The Capital Asset Pricing Model (CAPM; pronounced cap-M). Probably the most famous — and controversial — asset pricing model out there. Here’s everything you need to know about it, including how to use it.
First and foremost – at the risk of sounding too obvious, the CAPM is an asset pricing model (hence the name, Capital Asset Pricing Model). Genuinely gaining a solid command of the CAPM requires understanding asset pricing models well first.
That’s what we’ll start with.
Before we do though, if you’re new to the world of investment or finance, please see this post on Price, Risk, and Return first.
What are Asset Pricing Models?
First and foremost, what are asset pricing models?
Ultimately, they’re just tools that use a bit of math and some logic to determine the expected return of financial securities. To determine how much a security can earn us, should we invest in it.
You can use these models to estimate the expected return of stocks as well as other financial securities. For instance bonds; or even real estate for that matter.
In terms of how these models work… At a fundamental level, they rely on 3 things:
- Perfect information, and
- Efficient markets.
We’ll briefly discuss what these three things mean.
You’re probably already familiar with linearity. To give you the most simple example though, we have the equation of a straight line:
This says that any variable (or any dependent variable ) is dependent on an independent variable .
What this is doing is looking at the impact of on . So we’re saying that impacts .
You might think of as the grades you get in school (which are a function of , which might be “hard work” or the amount of time you spend studying, the quality of the learning materials, etc.
Alternatively you can think of as the salary you’re getting from employment. And that might be a function of say, your years of experience, or the qualifications that you have, etc.
So you can think of as anything that you’re trying to predict or explain, and is what will help you predict or explain .
That’s just a simple, very brief explanation of linearity. And Asset Pricing models rely quite heavily on this concept of linearity, as we’ll see in just a bit.
The other thing the asset pricing models rely on is perfect information.
And this pillar is a bit far-fetched or “out there”, if you will. It’s not necessarily reasonable. I personally don’t think it’s reasonable, but some people do.
It assumes that every single investor has exactly the same information as the other investors. All investors have exactly the same information.
And not only do they have the same information, they all have the right information. So there is no information asymmetry. There’s no wrong information. There’s no “fake news”, or there’s no ambiguity in the world in the context of information.
In my opinion, it’s a bit far fetched. But nevertheless, it’s what the models rely on. So that’s what we’re going to go with.
Last but not least, asset pricing models rely on efficient markets. This again sort of extends, and builds on the idea of perfect information.
If markets are efficient, it means that all the information is incorporated and reflected in the prices of securities as soon as it becomes available.
It means whenever any new information comes out about a stock or a company or a bond or a government or whatever it is, all of that information gets incorporated into securities immediately.
If you believe in efficient markets, you’re part of one group of people in finance.
And if you don’t, you’re likely a behavioural finance person. That’s just another group, or another school of thought in finance. It depends whether you believe in rational expectations or not, but for the context of asset pricing models, they rely on rationality and perfect information, linearity, and indeed efficient markets.
The logic behind asset pricing models
Let’s now dive a little deeper about the linearity aspect.
Recall the equation of a straight line:
And again you can think of y as whatever you like. So maybe the grades in school or the salary that you’re earning or the profit that you earn from your business.
You can think of as an output variable and as the input variable.
If we were to visualise the equation of a straight line, then it would look like this:
Notice the specific point where the line touches (or, starts from) the y axis.
That specific point — the point where the line starts — is what we call “alpha” (denoted ).
Alpha is the intercept term.
And it’s showing us the value of when is equal to zero.
At this point — at — you can see that the value of is equal to zero. Yet has some value.
So (or the intercept) is telling you the value of when .
Beta (denoted ) on the other hand, is the slope of the line.
And when you add the two, you end up with a value of which is equal to Alpha plus Beta times (or ).
Now this works if we have all of the data and everything is great and perfect.
If we think about applying this in the context of a simple regression with some sort of data, then your data points might look a bit like this:
Think of these observations as all the different hours that you spend studying, so whether you spend an hour a day, or three hours a day, or five hours a day, or whatever.
Or it might be the years of experience that you have working. Whatever you like. Let your imagination do the work.
With the regression, we want to plot a line that best fits.
For example, the line that best fits might be this:
And indeed and have the same interpretations as before. But now, because the line isn’t quite capturing all of the data points, we have what we call “error terms” (or epsilon; denoted ).
The line here predicts that the data points should be on the line.
But clearly there’s a whole host of data points that are not on the line.
The difference between any data point and the line is what we call an error term (often annotated as Epsilon, or ).
The equation for a line is now:
Generalised Asset Pricing Model
Previously, it was just our and . Now we also have this error term, .
Why am I talking about all of this in a finance related article?
Because the generalised asset pricing model essentially looks something like this:
is the expected return on a stock j.
is a ‘factor’ that impacts the expected return of the stock.
So where for you it might have been hard work impacting your grades, or experience impacting the salary, in the context of stocks, it might be for instance “the market”, or “oil prices”, or the kind of government that’s in place, etc. is some ‘factor’ that has an impact on the expected return on the stock.
here again is the intercept term, and as before, it’s the value of (in this case) expected return when the factor is equal to zero.
If is say, the market, then would tell us the expected return on the stock when the market return is equal to zero.
And here is looking at the impact of on the expected return of the stock.
It’s looking at the impact on when changes.
It tells us how much will increase or decrease by, if the market for instance were to increase by 1 percent
There is another major interpretation of the and we’ll talk about that in a bit.
But for now that’s pretty much all you need to know.
And as before, Epsilon () is the error term.
If we were to visualise this generalised asset pricing equation graphically, as before, you’d have something like this:
Pretty much identical to the equation of a straight line, with an error term.
Okay. So this is how asset pricing models tend to work in general. Now let’s look at the one you came to this post for to begin with.
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The Capital Asset Pricing Model (CAPM)
One of the most famous and indeed controversial models is what we call the Capital Asset Pricing Model (or CAPM; pronounced cap-M).
The CAPM works on the premise that the return on any asset is based exclusively on the asset’s relationship with the market.
In other words, the only thing that can impact stock returns is the market portfolio.
If you’re in the U.S., that might be the S&P 500; Japan, it might be the Nikkei; the U.K., it might be the FTSE, and so on and so forth, depending on where you are.
That’s as far as the practical aspects of the CAPM goes.
In theory, the market portfolio of the CAPM includes every single asset in the entire world. So it includes things like gold and silver, all your different commodities, as well as stocks, and real estate, and it’ll even include things like the vendors that you might find on market streets.
It really does include any and every single kind of financial asset and indeed real asset as well.
From a practical standpoint, it’s pretty much impossible to get a market of that sort. Or to get data of a sort of global market that encompasses every single kind of asset.
And so the only thing we can really rely on are national level markets, or maybe some international level markets. Because that’s what we have data for available quite freely / easily.
Importantly, the CAPM ignores every other factor that might influence stock returns.
It ignores the firm’s future prospects. As well as the culture of the firm. And the values of the management or the company as a whole. In fact, it ignores any other firm specific element.
Now maybe you’re thinking, that’s absolutely bonkers. I mean how on earth can we justify estimating stock returns or expected returns while ignoring all of this information?
Ultimately it does so because it relies on the power of diversification.
As long as you’re dealing with portfolios, individual risk of securities doesn’t really matter. This has, and can be, shown mathematically; but it’s beyond the scope of this particular article.
Just keep in mind that it’s not entirely unreasonable to ignore all these other firm specific elements when we’re trying to estimate the expected return of of a stock using the Capital Asset Pricing Model.
The logic behind the CAPM
Okay. That’s the notion behind the CAPM. And the premise of the CAPM.
But what does it look like as an equation?
Well, if we start with the generalised asset pricing equation we saw before, which was essentially just a regression equation (almost like a straight line). We said that it looks like this:
Under the CAPM, we’re not just interested in expected returns, but instead, we’re interested in what we call excess returns.
That’s the return over and above what you can get risk-free. It’s the return that you expect to earn in exchange for taking some level of risk.
How do we get excess returns? Well we subtract the risk-free rate from the expected return. In other words, it’s
The risk free rate is the rate that you can earn without any risk whatsoever. It’s a guaranteed return with no risk.
Theoretically, this is the rate that you earn if you were to lend money to the U.S. government.
So if you were to buy treasury securities or T-bills or T-bonds, then you’re essentially lending money to the government of the U.S. (or the U.S. Treasury). And the U.S. Treasury will pay you interest on those securities.
The reason it’s considered risk free is if the U.S. government is unable to pay you, that would mean that they default on their loan — on their legal obligation. And the likelihood of the U.S. defaulting on a loan is pretty slim. It’s quite very low.
Simplified Proof of the Capital Asset Pricing Model
Coming back to the generalised asset pricing model. Incorporating excess returns would transform it to this:
And in the case of the CAPM, the only factor which can impact / influence returns is the market portfolio. Strictly, the market excess return.
We can thus go from the generalised asset pricing equation to a more specific one which looks like this:
Notice that the “factor” is now . This is the excess market return. It’s the return that you earn on the market over and above what you can earn risk-free.
Now recall that we said the CAPM relies on the premise that the only thing that impacts stock returns is the market portfolio.
If the market is the only influencing factor in returns, then Alpha () must be equal to zero. And epsilon () must also be equal to zero.
Thus we end up with the following equation:
The excess expected return of a stock is equal to Beta () times the excess expected return the market.
Now, we don’t tend to write the CAPM in this way. We tend to take the risk-free rate to the other side (or strictly, adding to both sides).
The equation for the CAPM is thus:
Recall we said that the Beta is the slope. But there is an alternative interpretation, and this is a vital one.
Because the here is assessing the impact of the excess market return on the expected return on the stock, we define this as the systematic risk or the market risk of the stock.
The here is showing us how risky the stock is relative to the market.
Here’s the simplified proof again, in one slide:
Remember again that we just express the Capital Asset Pricing Model by modifying the equation ever so slightly. That is, adding the risk-free rate to both sides, so you end up with:
Applying the Capital Asset Pricing Model
Now that you’ve understood how asset pricing models work in general, and you now have a good understanding of one of the most simple and popular (albeit controversial) asset pricing models out there (the CAPM), we can go a little deeper and learn how to use the Capital Asset Pricing Model with some real world data.
Let’s go ahead now and apply this model with some real world data by estimating the expected return on Alphabet (previously known as Google).
Remember we need three things to use the CAPM. We need:
- The risk-free rate ()
- Beta of the stock (), and
- The expected return on the market ().
We can get all of this data thanks to the F.T.
So if we pop into the F.T. and search for Alphabet, we get the quote. And you can see Alphabet’s Beta right here:
We can get the data for the risk-free rate by heading over to markets.ft.com/data and scrolling down to the data for ‘Bonds & rates’. We want the yield on U.S. treasuries, here:
The last thing we need is the market return. We’re going to work with the S&P500 as the market portfolio. There’s two ways we can do this. One is to be a bit cheeky and just use the one year change as your return on the market which is what we’ve got here:
Alternatively, you can calculate the expected return using the mean or indeed state contingent weighted probabilities. These 2 concepts are outside the scope of this article, but you can learn all about it in our courses on Investment Analysis & Portfolio Management with Excel, or with Python.
Note that using the one year change is in fact quite crude. For instance, it relies on the idea that the only return that matters for the future is the most recent one. In other words, it ignores the past before the most recent observation.
Since we’ve got all the data we need, it’s now just a simple case of plugging in these numbers into our equation for the CAPM.
Note that we’ve followed the general consensus and used the ticker ‘GOOGL’ instead of the company name ‘Alphabet’.
Multiply the excess market return of by the beta of to get (approximately).
And that’s pretty much it!