How a statistical search for structure helps explain the evolution from one - factor CAPM to multifactor models of expected returns

Imagine looking at the monthly returns of thousands of stocks over many decades.

Every company is different. Every month is different. Prices rise and fall for countless reasons. The resulting dataset contains millions of individual observations.

At first glance, the stock market can appear almost impossibly complicated.

Statistics offers an intriguing question: Could much of that complexity be explained by a much smaller number of underlying dimensions?

That question lies behind an important family of statistical methods known as factor methods, and it also provides an intuitive way to understand one of the most important developments in modern financial economics: the work of Eugene Fama and Kenneth French.

Their research helped demonstrate that stock returns are not simply distributed randomly across companies. Instead, average returns appear to vary systematically along certain measurable dimensions.

Those dimensions ultimately became central to the investment philosophy associated with Dimensional Fund Advisors. But the history - and the statistics - are even more interesting than the name suggests.

What Is Factor Analysis?

In classical statistics, factor analysis is a technique for taking a large number of observed variables and attempting to explain their relationships using a smaller number of unobserved, or latent, factors.

Suppose, for example, we measured 30 different characteristics of thousands of individuals.

Some measurements might be strongly correlated with one another. Rather than treating all 30 as completely separate phenomena, factor analysis might discover that much of their common variation can be summarized by three or four underlying dimensions.

This is powerful because the human mind - and statistical models - can understand five dimensions much more easily than 5,000 separate observations.

There is one technical clarification worth making. Factor analysis normally does not divide variables into conventional dependent and independent variables in the way ordinary regression does. Its primary objective is to understand the common covariance structure among many observed variables.

That distinction becomes important when we get to Fama and French.

Fama and French Did Not Use Classical Factor Analysis

The Fama/French models are called factor models, but Fama and French did not simply feed thousands of stock returns into a conventional exploratory factor-analysis program and ask a computer to reveal hidden latent factors.

Their methodology was more economically structured.

They examined relationships between stock returns and observable company characteristics, formed portfolios, conducted cross-sectional statistical tests, and later constructed factor-mimicking portfolios whose returns could be used in regressions.

So classical factor analysis and Fama/French factor models should not be treated as identical statistical techniques. But the intellectual resemblance is striking.

For financial economists, the observations were stock returns. And the search for dimensions would fundamentally change asset pricing.

The Original One-Dimensional View: Beta

Before Fama and French, the dominant asset-pricing framework was the Capital Asset Pricing Model, or CAPM. Its central proposition was extraordinarily elegant. Stocks differ in their exposure to one systematic source of risk: the overall market.

That exposure is measured by beta. A stock with a beta of 1.0 moves approximately with the market. A beta above 1.0 indicates greater sensitivity to market movements; a beta below 1.0 indicates less.

Conceptually, CAPM reduced the enormous stock market to essentially one priced dimension of systematic risk: market beta. Everything else unique to an individual company could theoretically be diversified away. It was a beautiful model. The problem was that the data increasingly suggested that one dimension was not enough.

Then Researchers Found Size

One of the important challenges came from University of Chicago researcher Rolf Banz.

In 1981, Banz published evidence that company size helped explain differences in average stock returns beyond what market beta appeared to explain. Smaller companies had historically produced returns that were difficult to reconcile with the one-factor CAPM alone.

That same year, David Booth and Rex Sinquefield founded Dimensional Fund Advisors.

Their first strategy focused on small-company stocks. Booth and Sinquefield drew on the empirical work emerging from the University of Chicago, including Banz's research. Eugene Fama became involved with Dimensional from its beginning, while Kenneth French joined the firm's intellectual network later.

This chronology is important. Dimensional existed before the Fama/French Three-Factor Model. The company was not named after a model that had yet to be published.

Instead, the name reflected a broader way of thinking about investing: risk and return could have multiple dimensions. That idea would later receive extraordinary empirical support.

Fama and French Ask a Bigger Question

By the late 1980s and early 1990s, researchers had identified several company characteristics associated with differences in average stock returns.

• Market beta
• Company size
• Earnings relative to price
• Leverage
• Book equity relative to market equity

The problem was obvious. Were these five separate effects? Were several measuring essentially the same underlying phenomenon? Which variables actually contained independent information?

This is where the connection with multivariate statistical thinking becomes especially important. Instead of studying one variable at a time, Fama and French examined the variables jointly.

Their landmark 1992 paper, The Cross-Section of Expected Stock Returns, tested which characteristics retained explanatory power when competing with the others.

Their finding was surprisingly parsimonious. Two readily measurable variables - company size and book-to-market equity - captured much of the cross-sectional variation associated with several other variables they tested.

Complexity had been reduced. Many variables had become fewer dimensions.

Stock Risk Becomes Multidimensional

Fama and French drew a particularly important conclusion from their results.

They identified one possible dimension through company size and another through book-to-market equity. That sentence represents an important change in how investors could think about markets.

That was a major conceptual leap.

From Characteristics to Factors

The 1992 research focused heavily on explaining differences in average returns across stocks.

In 1993, Fama and French took the next major step. They constructed return factors designed to capture common movements associated with the dimensions they had identified.

The resulting Fama/French Three-Factor Model included:

Fama and French found that these three stock factors captured a substantial amount of common variation in stock returns and helped explain differences in average returns.

Now the word factor had a very concrete financial meaning. It wasn't merely a company characteristic. It was a measurable return series representing a systematic dimension of the market.

A Statistical Coordinate System for Stocks

One way to visualize the breakthrough is to think about a map.

A coordinate-system analogy: portfolios can be described by their exposure to systematic dimensions.

To locate a point geographically, we do not need to describe every feature surrounding it. We can specify a few coordinates: latitude, longitude, and elevation if necessary.

A multifactor model performs a somewhat analogous function for investments. Rather than describing a portfolio by listing every stock it owns, we can estimate its factor loadings:

• How much market exposure does it have?
• Does it behave more like small companies or large companies?
• Does it behave more like value stocks or growth stocks?

Later models added additional coordinates.

Mathematically, the Three-Factor Model can be expressed as:

The betas are sometimes called factor loadings. They tell us where a portfolio sits along each dimension. That is why the language of dimensions is so appropriate.

From Three Factors to Five

Financial science did not stop in 1993. Research continued to test which company characteristics were associated with differences in expected returns.

In 2015, Fama and French published their Five-Factor Model, adding two more factors:

• Profitability - RMW: Robust Minus Weak
• Investment - CMA: Conservative Minus Aggressive

Their research found that the five-factor model generally described average stock returns better than their earlier three-factor model, although - as every useful scientific model should - it also had documented limitations.

We can therefore visualize the evolution of the science:

Each development attempts to describe a complicated financial world using a relatively small set of systematic dimensions.

This Is the Connection to Factor Analysis

Now we can return to classical factor analysis. The methods are technically different, but their underlying objective has a clear family resemblance.

Classical Factor Analysis

Start with many correlated observed variables. Identify a smaller number of underlying factors that describe their common variation.

Fama/French Asset-Pricing Research

Start with thousands of stocks, company characteristics and returns. Determine whether a relatively small number of systematic variables and return factors can explain important differences in the behavior and average returns of those stocks.

Both are forms of dimensional reduction in thinking, even though their statistical implementations are different. The goal is not to memorize every individual observation. It is to discover the structure underneath the observations.

And That Brings Us to the Name "Dimensional"

There is a wonderful historical connection here - but we should tell it accurately.

Dimensional Fund Advisors was founded in 1981, more than a decade before the publication of the Fama/French Three-Factor Model.

So the Fama/French model did not produce the Dimensional name. Rather, the name anticipated the direction in which financial economics was heading.

"We call our name Dimensional because we think of investing in terms of risk and return, which have several dimensions." - David Booth

That may be the single most important sentence in understanding the firm's name. From its earliest days, Dimensional's approach was not simply: Buy an index.

It was: Use financial science to understand the dimensions of risk and expected return, and then build diversified portfolios around that evidence. The later Fama/French discoveries gave the word Dimensional even greater significance.

An Extraordinary Historical Reversal

There is a particularly interesting chronology here.

So rather than saying: Fama and French discovered factors, which caused Booth to call the company Dimensional, the historically accurate - and arguably more interesting - story is this:

Dimensional and Fama/French Grew Up Together

The relationship between the academic research and the investment firm was unusually close.

Booth studied under Fama at the University of Chicago and later served as his teaching assistant. Booth and Rex Sinquefield met through Fama's finance course. When Booth and Sinquefield founded Dimensional, Booth asked Fama to serve as a director. French later became a consultant and director as well.

The connection also flowed both directions.

Fama and French's 1992 paper acknowledges comments from both David Booth and Rex Sinquefield. Later Fama/French research has similarly acknowledged interactions with Dimensional researchers.

This is an unusual example of a continuing feedback loop between academic financial science and real-world portfolio implementation.

Research generated hypotheses. Data tested those hypotheses. New dimensions were identified. Investment strategies incorporated the evidence. Additional real-world experience generated new questions. And the research continued.

What "Dimensional" Means Today

Dimensional continues to use precisely this language.

The firm describes its equity strategies as emphasizing research-backed drivers - or dimensions - of higher expected returns, including company size, relative price and profitability. Its fixed-income framework similarly identifies dimensions associated with term, credit and currency.

This also reveals an important distinction.

Dimensional does not necessarily take every academic factor ever discovered and create a fund that mechanically maximizes exposure to it.

Hundreds of alleged factors have appeared in the academic literature.

Some may be statistical accidents. Some may disappear after publication. Some may have no convincing economic explanation. Some may be too expensive to capture after trading costs.

Financial science therefore requires more than finding correlations. A useful dimension should have compelling evidence, economic reasoning, persistence and practical investability.

From Thousands of Stocks to a Few Important Questions

This is perhaps the greatest practical contribution of factor-based thinking.

An investor looking at 5,000 stocks could ask 5,000 separate questions: Should I buy Apple? Should I buy Exxon? Should I buy this bank? Should I sell that technology company? What will its earnings be? What will its stock do next?

A dimensional investor asks a very different set of questions:

• How much equity risk should I take?
• How much exposure should I have to smaller companies?
• How much emphasis should I place on lower-priced value companies?
• Should profitability affect portfolio weights?
• How diversified should those exposures be?
• How reliably does the evidence support each dimension?

Those questions are much more compatible with inferential statistics. They do not require predicting individual companies. They require estimating relationships across large populations of securities.

Factor Models Do Not Make Markets Predictable

None of this means the Fama/French models can tell us which factor will outperform next year. Quite the opposite.

A factor is valuable precisely because it describes a systematic relationship across many observations - not because it provides a short-term forecasting signal.

A portfolio can have positive exposure to value and experience years when growth stocks outperform. It can emphasize profitable companies while less-profitable companies temporarily produce higher returns. It can hold small companies through a prolonged period of large-cap dominance.

The factor model describes the portfolio's exposures. It does not reveal next year's realized premiums.

That returns us to the central lesson of inferential statistics: We use samples to estimate relationships in a future that remains unknown.

Dimensions Are About Expected Returns, Not Guaranteed Returns

A factor model therefore changes the way we phrase an investment conclusion. Instead of saying, "Small stocks will outperform large stocks," we say, "The available evidence suggests differences in expected returns are associated with company size."

Instead of saying, "Value will beat growth next year," we say, "Relative price has historically contained information about differences in expected returns."

And instead of saying, "The Five-Factor Model predicts the future," we say, "The Five-Factor Model provides a framework for understanding systematic sources of differences in expected returns."

Those distinctions are not semantic. They are the essence of scientific investing.

From Factor Analysis to Financial Science

Classical factor analysis begins with a fascinating proposition: What if apparent complexity is being generated by a smaller number of underlying dimensions?

Financial economists asked essentially the same broad question about markets. The CAPM proposed one important dimension. Empirical evidence suggested that one was insufficient. Banz identified size.

Fama and French showed that size and relative price provided important independent information about average returns. Their Three-Factor Model formalized market, size and value. Their Five-Factor Model later added profitability and investment.

And throughout this evolution, Dimensional Fund Advisors built an investment philosophy around a remarkably appropriate word: Dimensional. Not because Fama and French's later research gave the company its name. But because the name captured an idea that subsequent research would make increasingly powerful: Investment returns are multidimensional.

The Bigger Lesson for Investors

There may be thousands of stocks. There may be millions of monthly return observations. There may be innumerable stories explaining why each security went up or down.

But financial science asks whether we can step back from those individual stories and identify broader systematic relationships.

That is the intellectual connection between factor analysis, multifactor asset-pricing models and dimensional investing.

• Reduce
• Find
• Measure the
• Identify meaningful
• Diversify broadly across

And never confuse an estimate of expected return with a prediction of the future.

The great achievement of Fama, French and the researchers who preceded and followed them was not discovering a formula that tells us which stocks will win.

It was something much more useful: They helped us see the structure hidden inside the randomness.

Key Takeaway

The Evolution: From One Dimension to Five

From One Dimension to Five

The historically important point is that the name "Dimensional" came first; the later Fama/French research increasingly formalized the multidimensional view of markets embedded in that name.

Selected References

Banz, R. W. (1981). The Relationship Between Return and Market Value of Common Stocks. Journal of Financial Economics, 9(1), 3-18. Fama, E. F., & French, K. R. (1992). The Cross-Section of Expected Stock Returns. Journal of Finance, 47(2), 427-465.

Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33(1), 3-56. Fama, E. F., & French, K. R. (2015). A Five-Factor Asset Pricing Model. Journal of Financial Economics, 116(1), 1-22.

Dimensional Fund Advisors. Corporate history and educational materials describing the firm's founding, research heritage, and the use of multiple dimensions of expected returns.

University of Chicago Booth School of Business. Historical materials on David Booth, Rex Sinquefield, Eugene Fama, Kenneth French, and the development of Dimensional Fund Advisors.

Important terminology note: The connection between classical statistical factor analysis and Fama/French factor models is conceptual rather than methodological. Fama and French did not use classical exploratory factor analysis to discover the factors in their asset-pricing models.

Important Information