a completely random manner.

As a “pro tip”, remember that having non-correlated assets in a portfolio of stocks can actually decrease portfolio risk. The reason and rationale for that are for another post perhaps. Although we do go over that in a lot of detail in our investment analysis and portfolio management course.

In a nutshell, though, the real effects of diversification are driven by holding low correlation or 0 correlation assets.

And incidentally, stocks that are either uncorrelated or negatively correlated, can be great for hedging strategies.

Here’s a quick summary of the core interpretation of the correlation of stocks…

Crucially though, and we really can’t stress this enough…

Correlation does NOT imply causality.

Correlation is NOT Causality

Just because two stocks move in the same direction, or in exactly opposite directions, doesn’t mean that one stock is causing the other stock to move.

It just so happens that they move in the same direction. Or in exactly different directions. Or in some sort of predictable manner / predictable pattern.

But the fact that they move in this manner or pattern doesn’t mean it’s being caused by one security’s movement.

It’s literally just some sort of coincidence.

There can be some reasonings or rationale, in some cases, however.

For instance, you would expect securities of certain industries to move together; just because they happen to be in the same industry.

And you would expect certain securities from different industries to move together, because although the industries are different, they may well be related.

So, you might expect that the oil and gas industry moves in some sort of manner with the airline industry.

Because the airlines rely heavily on oil.

And naturally, their costs are heavily influenced by the price of oil.

Obviously, this is ignoring the fact that they might well be using some sort of derivative instruments, like options, for instance, to control the volatility or the movement in oil prices.

But this is now digressing entirely from the point we’re trying to make, which is the fact that correlation does not imply causality.

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Spurious Correlations

And to show you why exactly correlation does not imply causality…

We’ve got this chart here, which plots out the per capita cheese consumption against the number of people who died by becoming tangled in their bedsheets!

We’ve actually adapted this spurious correlation chart from the original version which is available from Dr Vigan’s website.

Dr Vigan (pronounced Vegan) plots out various remarkably interesting examples of what are called “spurious correlations”. These are essentially correlations that can seem really high but don’t actually mean anything.

In this case, for instance, we’re looking at a chart of the amount of cheese consumption per year from 2000 through to 2009.

And there’s a very clear and strong correlation between that, and the number of people who died, by becoming tangled in their bedsheets for whatever reason.

Obviously, there’s no plausible reason as to why we should be seeing this “strong relationship”; it’s utterly meaningless!

It just so happens that these two variables are somehow correlated really strongly.

We sincerely hope that you take this chart as something to remember the fact that correlation does not imply causality.

Okay. So with that important disclosure and disclaimer out of the way, let’s now think about how to calculate stock correlation.

How to Calculate Stock Correlation

We can calculate stock correlation by dividing the covariance by the product of the standard deviations.

In other words, we can calculate stock correlation by…

$\rho_{j,k} = \frac{\sigma_{j,k}}{\sigma_j \times \sigma_k}$

And this is why we said you can think of the correlation as an extension of the covariance.

Because we need the covariance in order to estimate the asset correlation.

By taking the covariance of stock $j$ and $k$ , and scaling it (or dividing it) by the product of the two standard deviations, we essentially ensure that the correlation is bounded between $-1$ and $+1$ .

It ensures that we have a definitive range for the relationships between securities.

We provide a formal proof of why this is true in our Investment Analysis and Portfolio Management (with Python) course (and in the Excel version of the Investment Analysis course, too).

Correlation of Stock with Itself

Importantly, while the correlation between any two securities must be between $-1$ and $+1$ , the correlation of any asset with itself is always equal to $1$ .

This of course holds regardless of which asset class we’re looking at.

And that’s ultimately because the covariance of any asset with itself is always equal to its variance.

To see how this particular fact results in the correlation of any security being equal to one, let’s take a look at the equation for the correlation.

We now know that we calculate stock correlation between $j$ and $k$ like this…

$\rho_{j,k} = \frac{\sigma_{j,k}}{\sigma_j \times \sigma_k}$

If we were to consider the correlation of say, $j$ with itself (i.e., the correlation of an asset with itself), then it would look like this…

$\rho_{j,j} = \frac{\sigma_{j,j}}{\sigma_j \times \sigma_j}$

This is nothing but the equation for the correlation. The only difference is, rather than having two securities, we now just have one security.

Now, the covariance of any asset with itself is equal to its variance.

The equation thus changes to this…

$\rho_{j,j} = \frac{\sigma_{j}^2}{\sigma_j \times \sigma_j}$

And of course, we can see that if you take the standard deviation of $j$ and multiply it by the standard deviation of $j$ , that’s equivalent to taking the standard deviation of $j$ and squaring it.

The standard deviation squared is nothing but the variance.

And so our equation now simplifies to this…

$\rho_{j,j} = \frac{\sigma_{j}^2}{\sigma_j^2} = 1$

This is equal to $1$ because anything divided by itself is always equal to $1$ .

Okay. So that’s why the correlation of any asset with itself is always equal to one.

Let’s go ahead and calculate stock correlation now with an example.

Example on How to Calculate Stock Correlation

Imagine that you hold a portfolio of two stocks, and you have the following information:

The total risk or standard deviation for Betflix and Lotify are 24.39% and 31.94% respectively.

And the covariance between the two securities is 0.01792

Given this information, what is the correlation between Betflix and Lotify?

It’s a good idea to pause for a bit right now and see if you can solve this on your own.

We’re going to assume that you did that.

So let’s go ahead calculate stock correlation for these 2 securities together now.

We know that the correlation of stocks is estimated like this…

$\rho_{j,k} = \frac{\sigma_{j,k}}{\sigma_j \times \sigma_k}$

In our case, we can just substitute $j$ and $k$ with the tickers for Betflix and Lotify, so we can say that the correlation is estimated as…

$\rho_{BFLX,LOT} = \frac{\sigma_{BFLX,LOT}}{\sigma_{BFLX} \times \sigma_{LOT}}$

After that, it’s just a simple case of plugging in the numbers which we’ve got from the question.

$\rho_{BFLX,LOT} = \frac{0.01792}{0.2439 \times 0.3194}$

Solve for that, and you’ll find that the correlation between the two securities is approximately equal to 0.23.

$\rho_{BFLX,LOT} \approx 0.23$

Given a value of 0.23, we can say that the two securities are positively correlated.

But this is a pretty weak positive correlation. It’s not a particularly strong, positive correlation.

Strong vs Weak Correlation of Stocks

If we think of the correlation in absolute terms…

Generally speaking, at least in Finance, we would argue that a correlation of between say 0.01 and 0.5 is reasonably weak.

And a correlation of between say, 0.5 and 0.8 is relatively strong; but not particularly strong. So it’s not incredibly strong.

And generally speaking, a correlation that’s greater than or equal to 0.8, in absolute terms, would be considered strong.

It’s pretty rare to find stocks or indeed financial securities that are negatively correlated (but it’s certainly not impossible).

But it’s particularly rare, we would argue borderline impossible, to find securities that have a significantly strong negative correlation.

It’s pretty impossible to find two stocks that have a correlation of, say, $-0.80$ .

Wrapping Up – Correlation of Stocks

In summary, you learned that the correlation of stocks – similar to the covariance – measures the relationships between securities.

And that’s why we can think of the correlation as an extension of the covariance.

We learned that the correlation is bounded between negative one and plus one inclusive ( $-1 \leq \rho_{j,k} \leq +1$ )

And it’s this property of the correlation that allows it to overcome the major limitation of the covariance, in that the covariance is pretty hard to interpret.

With the correlation, we have a definitive range for the strength of relationships between securities.

Importantly, remember that correlation does not imply causality.

It just so happens that securities move either with each other or against each other.

But it’s not the case that one security is causing the other security to move with it (or indeed against it).

Of course, we learned that we can calculate stock correlation using this equation:

$\rho_{j,k} = \frac{\sigma_{j,k}}{\sigma_j \times \sigma_k}$

And you learned that the correlation of any security with itself is always equal to 1.

Hopefully, all of this makes sense. If any part of this article is not quite clear, please read it again. Or check out our course for a video walkthrough and quizzes to help you track your progress.

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What is the Correlation of Stocks?