Factor Models in Asset Pricing: A Comprehensive Literature Review (Up to 2024)

Introduction

Asset pricing factor models seek to explain differences in expected returns across assets using a set of common risk factors or characteristic-based factors. The Capital Asset Pricing Model (CAPM) introduced the idea that a single factor – market risk – should explain returns, but subsequent research uncovered numerous anomalies (patterns in returns unexplained by CAPM) such as size, value, and momentum. This prompted the development of multi-factor models in both academia and industry. In this review, we survey the evolution of factor models from the CAPM to the latest multifactor extensions, including academic models (e.g. Arbitrage Pricing Theory, Fama–French 3/5/6-factor models, Carhart’s 4-factor model, the q-factor model) and industry models (e.g. factor frameworks used by AQR and BlackRock). We also review prominent empirical anomalies – size, value, momentum, low volatility, profitability, investment, etc. – and how well these models account for them in the U.S. market. Key papers and original sources are cited throughout, and we include summary tables of models, their factors, empirical findings, and critiques.

Note: All tables and factor model descriptions refer to U.S. equity markets unless noted otherwise.

The Capital Asset Pricing Model (CAPM): Single-Factor Foundation

The CAPM of Sharpe (1964) and Lintner (1965) marks the birth of modern asset pricing theory. It posits that the market portfolio’s excess return is the sole factor explaining cross-sectional differences in expected returns. In the CAPM, an asset’s expected excess return is proportional to its beta (covariance with the market). The model implies that no other characteristic should systematically predict returns once market beta is accounted for. Despite its elegance, the CAPM has fared poorly empirically. Early tests (e.g. Black, Jensen, and Scholes 1972; Fama and MacBeth 1973) found the security market line to be flatter than theory predicts – low-beta stocks earned higher returns than CAPM would imply, and high-beta stocks earned lower returns. Moreover, starting in the late 1970s, researchers documented several variables besides beta that have predictive power for average returns, including firm size, earnings/price ratios, leverage, book-to-market ratios, and momentum. These findings indicate that market beta alone is an incomplete description of risk, contradicting CAPM’s core prediction.

Critically, CAPM has never been an empirical success. Fama and French (2004) conclude that the Sharpe–Lintner CAPM’s predictions are so violated by real-world return patterns that “the problems are serious enough to invalidate most applications of the CAPM”. For example, small-cap stocks and high book-to-market (“value”) stocks have had higher average returns than CAPM can justify using reasonable beta estimates. Likewise, momentum – the tendency for recent stock winners to continue outperforming losers – is not explained by CAPM at all. Another fundamental critique, raised by Roll (1977), is that the CAPM is untestable because the true market portfolio (which in theory includes all assets, not just stocks) is unobservable – any test of CAPM jointly tests a proxy for the market. These issues motivated the search for richer models with multiple risk factors that could better capture the cross-section of returns.

Table 1. Major Asset Pricing Factor Models (Academic and Industry)

Model (Year)	Factors Included	Key Publications/Proponents	Key Findings & Critiques
CAPM (Sharpe 1964; Lintner 1965)	Market (Beta)	Sharpe (1964); Lintner (1965)	Single-factor model (market). Empirically weak: beta explains little cross-sectional return variation. Fails to account for size, value, etc. Roll’s critique: untestable true market portfolio.
Arbitrage Pricing Theory (1976)	K unspecified factors (linear combination)	Ross (1976)	Multi-factor concept based on no-arbitrage. Factors could be macro or statistical. General framework that inspired later models. Critique: does not identify specific factors; many possible implementations.
Fama–French 3-Factor (1993)	Market; SMB (Size); HML (Value)	Fama & French (1992, 1993)	Size and value factors explain much of CAPM’s anomalies. Intercepts $\sim$0 for size–value sorted portfolios. Interpreted as additional risk factors. Critique: leaves momentum unexplained; debate over risk vs mispricing.
Carhart 4-Factor (1997)	Market; SMB; HML; MOM (Momentum)	Carhart (1997); Jagadeesh & Titman (1993)	Adds momentum factor (UMD) to FF3. Explains persistence in mutual fund returns (momentum-driven). Widely used for performance benchmarking. Momentum factor not clearly risk-based (prone to crashes).
Fama–French 5-Factor (2015)	Market; SMB; HML; RMW (Profitability); CMA (Investment)	Fama & French (2015)	Incorporates profitability and investment factors to capture “quality” and “investment” anomalies. Improves explanatory power (71–94% $R^2$). HML becomes redundant in U.S. sample, as its premium is absorbed by RMW and CMA. Still fails to explain momentum; some small-growth anomalies remain.
Fama–French 6-Factor (2018)	Market; SMB; HML; RMW; CMA; MOM	Fama & French (2018) (implicit)	Essentially FF5 + momentum. Acknowledges momentum’s role as a 6th factor. More comprehensive model for equities. Not formally a new theory, but used in practice to ensure no known anomaly omitted.
q-Factor Model (2015)	Market; ME (Size); IA (Investment); ROE (Profitability)	Hou, Xue, Zhang (2015)	Grounded in investment-based theory. Omits HML and MOM (value and momentum seen as byproducts of investment/profitability). Performs comparably or better than FF3/Carhart on many anomalies. Suggests many anomalies ($\approx$half of 80) are insignificant, and many others explained by these four factors. Critique: still leaves momentum and a few others as “exceptions.”
Mispricing 4-Factor (2015)	Market; Size; “Mispricing – Factor 1” (e.g. valuation anomalies); “Mispricing – Factor 2” (e.g. momentum-type anomalies)	Stambaugh & Yuan (2017)	Two composite mispricing factors constructed from 11 anomalies. Outperforms FF4 and FF5 in explaining a broad set of anomaly returns. Implies behavioral sources (sentiment related). Not as widely used as FF models, but highlights clustering of anomalies.
Barra Risk Model (1970s, 1980s) – Industry	Dozens of factors (style factors: value, growth, momentum, volatility, yield, etc., plus industry/sector factors)	Bar Rosenberg (Barra); MSCI Barra	An industry multi-factor risk model for portfolio risk attribution. Uses both fundamental and statistical factors (~40+ metrics). Focuses on risk exposure and covariance rather than expected return pricing. Widely used by practitioners for risk management. Critique: proprietary, not centered on explaining expected returns (though often aligned with known factors).
AQR Style Premia Model (2010s) – Industry	Core style factors: Value, Momentum, Carry, Defensive (Quality/Low-risk)	AQR Capital (Asness et al.)	Proprietary integration of academic factors across asset classes. Emphasizes factors that are “persistent, pervasive, robust”: value, momentum, carry, quality/low-volatility. Implemented in long/short portfolios. AQR research finds these factors historically profitable across markets. Critique: subject to implementation costs and potential crowding.
BlackRock Factor Model (2010s) – Industry	Style factors: Value, Size, Momentum, Quality, Low Volatility (and others like Credit, Liquidity for multi-asset)	BlackRock (Ang et al.)	Factor investing framework used in portfolio design. Similar to academic factors (value, size, etc.). Often used for tilting portfolios and risk budgeting. BlackRock emphasizes dynamic allocation to factors and across asset classes. Lacks published “model” paper; approach aligns with known factors.

Sources: Key details drawn from original papers and surveys.

Table 1 continued on next page if necessary.

Empirical Anomalies and Associated Factors in the U.S. Market

We now discuss the major empirical anomalies that have motivated factor models, focusing on those in U.S. equities. An “anomaly” here means a pattern in average returns not explained by the CAPM (and sometimes not by subsequent models until a new factor is introduced). Each anomaly usually corresponds to a factor in a model that “prices” that anomaly. Below we highlight each and summarize evidence, including which models account for them. Table 2 will then summarize these anomalies/factors, their definitions, and key sources.

Size Effect (Small vs. Large Stocks)

Definition: Small-cap stocks (companies with low market capitalization) have tended to earn higher average returns than large-cap stocks, even after adjusting for market beta. This is measured by the SMB factor (small minus big).

Discovery: The size effect was first documented by Rolf Banz (1981), who found that “on average, stocks of small companies have higher returns than those of larger companies” over the long run. The effect was economically significant in the U.S. from 1936–1975 in his study, and it could not be explained by CAPM betas (small firms didn’t have proportionally higher betas). The size premium became a puzzle for CAPM.

In Factor Models: Size is one of Fama–French’s original factors (SMB). In the FF3 model, SMB captures a premium for small stocks. Indeed, Fama–French showed SMB was significant in explaining the higher returns of small-cap portfolios. Most later models (Carhart, FF5, q-factor, etc.) also include a size factor or something similar (Hou et al. include a size factor; Stambaugh–Yuan include a size factor too). So the size anomaly is generally considered explained (in a descriptive sense) by including SMB in the model – small stocks have positive SMB loadings and thus no longer have unexplained alpha under the model.

Interpretations: Initially, size was thought to proxy for some unspecified risk – perhaps small firms are less diversified, more volatile, or more illiquid. Indeed, small stocks often are less liquid (trading costs, etc.), which could command a premium. Or they may be more sensitive to economic shocks (e.g. higher distress risk). However, the risk story isn’t crystal clear, and behavioral explanations exist (e.g. small stocks might be neglected or subject to different investor clienteles). An interesting note: after Banz’s publication, the size premium in the U.S. noticeably shrank. Research by the early 2000s found the size effect has been much weaker (essentially zero) since the 1980s, especially if one excludes micro-cap stocks. This raises the possibility that the original discovery was partly a historical anomaly that got arbitraged or was sample-specific. Some have even called the size effect “dead” (though it occasionally resurfaces, often in small/micro-cap rallies or in specific markets). In practice, SMB in the U.S. has had a modest average return and is correlated with value; it remains in models more for completeness and because in some international markets a size effect is still observed.

Recent Evidence: A 2011 literature review noted that after the 1980s the size effect largely disappeared in U.S. and UK data, possibly due to the widespread launch of small-cap funds arbitraging it. It’s also heavily concentrated in January in the U.S. (the “January effect”). Thus, while size was a groundbreaking anomaly in 1981, its status as a robust, compensatory factor is debated. Many models keep SMB for legacy reasons and because removing it doesn’t necessarily improve things (and small stocks do covary somewhat). Industry practice still often includes a size factor in risk models. But researchers remain cautious: the economic source of the size premium is not fully resolved.

Value Effect (High Book-to-Market vs. Growth)

Definition: “Value” stocks – typically defined as those with high book-to-market (B/M) ratios, or high earnings/price, or other measures of being “cheap” relative to fundamentals – have historically earned higher returns than “growth” stocks (low B/M, expensive stocks). The HML factor (high minus low B/M) quantifies this difference.

Discovery: Early evidence came from Sanjoy Basu (1977) who found P/E ratios predict returns (low P/E – a value signal – outperforms high P/E). Then in the 1980s, Rosenberg, Reid, and Lanstein (1985) explicitly showed B/M predicts returns (high B/M stocks did better). The definitive study was Fama and French (1992), which found book-to-market was the strongest predictor of cross-sectional stock returns among a host of variables, even subsuming the effect of P/E and leverage. By 1992 the “value premium” – the return edge of value over growth – was well-established in U.S. data (and subsequently found in many international markets too).

In Factor Models: HML (value factor) is a pillar of the FF3 model. Including HML explains why value stocks outperform growth: value stocks have high positive loading on HML, so their excess returns are captured as a risk premium to that factor, not as unexplained alpha. The three-factor model made the value effect mainstream – it was no longer an “anomaly” but rather evidence of a missing risk factor in CAPM. The value factor is also implicitly or explicitly present in many industry models (e.g. any “Value” style index). Even in the q-factor model, though HML is not literally included, the model will still price value portfolios through its other factors (as value companies often have certain investment/profitability profiles). The Stambaugh–Yuan mispricing model interestingly suggests that value is part of a broader mispricing cluster and perhaps could be replaced by a composite factor.

Interpretations: Why do value stocks outperform? Two broad camps: risk-based vs. behavioral. The risk-based story (supported by Fama and French) suggests value firms are fundamentally riskier – for example, they may be in financial distress or have weaker prospects, so they command a higher expected return as compensation. Value companies often have poor past performance or are in out-of-favor industries – conditions that might be related to macro risks (e.g. they could fare worse in a recession). There’s some supporting evidence: value stocks covary with different state variables (e.g. they might crash more in bad times). However, a competing narrative by Lakonishok, Shleifer, and Vishny (1994) argues that the value premium arises because investors make behavioral errors: they extrapolate past growth too far and become overly pessimistic about “cheap” stocks and overly optimistic about “glamour” growth stocks. Thus, value strategies exploit suboptimal investor behavior (over-extrapolation), not risk. This contrarian view sees the value premium as a correction to investor expectations. There is considerable evidence of such biases and of investor flows chasing growth, etc.

The truth could be a mix – value might be somewhat riskier and markets might initially underprice them. Over long horizons, value has provided a strong premium: Fama and French once noted a long-short HML portfolio in the U.S. earned over 5% annual premium historically. However, recent performance has challenged this: from 2007–2019, value underperformed dramatically (growth stocks, especially tech, soared). The 2010s were essentially a “lost decade” (or more) for value. This raised questions: Is the value premium dead, merely dormant, or was it a sample-specific phenomenon that has reversed? Some (e.g. Arnott et al. 2020) argue that value will recover, attributing the drought to unusual conditions (low interest rates favoring growth, accounting issues with intangibles that made many value stocks look worse than they are, etc.). Indeed, 2022–2023 saw a bit of value comeback. But it’s a reminder that even well-established factors can go through prolonged bad periods.

In Summation: The value effect is historically large and one of the most studied anomalies. It gave birth to the style of “value investing” and numerous smart beta funds. In factor models, HML is now somewhat overshadowed by profitability/investment factors (FF5) which partly subsume it, but many practitioners still include an explicit value factor for transparency. It remains a core part of the factor story – either as a risk premium for bearing distress risk or as a behavioral reward for contrarian investing.

Momentum Effect (Winner vs. Loser Stocks)

Definition: Stocks that have performed well in the recent past (e.g. past 6 to 12 months) tend to continue to perform well in the near future, and conversely, recent losers tend to continue underperforming. This is the momentum anomaly. A momentum factor (MOM or UMD) takes a long position in recent winners and a short position in recent losers, capturing this return spread.

Discovery: Jegadeesh and Titman’s 1993 Journal of Finance paper is the landmark study. They examined “relative strength” strategies and found that a 6-month formation, 6-month holding period momentum strategy yielded significant profits (~1% per month) in U.S. stocks. In their sample (1965–1989), buying past winners and selling losers produced an abnormal return even after accounting for risk by CAPM or other known factors at the time. They described these momentum profits as “quite profitable” on average. Subsequent research found momentum in many other markets and asset classes (Asness et al. 1997 showed it in Europe, Japan, etc., and even in country index returns; Moskowitz and Grinblatt 1999 found industry momentum). Momentum became known as “the premier market anomaly” because it was both highly significant and hard to reconcile with efficient markets.

In Factor Models: The Carhart four-factor model officially added momentum as a factor, which was a nod to momentum’s empirical importance. Carhart’s inclusion meant that any strategy or fund that was inadvertently loading on momentum (e.g. a growth fund might inadvertently hold recent winners) would no longer show alpha, as the model attributes the performance to a momentum risk premium. Indeed, Carhart found that momentum explained the persistence of mutual fund outperformance. Fama and French notably avoided momentum in their early models, but given momentum’s robustness, researchers often add MOM as an extra factor. As mentioned, Fama–French (2018) essentially endorse a six-factor model including momentum for a more complete description of returns.

Explanations: Momentum is notoriously difficult to explain with classical risk theory. Unlike value or size, it’s not obvious what risk would correspond to “recent winner” stocks that would justify extra return. In fact, momentum often implies buying stocks that have recently gone up (which could be growth or glamor stocks) and shorting those that went down (often distressed names), so it’s not simply capturing the same as value – it’s almost orthogonal to value (value and momentum historically are negatively correlated strategies, which is why they complement each other). Two main classes of explanations have been proposed:

• Behavioral: Most scholars lean towards behavioral explanations for momentum. One view (Jegadeesh and Titman; Daniel, Hirshleifer, Subrahmanyam 1998) is that momentum arises from investor underreaction to news – good news about a company’s prospects isn’t fully incorporated into the price initially, so the stock drifts upward for a while as the news gradually disseminates, and vice versa for bad news. Another complementary view is overreaction in the longer term (which could cause eventual reversals at horizons beyond 1 year – indeed, 3-5 year reversals have been documented, suggesting a pattern of underreaction short-term, overreaction long-term). Behavioral biases like anchor-and-adjust or herding can create momentum in prices. Additionally, there’s the aspect of investor psychology: chasing winners and dumping losers might itself push prices (creating a self-fulfilling momentum for some time).

• Risk-based: A few attempts at risk explanations exist. One is time-varying risk or dynamic risk – perhaps winner stocks happen to be riskier during certain periods (e.g. momentum crashes tend to occur in rebounds from bear markets, so maybe momentum carries a “crash risk” – it does very poorly in sudden regime shifts). For instance, momentum strategies suffered an infamous crash in early 2009: after the 2008 financial crisis, the beaten-down stocks (which momentum was short) rallied massively, causing momentum funds huge losses. That suggests momentum wins most of the time but occasionally pays the price with big drawdowns, which could be seen as compensation for bearing that crash risk. Some models (e.g. Barroso and Santa-Clara 2015) attempt to adjust momentum for its volatility to mitigate crash risk. Overall, while a risk story is not as straightforward as for other factors, one can argue momentum might be a reward for bearing the risk of abrupt trend reversals or liquidity crunches when everyone is on one side of the trade.

Current View: Momentum is still going strong as an anomaly. It has been observed for over 200 years in stock data (according to some long-run studies) and across many markets. Its premium in U.S. equities has been comparable to or larger than the value premium historically. But its occasional crashes and implementation challenges (higher turnover and trading costs) mean it’s not a free lunch. In factor models, momentum remains essential for capturing short-to-medium term return dynamics. Practitioners often pair momentum with value (the combination helps because their returns are negatively correlated, smoothing the ride). Momentum’s existence is a major challenge to pure efficient markets, and it’s one factor that even the originators of factor models (Fama and French) concede likely reflects behavioral mispricing rather than a risk factor – Eugene Fama has called momentum “the biggest embarrassment for efficient markets.” Regardless of cause, the momentum factor continues to be included in extended pricing models and is a staple in quantitative investing.

Low-Volatility and Low-Beta Effect

Definition: The low-volatility anomaly (also known as the low-beta anomaly) is the finding that stocks with lower risk (volatility or market beta) have had higher risk-adjusted returns than stocks with higher risk. In a CAPM world, higher beta should mean higher expected return, but empirically, high-beta stocks have underperformed what CAPM would predict, and low-beta stocks have overperformed. A factor proxy for this is the Betting-Against-Beta (BAB) factor or a general low-volatility factor: long low-beta assets, short high-beta assets.

Discovery: This anomaly has roots going back to the 1970s. Fischer Black (1972) theorized that if investors face borrowing constraints, high-beta stocks could be “overpriced” as constrained investors seek leverage via high-beta stocks instead of borrowing – implying a flatter security market line. Subsequently, Haugen and Heins (1975) found in U.S. data that risk (variance) seemed to have no positive relation with average return, which was provocative. More concrete evidence came later: Baker and Haugen (1991) showed low volatility stocks outperformed in many markets. The modern definitive study is often cited as Baker, Bradley, and Wurgler (2011) or Blitz and van Vliet (2007), who demonstrated that portfolios of low-volatility stocks produce market-like returns with much lower risk, thus achieving higher Sharpe ratios than high-vol stocks. The BAB factor introduced by Frazzini and Pedersen (2014) formalized a tradable strategy: they construct a factor that is long leveraged low-beta stocks and short high-beta stocks (beta-neutral overall). They found this BAB factor produces significant positive risk-adjusted returns across equities globally, as well as in other asset classes. In their sample, the BAB factor’s alpha is positive and significant even controlling for other factors – essentially confirming that low-beta assets outperform on a risk-adjusted basis while high-beta assets underperform.

In Factor Models: Traditionally, the standard academic factor models did not include a low-vol or low-beta factor. CAPM of course implies it’s not needed; FF3/FF4/FF5 didn’t add it explicitly. However, the low-beta anomaly is indirectly related to the flatter-than-expected security market line, which was noted as far back as Black (1972). Some argue the HML factor picks up a bit of this (because growth stocks often have higher beta, etc.), but not really – low-vol is somewhat distinct. The Frazzini–Pedersen BAB factor is sometimes considered an additional factor in extended models (especially in industry contexts). AQR and others often refer to a “defensive” factor (which covers low-beta or low-volatility investing, sometimes combined with quality metrics). In the AQR “style premia” taxonomy, defensive or quality investing is effectively capturing this low-risk effect (buy safe, quality stocks, short junky high-risk stocks). Some industry multifactor products include a specific low-volatility or “minimum volatility” factor tilt. Academic models have not canonized a low-vol factor in the way they did momentum or value, perhaps because it’s correlated with other factors and because its risk story was a bit non-standard.

Explanations: The Betting Against Beta model (Frazzini & Pedersen) offers a risk-based explanation involving leverage constraints. If many investors (like mutual funds or individuals) cannot or do not use leverage, but they want higher returns, they will overweight high-beta stocks to synthetically lever their portfolios. This excess demand drives up high-beta stock prices, lowering their future returns, and drives down low-beta stock prices (or at least leaves them appropriately priced with higher returns relative to their low risk). In equilibrium, high-beta assets can become “overpriced” (low alpha) and low-beta assets “underpriced” (high alpha). This leads to the flat or inverted risk-return relationship observed. Indeed, Frazzini and Pedersen found that in periods when leverage was constrained (e.g. high funding costs), the BAB factor did poorly (which fits the story that when leverage is scarce, high-beta gets bid up more, hurting BAB).

Behavioral explanations also exist: some argue that lottery-seeking behavior causes investors to irrationally bid up volatile stocks (seeking the small chance of a big payoff), leading those stocks to be overpriced and earn low returns on average. There’s evidence that retail investors prefer high-volatility, high-beta, low-priced “lottery” stocks, which could depress their future returns. Also, agency issues can cause institutional investors to avoid low-beta stocks (they might hug benchmarks, which forces them to hold high-beta names to not lag in bull markets).

Regardless of cause, the low-volatility effect is intriguing because it violates the basic notion of risk-return tradeoff. It’s essentially saying “safety wins” – a very counterintuitive result in finance. This has spurred a lot of interest in low-volatility funds (e.g. S\&P Low-Vol index, MSCI Min Vol indices). These strategies have done well in risk-adjusted terms historically, though one must be mindful they can lag in bull market phases (when high-beta stocks surge).

Current Status: The low-vol anomaly still appears in data and hasn’t been arbitraged away fully (perhaps due to constraints and because many investors benchmark to market indices, which are beta-weighted toward high-beta stocks). However, as more low-volatility products have launched, there’s some indication the premium narrowed a bit in recent years (an influx of capital into low-vol stocks can bid them up). Nonetheless, it remains an important factor in industry portfolios and a challenge to traditional theory. If one were to extend the academic factor model palette, a low-vol or low-beta factor would be a natural addition (some researchers have suggested a five-factor model with market, size, value, momentum, and low-vol as the main styles – in fact, many investment managers consider exactly these as the “big 5” equity styles). BlackRock, for example, explicitly manages a minimum volatility factor as one of its style factors.

Profitability (Quality) and Investment Factors

Profitability (Quality): As introduced earlier, profitability – often measured as gross profit or operating profit to assets – predicts returns (Novy-Marx 2013). The anomaly is that more profitable firms earn higher returns than less profitable firms, even controlling for other factors. This seems counterintuitive in a risk sense (one might think safer firms with steady profits would have lower returns), but it makes sense if investors under-appreciate the earnings power of “quality” firms. Robert Novy-Marx dubbed profitability “the other side of value” because value picks up weak firms being underpriced, whereas profitability picks up strong firms being underpriced. In any case, the profitability factor (RMW in Fama–French terms, or sometimes called a quality factor) has a clear positive premium: high-profitability stocks outperform low-profitability stocks on average.

Investment (Asset Growth): The investment anomaly is that firms with high asset growth or high capital investments tend to have lower subsequent returns, and firms with low investment (or even shrinking assets) have higher subsequent returns. This was shown by Titman, Wei, Xie (2004) and by Cooper, Gulen, Schill (2008) in terms of asset growth rates. The intuition might be that aggressive investment (especially if funded by equity issuance) can be an indicator of over-optimism or empire-building by managers – such firms often disappoint later, so their returns are low. Conversely, firms that restrain investment may be doing so due to lack of opportunities (which could correlate with value) or due to disciplined management, and they end up with higher returns. The CMA factor in FF5 captures this (Conservative minus Aggressive investment). Aharoni, Grundy, Zeng (2013) also documented the investment effect in a broad sample.

In Models: Both profitability and investment are explicitly in the Fama–French five-factor model, which validated that these factors have significant explanatory power. The q-factor model also has similar constructs (ROE and investment). These factors help explain a myriad of minor anomalies: for example, firms with high accruals (an earnings quality measure) have low returns – that is partially because high accrual firms tend to have low subsequent profitability (a quality issue). Or the “quality minus junk” factor popularized by Asness, Frazzini, and Pedersen (2019) combines profitability, growth stability, and low risk – capturing a broad quality premium, much of which overlaps with RMW. By including RMW and CMA, the FF5 model could explain anomalies like the accrual anomaly, the issuance/repurchase anomaly (firms that issue equity underperform, which relates to investment), and others that the FF3 left unexplained.

Interpretations: A risk-based interpretation offered by Fama–French is that more profitable firms and low-investment firms might actually be riskier in terms of the covariance structure of their cash flows. They tie it to the idea that in the dividend discount model, a stock’s sensitivity to profitability shocks or investment opportunities could be risk factors. However, many find a behavioral story plausible: investors might simply underestimate the persistent advantage of high profitability (perhaps thinking it will revert to the mean too fast, or ignoring it because they chase growth not profitability), so those stocks become undervalued and earn higher returns when the strong profits continue. Similarly, low-investment firms might signal management caution or lack of overinvestment, which investors don’t fully appreciate. Some also link high investment to potential overconfidence (managers overinvesting when times are good, leading to future disappointment).

Interestingly, the profitability and investment effects tie into “quality” investing – something practitioners often talk about. Quality typically means companies that are profitable, stable, and well-managed. There’s evidence that such companies’ stocks do well (relative to their risk). Asness et al. (2019, “Quality Minus Junk”) constructed a composite quality factor (including profitability, earnings stability, low leverage, etc.) and found a significant premium globally. This aligns with RMW (profitability) being a key driver and perhaps other aspects like stability overlapping with low-volatility.

Current Evidence: Both profitability and investment factors have held up in out-of-sample tests to a good degree, though their premiums are moderate (smaller than value or momentum historically). In the 2010s, one of the only factors that consistently worked was profitability/quality – as value lagged, many noticed that high quality (profitable, safe) stocks still did relatively well, which is consistent with a quality premium. Indeed, one analysis of FF5 factors 2010–2019 showed the profitability factor was the only one with a robust positive premium in that decade, whereas value and investment factors struggled or had weird performance (as mentioned, the CMA factor lost some of its edge post-2004). This suggests the market regime can affect these premiums: when growth stocks dominate, value suffers; when cheap financing is abundant, the payoff to conservative investment might diminish. Profitability (quality) seems more consistently rewarded over time.

In summary, profitability and investment anomalies reinforced that simple fundamentals do predict returns – profitable, efficient companies outperform unprofitable, profligate ones, which on the face of it sounds intuitive (good companies give good returns). The puzzle is why investors wouldn’t price that in – which circles back to either risk (maybe profitable firms have some hidden risk) or behavior (investors pay more attention to glamour or other metrics). Regardless, modern factor models account for these, and in practice many quantitative investors tilt towards high quality/profitable stocks and away from heavy-investment “growth” stocks to capture these premiums.

Table 2. Prominent U.S. Equity Market Anomalies and Factors

Anomaly (Factor)	Definition & Measurement	First Key Documentation	Factor Model Inclusion	Observations and Notes
Market (Beta)	Higher beta (covariance with market) stocks should give higher returns (CAPM). Empirically, the relationship is flatter than CAPM predicts.	CAPM: Sharpe (1964); Black et al. (1972)	All models include Market factor	The market factor is the baseline. However, low-beta stocks have done better than expected and high-beta worse (see Low-Vol anomaly). Black’s version of CAPM allowed a flatter slope.
Size (SMB)	Small-cap stocks outperform large-cap. Measured by market cap (SMB = returns of smallest minus largest stocks).	Banz (1981)	FF3, Carhart, FF5, q-factor, etc.	Size effect was strong pre-1980, but has weakened. SMB premium ~ historically ~3-4% annual, concentrated in micro-caps and in January. Could proxy for liquidity or other risks. Diminished after 1980s in U.S. likely due to arbitrage/fund flows.
Value (HML)	High book-to-market (“value”) stocks outperform low B/M (“growth”) stocks. HML = high B/M minus low B/M portfolio return. Sometimes measured via E/P, C/P etc.	Rosenberg et al. (1985); Fama & French (1992)	FF3, Carhart, (FF5 includes HML but it’s partly subsumed)	One of the strongest anomalies historically. Value premium ~5%/yr over growth long-term. Seen in U.S. and abroad. Interpretations: risk of distress vs. behavioral mispricing. Suffered long drought 2009–2019. In FF5, HML is “redundant” (captured by other factors), but many investors still target a value factor.
Momentum (MOM)	Recent 6-12 month winner stocks continue to outperform recent losers, over next ~3-12 months. Momentum factor = top-decile past winners minus bottom-decile losers.	Jegadeesh & Titman (1993)	Carhart 4-factor (added)； often added to FF models as 6th factor	A pervasive, high Sharpe anomaly. ~1%/month return in original study. Hard to explain via risk. Subject to occasional crashes (e.g. 2009). Suggests underreaction or other behavioral biases. Widely used in quant strategies; complements value (negative correlation).
Low Volatility / Low Beta (Defensive)	Low-beta or low-volatility stocks have higher risk-adjusted returns than high-beta/vol stocks. “Betting Against Beta” (BAB) factor = long low-beta, short high-beta (leveraged to beta-neutral).	Haugen & Heins (1975); Blitz & van Vliet (2007); Frazzini & Pedersen (2014)	Not in FF or Carhart; used in industry models (AQR Defensive, etc.)	Contradicts CAPM: high-beta stocks underperform. BAB factor earns positive alpha ~ “free lunch”. Explanation: leverage constraints and lottery demand – high-beta overpriced. Many low-volatility ETFs exist. Premium smaller after 2010 as strategy got popular, but still significant long-term.
Profitability (Quality) (RMW)	Stocks with high profitability (e.g. gross profit/assets or ROE) outperform those with low profitability. RMW factor = high-operating-profit minus low-profit stocks. Related to “quality” (earnings quality, margins).	Novy-Marx (2013) (gross profitability); Ball et al. (2015)	FF5 (added as RMW); q-factor (ROE factor); Quality-minus-Junk (Asness 2019)	Stronger profits → higher returns. Premium ~3-4%/yr in U.S. since 1960s. Risk story: profitable firms could be riskier in recessions? Behavioral: investors underprice boring profitable firms (prefer growth). In 2010s, profitability was one of few factors that performed positively. Often combined with other “quality” metrics (stability, low debt) in industry usage.
Investment (Asset Growth) (CMA)	Firms with high asset growth or heavy investment tend to underperform, while those with low investment (or shrinking assets) have higher subsequent returns. CMA factor = low investment minus high investment portfolios.	Titman, Wei, Xie (2004); Cooper et al. (2008); Aharoni et al. (2013)	FF5 (added as CMA); q-factor (Investment); partly in “quality” composite	“Empire builders” or firms issuing lots of equity/investing aggressively often see poor returns subsequently. Could be overinvestment or manager overconfidence. Low-investment firms (often value stocks) do better. This factor overlaps with value (value stocks often have low past investment). The premium historically ~ approx 4%/yr in FF data, but post-2013 it weakened. Risk story: high investment might correlate with lower risk but worse returns (inverse of expected), so behavioral angles likely.
Liquidity (Illiquidity)	Less liquid stocks (high trading costs, low volume) have higher expected returns to compensate liquidity risk. Liquidity factor often measured by Pastor-Stambaugh (2003) – returns of stocks sensitive to liquidity shocks.	Amihud & Mendelson (1986); Pastor & Stambaugh (2003)	Not in FF models; sometimes added in extensions	Illiquid stocks (e.g. small caps) show higher avg returns – partly why size premium may exist. A liquidity risk factor is separate: in liquidity crises, illiquid assets crash more. PS (2003) factor had significant premium. In practice, liquidity is both a factor and a trading cost consideration. Often not directly included in academic pricing models due to data issues, but important in applications.

Sources: See referenced studies for each anomaly; factor model attributions from Fama–French (1993, 2015), Carhart (1997), Frazzini–Pedersen (2014), etc.

Table 2 provides a simplified summary; many anomalies have additional nuances and related literature.

Comparative Performance of Factor Models and Critiques

Having surveyed the models and factors, we turn to how well these models perform in explaining asset returns and the critiques surrounding them.

Explaining the Cross-Section: Each progression – from CAPM to FF3 to Carhart 4 to FF5 (and beyond) – has generally improved the statistical fit to the cross-section of stock returns. For instance, the FF3 model achieved a zero-alpha fit for broad size and value sorted portfolios, which CAPM could not do. The FF5 model further reduced unexplained alphas for portfolios sorted on profitability and investment. However, no model thus far perfectly explains all anomalies. Fama and French (2015) report that the five-factor model, while capturing patterns related to size, value, profitability, and investment, is rejected by the formal GRS test – meaning there remain portfolios for which the model cannot account for average returns. Specifically, certain portfolios of small, growth-oriented firms still show non-zero intercepts (the “small growth anomaly” whereby small firms with low B/M and poor profitability did worse than the model expects). Adding momentum as a sixth factor addresses the momentum anomaly, leaving even fewer anomalies unexplained by the six-factor model.

Empirical head-to-head comparisons have been done. Hou, Xue, and Zhang (2015) showed their q-factor model often performs as well as FF3 or Carhart on many anomalies, and even better on some. They noted only a few exceptions where perhaps momentum or other niche anomalies still showed alpha. Stambaugh and Yuan (2017) demonstrated their mispricing factor model could price a collection of anomaly portfolios better than FF5 or q-factor in their tests. That said, FF5 and q-factor are quite competitive – different papers claim different “victors” depending on test assets and methodology. There is no consensus “best” model; rather, each has trade-offs. FF5 is simple and popular but redundant HML factor is a wrinkle. The q-factor aligns with a clean theory but excluding value and momentum may be seen as leaving something out. The mispricing model is intuitive for anomalies but less grounded in traditional theory (and hasn’t seen as wide adoption).

Model Robustness: A concern with factor models is stability – will the factor premia persist and will factor loadings maintain their predictive relationships? History has shown some factor premia vary over time. For example, the value premium has had multi-year droughts. The size premium virtually vanished for two decades. Momentum, while generally strong, had its crashes and occasional inversions. Profitability has been more stable but its premium might depend on the economic cycle. The low-volatility premium might shrink if too many investors crowd into it. So, while factor models describe the past well, their future performance is not guaranteed (though as long as investor behavior or fundamental risk persists, one might expect some continuation).

Out-of-sample tests and data-snooping critiques are serious: Many anomalies discovered in sample do not hold up out-of-sample or after publication. McLean and Pontiff (2016) found that anomaly returns drop by ~30-50% after the original publication, consistent with both data mining bias and some arbitrage capital flowing in. Harvey, Liu, Zhu (2016) raising the specter of hundreds of factors implies many were likely false positives. They argue one should use a stricter significance hurdle (like t > 3) to claim a new factor. This is a sobering critique: it suggests the literature may have overfit noise in the historical data, mistaking it for true patterns. For investors, it means chasing newly discovered factors can be dangerous – many will disappoint.

Risk versus Mispricing Debate: A core critique of multifactor models is whether the factors truly represent risk premia (i.e., investors earn these higher returns as compensation for bearing some systematic risk) or if they represent mispricing that persists due to behavioral or structural reasons. Fama and French’s stance has been to interpret factors as risk-based (at least initially for SMB and HML). Similarly, q-factor proponents tie factors to rational investment decisions. On the other hand, many behavioral finance researchers view factors like momentum, and even value, as arising from investor biases (overreaction, underreaction, etc.) rather than risks. The truth might vary factor by factor. For example, value could be partly risk (distressed companies) and partly mispricing (investor over-pessimism). Momentum is hard to rationalize as risk (some tried, e.g. “crash risk” or “prospect theory” preferences, but consensus leans behavioral). Low-volatility seems more constraint-driven than risk-driven. Profitability and investment factors could be seen as part of a rational ICAPM (investors care about earnings shocks, etc.) or as corrections to investor neglect of fundamentals. This debate matters: if factors are risk-based, they should persist (as long as the risk persists), and one can build asset pricing theory around them. If they’re mispricing, they might erode as investors catch on – or they might persist if limits to arbitrage (like costs, short-sale constraints, or institutional frictions) prevent arbitrageurs from fully eliminating them.

Performance in the 2010s: The last decade (2010–2019) was an interesting laboratory. According to a Robeco study, the Fama–French five factors “failed to deliver” during that decade – indeed, value had negative returns, size was flat, investment and momentum were weak, only profitability (and low-vol, outside FF5) did okay. This led to questions: Are factors just temporarily out of favor (a “factor winter”), or has something fundamentally changed (e.g. crowds arbitraging them)? The rebound of value in 2021–2022 suggests at least some mean reversion. Industry practitioners have started considering factor timing – dynamically tilting toward factors that are cheap or in favorable environments. BlackRock’s factor rotation approach, for example, looks at business cycle, relative valuations of factors, recent momentum of factors themselves, and factor dispersion to decide tilts. This acknowledges that factor performance is cyclical.

Multi-Collinearity and Redundancy: With many factors, often they are correlated. This can cause issues in regression-based models (e.g. distinguishing SMB vs HML contributions if small stocks are often value stocks). As noted, in FF5, HML became redundant. Some argue one could simplify models: perhaps the true underlying factors are fewer – e.g. some have proposed that investment and profitability essentially subsume value, so a three-factor model of market, size, and “quality” (profitability-investment) might suffice for many purposes (plus momentum as needed). Others have tried dimension reduction: concentration of a large set of anomalies into a few composite factors (like Stambaugh–Yuan did). These efforts suggest the factor zoo might be distilled into a smaller set of independent factors like Market, Value, Momentum, Quality, Low-Risk, and perhaps a few others like Term or Liquidity if across asset classes. In fact, many investment firms converge on a similar handful of styles.

Practical Challenges: When implementing factor models in the real world, several issues appear: trading costs and liquidity (some small-cap or microcap-based anomalies can’t be profitably traded once costs are considered – e.g. size premium might be mostly a microcap effect, which is costly to capture), turnover (momentum has high turnover, making actual net returns lower), shorting constraints (most academic factors are long-short, but many investors cannot short easily, so they use long-only or long–cash implementations, which yield lower returns). Also, factor crowding is a risk – if too many market participants pile into a factor trade (say everyone buys value stocks at the same time), that factor could become overvalued and lead to crashes or just muted future returns. There is evidence, for instance, that the value premium was partly arbitraged (or at least compressed) by the mid-2000s due to many value funds, and similarly momentum saw a huge crash which could be partly because so many quant funds were on the same trade in 2009 (a crowded trade unwinding). Thus, investors in factor strategies must monitor valuations of factors and not assume these are magic constants.

Model Selection: For an academic trying to explain returns, the choice of model might depend on the assets in question. For U.S. equities, FF5+MOM (6-factor) or q-factor are common choices now for tests. For international equities, sometimes a modified model including country or region factors is used. In fixed income or other asset classes, different factors apply (e.g. term and credit for bonds, carry for currencies, etc.). The spirit of factor modeling – identify broad drivers and fit a linear model – extends beyond equities, though our focus is equities.

New Frontiers: By 2024, researchers are also exploring machine learning methods to discover or combine factors, which can detect non-linear combinations or conditional factors (e.g. interactions between factors). There’s also interest in macro factors (like linking equity factors to macroeconomic risks). But an important caution is to avoid “factor fishing” with too flexible tools, which could just overfit data even more. The lesson from the factor zoo is to prioritize factors that are robust, have economic intuition, and appear in multiple markets and periods.

Conclusion

Over the past several decades, asset pricing has evolved from the simplicity of the single-factor CAPM to a rich tapestry of multi-factor models. Both academia and industry now embrace factor-based thinking: returns are driven not by idiosyncratic quirks of each stock, but by exposure to broad factors – be they market risk, value vs. growth, size, momentum, quality, low volatility, or others. The literature up to 2024 has catalogued hundreds of potential factors, but a central set of them consistently emerges as the most important and persistent in the U.S. equity market. The Fama–French family of models (3, 5, and now 5+momentum) encapsulate a large portion of these known effects, and alternative formulations like the q-factor or mispricing factor models offer different perspectives on how to parsimoniously explain returns.

Empirically, factor models have been successful in explaining much of the cross-sectional variation in stock returns – a considerable achievement given CAPM’s early shortcomings. They provide investors and researchers a framework to understand why certain portfolios (e.g. value stocks or momentum strategies) outperform and to adjust performance for “style” exposures. They have also blurred the line between alpha (skill) and beta (risk exposure) – what was once considered stock-picking alpha can often be attributed to factor loadings (for instance, a fund manager outperforming might just be loading on momentum or value). This has fueled the rise of smart beta and factor investing strategies, where investors deliberately seek these systematic premiums at lower cost.

Yet, factor models are not without critiques and they continue to be refined. Questions remain whether the observed premiums truly compensate risk or are anomalies that will shrink. The effectiveness of these models can vary by market regime (as seen in the 2010s when several factors struggled). Moreover, with so many factors discovered, issues of data mining loom large. Researchers now take more care in validating factors out-of-sample and adjusting for multiple hypothesis testing.

In industry practice, factor models like those from AQR and BlackRock show that the ideas from academia are being put into action: portfolios are built with deliberate tilts towards value, momentum, quality, low volatility, etc., and risk management systems (e.g. MSCI Barra) decompose portfolio risks into factor exposures. These models are used to explain performance (e.g. a fund lagged because it was underweight the momentum factor during a momentum rally, etc.) and to guide allocation (factor timing and strategic factor allocation).

As of 2024, the consensus is that a small number of pervasive factors explain a lot of asset price behavior: for equities, one might cite Market, Value, Size, Momentum, Profitability/Quality, Investment, Low Risk, and maybe Liquidity as the major ones – though some of these are correlated and might not all be needed simultaneously. The exact model one uses can be chosen based on context; for academic asset pricing tests, the Fama–French five-factor plus momentum model is often a solid default, whereas an investor implementing a strategy might pick the most rewarded factors net of costs (often value, momentum, quality, low vol).

In conclusion, factor models have greatly enhanced our understanding of asset returns by moving beyond the CAPM’s one-dimensional view. They bring to light the various dimensions along which assets are priced. Each new model has addressed weaknesses of the previous, but also raised new questions (e.g. what is the economic rationale of this new factor?). The literature up to 2024 reflects a mature yet still evolving field: while the core factors are well-established, researchers are actively examining how these premiums change over time, interact with each other, and whether new factors (or better formulations of old ones) can further improve the explanation of returns. Future research may unify some of these factors, incorporate macroeconomic risks, or leverage advanced techniques to detect when factor premia are robust or are likely to mean-revert.

For practitioners, the factor paradigm has shifted investing toward a more systematic approach – factor investing – which can be seen as a middle ground between pure passive indexing and pure active stock picking. By understanding factor models, investors can make informed decisions about where returns are coming from and what risks they are truly exposed to. As the U.S. market and global economy evolve, factors themselves may evolve (for example, if the economy changes structurally, new types of factors might gain prominence or old ones might wane). Thus, the ongoing dialogue between academic research and real-world data will continue to refine these models. The factor framework, however, is likely to remain a cornerstone of asset pricing, providing a structured way to decompose and explain the otherwise complex variations in asset returns.

References (Key Sources Cited):

• Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance. (Origin of CAPM)
• Fama, E., & French, K. (1992, 1993). The cross-section of expected stock returns; Common risk factors in the returns on stocks and bonds. J. Finance; J. Financial Econ. (Introduced size and value factors)
• Carhart, M. (1997). On persistence in mutual fund performance. Journal of Finance. (Carhart 4-factor model with momentum)
• Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance. (Momentum anomaly)
• Novy-Marx, R. (2013). The other side of value: The gross profitability premium. Journal of Financial Economics. (Profitability anomaly)
• Fama, E., & French, K. (2015). A five-factor asset pricing model. Journal of Finance. (FF5 model: adds profitability and investment)
• Hou, K., Xue, C., & Zhang, L. (2015). Digesting anomalies: An investment approach. Review of Financial Studies. (q-factor model: market, size, investment, ROE)
• Stambaugh, R., & Yuan, Y. (2017). Mispricing factors. Review of Financial Studies. (Two mispricing factors model)
• Frazzini, A., & Pedersen, L.H. (2014). Betting against beta. Journal of Financial Economics. (Low-beta anomaly and BAB factor)
• Harvey, C., Liu, Y., & Zhu, H. (2016). …and the cross-section of expected returns. Review of Financial Studies. (Multiple testing and factor zoo critique)
• Asness, C., Moskowitz, T., & Pedersen, L.H. (2013). Value and momentum everywhere. Journal of Finance. (Shows value and momentum work in various markets – illustrating generality of these factors)
• Ang, A. (2014). Asset Management: A Systematic Approach to Factor Investing. (Book consolidating academic and industry perspective on factor investing)
• Plus various industry white papers: e.g., AQR (Moskowitz et al. 2019 “Fact, Fiction, and Factor Investing”), BlackRock (Ang 2017, etc.), which discuss practical implementation of factor models.