Letting the Data Name the Factors: Roll and Ross Test the APT

Stephen Ross's Arbitrage Pricing Theory was a beautiful piece of reasoning with an enormous practical hole in it. It said that expected returns are driven by exposure to a small number of common factors, and it derived that conclusion from a much weaker set of assumptions than the CAPM needed: essentially just that there are no free lunches available in a large, diversified market.

But it did not say what the factors were. Or how many there were. That is not a small omission. It is like a theory of nutrition that says "there are a few essential vitamins" without naming any of them.

Roll and Ross set out, in 1980, to answer the empirical question the theory had ducked. Their method was to stop guessing and let the data tell them.

The problem: a theory with unnamed variables cannot be tested

The CAPM at least tells you exactly what to measure: sensitivity to the market. You can go and do it, and Roll's own critique notwithstanding, you can produce a number.

The APT gives you a functional form with blanks in it. Expected return equals risk-free rate plus a sum of factor exposures times factor premia, for some unspecified factors. Fill in the blanks however you like and you can fit almost anything, which means the theory as stated is nearly untestable. Roll, of all people, was acutely sensitive to untestable theories.

The key idea, via analogy

Imagine you have recordings of a thousand people talking in a crowded restaurant, one microphone per person, and you want to know what the shared background noises are. You do not know in advance whether it is music, or a passing train, or the kitchen. You just have a thousand messy signals.

There is a statistical technique for this. Look at how the signals move together. If hundreds of microphones all show a surge at the same instant, there is a common source. Analyze the pattern of co-movement across all of them and you can extract the handful of underlying sources that explain most of the shared variation, without ever knowing in advance what those sources are. You get the sources first, and only afterwards do you try to figure out what they physically were.

That is factor analysis, and it is exactly what Roll and Ross did with stock returns. Their procedure:

1. Take a large collection of individual stocks and their returns over the period they studied, the 1960s and early 1970s.
2. Compute how all of them co-move with each other, which produces a huge covariance structure.
3. Run factor analysis on that structure to extract the small number of unobserved common factors that account for most of the co-movement, along with each stock's sensitivity to each factor.
4. Then, crucially, test whether those factor sensitivities actually predict average returns. It is not enough for a factor to explain why stocks move together. To be a priced risk factor, being exposed to it must earn you a premium.

Their conclusion: they found evidence of at least three, and probably four, priced factors in the data. Stocks' exposures to these statistically extracted factors did help explain their average returns, which is what the APT requires.

They also ran a sharp test against a rival explanation. Does a stock's own volatility, its total standard deviation, add anything once you know its factor exposures? The APT says it should not: idiosyncratic risk is diversifiable and therefore unpriced. And that is what they found. Own standard deviation was correlated with average return in a raw sense, but once factor loadings were accounted for, it added no further explanatory power. That is a genuine, non-obvious victory for the theory.

Why it mattered

• It made the APT into an empirical research programme. After 1980, "how many factors are there?" was a question with a method attached, and factor analysis of returns became a standard tool.
• It is the ancestor of statistical factor models. Every risk model that extracts principal components from a return covariance matrix, which is most of the commercial risk systems in use, is doing a modernized version of what Roll and Ross did here.
• It showed idiosyncratic risk is not priced. This is one of the cleanest empirical confirmations of a central prediction shared by the CAPM and APT, and it is worth remembering when someone tries to sell you a high-volatility stock on the grounds of its volatility alone.
• It offered an escape from Roll's critique. Because the APT does not require identifying the true market portfolio, it sidesteps the untestability trap Roll himself had built three years earlier. That is a delicious piece of intellectual history.

The honest limitations

• The factors have no names. Factor analysis returns mathematical constructs, not economic ones. You get "factor one" and "factor two," and interpreting them is an act of storytelling. This is deeply unsatisfying, and it is what pushed Chen, Roll and Ross to try again in 1986 using named macroeconomic variables instead.
• The number of factors is not well identified. How many factors you "find" depends on your sample size, the group of stocks you use, and the statistical criteria you pick. Dhrymes, Friend and Gultekin published a well-known critique arguing that the apparent number of factors grew mechanically with the number of securities in the group, which is a serious problem, and Roll and Ross replied. The debate was never fully resolved.
• Factor analysis is rotation-invariant. Any set of factors can be mathematically rotated into a different set that fits equally well. So the specific factors are not unique, only the space they span is. That further undermines any attempt to name them.
• It cannot rule out a single-factor world. If the true model is a well-specified CAPM with a proper market portfolio, you might still find multiple statistical factors due to sampling noise and industry clustering. Distinguishing the hypotheses is hard.

The one-line takeaway

Roll and Ross took a theory that insisted returns are driven by a few unnamed common factors and used statistical factor analysis to let the data reveal them, finding three or four, while confirming the key prediction that a stock's own volatility earns you nothing.