Investing > Arbitrage Pricing Theory Explained

Arbitrage Pricing Theory Explained

Although decades old, arbitrage pricing theory is more relevant than ever— we’re about to find out why.

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Updated June 17, 2022

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Can you tell whether or not an asset is overpriced or undervalued?

By now, it is obvious that these things cannot be determined at first glance. Stock recommendations, keeping up with the news, and newsletters are great—but they only go so far. Investing in the stock market is risky, and there is always a lot on the line—to truly make it work, an investor must use something a bit more concrete.

This is where the arbitrage pricing theory comes in. This method of analysis hails from the 1970s, as was developed as an improvement upon the capital asset pricing model. The question of whether or not an asset is overvalued or undervalued is an important one—both of these possibilities present opportunities to make money. 💰

To make things even more high-stakes, the stock market is held to be quite expensive overall nowadays. Making the wrong call and buying an overvalued and risky stock can easily end up costing investors a lot of money.

Arbitrage pricing theory uses multiple factors when looking at an asset—including expected returns and macroeconomic or systematic risk factors. Once all of that information is gathered, analyzed, and put through the math-grinder (a rather scary-looking formula), investors are left with a rather simple point of data—the expected return of an asset.

Once an investor knows the expected return of an asset, they can compare it with alternatives, determine whether or not it is fairly priced, and take steps to profit from any incongruencies in pricing.

Fair warning—this is a financial theory, and it falls under academic finance. This stuff involves a lot of complex math, and without a tool purposely built for using the APT, it is unlikely that it can be applied as is to retail investing. 

However, even without a complete mastery of everything involved, the principles behind the theory can still be applied to retail investing. Macroeconomic factors, expected returns, and sensitivity to various factors—this is the kind of thinking that all serious investors should apply. 🤓

What you’ll learn
  • What is Arbitrage in Finance?
  • Understand Arbitrage Pricing Theory
  • How Does APT Work
  • Arbitrage Pricing Theory - Formula
  • An Example of APT
  • APT vs. CAPM
  • Pros and Cons
  • Conclusion
  • FAQs
  • Get Started with a Stock Broker

What is Arbitrage in Finance? 📚

Before we move on, we should start by demystifying one of the terms related to today’s topic that isn’t often mentioned—and that is arbitrage. Arbitrage is a risk-free method of investing—and while that might sound far-fetched, it isn’t.

Arbitrage is accomplished by purchasing and selling the same security but on two different markets. The idea behind the approach is simple—if there are discrepancies in the market, and the same asset is listed at different prices, the difference between the prices can be pocketed as profit. Those who utilize this approach are called arbitrageurs, and arbitrage can be used for a variety of investments, including currencies, commodities, and securities.

While it might sound great on paper, arbitrage usually has very low profit margins, and finding and taking advantage of such market inefficiencies takes both skill and time. Arbitrage is perfectly legal—in fact, it is welcomed by regulatory bodies worldwide, as it serves to highlight and ameliorate market inefficiencies.

Being a low-risk strategy, arbitrage also brings in small profits per trade—in practice, most arbitrageurs are large financial institutions or funds that can use both high-volume and high-frequency trading, together with the necessary infrastructure to operate in multiple markets and keep track of arbitrage opportunities.

So, all in all, pure arbitrage is mostly out of reach for retail investors—who simply cannot compete with advanced algorithms, access to dealing desks, and who wouldn’t be able to make profitable trades due to commissions. Several other types of arbitrage, such as risk arbitrage, are available to retail investors—but that’s not the topic of this guide.

The concept of arbitrage was instrumental in developing arbitrage pricing theory—one of the most utilized frameworks for asset pricing and an essential tool for value investors. Although the way arbitrage is used in this theory is slightly different, understanding the key concept is still essential.

What Exactly is Arbitrage Pricing Theory? 👨‍🏫

The arbitrage pricing theory is a pricing model for assets that was first defined by American economist Stephen Ross, an MIT Sloan School of Management professor, in 1976. It was designed to improve upon earlier work—namely, the capital asset price model (CAPM) that was introduced in the 1960s.

The CAPM had (and still has) a couple of drawbacks that we will cover later—chief among them the fact that it only took one variable into account. The APT improves on this by factoring in various things that can affect a stock’s price—mainly macroeconomic factors such as inflation, interest rates, risk-free rates, index performances, gold prices, and GDP growth, among others.

Because it takes into account multiple variables, APT is generally much more accurate than its predecessor—on top of that, it is more suitable for long-term investments. The variables used in the model can also be changed and customized by investors if they think other factors are relevant to the price of the asset being evaluated.

So, this is all well and good, but saying that something is an “asset pricing model”, while true, doesn’t exactly explain everything. The APT is used to determine an asset’s fair price and expected returns—once an investor has that knowledge, they can use it to profit if the current asset price does not match its fair price. 

However, that’s still a bit too general—but to really explain how everything works, we have to take a step back and look at some of the groundwork and the assumptions behind this theory. Once the most important terms have been cleared up, everything will fall into place—and it becomes apparent that all these highly-technical sounding terms actually describe a rather simple and logical theory.

Assumptions in the Theory 📖

Like all financial theories, the APT relies on a certain number of baseline assumptions, without which the model simply wouldn’t work. There is an important note to be made here—like most financial theories, the APT is based on a set of hypothetical conditions that aren’t realistic—so, does that invalidate everything and turn the theory into a simple thought exercise?

Well, no, not exactly—although economics as a science always strives toward objectivity and empiricism, there is no way to construct a theory like this without hand waving away some of the issues that do occur in financial markets worldwide. 

This doesn’t invalidate the theory—but it does explain why one can’t simply use it and win at the stock market—the theory is solid, but those pesky assumptions lead to unexpected results.

Okay—now for the actual assumptions. Much like its predecessor, CAPM, the APT assumes that investors are risk-averse—however, APT is a bit more relaxed in this regard. Both theories hold that returns follow a linear pattern by way of mean reversion.

Graph illustraing the mean reversion theory
The theory of mean reversion is an important element of the arbitrage pricing theory.

Both theories also assume that the market is perfect—there is unlimited supply and demand, perfect competition, all investors have access to the same information, and all of them seek to maximize profit.

However, the APT does hold that the market sometimes does act inefficiently—by mispricing securities. This is a very important point—commit it to memory, because we’ll come back to it later.

On top of that, both theories assume that the market is frictionless—meaning that taxes on assets and transaction costs do not apply (we wish).

Arbitrage in APT 👩‍⚖️

As we’ve mentioned, arbitrage in the APT doesn’t really work like arbitrage in most other models, theories, or cases. In fact, this is the reason why the theory got its name. So, let’s take it from the top.

Whereas in other cases, arbitrage refers to the simultaneous purchasing and selling of a security in two markets (buying in a market where the asset is cheaper and selling where it is more expensive) in order to profit from those small differences, arbitrage in APT refers to a directional trade. If this doesn’t yet make sense, don’t worry,—we’ll backtrack a bit now.

Remember when we talked about the assumptions behind the theory just a few moments ago? In an efficient market, arbitrage as it is generally understood could never happen. 

However, APT does state that markets sometimes misprice securities. So, if an investor determines that a security is mispriced, they can place a directional trade—if it is overvalued, the asset can be shorted, and if it is an undervalued asset, it can (and should) be bought. If the model holds true, then this is almost risk-free—akin to arbitrage except that there is still some degree of risk.

So, what’s the catch? Well, according to the APT, the added buying and selling pressure created by these directional trades essentially revert things back to being balanced—arbitrage, as understood by APT, exists, it is taken advantage of, and then it disappears—hence the name of the theory.

How Does APT Work? 👷‍♂️

As established, the ATP works by identifying an asset’s fair price by factoring in expected returns and various macroeconomic risk factors. Once an asset’s fair price is determined, it is simple to figure out how much the current price of the asset is different from it.

With that information, the process of value investing becomes much simpler. If a stock is sufficiently undervalued, and other metrics look promising, then applying the theory is simply a matter of executing a trade and waiting for the right opportunity to resell the stock.

To use an example, if a stock is trading at $70, but arbitrage pricing theory leads to the conclusion that its fair price is $85, then a profitable trading opportunity is present—all one has to do is purchase the stock and wait for the eventual price correction. Although this is the most straightforward way to utilize the APT, it isn’t the only one.

On the other hand, if a stock is overvalued, then an opportunity for short-selling is at hand. A stock that is trading at $35 that has a fair market value of $25 will eventually drop in price to $25—borrowing the stock, selling it, and returning it at a later date when it is less expensive is a simple, straightforward way to profit from shorting.

The important thing to keep in mind is that, like with everything in finance, there is more than one way to skin a cat, and there are no one-size-fits-all solutions and silver bullets. The art of properly researching stocks is a skill that takes lifelong practice, and while the APT is a great model, it is only one part of a seasoned investor’s toolkit. 

The APT is a highly-customizable approach in that it is left up to investors what inputs and factors to consider—but even more than that, it should be used in conjunction with other methods. 

The actionable data taken from APT should be contrasted and used together with methods such as fundamental analysis, discounted cash flow analysis, technical analysis, analyzing earnings reports, and looking at assets through the lens of various ratios such as price-to-earnings, and various metrics such as income, liquidity, and leverage ratios. To make this process of analysis easier, investors can look into high-level tools designed to analyze stocks.

Arbitrage Pricing Theory—Formula 🧮

Finally, let’s move on to the actual mathematics behind the arbitrage pricing theory. The formula that is used to calculate expected returns might seem daunting at first glance, but it actually isn’t that complicated. To explain it as best as possible, let’s first take a look at the formula itself, followed by an explanation of the various inputs and other factors that go into it.

The Mathematical Model 📗

To understand the mathematical model of the theory, we have to figure out what its main components are. There are four of them, and they are:

  • ☑ ER(x) – Expected return of the asset
  • ☑ RF– Current riskless rate of return
  • ☑ βn (Beta) – The asset’s price sensitivity to factor n
  • ☑ RPn – The risk premium associated with factor n

The equation used for the arbitrage pricing theory goes as follows:

ER(x) = RF + β1RP1 + β2RP2 + β3RP3….

β1 and RP1, in this case, represent the price sensitivity to the first factor and the risk premium associated with the first factor, respectively. In this example, we wrote down three factors—but the model can factor in a lot more—the equation could easily continue to include + β4RP4 + β5RP5 and so forth.

Inputs 👨‍💻

As mentioned, the inputs or macroeconomic factors that are considered in the model are left up to the investor. However, there are a couple of key inputs that generally have a significant effect on all asset classes, so they are always included.

These inputs are:

  • ☑ Changes in the inflation rate
  • ☑ Gross domestic product (GDP) and gross national product (GNP) growth
  • ☑ Changes in interest rates
  • ☑ Exchange rates for different currencies
  • ☑ Market sentiment
  • ☑ Changes in the yield curve
A graph representing the effects of just one macroeconomic factor, inflation, on stock prices.
Macroeconomic factors must be taken into consideration when using the APT, with inflation being one of the most important ones. Without these inputs, we cannot calculate an accurate expected return.

On top of that, there are a variety of other factors that may or may not have an effect on the asset that is being analyzed. These factors include:

  • ☑ The price of commodities (particularly gold and oil)
  • ☑ Changes in the industrial production index (IPI)
  • Corporate bond spreads
  • ☑ Short-term interest rates
  • ☑ The difference between long-term interest rates and short-term interest rates
  • ☑ The performance of a well-diversified and large stock index (think S&P 500 or NYSE composite)

Beta 🅱

A metric that is used quite often in investing, beta is a measure of volatility. In other words, asset beta represents the sensitivity of an asset to a certain benchmark. Volatility’s main utility is in figuring out how stable a stock is.

To quickly explain, a stock with a beta of 1 moves the same way as the benchmark does. If a stock has a beta of 1 in relation to the S&P 500, it will move in tandem with it. In this case, if the index rises by 3%, the stock will rise by 3%—if the index drops by 4%, the stock will likewise drop 4%. Beta can, of course, be equal to less than 1 and it can exceed 1—stocks with high beta tend to outperform the market when rising, but they also tend to suffer greater losses in a downturn.

Beta focuses on systematic risk—the risk that is immanent to the entire market and cannot be diversified away. In contrast with this, unsystematic risk is the risk that applies only to a certain security.

In the arbitrage pricing theory, beta is extensively used—every input that is put into the model has to have a corresponding beta. In the APT, beta represents how volatile or sensitive a security is to specific factors.

Risk-Free Return 💵

The risk-free rate of return is the rate of return that can be achieved with an investment that has zero risk. Of course, investments with zero risks don’t actually exist, as even the safest investment carries some small degree of risk, but the risk-free return rate is nevertheless a useful metric in investing.

In the arbitrage pricing theory, the risk-free return rate is found at the beginning of the mathematical formula and is used as a baseline to which all the other risk premiums are added to size up expected returns.

Although it might seem like an abstract concept due to its theoretical nature, finding the risk-free return rate is actually quite simple—in fact, it is the simplest part of the APT. To get the risk-free return rate, simply use the return rate of a three-month U.S. treasury bill. There is virtually no chance of the U.S. government defaulting on such short-term debt instruments—so in practice, it is the closest possible thing to a risk-free investment.

The return rate of a three-month T-bill is great for U.S.-based investors—however, investors in Europe might consider using the returns of short-term debt instruments belonging to other countries with high credit ratings, if those rates of return are more relevant to the investment being analyzed.

Risk Premium 🌟

Risk and reward always go hand in hand—no one is prepared to tolerate more risk without greater returns. The risk premium is a measure used to quantify the returns that investors expected in excess of the risk-free rate, as compensation for taking on more risk.

To put it in more practical terms, consider the choice between buying risk-free three-month T bills and investing in stocks. The T bills are practically risk-free—but because of this, they offer humble returns. Investing in stocks, on the other hand, comes with a certain degree of risk—the share price can drop, and investors can end up losing money. 

To compensate investors for taking on that risk, the potential returns of stocks have to be higher—otherwise, everyone would just invest in T-bills. To calculate the risk premium of something, we simply take the expected returns, and subtract the risk-free rate from them—the difference is equal to the risk premium. 

To use an example, if the risk-free rate is 3%, and a stock has an expected return of 9%, then the risk premium, the compensation expected for taking on the riskier investment, is 6%. In the arbitrage pricing theory, just as is in the case of volatility, the risk premium of every factor considered has to be taken into account.

An Example of How APT Can Be Used 📝

To cap off the practical part of this guide, let’s put everything that we’ve talked about into practice using a hypothetical example. To demonstrate the formula, let’s say that the salient factors for the asset being analyzed are:

  • ☑ GDP growth—RP=2%, β=0.5
  • ☑ Inflation rate—RP=1.3%, β=0.9
  • ☑ Changes in interest rates—RP=2.2%, β=0.4
  • ☑ S&P 500 returns—RP=4%, β=0.3
  • ☑ Oil prices—RP=6%, β=1.2
  • ☑ RIsk-free rate = 2.5%

With these six factors in play, in order to get the expected returns of the asset being analyzed, we simply have to plug all of this data into the formula, which is, in this case: 

ER(x) = RF + β1RP1 + β2RP2 + β3RP3 + β4RP4 + β5RP5 + β6RP6

In other words, ER(x), or the expected return 

= 2.5% + (0.5 x 2%) + (0.9 x 1.3%) + (0.4 x 2.2%) + (0.3 x 4%) + (1.2 x 6%) 

= 2.5% + 1% + 1.17% + 0.88% + 1.2% + 7.2% 

= 13.95%

APT vs. CAPM—Compared ⚔

As we’ve mentioned, the arbitrage pricing theory was developed as an addition or alternative to another, earlier theory.

The capital asset pricing model was developed by economists including Jack Treynor and William F. Sharpe (yes, the same people the Treynor ratio and Sharpe ratio were named after). It was used to determine how much return an asset ought to provide for the given amount of risk.

However, like a lot of financial theories, the CAPM was eventually found to be a bit lacking—for one, it relied far too much on hypotheticals. CAPM works with efficient market portfolios and assumes that returns follow a normal distribution—which is questionable, to say the least.

As far as differences go, the main one is complexity—whereas CAPM only uses one factor, expected market return, APT uses a bunch—and all of those factors come with their own beta.

The mathematical formula and elements of the capital asset pricing model
Generally, the CAPM is better suited toward short-term purposes, but struggles in the long run—whereas the APT isn’t all that great for short-term purposes, but does much better in the medium to long terms.

All in all, the APT is considered to be a much more accurate approach when compared to the CAPM, mainly due to the presence of multiple factors that are included in the formula. This, along with the fact that it is a bit less reliant on hypotheticals, makes it a much more customizable, tweakable, and versatile approach.

However, credit where credit is due—because of the number of factors that APT takes into account, using it for a portfolio is an almost-unworkable nightmare. In contrast, using the CAPM to assess the risk and return of a portfolio is quite easy. 

This doesn’t just apply to portfolios; although simpler and much less accurate, the CAPM is much easier to use—relying only on expected market return makes this model much more practical.

Pros and Cons of Arbitrage Pricing Theory ⚖

At the end of the day, although arbitrage pricing theory and the capital asset pricing model have quite a few differences, it is clear that APT is the preferred choice of most investors. So, with that being said, let’s take a closer, more focused look at just APT, and the advantages and disadvantages associated with this approach.

First and foremost, being a multi-factor model, the APT takes a lot of factors into consideration. Although we’ll explain why this is a mixed bag a bit later, it is also a real advantage—allowing for detailed analysis, as well as tweaking and customization to fit particular needs.

The assumptions that APT rests on, primarily arbitrage-free pricing, result in a fairly accurate expectation of asset returns. The focus on external or exogenous factors makes APT easier to apply to a wide spectrum of assets. 

On the other hand, there are some limitations associated with APT that are hard to ignore. First and foremost, it can be a bit impractical—the sheer number of factors that can affect an asset’s price is enormous and hard to quantify. This, in turn, means that there is a lot of data to input and parse through when using APT.

On top of that, APT requires that investors not only take a lot of factors into account, but also how those factors impact asset price, and how sensitive these factors are to change. To make matters slightly more difficult, even finding out exactly how a macroeconomic factor affects asset price is hard to do. However, a lot of the leading stock brokerages offer research tools designed to make using APT much easier.

Keep in mind, however, that nothing is static—all of these factors are subject to change, and so is the sensitivity of the asset price to those factors. To put it simply, the factors that affect security prices can become more important, or less important, and new ones can appear.

Conclusion 🏁

The arbitrage pricing theory is a rare breed—while it is as complex as other financial theories, it can also be applied by retail investors with a few caveats and modifications, and it allows for a certain amount of creative thinking when it comes to determining the input factors.

The APT is a tried and tested method that value investors have used for years to identify which stocks are overpriced and which stocks are undervalued. That’s something that will surely come in handy in the so-far turbulent 2020s—but more than that, APT serves as a  great jumping-off point for investors who are really interested in learning how to perform thorough research.

Arbitrage Pricing Theory: FAQs

  • What Does Arbitrage Pricing Theory Mean?

    Arbitrage pricing theory is a mathematical model used to determine the expected returns of an asset, thereby also determining whether or not it is undervalued or overvalued.

  • What Are the Benefits of Arbitrage Pricing Theory?

    Arbitrage pricing theory uses multiple factors, making it versatile and applicable to a variety of asset classes, and the theory tends to be more accurate than CAPM. Arbitrage pricing theory allows investors to identify trading and investing opportunities by evaluating assets through the lens of value investing.

  • Who Invented Arbitrage Pricing Theory?

    Arbitrage pricing theory was first proposed by American economist Stephen Ross in 1976 who published it in the academic peer-reviewed journal of economic theory.

  • Is Arbitrage Pricing Theory Better than CAPM?

    Arbitrage pricing theory is better than CAPM for most intents and purposes—however, there are a couple of specific situations in which using the CAPM is either better or just as good as using APT—on top of being simpler to implement.

  • How Do You Determine if There is an Arbitrage Opportunity?

    In terms of the APT, “arbitrage” opportunities exist whenever an asset is undervalued or overvalued—in general terms, arbitrage opportunities exist when the same asset can be bought in one market and resold in another at a higher price, thus essentially creating a risk-free trade.

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