This article compares two leading asset pricing models: Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT): I argue that while APT is consistent with the data available to test asset pricing theories, CAPM is not. In arriving at this conclusion, emphasis is placed on distinguishing between the unconditional (relatively incomplete) information that economists must use to estimate asset pricing models and the conditional (complete) information that investors use in making portfolio decisions that determine asset prices. The empirical work so far indicates that APT is unlikely to produce a simple equation that explains the differences in risk premium well with some parameters. If CAPM is true, it will provide such an equation.

Financial Asset Pricing Theory provides ‘a comprehensive overview of classic and current research in theoretical asset pricing. Asset pricing is developed around the concept of state price deflation which links the price of an asset to its future (risk) profit and thus includes how to adjust to both time and risk in asset valuation.’ The willingness of a utility-maximizing investor to shift consumption over time determines a country’s price-deflation factor that provides a link between optimal consumption and asset prices that leads to a depreciation-based capital asset pricing model (CCAPM)

A simple version of the CCAPM cannot explain various realities of asset pricing, but the “puzzles” of asset pricing can be resolved through a number of recent additions that include habit formation, recursive utilities, multiple consumer goods, and long-term depreciation risk. Valuation techniques and other modeling methods (such as factor models, range structure models, risk-neutral valuation, option pricing models) are explained and correlated with a country’s price deflator.

**Arbitrage Pricing Theory (APT)**

What Is the Arbitrage Pricing Theory (APT)?

Arbitrage pricing theory (APT) is a multi-factor asset pricing model based on the idea that an asset’s returns can be predicted using the linear relationship between the asset’s expected return and a number of macroeconomic variables that capture systematic risk. It is a useful tool for analyzing portfolios from a value investing perspective, in order to identify securities that may be temporarily mispriced.

Arbitrage Pricing Theory

The Formula for the Arbitrage Pricing Theory Model Is

E(R)i=E(R)z+(E(I)−E(R)z)×βnwhere: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\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)×βnwhere: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 by using linear regression. In general, historical securities returns are regressed on the factor to estimate its beta.

How the Arbitrage Pricing Theory Works

The arbitrage pricing theory was developed by the economist Stephen Ross in 1976, as an alternative to the capital asset pricing model (CAPM). Unlike the CAPM, which assume markets are perfectly efficient, APT assumes markets sometimes misprice securities, before the market eventually corrects and securities move back to fair value. Using APT, arbitrageurs hope to take advantage of any deviations from fair market value.

However, this is not a risk-free operation in the classic sense of arbitrage, because investors are assuming that the model is correct and making directional trades—rather than locking in risk-free profits.

Mathematical Model for the APT

While APT is more flexible than the CAPM, it is more complex. The CAPM only takes into account one factor—market risk—while 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, which means investors will have varying results depending on their choice. However, four or five factors will usually explain most of a security’s return. (For more on the differences between the CAPM and APT, read more about how CAPM and arbitrage pricing theory differ.)

APT factors are the systematic risk that cannot 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.

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 using the linear relationship between the asset’s expected return and a number of macroeconomic variables that capture systematic risk.

Unlike the CAPM, which assume markets are perfectly efficient, APT assumes markets sometimes misprice securities, before the market eventually corrects and securities move back to fair value.

Using APT, arbitrageurs hope to take advantage of any deviations from fair market value.

Example of How Arbitrage Pricing Theory Is Used

For example, the following four factors have been identified as explaining a stock’s return and its sensitivity to each factor and the risk premium associated with each factor have been calculated:

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%

Using the APT formula, the expected return is calculated as:

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