# Arbitrage Pricing Theory (APT): Formula and How It's Used

> **Definition**
>
> Asset pricing theory (APT) is a model that forecasts the potential returns of an asset based on its expected returns in tandem with several other factors.

## What Is the Arbitrage Pricing Theory (APT)?

The arbitrage pricing theory (APT) is a multi-factor asset pricing model. It's based on the idea that an asset's returns can be predicted using the linear relationship between the asset’s expected return and several macroeconomic variables that capture systematic risk. It's a useful tool for analyzing portfolios from a [value investing](https://www.investopedia.com/terms/v/valueinvesting.asp) perspective to identify securities that may be temporarily mispriced.

### Key Takeaways

*   Arbitrage pricing theory (APT) is a multi-factor asset pricing model based on the idea that an asset's returns can be predicted.
*   APT uses the relationship between the asset’s expected return and numerous macroeconomic variables that reflect systematic risk.
*   APT assumes that markets sometimes misprice securities before the market eventually corrects and securities move back to fair value.
*   Arbitrageurs hope to take advantage of any deviations from fair market value using APT.

![Arbitrage Pricing Theory (APT)](https://www.investopedia.com/thmb/DHVCAy-LJDEVICmthLo1zorGpKA=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/apt.asp-final-e1723d0571a1424bbd76d6c7651e7e44.png)

*— Investopedia / Mira Norian*

## How the APT Works

The arbitrage pricing theory was developed by economist Stephen Ross in 1976 as an alternative to the [capital asset pricing model (CAPM)](https://www.investopedia.com/terms/c/capm.asp). The CAPM assumes markets are perfectly efficient but the APT assumes that markets sometimes misprice securities before the market eventually corrects and securities move back to fair value. Arbitrageurs hope to take advantage of any deviations from fair market value using APT.

This isn't a risk-free operation in the classic sense of [arbitrage](https://www.investopedia.com/terms/a/arbitrage.asp), however, because investors are assuming that the model is correct and making directional trades rather than locking in risk-free profits.

## The APT Model Formula

$\begin{aligned} &\text{E(R)}_\text{i} = E(R)_z + (E(I) - E(R)_z) \times \beta_n\\ &\textbf{where:}\\ &\text{E(R)}_\text{i} = \text{Expected return on the asset}\\ &R_z = \text{Risk-free rate of return}\\ &\beta_n = \text{Sensitivity of the asset price to macroeconomic} \\ &\text{factor}\textit{ n}\\ &Ei = \text{Risk premium associated with factor}\textit{ i}\\ \end{aligned}$

The beta coefficients in the APT model are estimated using linear regression. Historical securities returns are generally regressed on the factor to estimate its beta.

## Mathematical Model for the APT

The APT is more flexible than the CAPM but it's more complex. The CAPM only takes into account one factor: market risk. The APT formula has multiple factors and it takes a considerable amount of research to determine how sensitive a security is to various macroeconomic risks.

The factors as well as how many of them are used are subjective choices. Investors will have varying results depending on their choice. Four or five factors will usually explain most of a security's return, however.

> **Fast Fact**
>
> Read more about how CAPM and arbitrage pricing theory differ for more on the differences between the CAPM and APT.

APT factors are the systematic risk that can't be reduced by the diversification of an investment portfolio. The macroeconomic factors that have proven most reliable as price predictors include unexpected changes in inflation, [gross national product](https://www.investopedia.com/terms/g/gnp.asp) (GNP), corporate bond spreads, and shifts in the yield curve. Other commonly used factors are [gross domestic product](https://www.investopedia.com/terms/g/gdp.asp) (GDP), commodities prices, market indices, and exchange rates.

## Example of How APT Is Used

These four factors have been identified as explaining a stock's return, its sensitivity to each factor, and the risk premium associated with each factor:

*   Gross domestic product (GDP) growth: _ß_ = 0.6, RP = 4%
*   Inflation rate: _ß_ = 0.8, RP = 2%
*   Gold prices: _ß_ = -0.7, RP = 5%
*   Standard and Poor's 500 index return: _ß_ = 1.3, RP = 9%
*   The risk-free rate is 3%

The expected return using the APT formula is calculated as:

*   Expected return = 3% + (0.6 x 4%) + (0.8 x 2%) + (-0.7 x 5%) + (1.3 x 9%) = 15.2%

## What's the Difference Between CAPM and Arbitrage Pricing Theory?

The main difference is that CAPM is a single-factor model while the APT is a multi-factor model. The only factor considered in the CAPM to explain the changes in the security prices and returns is the market risk. The factors can be several in the APT.

## What Are the Limitations of APT?

The main limitation of APT is that the theory doesn't suggest factors for a particular stock or asset. One stock could be more sensitive to one factor than another and investors have to be able to perceive the risk sources and sensitivities.  
  

## What Is the Main Advantage of APT?

The main advantage of APT is that it allows investors to customize their research because it provides more data and it can suggest multiple sources of asset risks.

## The Bottom Line

Arbitrage pricing theory (APT) is a multi-factor asset pricing model based on the idea that an asset's returns can be forecasted with the linear relationship between an asset’s return and several macroeconomic factors that affect the asset’s risk.

Arbitrage pricing theory assumes that markets sometimes misprice securities before they're corrected and move back to fair value.

_The comments, opinions, and analyses expressed on Investopedia are for informational purposes online. Read our [warranty and liability disclaimer](https://www.investopedia.com/legal-4768893#toc-warranty-and-liability-disclaimer) for more info._  

> **Article Sources**
>
> 1.  Reinganum, Marc R. “[The Arbitrage Pricing Theory: Some Empirical Results](https://www.jstor.org/stable/2327013).” _The Journal of Finance_ 36, no. 2, May 1981, pp. 313–21.
>     
> 2.  CFI Education. "[Arbitrage Pricing Theory](https://corporatefinanceinstitute.com/resources/wealth-management/arbitrage-pricing-theory-apt/)."