Abstract
 Top of page
 Abstract

 1 Background
 2 Expectations Models in Finance
 3 The Fama and French Model
 4 Beyond the Fama and French ThreeFactor Model
 5 Conclusion
 References
The capital asset pricing model (CAPM) states that assets are priced commensurate with a tradeoff between undiversifiable risk and expectations of return. The model underpins the status of academic finance, as well as the belief that asset pricing is an appropriate subject for economic study. Notwithstanding, our findings imply that in adhering to the CAPM we are choosing to encounter the market on our own terms of rationality, rather than the market's.
 Top of page
 Abstract

 1 Background
 2 Expectations Models in Finance
 3 The Fama and French Model
 4 Beyond the Fama and French ThreeFactor Model
 5 Conclusion
 References
Modern academic finance is built on the proposition that markets are fundamentally rational. The foundational model of market rationality is the capital asset pricing model (CAPM). The implications of rejecting market rationality as encapsulated by the CAPM are very considerable. In capturing the idea that markets are inherently rational, the CAPM has made finance an appropriate subject for econometric studies. Industry has come to rely on the CAPM for determining the discount rate for valuing investments within the firm, for valuing the firm itself, and for setting sales prices in the regulation of utilities, as well as for such purposes as benchmarking fund managers and setting executive bonuses linked to adding economic value. The concept of market rationality has also been used to justify a policy of arm'slength market regulation—on the basis that the market knows best and that it is capable of selfcorrecting. Nevertheless, we consider that in choosing to attribute CAPM rationality to the markets, we are imposing a model of rationality that is firmly contradicted by the empirical evidence of academic research.
In Fisher Black and the Revolutionary Idea of Finance, Mehrling (2007) considers the CAPM as the ‘revolutionary idea’ that runs through finance theory. He recounts the first major step in the development of modern finance theory as the ‘efficient markets hypothesis’, followed by the second step, which is the CAPM. While the efficient market hypothesis states that at any time, all available information is imputed into the price of an asset, the CAPM gives content to how such information should be imputed. Simply stated, the CAPM says that investors can expect to attain a riskfree rate plus a ‘market risk premium’ multiplied by their exposure to the market. Mehrling presents the model formally as:
 (1)
where E(R_{j}) is the expected return on asset j over a single timeperiod, r_{f} is the riskless rate of interest over the period, E(R_{M}) is the expected return on the market over the period, and β_{j} identifies the exposure of asset j to the market.
In Mehrling's account, Black (1972) recognized that a rational market effectively requires the CAPM. As Black saw it, if the market of all assets offers investors a ‘risk premium'—[E(R_{M}) − r_{f}]—in compensation for bearing risk exposure, then, all else being equal, each individual stock, j, must rationally offer a risk premium equal to β_{j}.[E(R_{M}) − r_{f}], since β_{j} measures the asset's individual exposure to market risk. Market frictions (limited access to borrowing at the riskfree rate, for example) might imply adjustments, but, at the core, the CAPM must maintain (Black, 1972).
Nevertheless, we argue that the CAPM fails as a paradigm for asset pricing. To this end, we show, first, how a reexamination of the research of Black et al. (1972), which did much to lay the empirical foundation for the CAPM, reveals that the data do not actually provide a justification of the CAPM as claimed, but rather constitute confirmation of the null hypothesis, namely that investors impose a single expectation of return on assets. Researchers, however, did not wish to abandon the core paradigm of market rationality. Such paradigm, after all, justified the status of finance as a subject worthy of ‘scientific inquiry’. Second, we show that though the evidence now obliges academics to admit the ineffectiveness of beta, the impression remains that the CAPM (in some adjusted form) is core to the empirical behaviour of markets. Fama and French, for example, resolutely defend their threefactor model (which currently stands as the industrystandard alternative to the CAPM) as a multidimensional risk model of asset pricing. Nevertheless, they concede that the average return for an asset over multiple periods is insensitive to its beta. This fact alone suggests that markets might be unable to price risk differentially across assets.
There is a correspondence here with the observation of the scientific philosopher Thomas Kuhn (1962), who states that facts always serve to justify more activity without ever seriously being allowed to threaten the paradigm core. In Kuhn's view, ‘normal science’ generally consists of a protracted period of adjustments to the surrounding framework of a central paradigm with ‘addon’ hypotheses aimed at defending the central hypothesis against various ‘anomalies’. The continued defence of the CAPM—adding more factors to the CAPM to explain more anomalies—has led the singlefactor CAPM model to become the threefactor model of Fama and French. To this model are added additional factors for idiosyncratic volatility, liquidity, momentum, and so forth, all of which typify Kuhn's articulation of ‘normal science.’
If the CAPM must be rejected, we are obliged to return to a view of markets as predating the introduction of the CAPM. Namely, that markets respond generally positively to good news, and negatively to bad news, but wherein Keynesian crowd psychology as each investor looks to other investors inevitably influences the reaction, which may take on a degree of optimism or pessimism that disconnects from the fundamental news. Markets may indeed be capable of selfcorrection ‘in the longterm’, but this may be of little compensation to members of society enduring losses and the negative impact on the economy in the meantime. Such a view of markets would imply that a research agenda aimed at understanding market fallibility and their potential for selfdestruction, rather than aimed at enriching an account of ‘markets in equilibrium’, provides a more useful contribution to policy making. In effect, the paradigm of the CAPM and efficient markets may need to be replaced with a paradigm of markets as vulnerable to capricious behaviour.
1 Background
 Top of page
 Abstract

 1 Background
 2 Expectations Models in Finance
 3 The Fama and French Model
 4 Beyond the Fama and French ThreeFactor Model
 5 Conclusion
 References
By the late 1950s, the prestige of the natural sciences had encouraged the belief that the modelling of decisionmaking and resource allocation problems could be identified through the elaboration of optimization models and the general extension of techniques from applied mathematics. Into this environment, Modigliani and Miller (1958, 1963) ushered their agenda for the modern theory of corporate finance. Thus the discipline was transformed from an institutional normative literature—motivated by and concerned with topics of direct relevance to practitioners (such as technical procedures and practices for raising longterm finance, the operation of financial institutions and systems)—into a microeconomic positive science centred about the formation and analysis of corporate policy decisions with reference to perfect capital markets. A capital market where prices provide meaningful signals for capital allocation is an important component of a capitalist system. When investors choose among the securities that represent ownership of firms' activities, they can do so under the assumption that they are paying fair prices given what is known about the firm (Fama, 1976). The foundations of modern finance theory embrace such a view of capital markets. The underlying paradigm asserts that financial capital circulates to achieve those rates of return that are most attractive to its investors. In accordance with this principle, prices of securities observed at any time ‘fully reflect all information available at that time’ so that it is impossible to make consistent economic profits by trading on such available information (e.g., Modigliani and Miller, 1958; Fama, 1976; or Weston, 1989).
The efficient market hypothesis—the notion that market prices react rapidly to new information (weak, semistrong or strong form)—is claimed to be the most extensively tested hypothesis in all the social sciences (e.g., Smith, 1990). Consistent with the efficient market hypothesis, detailed empirical studies of stock prices indicate that it is difficult to earn abovenormal profits by trading on publicly available data because they are already incorporated into security prices. Fama (1976) reviews much of this evidence, though the evidence is not completely onesided (e.g., Jensen, 1978). Yet even allowing that empirical research has succeeded in broadly establishing that successive share price movements are systematically uncorrelated, thus establishing that we are unable to reject the efficient market hypothesis, this does not describe how markets respond to information and how information is impounded to determine share prices. That is to say, the muchvaunted efficient market hypothesis does not in itself enable us to conclude that capital markets allocate financial resources efficiently. If we wish to claim allocative efficiency for capital markets, we must show that markets not only rapidly impound new information, but also meaningfully impound that information.
The variant of the efficient market hypothesis that encapsulates such efficient allocation is the capital asset pricing model (CAPM). The CAPM has dominated financial economics to the extent of being labelled ‘the paradigm’ (Ross, 1978; Ryan, 1982). Since its inception in the early 1960s, it has served as the bedrock of capital asset pricing theory and its application to practitioner activities. The CAPM is based on the concept that for a given exposure to uncertain outcomes, investors prefer higher rather than lower expected returns. This tenet appears highly reasonable, and following the inception of the CAPM in the late 1960s, a good deal of empirical work was performed aimed at supporting the prediction of the CAPM that an asset's excess return over the riskfree rate should be proportional to its exposure to overall market risk, as measured by beta.
The underlying intuition of the CAPM has appealed forcibly to practitioners in the fields of finance and accounting. At universities, future practitioners are inculcated with the notion of the CAPM and its attendant ‘beta’. Management accountants are likely to instinctively determine an acceptable discount rate in terms of the CAPM and a ‘project beta’ when discounting. Corporate and fund management performances are measured in terms of ‘abnormal’ returns, where ‘abnormal’ is relative to a CAPMdetermined return.
Early tests of the CAPM showed that higher stock returns were generally associated with higher betas. These finding were taken as evidence in support of the CAPM while findings that contradicted the CAPM as a completely adequate model of asset pricing did not discourage enthusiasm for the model.1 Miller and Scholes (1972), Black et al. (1972) and Fama and McBeth (1973) also demonstrate a clear relationship between beta and asset return outcomes. Nevertheless, the returns on stocks with higher betas are systematically less than predicted by the CAPM, while those of stocks with lower betas are systematically higher. In response, Black proposed a twofactor model (with loadings on the market and a zerobeta portfolio). Thus the claim was made that the CAPM could be fixed by substituting the riskfree rate in the model with the rate of return on a portfolio of stocks with zero beta.
Controversially, Fama and French (1992) show that beta cannot be saved. Controlling for firm size, the positive relationship between asset prices and beta disappears. Additional characteristics such as firm size (Banz, 1981), earnings yield (Basu, 1983), leverage (Bhandari, 1988), the firm's ratio of book value of equity to its market value (Chan et al., 1991), stock liquidity (Amihud and Mendelson, 1986), and stock price momentum (Jegadeesh and Titman, 1993) now appear to be important in describing the distribution of asset returns at any particular time. The Fama and French (1996) threefactor model identifies exposures to differential returns across high and low booktomarket stocks and across large and small firms to the CAPM as proxies for additional risk factors. As is often remarked, the model derives from a fitting of data rather than from theoretical principles. Black (1993) considered the then fledgling Fama and French threefactor model as ‘data mining’. Although Fama and French have decried the capacity of beta, they nevertheless insist that their two factors are ‘additional'—designed to capture ‘certain anomalies with the CAPM’. Formally, their model is presented as a refinement in the spirit of the CAPM. The trend of adding factors to better explain observed price behaviours has continued to dominate asset pricing theory. Subrahmanyam (2010) documents more than 50 variables used to predict stock returns. Nevertheless, the CAPM remains the foundational conceptual building block for these models. The threefactor model of Fama and French (1993, 1996), and the Carhart model (1997) which adds momentum exposure as a fourth factor, are now academically mainstream.
2 Expectations Models in Finance
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 Abstract

 1 Background
 2 Expectations Models in Finance
 3 The Fama and French Model
 4 Beyond the Fama and French ThreeFactor Model
 5 Conclusion
 References
In what is generally recognized as the first methodologically satisfactory test of the CAPM, Black, Jensen and Scholes (1972) (hereafter, BJS) find that there is a positive relation between average stock returns and beta (β) during the pre1969 period. BJS, however, recognized that although this observation might be interpreted as encouraging support for the CAPM, it is not actually sufficient to substantiate the CAPM. Insightfully, they recognized that even if it were the case that beta is actually ignored by investors, beta would still be captured in the data of stock returns as β_{j}.[R_{M} − E(R_{M})], where β_{j} is the beta for a stock j, and [R_{M} − E(R_{M})] represents the actual market return (R_{M}) over what it was expected to be (E(R_{M})). To see where the β_{j}.[R_{M} − E(R_{M})] term comes from, consider that a researcher wishes to test the null hypothesis that investors actually ignore beta and simply seek those stocks offering the highest returns, with the outcome that all stocks are priced to deliver the same expected return, say 10%, in a given year. Now, suppose that the actual market return for this year turns out to be 18%. In accordance with the null hypothesis model (all stocks are priced to deliver the same return), should the researcher now expect to find that outcome returns for this year are distributed around 18% and that beta has no explanatory role? Surprisingly, the answer is ‘no’. Consider, for example, that Stock A has a sensitivity to the market described by its beta of 1.5, and Stock B has a sensitivity to the market described by its beta of 0.5. BJS argue that the researcher should expect to find that each stock has achieved a return equal to the initial expectation (10%) plus the ‘surprise’ additional market return (8% = 18% − 10%) multiplied by that stock's beta (defined as an asset's return sensitivity to the market return). In other words, the researcher expects to find that the outcome return for Stock A is 10% + 1.5*8% = 22%, and for Stock B is 10% + 0.5*8% = 14%, even though both stocks were priced to give the same expected outcome of 10%.
Thus for BJS, the outcome regression equation to test a hypothesis for the expectation of return, E(R_{j}), for assets j against the actual outcome returns, R_{j}, for the assets, becomes:
 (2)
where E(R_{j}) formulates the model to be tested (e.g., the righthand side of the CAPM expression in equation ((1)) and [R_{M} − E(R_{M})] is the ‘unexpected’ excess market return multiplied by the asset beta (β_{j}) and ε_{j} allows an error term (cf. BJS, equation ((3))). Note again that the β_{j} term here does not depend on any assumptions regarding investor expectations.
In seeking to test the CAPM, BJS therefore formed their appropriate regression equation by substituting the CAPM equation for E(R_{j}) as equation ((1)) into equation ((2)) to give:
The E(R_{M}) terms cancel out and the required regression equation of the excess asset return R_{j} − r_{f} on the excess market return (R_{M} − r_{f}) becomes:
 (3)
A significant advantage of the regression equation is that its inputs are observable output data and not ‘expectations’.
BJS (1972) and Black (1993) apply equation ((3)) to the data following a doublepass regression method so as to achieve a number of testable predictions. Thus they founded the elements of the methodology that underpins all subsequent tests of asset pricing models. The method can be explained briefly. In a first pass, equation ((3)) is run as a timeseries regression of each stock's monthly excess return (R_{j,t} − r_{f}) at time t on the monthly excess market return (R_{M,t} − r_{f}) for that month so as to determine each stock's beta (β_{j}) as the ‘slope’ of the regression:
 (4)
where α_{j} denotes the ‘intercept’ of the regression and ε_{j,t} are the regression error terms, which are expected to be symmetrically distributed about zero (cf. BJS, equation ((6))). The stocks are then ranked by their beta and 10 decile portfolios are partitioned from lowest beta to highest beta stocks. In this way, an average intercept (α_{P}) and average slope (that is, beta, β_{P}) may be assigned to each portfolio. We can see that if the CAPM of equation ((1)) is well specified in describing expectations, the intercepts α_{P} for each portfolio should be close to zero. In the second pass, equation ((3)) can now be run as a single crosssection regression of the excess portfolio returns (R_{P} − r_{f}) on the portfolio betas (β_{P}) (as determined in the prior timeseries regression as the explanatory variable):
 (5)
(cf. BJS, equation ((11))). Again, if the CAPM of equation ((1)) is well specified, the intercept γ_{0} term should be statistically indistinguishable from zero, and the coefficient γ_{1} on the β_{P}'s should identify the average excess market return, (R_{M} − r_{f} ).
In the timeseries regressions, the BJS studies determine that the intercept α_{P}s are consistently negative for the highrisk portfolios (β > 1) and consistently positive for the lowrisk portfolios (β < 1). In the crosssectional regression, they find that the intercept is positive and the slope is too low to be identified with an average excess market return, (R_{M} − r_{f} ). Both pass regressions therefore contradict the CAPM.
As highlighted in Mehrling's biography 2007), Black realized that without some meaningful version of the CAPM, markets cannot be held to be rational. As Black (1993) explained, if the market does not appropriately reward beta, no investor should invest in highbeta stocks. Rather, the investor should form a portfolio with the lowest possible beta stocks and use leverage to achieve the same market exposure but with a superior return performance as compared with a highbeta stock portfolio.
The simplest way to make the CAPM fit the data is to replace the riskfree rate, r_{f} (typically the return on shortterm U.S. Treasury bonds) with some larger value, R_{z}, since that would adjust the intercepts and explain the lower slope of the crosssectional regressions. In fact, BJS use the data to calculate the required substitute rate, R_{z}, that offers the best fit. As Mehrling's biography recalls, ‘the R_{z} term was a statistical fix in search of a theoretical explanation’ (p. 114). Accordingly, Black proposed his version of the CAPM as:
 (6)
where R_{z} is postulated as representing the return on a portfolio that has zero covariance with the return on the market portfolio. Black argued that the model is consistent with relaxing the assumption of the existence of riskfree borrowing and lending opportunities.
The test of whether the data are being generated by the process of equation ((6)) is that of whether the actual outcome returns are explained by the regression equation ((3)) with the standard riskfree rate r_{f} replaced by R_{z}:
which (because we wish to maintain the regression format of a dependence of R_{j} − r_{f} on R_{M} − r_{f} as the independent variable) can be rewritten as:
That is, the firstpass timeseries regressions of the excess return (R_{j} − r_{f}) on the excess market return (R_{M} − r_{f}) now has predicted intercepts α_{P} for the portfolios as:
 (7)
where R_{z} is the average excess mean return on the zerobeta portfolio over the period. Equation ((7)) (and therefore equation ((6))) could therefore be declared consistent with the JBS findings that the intercept α_{P}s are increasingly negative (positive) with increasing (decreasing) betas from the base β_{P} = 1. Additionally, the secondpass crosssection regressions of the portfolio returns (R_{P}) on the portfolio betas (β_{P}) as equation (5) above:
 (8)
now predicts:
 (9)
which is consistent with the determinations of JBS of a positive intercept and a slope that understates the excess market return.
Suppose, however, that we insist on testing the possibility that investors contravene Black's CAPM and can be modelled as adhering to our (heretical) null hypothesis that all assets j have the same expected rate of return, E(R_{j}), which is then necessarily that of the market, E(R_{M}):
 (10)
How do the regressions separate the hypotheses as preferable explanations of the data? To test the equation ((10)) hypothesis, we would form the regression equation (with equation ((2))) as:
 (11)
Note again how β_{j} above identifies the ‘drag’ of the excess market return on the return on asset j. Equation ((11)) (again for the purpose of expressing a preferred regression dependence of R_{j} − r_{f} on R_{M} − r_{f} as the independent variable) can be rewritten as:
The firstpass timeseries regressions should now have the intercept α_{P}:
 (12)
and the secondpass crosssection regressions (equation ((5))):
 (13)
should reveal the parameters γ_{0} and γ_{1} as:
 (14)
Thus we find that the difference in predictions between the traditional CAPM (equation ((1))), Black's CAPM (equation ((6))) and the null hypothesis model of equation ((10)) are as follows. For the traditional CAPM hypothesis the predictions (as above) are:
for Black's CAPM hypothesis:
 (15)
 (16)
and for the null hypothesis:
 (17)
 (18)
Thus, the original CAPM hypothesis predicts R_{z} = r_{f}, Black's CAPM hypothesis predicts R_{z} = a value greater than r_{f} but less than E(R_{M}). The null hypothesis predicts R_{z} = E(R_{M}). So what do the data say? BJS actually observe:
This (the beta factor, R_{z}) seems to have been significantly different from the riskfree rate and indeed is roughly the same size as the average market return (R_{M}) of 1.3 and 1.2% per month over the two sample intervals (1948–57 and 1957–65) in this period. (p. 82, emphasis added)
In other words, the BJS results validate the null hypothesis of equation ((10)) in favour of either the CAPM of equation ((1)) or Black's CAPM of equation ((6))! This is an extraordinary observation; the evidence from the beginning has always been squarely against the notion that investors set stock prices rationally in relation to stock betas. Such a revelation, however, would have fundamentally undermined the determination of finance to be accepted as a domain of economics with its study of ‘efficient markets’ in terms of econometric techniques.
3 The Fama and French Model
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 Abstract

 1 Background
 2 Expectations Models in Finance
 3 The Fama and French Model
 4 Beyond the Fama and French ThreeFactor Model
 5 Conclusion
 References
Fama and French (hereafter, FF) have been aggressive in pronouncing the ineffectiveness of the relation between beta (β) and average return (see also, Reinganum, 1981, and Lakonishok and Shapiro, 1986). They commence their 1992 paper with the pronouncement that ‘when the tests allow for variation in β that is unrelated to size, the relation between β and average return is flat, even when β is the only explanatory variable’. They find, however, that two other measured variables, the market equity value or ‘size’ of the underlying firm (ME) and the ratio of the book value of its common equity to its market equity value (BE/ME), ‘provide a simple and powerful characterization of the crosssection of average stock returns for the 1963–90 period’ (FF (1992), p. 429) and conclude that ‘if stocks are priced rationally, the results suggest that stock risks are multidimensional’ (p. 428).
As BJS (1972) before them, Fama and French realize the necessity of retaining a riskbased model of asset pricing. In the absence of such a model, the rational integrity of markets is undermined. In their 1996 paper, Fama and French place their model squarely in the tradition of the CAPM, stating that ‘this paper argues that many of the CAPM averagereturn anomalies are related, and that they are captured by the threefactor model in Fama and French (1993)’. The model says that the expected return on a portfolio in excess of the riskfree rate [E(R_{j}) − r_{f}] is explained by the sensitivity of its return to three factors: (a) the excess return on a broad market portfolio (R_{M} − r_{f}); (b) the difference between the return on a portfolio of small firm stocks and the return on a portfolio of large firm stocks (E(R_{SMB}), small minus big); and (c) the difference between the return on a portfolio of highbooktomarket stocks and the return on a portfolio of lowbooktomarket stocks (E(R_{HML}), high minus low). Specifically, the expected return on portfolio j is,
 (FF1)
where E(R_{M}) − r_{f}, E(R_{SMB}), and E(R_{HML}) are expected premiums, and the factor sensitivities or loadings, b_{j}, s_{j} and h_{j}, are the slopes in the timeseries regression,
 (FF2)
where α_{E} and ε_{E} represent, respectively, the intercept and error terms of the regression.
In seeking to establish their model as a strictly riskbased model, Fama and French argue that the size of the underlying firm and the ratio of the book value of equity to market value are ‘riskbased’ explanatory variables, with the former a proxy for the required return for bearing exposure to small stocks, and the latter a proxy for investors' required return for bearing ‘financial distress’, neither of which are captured in the market return (FF, 1995). They also claim that their model provides both a resolution of the CAPM (FF, 1996) and a resolution of prior attempts to generalize a riskbased model of stock prices:
At a minimum, the available evidence suggests that the threefactor model in (FF 1) and (FF 2) (see above), with intercepts in (FF 2) equal to 0.0, is a parsimonious description of returns and average returns. The model captures much of the variation in the crosssection of average returns, and it absorbs most of the anomalies that have plagued the CAPM. More aggressively, we argue in FF (1993, 1994, 1995) that the empirical successes of (FF 1) suggest that it is an equilibrium pricing model, a threefactor version of Merton's (1973) intertemporal CAPM (ICAPM) or Ross's (1976) arbitrage pricing theory (APT). In this view, R_{SMB} and R_{HML} mimic combinations of two underlying risk factors or state variables of special hedging concern to investors. (FF, 1996, p. 56)2
We nevertheless observe an inherent contradiction between, on the one hand, Fama and French's repeated denouncement of β, and on the other hand their inclusion of β as an explanatory variable in their model. In testing their model, FF (1996) form 25 (5 × 5) portfolios on ‘booktomarket value’ and ‘firm size’. Crucially, they do not form portfolios on β, with the outcome that the b_{j} coefficients of the 25 portfolios are all very close to 1.0 (none diverge by more than 10% as shown in Table 1 of FF, 1996). In effect, the Fama and French threefactor model has made redundant β as an explanatory variable, which makes sense given their studies confirming that beta has little or no explanatory power. But thereby we have a disconnect between the FF threefactor model and the CAPM: whereas the CAPM states that all assets have a return equal to the riskfree rate ‘as a base’ plus a market riskpremium multiplied by the asset's exposure to the market, the FF threefactor model states that all stocks have the market return ‘as a base’ plus or minus an element that depends on the stock's sensitivity to the differential performances of high and low bookvaluetomarketequity stocks and big and small firm size stocks. The FF model might equally (and more parsimoniously) be expressed as a ‘two’ rather than a ‘three’ factor model:
But to express it thusly would be to concede that investor rationality, as captured by the CAPM, is now abandoned, whereas by allowing the loading b_{j} coefficients on the excess market return [E(R_{M}) − r_{f}] to remain in the model, a formal continuity with the CAPM and the illusion that the threefactor model can be viewed as a refinement of the CAPM is maintained.
The Fama and French model states that U.S. institutional and retail investors (a) care about market risk but (b) do not appear to care about how such risk might be magnified or diminished in particular assets as captured by their beta (thereby contradicting the CAPM), while (c) simultaneously appearing to care about the booktomarket equity ratio and the firm size of their stocks. But if sensitivity to market risk as captured by beta does not motivate investors, it is, on the face of it, difficult to envisage how the booktomarket equity and firm size variables can be expected to motivate them. Lakonishok et al. (1994) argue that the Fama and French risk premiums are not risk premiums at all, but rather the outcome of mispricing. They argue that investors consistently underestimate future growth rates for ‘value’ stocks (captured as high markettobook equity value), and therefore underprice them. This results in value stocks outperforming growth stocks. Dichev (1998) and Campbell et al. (2008) also provide evidence against the Fama and French premiums as proxies for risk premiums by showing that the risk of bankruptcy is negatively rather than positively related to expected returns. If the Fama and French booktomarket premium proxies for distress risk, it should be the case that distressed firms have high booktomarket values, which they find not to be the case.3 From another perspective, Daniel and Titman (1997) provide evidence against the premiums as risk premiums by finding that the return performances of the Fama and French portfolios do not relate to covariances with the risk premiums as the Fama and French model dictates, but, rather, relate directly to the booktomarket and size of the firm as attributable ‘characteristics’ of the stock.
4 Beyond the Fama and French ThreeFactor Model
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 Abstract

 1 Background
 2 Expectations Models in Finance
 3 The Fama and French Model
 4 Beyond the Fama and French ThreeFactor Model
 5 Conclusion
 References
The riskreturn rationality of the CAPM and Fama and French threefactor models has stimulated a very substantial volume of asset pricing literature aimed at testing the models and recording an ‘anomaly’ when a new variable adds to the description of the crosssection of ex post stock returns. The new variable may then be incorporated in an extended FF threefactor model. For example, the ‘value’ and ‘size’ effects in the FF threefactor model have been augmented by a stock return momentum effect which is now viewed as a standard variable in asset pricing models (the Carhart, 1997, model). The model captures the observation that stocks that have recently performed well are likely to continue such performance for a period. Since Jegadeesh and Titman (1993) demonstrated a momentum effect based on three to 12 months of past returns, the effect and its relation to other variables has spurred considerable research effort. Grinblatt and Moskowitz (2004) explore the effect in terms of a dependence on whether the returns are achieved discretely or more or less continuously, while Hong et al. (2000) relate the momentum effect negatively to firm size and analyst coverage. Chordia and Shivakumar (2002) argue that momentum profits in the U.S. can be explained by business cycles; which finding is elaborated by Griffin et al. (2003) and Rouwenhorst (1998), who report evidence of momentum internationally; while Heston and Sadka (2008) report how winner stocks continue to outperform the same loser stocks in subsequent months.
Although it is possible to conjecture how momentum may come about as an outcome of a stock's attractiveness continuing to build on its recent performance, it is difficult to justify stock return momentum (which, in effect, offers a level of predictability for a stock's price movement) as an inherent risk factor. The challenge has recently been recognized by Fama and French (2008), who indicate that mispricing may need to be incorporated in asset pricing explanations (with the momentum effect allowed to differentiate across firm size).4
A good deal of research has also been aimed at replacing the Fama and French ‘high booktoequityvalue’ and ‘small firm size’ explanatory variables with economic variables that appear to relate more naturally to investors' concerns. As examples of the work in this area, Petkova (2006) shows that a factor model that incorporates the term and credit spreads of bonds makes redundant the Fama and French (1993) risk proxies for the Fama and French 25 portfolios sorted by size and booktomarket. Also working with the Fama and French 25 portfolios, Brennan et al. (2004) report that the real interest rate along with the Sharpe ratio (of market excess return to market standard deviation) describe the expected returns of assets in equilibrium. Again with reference to the Fama and French portfolios, Da (2009) reports that the expected return of an asset is the outcome of the asset's covariance with aggregate consumption and the time pattern of market cash flows; and Campbell and Vuolteenaho (2004) also argue for focusing on an asset's covariance with market cash flows as the important risk factor. And Jagannathan and Wang (1996) argue that a conditional CAPM where betas are allowed to vary with the business cycle works well when returns to labour income are included in the total return on the market portfolio (which is supported by Santos and Veronesi, 2006, who show that the labour income to consumption ratio is a useful descriptor of expected returns). It comes, of course, as no surprise that aspects of the economy relate to stock price formation. Nor is it a surprise that the relations are evident as covariances in the data of stock price returns. This is in fact what we expect (as clarified in Section 'The Fama and French Model'). Such observations need ‘not cause a ripple’ (Cochrane, 2005, p. 453).
Stock returns have also been related to ‘microfinance'—the institutional mechanics of trading equities. Thus, Amihud and Mendelson (1986) relate asset returns to stock liquidity, measured for example by the quoted bid–ask spread. Liquidity is promoted as an explanatory variable in understanding asset returns by Chordia et al. (2002, 2008) and Chordia et al. (2005). More recently, studies have begun to identify crosssectional predictability with frictions due to the cognitive limitations of investors (e.g., Cohen and Frazzini, 2008; Chordia et al., 2009).5 But again, it is difficult to see how such variables might be interpreted as proxies for ‘risk’ factors. Fama and French (2008) have reported accruals, stock issues and momentum as robustly associated with the crosssection of returns, while Cooper et al. (2008) argue that growth in assets predicts returns. Haugen and Baker (1996) consider past returns, trading volume, and accounting ratios such as return on equity and price/earnings as the strongest determinants of expected returns, and go so far as to report that they find no evidence that risk measures (such as systematic or total volatility) are influential in the crosssection of equity returns.
As we stress, the integrity and rationality of markets in a CAPM sense is founded not on covariances of market returns with economic or psychological considerations, or with market institutional (liquidity attributes) considerations, but on their ability to monitor and price risk. Indeed, it is now the convention for models not to make the claim to be ‘asset pricing’ models in the riskreturn sense, but rather to be ‘factor’ models. The identification of the correlation of a variable with asset returns is then presented as either an ‘anomaly’ or as the demonstration that the variable is ‘priced’ by the markets.6 This is what Black meant by saying that the exercise amounts to data mining.
Even if we have failed to identify and quantify the essential riskreturn relationship of the markets, we can at least claim that we have acquired a fairly detailed description of correlations in asset pricing over a sustained period of stock market history. Yet, interesting though the findings undoubtedly are, the findings can be questioned as satisfactorily generalizing the functioning of markets. With regard to value stocks—which constitute the dominant factor in the Fama and French model—Malkiel (2004) observes:
While there appear to be long periods when one style seems to outperform the other, the actual investment results over a more than 65year period are little different for value and growth mutual funds. Interestingly, the late 1960s through the early 1990s, the period Fama and French use to document their empirical findings may have been one of the unique long periods when value stocks outperformed growth stocks. (p. 132)
With reference to actual funds of small firm capitalization, Malkiel (2004) also observes that periods of small firm outperformance are followed by periods of underperformance. On the whole, he finds no consistency of performance that points to a dependable strategy of earning excess returns above the market, quite independent of any risk consideration. Reflecting Malkiel's observation, Cochrane (2005) also recognizes caution in making definite conclusions due to the difficulty of measuring average returns with statistical meaningfulness.
5 Conclusion
 Top of page
 Abstract

 1 Background
 2 Expectations Models in Finance
 3 The Fama and French Model
 4 Beyond the Fama and French ThreeFactor Model
 5 Conclusion
 References
The capital asset pricing model (CAPM) captures the idea that markets are essentially rational and are an appropriate subject for scientific inquiry. Unfortunately, the facts do not support the CAPM. The additional variables brought in to describe the distribution of asset returns generally resist interpretation as contributing to a risk–return relation. For this reason, we cannot interpret more recent models as refinements of a fundamentally robust risk–return relation. Rather, they represent a radical departure from the essential risk–return premise of the CAPM. Nevertheless, the impression is often given that the CAPM model of rational markets has simply paved the way for more sophisticated models. This is unfortunate. A good deal of finance is now an econometric exercise in mining data either for confirmation of a particular factor model or for the confirmation of deviations from a model's predictions as anomalies. The accumulation of explanatory variables advanced to explain the crosssection of asset returns has been accelerating, albeit with little overall understanding of the correlation structure between them. We might consider that the published papers exist ‘on the periphery of asset pricing’. They show very little attempt to formulate a robust riskreturn relationship that differentiates across assets.
We might query why academic finance should be given to such a colossal commitment to data mining. In a review of the U.S. CRSP (Center for Research in Security Prices) data base, The Economist magazine (20–26 November 2010) observes that a reason for the high level of ‘data mining’ is the opportunity that the CRSP database offers financial economists (it estimates that more than onethird of published papers in finance represent econometric studies of the data base). Robert Shiller in the same article in The Economist is quoted as saying that with the creation of the CRSP data base, economists have been led to believe that finance has become scientific, and that conventional ideas about investing and financial markets—and about their vulnerabilities—are out of date. He adds that to have seen the financial crisis coming, it would have been better to ‘go back to oldfashioned readings of history, studying institutions and laws. We should have talked to grandpa.’
Without the CAPM, we are left with a market where stock prices generally respond positively to good news and negatively to bad news, with market sentiment and crowd psychology playing a role that is never easy to determine, but which at times appears to produce tipping points, sending the market to booms and busts. Which is how markets were understood prior to the CAPM. In a nonCAPM world, the practitioner needs to understand how markets function in disequilibrium, as well as in equilibrium, with the caveat that history never repeats itself exactly. As market trends consolidate, we are naturally seduced into considering that they represent ‘the way the market works’. But a market trend can prove a fickle friend. We venture that it is in this sense that markets are ultimately risky.
The implications of not having a ‘scientific’ model of share prices are considerable. Derivation of the appropriate discount factor for valuation of cash flows requires such a model. Without a rationalized discount factor, attempts to value a firm, its projects, or impose fair prices for regulated industries, or to set realistic benchmarks for fund managers and for managers seeking bonuses, will have even more the appearance of ‘guesstimating’. For academics, an inexact science becomes even more inexact. For professionals, the image of professional expertise in controlling risk may be compromised. Ultimately, however, we must seek to understand markets on their own terms and not on our own.