Arbitrage Pricing Theory: It's Not Just Fancy Math (2024)

Arbitrage pricing theory (APT)is an alternative to the capital asset pricing model (CAPM) for explaining returns of assets or portfolios. It wasdeveloped by economistStephen Ross in the 1970s.Over the years, arbitrage pricing theory has grown in popularity for its relatively simpler assumptions. However, arbitrage pricing theoryis a lot more difficult to apply in practice because it requires a lot of data and complex statistical analysis.

Let's see what arbitrage pricing theory is and how we can put it into practice.

What Is APT?

APTis a multi-factor technical model based on the relationship between a financial asset's expected return and its risk. The model is designed to capture the sensitivity of the asset's returns to changes in certain macroeconomic variables. Investors and financial analysts can use these results to help price securities.

Inherent to the arbitrage pricing theory is the belief that mispriced securities can represent short-term, risk-free profit opportunities. APT differs from the more conventionalCAPM, which uses only a single factor. Like CAPM, however, the APT assumes that a factor model can effectively describe the correlation between risk and return.

3 Underlying Assumptions of APT

Unlike the capital asset pricing model, arbitrage pricing theory does not assume that investors hold efficient portfolios.

The theory does, however, follow three underlying assumptions:

• Asset returns are explained by systematic factors.
• Investors can build a portfolio of assets where specific risk is eliminated through diversification.
• No arbitrage opportunity exists among well-diversified portfolios. If any arbitrage opportunities do exist, they will be exploited away by investors. (This is how the theory got its name.)

Assumptions of theCapital Asset Pricing Model

We can see that these are more relaxed assumptions thanthose of the capital asset pricing model. That model assumes that all investors hold hom*ogeneous expectations about mean return and variance of assets. It also assumes that the same efficient frontier is available to all investors.

For a well-diversified portfolio, a basic formula describing arbitrage pricing theory can be written as the following:

$\begin{aligned} &E(R_p) = R_f + \beta_1 f_1 + \beta_2 f_2 + \dotso + \beta_n f_n \\ &\textbf{where:}\\ &E(R_p)=\text{Expected return}\\ &R_f=\text{Risk-free return}\\ &\beta_n=\text{Sensitivity to the factor of }n\\ &f_n=n^{th}\text{ factor price}\\ \end{aligned}$​E(Rp​)=Rf​+β1​f1​+β2​f2​+…+βn​fn​where:E(Rp​)=ExpectedreturnRf​=Risk-freereturnβn​=Sensitivitytothefactorofnfn​=nthfactorprice​

R_fis the return if the asset did not have exposure to any factors, that is to say, all

$\beta_n = 0$βn​=0

Unlike in the capital asset pricing model, the arbitrage pricing theory does not specify the factors. However, according to the research of Stephen Ross and Richard Roll, the most important factors are the following:

• Change in inflation
• Change in the level of industrial production
• Shifts in risk premiums
• Change in the shape of the term structure of interest rates

According to researchers Ross and Roll,if no surprise happens in the change of the above factors, the actual return will be equal to the expected return. However, in case of unanticipated changes to the factors, the actual return will be defined as follows:

$\begin{aligned} &R_p = E(R_p) + \beta_1 f'_1 + \beta_2 f'_2 + \dotso + \beta_n f'_n + e \\ &\textbf{where:}\\ &\begin{aligned} f'_n=&\text{ The unanticipated change in the factor or}\\ &\ \text{ surprise factor}\end{aligned}\\ &e=\text{The residual part of actual return}\\ &7\% = 2\% + 3.45*f_1 + 0.033*f_2\\ &f_1= 1.43\%\\ &f_2= 2.47\%\\ &E(R_i) = 2\% + 1.43\%*\beta_1 + 2.47\%*\beta_2\\ \end{aligned}$​Rp​=E(Rp​)+β1​f1′​+β2​f2′​+…+βn​fn′​+ewhere:fn′​=​Theunanticipatedchangeinthefactororsurprisefactor​e=Theresidualpartofactualreturn7%=2%+3.45∗f1​+0.033∗f2​f1​=1.43%f2​=2.47%E(Ri​)=2%+1.43%∗β1​+2.47%∗β2​​

Note that f'_n is the unanticipated change in the factor or surprise factor, e is the residual part of actual return.

EstimatingFactor Sensitivities and Factor Premiums

How we can actually derive factor sensitivities? Recall that in the capital asset pricing model, we derived asset beta, which measuresasset sensitivity to market return, by simply regressing actual asset returns against market returns. Deriving the factors' beta is pretty much the same procedure.

For the purpose of illustrating the technique of estimatingß_n (sensitivity to the factor n)and f_n (the nth factor price),let'stake the and the NASDAQComposite Total Return Index as proxies for well-diversified portfolios for which we wish to find ß_nandf_n. For simplicity, we'll assume that we know R_f (the risk-free return)is 2%. We'll also assume that the annual expected return of the portfolios are 7% for the S&P500 Total Return Index and 9% for the NASDAQ Composite Total Return Index.

Step 1: Determine Systematic Factors

We have to determine thesystematic factors by which portfolio returns are explained.Let’s assume that the real gross domestic product (GDP) growth rateand the 10-year Treasury bond yield change are the factors that we need.Since we have chosen two indices with large constituents, we can be confident that our portfolios are well diversified with close to zero specific risk.

Step 2: Obtain Betas

We ran aregression on historical quarterly data of each index against quarterly real GDP growth rates and quarterly T-bond yield changes. Note that because these calculations are for illustrative purposes only, we will skip the technical sides of regression analysis.

Here are the results: