In this article, we’re going to explore asset pricing models. Let’s get into it.
What are Asset Pricing Models?
Asset Pricing Models, as the name sort of suggests, are models that help us price assets.
Put differently, they help us determine how much something is worth.
And while most people tend to think of “worth” in dollar terms (i.e., in money terms, regardless of the currency), asset pricing models think of “worth” more in terms of risk vs. return payoffs.
In other words, asset pricing models help us determine the price of risk. Or the price of taking on a risk in order to earn a reward.
Essentially, they are tools that use some math, as well as logic, to determine the expected return of financial securities. Notice that we described them as tools to price risk just earlier; and now we’re saying it’s about expected returns.
You need to stop reading this article right now. And check out our post on Investment Fundamentals: Price, Risk, and Return. Feel free to come back here once you’ve read that!
We’re going to assume that you “get it” so far. So let’s go ahead and explore the fundamental assumptions of Asset Pricing Models.
At a fundamental level, asset pricing models rely on 3 things:
- Perfect information, and
- Efficient markets.
Let’s explore these 3 aspects in greater depth.
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.
This Article features concepts that are covered extensively in our course on Investment Analysis & Portfolio Management (with Excel®).
If you’re interested in learning how to apply Asset Pricing Models for your own investments while working with real world data, then you should definitely check out the course.
The intuition behind asset pricing models
Let’s now dive a little deeper about the linearity aspect.
Equation of a Straight Line
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.
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Asset Pricing Interpretations
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. Let’s now explore the different types of Asset Pricing Models.
Types of Asset Pricing Models
There are a whole host of Asset Pricing Models out there. Far more than we can cover in a single post.
But if one were to categorise all the asset pricing models into, say, two groups…
Then we can think of them as either:
- Single factor models
- Multi-factor models
Single Factor Asset Pricing Models
Single factor models – again, as the name kind of suggests – are models with a single factor.
According to these models, the returns of all securities (stocks, bonds, etc) can be explained by one single factor.
They suggest or hypothesise that everything depends on just one single factor.
Perhaps the most famous – and controversial – single factor asset pricing model is the Capital Asset Pricing Model (or CAPM; pronounced cap-M). More on that in a bit.
Multi-factor models suggest or hypothesise that the returns of securities are explained by multiple factors.
So there’s more than just a single, all encompassing factor.
We’ll talk a lot more about these further down.
Exploring Asset Pricing Models
We’re now going to explore the most famous asset pricing models of all time. Starting with the most famous, and perhaps most controversial one, the CAPM.
Capital Asset Pricing Model (CAPM)
One of the most famous and indeed controversial asset pricing models out there is the Capital Asset Pricing Model (or the CAPM) which ultimately says that the only thing that impacts returns is the market portfolio. That is, the overall stock market.
The CAPM formula looks like this…
Where reflects the expected return on a stock
refers to the risk free rate of return (aka “risk free rate”). reflects systematic risk / market risk, and reflects the expected return on the market portfolio.
Now, just for the record, please don’t call it the CAPM model. The M in CAPM refers to Model already. Calling the CAPM “CAPM Model” is like calling Chai, “Chai Tea”. Chai is tea. It’s not tea tea, now is it…
We digress. The point is…
It’s not just the market though
The CAPM has some major limitations. And we’ve got a whole article dedicated just for the CAPM. Over there, we talked about the fact that there’s no sort of general consensus on how we measure expected return in the market. Or even what we consider to be the real risk free rate for instance.
But perhaps even more importantly, we kind of know that the market is not the only factor that impacts stock returns.
Thus, while the CAPM posits that it’s largely only systematic risk / market risk that matters (captured by the beta value), the reality is a little different.
Firm specific risk or unsystematic risk (aka ) can matter, and can have an impact, albeit not all that much in a well diversified portfolio.
But aside from unsystematic risk, there are other factors that can play a role in explaining asset prices.
And we know this thanks to the more advanced asset pricing models.
The other models work on the premise that stock returns are impacted by two or more factors; not just the market portfolio.
In other words, they hypothesise and suggest that there are other things that impact a stock returns or expected returns of stocks.
Now, the biggest framework for this notion is “Arbitrage Pricing Theory”
Arbitrage Pricing Theory (APT)
This theory suggests that there’s multiple factors that affect returns. Recall that earlier in this post we said that Multi-Factor models are 1 of 2 types of asset pricing models.
They’re based almost entirely on Arbitrage Pricing Theory (or APT)
And so you have a generalised multi-factor model, which looks a bit like this:
It essentially says that the expected return on any stock is equal to the risk free rate, plus a whole host of other factors.
here is looking at the impact of factor one () on the expected return of . And then there’s another factor, .
And the impact of is assessed or evaluated by .
So you can have however many factors as you want, and you go all the way up to, and including the nth factor, , whose impact on the expected return is measured by
Now, we appreciate, this seems rather abstract.
So let’s consider what the classic example of APT models look like.
Classic APT Model
One major, classic example of APT says that the expected return on a stock is impacted by the market (consistent with the CAPM); but it’s also impacted by oil prices, as well as interest rate differentials, which is a very fancy and sophisticated word for essentially saying the difference between the long-term interest rate and the short-term interest rate.
Mathematically, the classic example looks like this…
Where refers to the Price of Oil. And reflects the interest rate differential.
Importantly, this is not the be all and end all of asset pricing models.
Some consider other macroeconomic factors. For instance, you might include things like GDP. Or unemployment rates; or the level of productivity – whatever you like.
The bottom line of this multi-factor model – or any multi-factor model – is stock returns are impacted by a few things; not just the market risk premium.
Now, if you look closely at the equation for the “classic” multi-factor model, you can see that the first part of the right hand side displays .
Essentially then, you can see that this first part of the right hand side of the equation is nothing but the Capital Asset Pricing Model!
And then we’re just extending the CAPM, and saying that there are also these other 2 factors.
You’ll see a similar approach across most other asset pricing models, too.
Reminder: this Article features concepts that is covered extensively in our course on Investment Analysis & Portfolio Management (with Excel®).
If you’re interested in learning how to apply asset pricing models for your own investments and work with real world data, then you should definitely check out the course.
Fama French 3 Factor Model
Perhaps the most popular, or at least one of the most popular multi-factor models is the Fama-French (1992) 3 Factor Model.
It’s still quite popular today, despite being published all the way back in 1992!
And if you’re doing research in finance, then you need to demonstrate that any other factor that you think has an impact on returns, displays that impact in the presence of the Fama-French 3 Factor Model.
The 3 Factor model says that stock returns are affected by the:
- market portfolio (as with the CAPM),
- size of the firm (whether it’s a small company versus a big company), and
- nature of the firm (whether it’s a “value firm” or a “growth firm”)
Note that in this context, “nature of the firm” is not a generally accepted term. It’s just what we like to call it. Officially, it’s called the “value premium”.
Here’s what the model looks like…
Here refers to “Small minus Big”.
We take the returns of small firms and substract the returns of large firms. This is the “size premium”.
The idea’s that small firms tend to earn higher returns vis-a-vis large firms. And this is intuitive, right?
Because if you think back to our sister article which explored the relationships between Price, Risk and Return, we said that the expected return of stocks increases as the risk increases.
Now of course, smaller firms tend to be more risky than larger firms.
So you’d expect the returns on smaller firms to be greater than the returns on bigger firms.
And so the factor is trying to show how the size of the firm impacts its expected returns.
Next up, you’ve got the factor, which is “High minus Low”, and this is the “value premium”.
The factor looks at the returns of high Book to Market (BTM) firms vs. the returns of low Book to Market (BTM) firms.
Low BTM firms would be for instance, technology companies; because they generally tend to be high growth companies. They might not have a lot of tangible assets in place. In other words, they’ll tend to have intangibles, and assets that may not show up on financial statements..
So if you think about Facebook, for example, their biggest asset is the number of users that they have.
And unfortunately, present day accounting standards mean that things like the number of users simply don’t show up as an asset in financial statements.
Thus, as a result, Facebook will inevitably end up having a low book to market.
Now, you would expect that Growth firms tend to do better than Value firms…
But the evidence suggests the contrary.
In fact, Value firms tend to outperform Growth firms. That’s at least what the empirical evidence shows.
And so the factor is incorporating the nature of the firm in a sense, right?
It’s saying, well, how does the fact that this is a high book to market firm or low book to market firm…
How does that impact the expected return on a given stock?
So that’s the Fama French 3 Factor model. Pretty groundbreaking, at least back in 1992.
5 years after it was published though, another seminar paper was published. And a model was created, the core factor of which is now the basis of many an investor’s strategy.
You may not know much about the man who invented it, but we’ll bet that you’ve probably heard of the factor – momentum.
In 1997, the Carhart 1997 4 factor model was made available to the world.
Carhart 4 Factor Model
Carhart (1997) introduced a what’s now called “momentum portfolio”.
Momentum essentially shows that stocks that have consistently delivered higher returns – “winner stocks” – will continue to do well in future.
And stocks that have consistently delivered low returns – “loser stocks” – continue to deliver low returns.
Here’s the model…
Do you see it?
Yeah. Much of the right hand side of the equation is in fact identical to the Fama French 3 Factor model.
here refers to the Market Risk Premium, which is nothing but .
The Carhart 4 extends the Fama French 3 factor model.
The new factor, , reflects “Winners minus Losers”.
Strictly speaking, Carhart called it a factor – referring to momentum.
You’ll find some people calling it “winner minus loser” and others calling it “the factor”, or just “Momentum”.
Regardless of what you call it, Carhart’s 4th factor looks at the impact of momentum on stock returns.
Now, just like Carhart added one new factor to create a new model, we can of course do the same thing.
And in fact, Farma French did so fairly recently, publishing the Fama French 5 factor model in 2015.
But instead of extending the Carhart 4 factor model, they opted to extend their own 3 factor model by adding 2 new factors.
And ignored the Momentum factor, which was, perhaps not as constructive as including it would’ve been.
We digress. The point is…
New Asset Pricing Models can be created by:
- Creating or adding as little as 1 new factor
- Modifying an existing factor (e.g. measuring it slightly differently)
And using this approach, we’ve gone from a single factor model (the CAPM) – nice and elegant – to a 3 factor model, then a 4 factor model, followed by a 5 factor model.
Perhaps you’re wondering…
Are you now going to tell me about a six factor model?
And where does it end?
Beyond 3, 4, and 5 Factor Models
Harvey, Lou and Xu (2016) found that there’s 300 odd factors reportedly impacting returns.
300. Yes, three hundred. That’s not a typo!
So it doesn’t quite end that a six factor model.
You can have a 300 factor model.
But in our opinion, this is arguably over complicating things, for no reason; and it’s not quite elegant.
Finance is a beautiful subject.
It’s a beautiful area.
And doing something like having 300 different factors in one model perhaps distorts the elegance of this field.
But beauty and elegance aside, there are serious concerns about methodology and statistical tests for new factors.
Because it seems like there’s no end to explaining returns!
So we can’t actually say anything with total certainty.
And truth be told, that might just be the case.
In other words, it might well be the case that we can’t explain stock returns neatly in a single model.
And it is indeed a lot more complex than that.
So we have to use a battery of different tools and techniques to try and evaluate investments or analyse investments and manage our portfolios.
This is one of the reasons we focused on giving you a battery of different tools and show you a variety of ways of analysing investments and managing portfolios in our course on Investment Analysis & Portfolio Management (with Excel); and in the Python based iteration of the same course..
The idea’s that once you learn and master these techniques, you can apply different tools in different combinations…
And ultimately test it out for yourself.
Alternative influences on returns
Use these tools with your own investment thesis, and your own investment philosophies to see what works best for you.
Because ultimately even the 3 factor, 4 factor and 5 factor models we looked at aren’t necessarily the be all and end all in Finance.
For instance, stock returns have been shown to be influenced by:
- the day of the week (Gibbons and Hess, 1981)
- the weather (Hirshleifer and Shumway, 2003; Shu and Hung, 2009; Goetzmann et al., 2014)
- lunar cycles (Dichev and Janes, 2003)
- seasonality (Kamstra et al., 2003)
- liquidity (Pástor and Stambaugh, 2003)
- investor sentiment (Baker and Wurgler, 2006)
- religion (Callen and Fang, 2015)
- football (soccer) results (Edmands et al., 2007)
The point is…
There’s a whole host of factors that might influence the stock return and in fact, do influence the stock return.
So when you’re evaluating investments, you want to bear in mind that any model you’re looking at is unlikely to be the “true model” or the “best model”.
We do hope however, that this article inspires you in some way to perhaps join us as researchers and finance, as practitioners and finance, to try and identify more parsimonious, simple models.
Models that are more scalable. And those that better reflect or better explain stock prices.
Globally, the search for a one size fits all asset pricing model still continues.
All right, hopefully you’ve enjoyed this detailed overview of asset pricing models.
In summary, you learnt that asset pricing models are tools that use math and logic to determine the expected return of financial securities.
They rely on linearity, perfect information, and efficient markets as the fundamental assumptions.
Perhaps more importantly, you saw how the framework for multi-factor models – Arbitrage Pricing Theory (APT) – works on the premise that multiple factors affect stock returns; not just the market portfolio.
You saw examples of multi-factor models, including the Farma French 3 Factor Model, the Carhart 4 factor model, as well as the Fama French 5 factor model.
And you saw how it kind of starts from the CAPM and then extends to include other factors.
Of course, you saw that it doesn’t end with these five factor models. There are papers and research out there with models that have hundreds of different factors.
And so this is a field, an area of finance, that still continues to be researched quite heavily and quite actively.
We’ve included the references that we mentioned in this particular article below. So if you are interested in any of the areas, and if you want to dig in deeper, do give them a read.
Do you want to leverage the power of Excel® and learn how to rigorously analyse investments and manage portfolios?
Callen, J. L. & Fang, X., 2015. Religion and Stock Price Crash Risk. Journal of Financial and Quantitative Analysis, 50(1-2), pp. 169-195.
Carhart, M. M., 1997. On Persistence in Mutual Fund Performance. The Journal of Finance, 52(1), pp. 57-82.
Dichev, I. D. & Janes, T. D., 2003. Lunar Cycle Effects in Stock Returns. The Journal of Private Equity, 6(4), pp. 8-29.
Edmands, A., García, D. & Norli, Ø., 2007. Sports Sentiment and Stock Returns. The Journal of Finance, 62(4), pp. 1967-1998.
Fama, E. F. & French, K. R., 1992. Liquidity Risk and Expected Stock Returns. The Journal of Finance, 47(2), pp. 427-465.
Fama, E. F. & French, K. R., 2015. A five-factor asset pricing model. Journal of Financial Economics, 116(1), pp. 1-22.
Gibbons, M. R. & Hess, P., 1981. Day of the Week Effects and Asset Returns. The Journal of Business, 54(4), pp. 579-596.
Goetzmann, W. N., Kim, D., Kumar, A. & Wang, Q., 2014. Weather-Induced Mood, Institutional Investors, and Stock Returns. The Review of Financial Studies, 28(1), pp. 73-111.
Harvey, C.R., Liu, Y., & Zhu, H., 2016. … and the Cross-Section of Expected Returns. The Review of Financial Studies, 29(1), pp. 5-68.
Hirshleifer, D. & Shumway, T., 2003. Good Day Sunshine: Stock Returns and the Weather. The Journal of Finance, 58(3), pp. 1009-1032.
Kamstra, M. J., Kramer, L. A. & Levi, M. D., 2003. Winter Blues: A SAD Stock Market Cycle. American Economic Review, 93(1), pp. 324-343.
Pástor, L. & Stambaugh, R. F., 2003. Liquidity Risk and Expected Stock Returns. Journal of Political Economy, 111(3), pp. 642-685.
Shu, H.-C. & Hung, M.-W., 2009. Effect of wind on stock market returns: evidence from European markets. Applied Financial Economics, 19(11), pp. 893-904.