Arbitrage Pricing Theory and Multi-Factor Equity Models

The Arbitrage Pricing Theory Framework

The CAPM reflects a single source of systematic risk: variation in overall stock market returns. However, empirical research has consistently shown that market risk is not the only source of risk recognized by investors. A wide array of academic and practitioner research has searched for additional systematic risk factors that command risk premiums.
Arbitrage pricing theory (APT) provides a general framework for multi-factor models. Rather than a single risk factor, APT allows for multiple sources of systematic risk that affect equity returns:

Arbitrage pricing theory (APT) provides a general framework for multi-factor models. Rather than a single risk factor, APT allows for multiple sources of systematic risk that affect equity returns:

$$ E(r_i) = r_f + \beta_{i,1} E(f_1) + \beta_{i,2} E(f_2) + \dots + \beta_{i,k} E(f_k) $$

Where

\(k\) denotes the number of systematic factors,

\(\beta_{i,j}\) is the sensitivity (or factor loading) of asset \(i\) to factor \(j\),

\(E(f_j)\) is the expected risk premium associated with factor \(j\).

Each factor premium represents the additional return investors require for bearing one unit of that specific systematic risk, and each beta measures the asset’s exposure to that risk.

The \(\text{APT}\) does not specify the identity or number of factors—this is a key distinction from the \(\text{CAPM}\). The theory only posits that systematic risk is likely to be reflected by multiple factors rather than a single market factor. The factors themselves are theoretical constructs that must be identified empirically. Just as in the \(\text{CAPM}\), the average investor holds a diversified portfolio, so only systematic risk commands a premium. The significant difference is that \(\text{APT}\) recognizes that systematic risk may arise from multiple economic forces, not just stock market fluctuations.

Multi-Factor Models in Practice

Research into building multi-factor models is a constantly evolving process. Factors may be constructed using data from three primary sources: company stock market data (e.g., returns, market capitalization), macroeconomic data (e.g., inflation, industrial production), and company accounting data (e.g., earnings, book values). By the early 2020s, published research had proposed and found positive results for hundreds of different factors. Many of these may be products of data mining or spurious correlation, so most practitioners and scholars believe that the population of factors can be distilled to a small set that effectively captures all relevant systematic risks.

The Four-Factor Model. One practical example is a four-factor model that uses the following factors to explain stock returns:

$$ E(r_i) = r_f + \beta_{i,\text{Market}} \times E(r_m – r_f) + \beta_{i,\text{Size}} \times E(\text{SMB}) + \beta_{i,\text{Value}} \times E(\text{HML}) + \beta_{i,\text{Momentum}} \times E(\text{UMD}) $$

Where

\(\beta_{i,\text{Size}}\) = the factor loading of asset or portfolio \(i\) on the size risk premium, represented by \(E(\text{SMB})\).

\(E(\text{SMB})\) denotes the anticipated payoff from a dollar-neutral strategy that buys small-capitalization stocks and shorts large-capitalization stocks in equal notional amounts.

\(\beta_{i,\text{Value}}\) = the sensitivity of asset \(i\) to the value factor, captured by \(E(\text{HML})\).

\(E(\text{HML})\) is the expected outcome of a zero-cost portfolio that takes long positions in equities with high book-to-market ratios (low price-to-book, or “value” stocks) and short positions in those with low book-to-market ratios (high price-to-book, or “growth” stocks).

\(\beta_{i,\text{Momentum}}\) = the exposure of asset \(i\) to the momentum factor, measured by \(E(\text{UMD})\).

\(E(\text{UMD})\) reflects the expected return on a self-financing portfolio that goes long on past high-return performers and shorts past low-return performers.

For any given stock or portfolio, the respective factor betas can be derived by compiling historical return data for the specified risk factors and running a regression of the asset’s returns against those factor series. Unlike the market beta, which has a cross-sectional average close to \(1.0\), the mean beta for each of the three additional factors is expected to be zero.

Comparing CAPM and Multi-Factor Model Results

When a multi-factor model includes the market factor, the estimated market beta is unlikely to equal the market beta from a single-factor \(\text{CAPM}\) regression. The inclusion of additional factors changes the coefficient on the market factor because the extra factors absorb some of the variation previously attributed to the market. However, the direction of the change cannot be predicted—the market beta may increase or decrease depending on the correlations between the stock’s returns and the other factors.

More importantly, even if the market beta remains unchanged, the required return estimate from a multi-factor model will typically differ from the CAPM estimate. Stocks with positive loadings on factors with positive risk premiums will have higher required returns in the multi-factor model. Stocks with negative loadings on those same factors will have lower required returns. This difference reflects the additional information captured by the multi-factor framework about the stock’s systematic risk exposures.

Interpretation of Factor Betas. In the four-factor model, the average value for SMB, HML, and UMD betas across all stocks is zero by construction. Therefore, negative beta estimates are quite common for these factors. A negative HML beta, for example, indicates that a stock behaves like a growth stock, it tends to perform poorly when value stocks outperform growth stocks. A negative SMB beta indicates a large-cap stock that tends to underperform when small caps do well.

Applications of Multi-Factor Models

Multi-factor models serve several important purposes in equity investing beyond discount rate estimation. They are used in portfolio construction to create factor-tilted strategies that target specific risk premiums. They are employed in performance attribution to decompose portfolio returns into contributions from systematic factors versus manager skill. They are also used in risk management to assess a portfolio’s exposure to different sources of systematic risk.

The search for the best set of factors is an ongoing exercise among both investment practitioners and academic researchers. Analysts should follow developments regarding newly discovered factors as well as those that may have been shown not to reliably explain returns. While the four-factor model described here has strong empirical support, it is not the only specification, and the factor landscape continues to evolve.

Question

An analyst is estimating the expected return for Equity \(P\) using a multifactor model that incorporates separate risk premiums for market, size, value, and momentum factors. The risk-free rate is treated as the intercept term in the model. Selected data for Equity \(P\) are presented below:

$$ \begin{array}{l|l|l}
\textbf{Factor} & {\textbf{Expected Risk} \\ \textbf{Premium}} & {\textbf{Equity Portfolio}\\ \textbf{Factor Beta}} \\
\hline
\text{Market} & 5.0\% & 1.20 \\
\hline
\text{Size (SMB)} & 2.5\% & 0.80 \\
\hline
\text{Value (HML)} & 5.0\% & -0.40 \\
\hline
\text{Momentum (UMD)} & 2.0\% & 0.50 \\
\hline
\text{Risk-free rate } (r_f) & \text{—} & 2.0\% \\
\end{array} $$

Based on the multifactor model, the expected return for Equity P is closest to:

1. 7.0%
2. 9.0%
3. 13.0%

Solution.

Correct Answer is B.

The expected return is the sum of five distinct components:

1. Risk-free rate (intercept):$$ r_f = 2.0\% $$
2. Market factor contribution:$$ \beta_{i,\text{Market}} \times E(r_m – r_f) = 1.20 \times 5.0\% = 6.0\% $$
3. Size factor (\(\text{SMB}\)) contribution:$$ \beta_{i,\text{Size}} \times E(\text{SMB}) = 0.80 \times 2.5\% = 2.0\% $$
4. Value factor (\(\text{HML}\)) contribution:$$ \beta_{i,\text{Value}} \times E(\text{HML}) = -0.40 \times 5.0\% = -2.0\% $$
5. Momentum factor (\(\text{UMD}\)) contribution:$$ \beta_{i,\text{Momentum}} \times E(\text{UMD}) = 0.50 \times 2.0\% = 1.0\% $$

Total expected return:

$$ E(r_p) = 2.0\% + 6.0\% + 2.0\% – 2.0\% + 1.0\% = 9.0\% $$