What is Systematic Risk (aka Beta)? How to Calculate Beta?

In this article we’re going to explore what is Systematic Risk (aka Market Risk, or “Beta”). We’re also going to learn how to calculate beta (or systematic risk) from scratch.

Just a quick recap though, remember that the generally accepted measure for total risk is the standard deviation of a stock.

Now, as far as this article goes, you need to be quite comfortable with the variance and standard deviation formula. If you’re not familiar with these, we’d encourage you to check our article on How to Measure the Risk of a Stock first, and then come back to this one.

Alright, with that out of the way. Let’s now explore systematic risk.

What is Systematic Risk?

What is systematic risk, fundamentally? In a nutshell, systematic risk is an aggregate market-level risk that impacts all businesses and all companies in one way or another.

This is different from unsystematic risk, or firm-specific risk, which affects firms individually, but not collectively.

In the financial market, we can think of Systematic Risk or “Market Risk” as a measure of how much the stock is impacted by the overall market.

For example, this would mean how much a company like Amazon Inc. is impacted by the S&P 500 or indeed the NASDAQ or the Dow Jones or any of the stock market index, as well as how much it’s impacted by a recession, or a boom, or inflation, or other forms of “market risk”.

Why is it non-diversifiable risk?

Notice that each of these systematic risk factors is totally outside our control, in that we can’t influence them in any way.

We can’t get rid of them on our own, either. And that’s why systematic risk is also referred to as “non-diversifiable” or undiversifiable risk.

Importantly, these aspects of the market risk are incorporated assuming that the overall stock market index is aware or reflective of these other risks.

And that’s a fairly reasonable assumption, right?

If we think about the stock market index as a whole, it’s likely reflecting whether an economy is in a recession or whether it’s in a boom or the overall state of the economy and the macroeconomic factors that affect it.

For the most part, the financial market as a whole kind of knows where the economy is.

Systematic Risk Jargon Buster

We’ve already used a variety of terms to refer to systematic risk. And this is consistent with other areas of Finance – there are a whole host of different terms for the same thing.

Here’s a simple jargon buster to bring you up to speed.

Systematic risk is also called:

non diversifiable risk
undiversifiable risk
systemic risk
market risk
non idiosyncratic risk

CAPM and Systematic Risk

Now, if you recall, from our sister post on the Capital Asset Pricing Model Explained, we saw that the CAPM is one of the most popular asset pricing models.

It works on the premise that the expected return for any stock is only impacted by the market exclusively.

And the CAPM allows us to estimate the expected return on a stock as…

$E[r_j] = r_f + \beta (E[r_m] - r_f)$

Here $E[r_j]$ reflects the expected return on a stock $j$ .

$r_f$ refers to the risk-free rate of return (or simply just the risk-free rate).

$E[r_m]$ reflects the expected return on the market portfolio (aka expected market return).

And last but certainly not the least, Beta ( $\beta$ ) here represents the stock’s systematic risk or the market risk.

Let’s now think about how we actually measure it.

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How to calculate beta

The beta can be calculated quite simply by using this formula:

$\beta_j = \frac{\sigma_{rj,rm}}{\sigma^2_{rm}}$

But what does this actually mean? And how to interpret beta? To understand what the beta formula above means, let’s get back to the basics.

The Beta represents the so-called systematic risk. Or put differently, it represents the stock’s exposure to the market risk.

In other words, the Beta evaluates and measures how excess expected return on the market ( $E[r_m] - r_f$ ) impacts the expected return on the stock ( $E[r_j]$ ).

Thus, in thinking about how to calculate systematic risk, it’s important to know how to calculate beta.

Because the beta looks at the stock’s individual exposure to the market.

How does it do that?

It does so by measuring the relationship between the stock and the market, relative to the overall market risk.

It’s quantifying the relationship between any security or any firm or any stock, with that of the market; and it’s scaling it by the overall market risk.

And you’ll see what I mean when we look at the formula for the beta (aka systematic risk formula) in just a bit.

But essentially, what we need to know is that the Beta comprises of two parts:

the relationship between the stock and the market portfolio, and
the overall market risk

Measuring overall market risk

If you know how to measure the total risk of a stock, then you already know how to measure the “overall market risk”.

Because “overall market risk” is nothing but the volatility of the market. And that in turn can be estimated by calculating the variance of the market.

The variance of the market can be estimated as…

$\sigma^2_j = \frac{1}{n-1}\sum_{t=1}^n (r_m - E[r_m])^2$

The part we don’t know how to measure just yet is indeed the relationship between the stock and the market.

Measuring the relationships between securities

Now, the relationship between any two assets can be measured by the Covariance, as well as the Correlation.

We’re not going to learn about the correlation in this article because we don’t really need to.

If you are interested though, feel free to read this article which explains the Correlation in great detail.

But as far as this article goes, we’re going to be focusing on the covariance as our measure of security relationships.

The covariance measures the relationships between any two securities, and it does so by estimating their co-variability.

The formula for the covariance looks a bit messy. It’s this beauty right here…

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

This is a general/generic equation for the covariance. We’re interested in the covariance between a stock and the overall market. That looks like this…

$\sigma_{rj,rm} = \frac{1}{n-1}\sum_{t=1}^n (r_j - E[r_j])(r_m - E[r_m])$

Not hugely different to the previous one now, is it? We’ve just changed the subscript $k$ to $m$ to reflect the fact that we’re dealing with a stock $j$ and the market $m$ .

Even then, this equation is not significantly different to something we’ve already seen, right?

Exploring the covariance

It’s very similar to the variance, except rather than looking at the variability of an asset with itself, we’re looking at the variability of one asset with the variability of another asset.

Just a quick note – in case you’re wondering “hang on, that $\sigma$ symbol was what we used for the standard deviation, right?!”…

Note that the standard deviation will have just one subscript.

For the covariance, we have two subscripts. In our case, $rj$ and $rm$ . And when you have two subscripts for $\sigma$ , you’re dealing with the covariance, not the standard deviation.

Okay. So now we know how to measure the overall risk of the market by the variance of the market ( $\sigma^2_{rm}$ ). And we know how to quantify the relationship between the security and the market ( $\sigma_{rj,rm}$ ).

In other words, we have the two measures that we need in order to estimate the market risk of a stock. To estimate the Beta.

Systematic Risk Formula (Beta Formula)

And this takes us to the systematic risk formula (aka Beta formula):

$\beta_j = \frac{\sigma_{rj,rm}}{\sigma^2_{rm}}$

The Beta for stock $j$ is calculated by taking the covariance of $rj$ and $rm$ and dividing it, or scaling it by the variance of the market ( $\sigma^2_{rm}$ ).

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In other words, systematic risk displays the relationship between the stock and the market, relative to the overall market risk.

If we were to open up this equation, then it would look something like this…

$\beta_j = \frac{\sigma_{rj,rm}}{\sigma^2_{rm}} \equiv \frac{\frac{1}{n-1}\sum_{t=1}^n (r_j - E[r_j])(r_m - E[r_m])}{\frac{1}{n-1}\sum_{t=1}^n (r_m - E[r_m])^2}$

You have the covariance between any stock $j$ and the market $m$ , divided by the variance of the market.

If this equation is freaking you out right now, please don’t let it freak you out.

Because again, this is something you already know, right?

We’ve talked about the variance in quite a lot of detail when we looked at estimating the total risk of a stock.

The only new thing here is this covariance, which again, is actually quite very similar to the variance.

Now, importantly, this is the Beta of a given stock.

The market itself also has its own Beta.

Just a quick note – if the equation is still freaking you out, pause. Breathe. Calm down.

Take a look at our Financial Math course on Financial Math Primer for Absolute Beginners. You can thank us later!

We’re going to assume that you’re more or less alright otherwise.

How to calculate beta of the market

Now that you know how to calculate beta for an individual stock, let’s look at how to calculate beta for the market as a whole.

Crucially, the Beta of the market will always be equal to one.

$\beta_m = 1$

Why is that the case?

Well, it’s actually very easy to see. We know that the (expanded) formula for the systematic risk / Beta of a stock is…

$\beta_j = \frac{\frac{1}{n-1}\sum_{t=1}^n (r_j - E[r_j])(r_m - E[r_m])}{\frac{1}{n-1}\sum_{t=1}^n (r_m - E[r_m])^2}$

If we were to look at the Beta of the market, then you’d have something like this…

$\beta_j = \frac{\frac{1}{n-1}\sum_{t=1}^n (r_m - E[r_m])(r_m - E[r_m])}{\frac{1}{n-1}\sum_{t=1}^n (r_m - E[r_m])^2}$

Now, this is actually very similar to the equation above.

The only difference is that rather than $(r_j - E[r_j])$ , we have $(r_m - E[r_m])$

This is now the covariance of the market, with the market.

Because we’re looking at the relationship of the market, with the market.

And that sounds really strange, of course, but I mean, this is precisely what the Beta is, right?

The Beta looks at the impact of the market on a security. If you’re looking at the Beta of the market, then you’re looking at the impact of the market on the market.

So what you have now is the covariance of the market with the market, divided by the variance of the market.

Now, the covariance of any asset with itself is equal to its variance. So essentially what you have then is the variance of the market divided by the variance of the market.

$\beta_m = \frac{\frac{1}{n-1}\sum_{t=1}^n (r_m - E[r_m])^2}{\frac{1}{n-1}\sum_{t=1}^n (r_m - E[r_m])^2} \equiv \frac{\sigma^2_{rm}}{\sigma^2_{rm}}$

And of course, anything divided by itself is equal to one. Therefore…

$\beta_m = \frac{\sigma^2_{rm}}{\sigma^2_{rm}} = 1$

Thus, the Beta of the market by definition must always be equal to one.

How to Interpret Beta

Now that you know how to calculate beta for individual stocks as well as the market as a whole, let’s think about how to interpret beta.

Using the Market Beta

What’s more important, and indeed powerful in a sense, is that because the market Beta is always equal to one, we can use it as a benchmark to compare other stocks. We can use it to compare the level of the market risk of a given stock.

In other words, what we can say is that if the Beta of any stock $j$ is lesser than the Beta of the market, then the stock is less risky, relative to the market.

So it’s less volatile in comparison to the market.

And similarly, if the Beta for stock $j$ is greater than the Beta of the market, then the stock is riskier than the market.

Lastly, if the Beta of a stock $j$ is equal to the Beta of the market, then the stock is as risky as the market.

Now because the Beta of the market is always equal to one, we have a nice and clear, numerical benchmark to work with.

So if the Beta of any stock is lesser than one, then the stock is less risky relative to the market.

And if it’s greater than one, it’s riskier than the market.

Finally, if it’s equal to one, then it’s as risky as the market.

Systematic Risk and Expected Returns

And remember when we looked at the relationships between Price, Risk, and Return, we said that if the risk increases, then the expected return increases.

And if the risk decreases, then the expected return decreases.

Similarly then, if the stock is less risky than you’d expect the return to be lower than the market.

And if the stock is riskier – so if the Beta is greater than one – then since it’s riskier than the market, we’d expect the return to be greater than the market.

And indeed, if the is equal to one, then we’d expect the return on the stock to be pretty much the same as that of the market.

Now, importantly, this is what we expect. The empirical evidence does actually show that sometimes this doesn’t quite work.

Sometimes, you have “low beta stocks” (typically those with Betas lower than 1) delivering returns that are greater than the market.

There’s in fact a theme called “betting against beta”, with a pretty vast research literature on it. So if you’re interested, you can give those types of research papers a read.

The point is, this is what we expect to happen. But don’t be surprised if you find low beta stocks delivering a high return. And indeed high Beta stocks – those with a Beta greater than one – delivering lower returns.

We really can’t stress the importance of this enough. Especially because the beta is often used by many an investor for asset allocation decisions.

Beta and Asset Allocation

Specifically, because a diversified portfolio is not affected by nonsystematic risk / firm specific risk, the only thing that matters is systematic risk. In fact, that is the only thing you as an investor are rewarded for.

The effects of diversification mean that most if not all business risk / nonsystematic risk is eliminated. The only risk that remains is systematic risk.

And this is why investors place much importance on managing the portfolio beta, for example.

The investment decision here revolves around choosing an individual security to add to the portfolio based largely, sometimes almost exclusively on, the stock’s beta.

It’s still worth treating that strategy with some caution, however. Even though, as has been proven mathematically, firm specific risk tends to 0 as the number of assets in a portfolio increase.

And the only thing that remains, and needs to be somehow “managed” is systematic risk.

Wrapping Up – What is Systematic Risk

All right, in summary, we’ve learned what is systematic risk (aka market risk of a stock), and that it is measured by the Beta.

The Beta displays the relationship between the stock as well as the market, relative to the overall market risk.

We learned how to calculate beta from scratch. Specifically, we saw that the beta formula is made up of two parts.

The overall market risk is captured by the variance or volatility ( $\sigma_2_{rm}$ ), and the relationship between the stock and the market is proxied by the covariance ( $\sigma_{rj}{rm}$ ) which is looking at the co-variability between the asset and the market.

We also learned that since the Beta of the market is always equal to one, we can use it as a benchmark to compare other stocks.

So remember, if the Beta of a stock is greater than one, then it’s riskier relative to the market.

If it’s less than one, it’s less risky compared to the market.

And if it’s equal to one, then it’s as risky as the market.

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What is Systematic Risk (aka Beta)? How to Calculate Beta?