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Exploring the World of Factors

Understanding factors leads to better investment strategies.

Gideon Magnus, 10/08/2012


Why do some assets yield higher returns than others? Factors provide an explanation. In a nutshell, factors represent the risks that people care about. The stronger an asset is related to a factor, the riskier it is, and because of this, people will demand a higher rate of return. In order to understand factors, I provide a basic review of asset pricing theory, discuss the most important empirical findings in the academic literature, and explain why people should care about factors when deciding on an investment strategy.

Asset Pricing Theory
One basic prediction of asset pricing theory is that assets are more valuable if they are expected to yield higher future payments. In other words, they (directly or indirectly) give the owner the right to consume more goods and services. A stock’s payments, for example, are its dividends. Another prediction is that assets are more valuable if these payments come sooner rather than later.

A crucial characteristic of any asset is its risk— that is, uncertainty about its future payments. In general, people dislike uncertainty. But if an asset’s payments are more uncertain, will it be less desirable, priced lower, and have higher expected returns? An important prediction of asset pricing theory is that this is not necessarily the case: A risky asset should not be less valuable than a riskless asset, provided its risks are diversifiable.

Diversification means costless elimination of risk. Holding an asset with diversifiable risk should not increase the uncertainty of a portfolio; therefore, people should be equally happy to hold this asset and an asset that has no risk whatsoever. The risk of the overall portfolio is identical.

In contrast, suppose an asset’s payments are lower than expected exactly when your portfolio is doing poorly. In that case, adding even a small amount of this asset to your portfolio will increase the risk of your portfolio. This makes an asset less attractive.

Again, asset pricing theory predicts that the more undiversifiable risk an asset has the lower its price and higher its returns will be. High undiversifiable risk means that an asset’s performance is strongly correlated with the performance of a sensibly chosen portfolio. It is important to note that I am interpreting the term portfolio broadly. Think of your portfolio as consisting of the full collection of claims that enable you to consume. Consumption, likewise, should be broadly interpreted to mean anything that people value. An important component of many people’s (broad) portfolios is their capacity to earn income from a job.

An unconventional example is the weather. Every day, this “asset” yields a “payment” that we “consume.” Some days this payment gives us more pleasure than others. Hypothetically, financial assets that perform poorly whenever the weather is bad might be valued less. It is doubly frustrating if stocks are down when it is raining.

Theory provides a framework for thinking about asset prices. Two questions emerge:

  1. Can we identify the undiversifiable risks that people care about?
  2. How much do people need to be compensated for being exposed to these risks?

This is where factors enter the picture. Factors are variables that represent the risks that people care about. The stronger any asset is related to a factor, the riskier it is, and so people will demand a higher rate of return.

Factor Models: Empirical Findings
A vast amount of empirical research in finance is concerned with the question: Can we find a set of factors that seem to explain asset returns? The most common approach is to consider portfolios of tradable assets as the candidate factors. A key advantage of this is that it allows us to think about risk compensation in a straightforward way. When factors are portfolios, the factor risk premium is the additional return a factor can be expected to yield over a risk-free asset.1

A “pure” identification of factors and their risk premiums would require uncovering what people believe to be the full joint probability distribution of all future payments of all assets, both now and in the past. This is infeasible. We cannot accurately measure these probability distributions; in other words, we can’t easily measure people’s expectations.

We can, however, readily observe historical realized asset returns. The question, then, is whether historical realized returns tell us anything about the historical subjective probability distributions of payments. The answer is yes, but as I will discuss below, only if we are willing to make an important additional assumption.

The Capital Asset Pricing Model
The Capital Asset Pricing Model is the most famous factor model. Developed in the 1960s by Jack Treynor, William Sharpe, John Lintner, and Jan Mossin, the CAPM assumes that all assets in the economy are tradable. People do not have jobs. Rather, they simply own different types of capital that produce consumable payments. People are assumed to be very similar, but some may be more risk-averse than others.

A famous result of the CAPM is that everyone holds portfolios consisting of two parts. People put some fraction of their wealth into the “market” portfolio, which is a value-weighted portfolio of all risky assets. They then put the rest of their wealth into a risk-free asset. People who dislike risk hold more of the risk-free asset (and less of the risky market portfolio). But importantly, everyone holds the same risky portfolio.

In the CAPM, there is one factor: the market portfolio. A central CAPM prediction is that the expected excess return of any asset equals the expected excess return of the market portfolio multiplied by a number, called beta:

E(Ri) -  = βi x (E(RM) - Rf)
expected excess return of asset i
expected excess return of market

The excess return of an asset is its return minus the return on a risk-free asset, denoted Rf. Beta represents the “exposure” of an asset to the market factor, and the expected excess return of the market represents the factor risk premium.

Let us think about the intuition behind this equation. An asset with a higher beta is expected to yield a higher return. The higher return is a compensation for the asset’s exposure to the factor. Whenever the market declines, everyone is poorer and unhappier, but the high-beta assets are the ones that do the most damage.

How can we test this equation, given that we do not observe expectations? Let us rewrite the equation as follows:

Ri - Rf = βi(RM-Rf) + εi

forecasting error

Note the difference: We now have returns, whereas before we had expected returns. You may recognize this as a linear regression model.2 If we wish to uncover (historical) betas from returns data, we need to assume that 1) forecasting errors are zero on average and 2) that they are not related to excess market returns. Put differently, people’s expectations of returns are correct on average, and their incorrectness is not related to excess market returns.

Let us add one final ingredient to the equation, a parameter called alpha:

Ri - Rf = αi + βi(RM-Rf) + εi

alpha = 0 

Alpha is the intercept term in this regression. An empirical test of any factor model (including the CAPM) is whether the alphas for all assets are plausibly equal to zero.

Alternative Factor Models
The CAPM was a great success and remains to this day a useful benchmark for understanding asset returns, in particular for stocks. Not surprisingly, however, researchers have sought and found alternative factor models that are now widely accepted as superior to the CAPM. Departures from the CAPM are sometimes called anomalies. This is a bit of a misnomer, because it should not be surprising at all that the CAPM fails to account for certain return patterns. Recall that the assumptions underlying the CAPM are pretty strong, so it would be surprising if CAPM were in fact found to be “true.”

The most famous alternative factor model is the Fama-French model.

Fama-French Model
In 1993, Eugene Fama and Kenneth French introduced a model with three factors. The first factor is a measure of the market portfolio, as in the CAPM. The second factor is a portfolio that is long on companies with a high book-tomarket ratio and short on companies with a low book-to-market ratio. This factor is called high-minus-low (HML). Companies with high book-to-market ratios—companies with high debt-to-equity ratios—are called value companies, while those with a low book-tomarket are called growth. The third factor is a portfolio long on companies with relatively small market capitalization and short on those with a big market capitalization. This factor is called small-minus-big (SMB).

When running regression analyses using these three factors, Fama and French found that,to a considerable extent, the shortcomings of the CAPM disappeared. Again, the main test is whether alphas are plausibly different from zero.

Additional factors
The size and value factors are now widely accepted and form the basis of most factor models. Both practitioners and academics continue to look for systematic return patterns that cannot be explained by the Fama- French model alone. John Cochrane (2011) lists “momentum, accruals, equity issues and other accounting-related sorts, beta arbitrage, credit risk, bond and equity market-timing strategies, foreign-exchange carry trade, put option writing, and various forms of ‘liquidity provision’.”3

Momentum4 is the most widely recognized additional factor and helps explain why stocks that have done well in the recent past often continue to outperform stocks that have recently done poorly. The reason for this premium is not well understood. It has been argued that momentum is difficult to exploit due to the fact that it requires frequent trading, which increases transaction costs5.

Factor Investing in Practice
We turn now to the question: How might an investor use any of the insights from asset pricing research? But first we need to discuss an important caveat. Asset pricing theory does not really offer any practical guidance on how to value an asset. In other words, it does not say which information to examine if one wishes to forecast future payments or (equally important) the relationship between payments of different assets.

In short, it does not offer a method for examining investment opportunities and evaluating whether they are worthwhile.

Rather, it takes it as given that investors have somehow acquired this knowledge and made their decisions based on this. There is, therefore, a notable circularity to the idea that investors should incorporate lessons from asset pricing theory, because asset pricing theory starts with the premise that investors have all requisite knowledge to value assets. In other words, investors have already made optimal choices.

This does not mean that insights from asset pricing research are useless. For example, asset pricing theory is based on a set of behavioral assumptions, which may have some normative merit (for example, do not evaluate any asset in isolation). But at a minimum, insights from asset pricing should not be used as a substitute for examining the fundamental question faced by anyone considering buying an asset: Is it worth its price?

The CAPM predicts that everyone holds a combination of a risk-free asset and the market portfolio. By definition, the average investor holds the market portfolio, at least if we weigh investors by the wealth they have invested in risky assets. A reasonable question to ask, then, is why not simply hold a combination of a broad market index and a risk-free asset? Perhaps you believe you have some special knowledge about the value of an asset and are convinced that other investors will come around to this view in the future. But some degree of modesty is warranted. Are you really sure your information is superior?

There is a better justification for making the composition of your risky asset holdings different from the average—namely when your broad personal portfolio is different from the average. Put another way, do you face certain risks that are different from the average investor, or vice versa? For example, do you have a job? People with a job have a markedly different broad portfolio from people who are unemployed (retirees being a prime example). Suppose the average investor is employed. His income is a nontradable asset, but its returns could be more related to some tradable assets than others. Therefore, he will demand a higher premium from assets that move together with his income. If you are unemployed, you might want to consider a portfolio that is tilted toward these assets. Because you don’t have a job, these risks are more acceptable to you.

Also, consider also the industry you work in. It would be nice if you could partially hedge your income risk. To do this, you should tilt away from investing in companies that are highly correlated with your income. Suppose you work in the auto industry. If the auto industry unexpectedly hits a rough patch and you lose your job, then at least your portfolio has not lost as much value as it otherwise would have.

The value and size premiums suggest that you can get a better-performing portfolio if you tilt away from the market. But keep in mind that there is probably a reason why value and size offer such premiums. If everyone tried to tilt toward size and value, then the tilted portfolios would become the market, and the premiums would disappear.

Fama and French speculated that the value premium exists because value companies do particularly poorly during recessions. This is a sensible justification. If certain stocks perform particularly poorly in recessions, then investors, who typically have jobs, will be more hesitant to hold them. Other researchers have speculated that many investors hold a sizable fraction of their wealth in small privately held business ventures.6 If the fortunes of these small companies are sensitive to the state of the economy, then these investors will shy away from assets that share this characteristic.

Factors Explained
Academic research has been very successful at deepening our understanding of asset prices and returns, but of course, much work remains to be done. Factors are a central organizing concept and represent the risks that people care about and the compensation they require for bearing these risks. They explain why some assets yield higher returns than others. Keeping certain caveats in mind, factors are useful for making better investment decisions.


1 Asset pricing theory predicts that there will always exist portfolios of traded assets that can be used as factors, even if not all assets in the economy are tradable. The question is to figure out what these factor mimicking portfolios are.
2 The forecasting error is given by εi = βi(RM-Rf) + R- E(Ri)
3 Cochrane, John (2011), “Discount Rates,” Journal of Finance, vol. 66, no. 4.
4 Jegadeesh, Narasimhan, and Sheridan Titman (1993), “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency, Journal of Finance, vol 48, no. 1.
5 Carhart, Mark (1997), “On Persistence in Mutual Fund Performance,” Journal of Finance, vol. 52, no. 1.
6 Heaton, John, and Deborah Lucas (2000), “Portfolio Choice and Asset Prices: The Importance of Entrepreneurial Risk,” Journal of Finance, vol. 55, no. 3.

Gideon Magnus, director of quantitative research at Morningstar.

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