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

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 perspective to identify securities that may be temporarily mispriced.

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). 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, 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

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

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.

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 (GNP), corporate bond spreads, and shifts in the yield curve. Other commonly used factors are gross domestic product (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%

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 for more info.