Market anomaly

A market anomaly in a financial market is predictability that seems to be inconsistent with (typically risk-based) theories of asset prices.[1] Standard theories include the capital asset pricing model and the Fama-French Three Factor Model, but a lack of agreement among academics about the proper theory leads many to refer to anomalies without a reference to a benchmark theory (Daniel and Hirschleifer 2015[2] and Barberis 2018,[3] for example). Indeed, many academics simply refer to anomalies as "return predictors", avoiding the problem of defining a benchmark theory.[4]

Academics have documented more than 150 return predictors (see List of Anomalies Documented in Academic Journals). These "anomalies", however, come with many caveats. Almost all documented anomalies focus on illiquid, small stocks.[4] Moreover, the studies do not account for trading costs. As a result, many anomalies do not offer profits, despite the presence of predictability.[5] Additionally, return predictability declines substantially after the publication of a predictor, and thus may not offer profits in the future.[4] Finally, return predictability may be due to cross-sectional or time-variation in risk, and thus does not necessarily provide a good investment opportunity. Relatedly, return predictability by itself does not disprove the efficient market hypothesis, as one needs to show predictability over and above that implied by a particular model of risk.[6]

The four primary explanations for market anomalies are (1) mispricing, (2) unmeasured risk, (3) limits to arbitrage, and (4) selection bias.[4] Academics have not reached a consensus on the underlying cause, with prominent academics continuing to advocate for selection bias,[7] mispricing,[3] and risk-based theories.[8]

Anomalies can be broadly categorized into time-series and cross-sectional anomalies. Time-series anomalies refer to predictability in the aggregate stock market, such as the often-discussed Cyclically Adjusted Price-Earnings (CAPE) predictor.[9] These time-series predictors indicate times in which it is better to be invested in stocks vs a safe asset (such as Treasury bills). Cross-sectional anomalies refer to the predictable out-performance of particular stocks relative to others. For example, the well-known size anomaly[10] refers to the fact that stocks with lower market capitalization tend to out-perform stocks with higher market capitalization in the future.

Explanations for anomalies

Mispricing

Many, if not most, of the papers which document anomalies attribute them to mispricing (Lakonishok, Shelifer, and Visny 1994,[11] for example). The mispricing explanation is natural, as anomalies are by definition deviations from a benchmark theory of asset prices. "Mispricing" is then defined as the deviation relative to the benchmark.

The most common benchmark is the CAPM (Capital-Asset-Pricing Model). The deviation from this theory is measured by a non-zero intercept in an estimated security market line. This intercept is commonly denoted by the Greek letter alpha:

where is the return on the anomaly, is the return on the risk-free rate, is the slope from regressing the anomaly's return on the market's return, and is the return on the "market", often proxied by the return on the CRSP index (an index of all publicly traded U.S. stocks).

The mispricing explanations are often contentious within academic finance, as academics do not agree on the proper benchmark theory (see Unmeasured Risk, below). This disagreement is closely related to the "joint-hypothesis problem" of the efficient market hypothesis.

Unmeasured risk
Main article: Risk factor (finance)

Among academics, a common response to claims of mispricing was the idea that the anomaly captures a dimension of risk that is missing from the benchmark theory. For example, the anomaly may generate expected returns beyond those measured using the CAPM regression because the time-series of its returns are correlated with labor income, which is not captured by standard proxies for the market return.[12]

Perhaps the most well-known example of this unmeasured risk explanation is found in Fama and French's seminar paper on their 3-factor model: "if assets are priced rationally, variables that are related to average returns ... ..., must proxy for sensitivity to common (shared and thus undiversifiable) risk factors in returns. The [3-factor model] time-series regressions give direct evidence on this issue."[13]

The unmeasured risk explanation is closely related to the shortcomings of the CAPM as a theory of risk as well as shortcomings of empirical tests of the CAPM and related models. Perhaps the most common critique of the CAPM is that it is derived in a single period setting, and thus is missing dynamic features like periods of high uncertainty. In a more general setting, the CAPM typically implies multiple risk factors, as shown in Merton's Intertemporal CAPM theory. Moreover, the ICAPM generally implies the expected returns vary over time, and thus time-series predictability is not clear evidence of mispricing. Indeed, since the CAPM cannot at all capture dynamic expected returns, evidence of time-series predictability is less often regarded as mispricing as compared to cross-sectional predictability.

Empirical shortcomings primarily regard the difficulty in measuring wealth or marginal utility. Theoretically, wealth includes not only stock market wealth, but also non-tradable wealth like private assets and future labor income. In the consumption CAPM, (which is theoretically equivalent to Merton's ICAPM), the proper proxy for wealth is consumption, which is difficult to measure (Savov 2011,[14] for example).

Despite the theoretical soundness of the unmeasured risk explanation, there is little consensus among academics about the proper risk model over and above the CAPM. Propositions include the well-known Fama-French 3-Factor Model, Fama-French-Carhart 4-factor model, Fama-French 5-factor model, and Stambaugh and Yuan's 4-factor model.[15][16][17] These models are all empirically-oriented, rather than derived from a formal theory of equilibrium like Merton's ICAPM.

Limits to arbitrage
Main article: Limits to arbitrage

Anomalies are almost always documented using closing prices from the CRSP dataset. These prices do not reflect trading costs, which can prevent arbitrage and thus the elimination predictability. Moreover, almost all anomalies are documented using equally-weighted portfolios,[4] and thus require trading of illiquid (costly-to-trade) stocks.

The limits to arbitrage explanation can be thought of as a refinement of the mispricing framework. A return pattern only offers profits if the returns it offers survives trading costs, and thus should not be considered mispricing unless trading costs are accounted for.

A large literature documents that trading costs greatly reduce anomaly returns. This literature goes back to Stoll and Whaley (1983) and Ball, Kothari, and Shanken (1995).[18][19] A recent paper that studies dozens of anomalies finds that trading costs have a massive effect on the average anomaly (Novy-Marx and Velikov 2015).[5]

Selection bias
Main article: Selection bias

The documented anomalies are likely the best performers from a much larger set of potential return predictors. This selection creates a bias and implies that estimates of the profitability of anomalies is overstated. This explanation for anomalies is also known as data snooping, p-hacking, data mining, and data dredging, and is closely related to the multiple comparisons problem. Concerns about selection bias in anomalies goes back at least to Jensen and Bennington (1970).[20]

Most research on selection bias in market anomalies focuses on particular subsets of predictors. For example, Sullivan, Timmermann, and White (2001) show that calendar-based anomalies are no longer significant after adjusting for selection bias.[21] A recent meta-analysis of the size premium shows that the reported estimates of the size premium are exaggerated twofold because of selection bias.[22]

Research on selection bias for anomalies more generally is relatively limited and inconclusive. McLean and Pontiff (2016) use an out-of-sample test to show that selection bias accounts for at most 26% of the typical anomaly's mean return during the sample period of the original publication. To show this, they replicate almost 100 anomalies, and show that the average anomaly's return is only 26% smaller in the few years immediately after the end of the original samples. As some of this decline may be due to investor learning effects, the 26% is an upper bound.[4] In contrast, Harvey, Liu, and Zhu (2016) adapt multiple testing adjustments from statistics such as the False Discovery Rate to asset pricing "factors". They refer to a factor as any variable that helps explain the cross-section of expected returns, and thus include many anomalies in their study. They find that multiple-testing statistics imply that factors with t-stats < 3.0 should not be considered statistically significant, and conclude that most published findings are likely false.[23]

List of anomalies documented in academic journals
DescriptionAuthor(s)YearJournalBroad Category
Change in capital investment, industry adjustedAbarbanell and Bushee1998The Accounting ReviewCross-Sectional
Gross Margin growth over sales growthAbarbanell and Bushee1998The Accounting ReviewCross-Sectional
Sales growth over inventory growthAbarbanell and Bushee1998The Accounting ReviewCross-Sectional
Sales growth over overhead growthAbarbanell and Bushee1998The Accounting ReviewCross-Sectional
Operating Cash flows to priceDesai, Rajgopal, and Benkatachalam2004The Accounting ReviewCross-Sectional
Earnings ForecastElgers, Lo, and Pfeiffer2001The Accounting ReviewCross-Sectional
Growth in Long term net operating assetsFairfield, Whisenant and Yohn2003The Accounting ReviewCross-Sectional
Earnings SurpriseFoster, Olsen and Shevliln1984The Accounting ReviewCross-Sectional
Percent Operating AccrualsHafzalla, Lundholm, and Van Winkle2011The Accounting ReviewCross-Sectional
Percent Total AccrualsHafzalla, Lundholm, and Van Winkle2011The Accounting ReviewCross-Sectional
Real dirty surplusLandsman et al.2011The Accounting ReviewCross-Sectional
Taxable income to incomeLev and Nissim2004The Accounting ReviewCross-Sectional
Piotroski F-scorePiotroski2000The Accounting ReviewCross-Sectional
AccrualsSloan1996The Accounting ReviewCross-Sectional
Asset TurnoverSoliman2008The Accounting ReviewCross-Sectional
Change in Asset TurnoverSoliman2008The Accounting ReviewCross-Sectional
Change in Noncurrent Operating AssetsSoliman2008The Accounting ReviewCross-Sectional
Change in Net Working CapitalSoliman2008The Accounting ReviewCross-Sectional
Change in Profit MarginSoliman2008The Accounting ReviewCross-Sectional
Profit MarginSoliman2008The Accounting ReviewCross-Sectional
Abnormal AccrualsXie2001The Accounting ReviewCross-Sectional
Earnings ConsistencyAlwathainani2009British Accounting ReviewCross-Sectional
Deferred RevenuePrakash and Sinha2012Contemporary Accounting ResearchCross-Sectional
Sales-to-priceBarbee, Mukherji, and Raines1996Financial Analysts' JournalCross-Sectional
earnings / assetsBalakrishnan, Bartov, and Faurel2010Journal of Accounting and EconomicsCross-Sectional
Net debt financingBradshaw, Richardson, and Sloan2006Journal of Accounting and EconomicsCross-Sectional
Net equity financingBradshaw, Richardson, and Sloan2006Journal of Accounting and EconomicsCross-Sectional
Net external financingBradshaw, Richardson, and Sloan2006Journal of Accounting and EconomicsCross-Sectional
Net Operating AssetsHirschleifer, Hou Teoh, and Zhang2004Journal of Accounting and EconomicsCross-Sectional
Change in depreciation to gross PPEHolthausen Larcker1992Journal of Accounting and EconomicsCross-Sectional
Change in equity to assetsRichardson, Sloan Soliman and Tuna2005Journal of Accounting and EconomicsCross-Sectional
Change in current operating assetsRichardson, Sloan Soliman and Tuna2005Journal of Accounting and EconomicsCross-Sectional
Change in current operating liabilitiesRichardson, Sloan Soliman and Tuna2005Journal of Accounting and EconomicsCross-Sectional
Change in financial liabilitiesRichardson, Sloan Soliman and Tuna2005Journal of Accounting and EconomicsCross-Sectional
Change in long-term investmentRichardson, Sloan Soliman and Tuna2005Journal of Accounting and EconomicsCross-Sectional
Enterprise component of BMPenman, Richardson, and Tuna2007Journal of Accounting ResearchCross-Sectional
Leverage component of BMPenman, Richardson, and Tuna2007Journal of Accounting ResearchCross-Sectional
Net debt to pricePenman, Richardson, and Tuna2007Journal of Accounting ResearchCross-Sectional
Change in TaxesThomas and Zhang2011Journal of Accounting ResearchCross-Sectional
IPO and no R&D spendingGou, Lev, and Shi2006Journal of Business, Finance and AccountingCross-Sectional
Change in capex (two years)Anderson and Garcia-Feijoo2006Journal of FinanceCross-Sectional
Idiosyncratic riskAng, Hodrick, Xing, and Zhang2006Journal of FinanceCross-Sectional
Junk Stock MomentumAvramov, Chordia, Jostova, and Philipov2007Journal of FinanceCross-Sectional
Maximum return over monthBali, Cakici, and Whitelaw2010Journal of FinanceCross-Sectional
Consensus RecommendationBarber, Lehavy, McNichols, and Trueman2001Journal of FinanceCross-Sectional
Down forecast EPSBarber, Lehavy, McNichols, and Trueman2001Journal of FinanceCross-Sectional
Up ForecastBarber, Lehavy, McNichols, and Trueman2001Journal of FinanceCross-Sectional
Earnings-to-Price RatioBasu1977Journal of FinanceCross-Sectional
PriceBlume and Husic1972Journal of FinanceCross-Sectional
Net Payout YieldBoudoukh, Michaely, Richardson, and Roberts2007Journal of FinanceCross-Sectional
Payout YieldBoudoukh, Michaely, Richardson, and Roberts2007Journal of FinanceCross-Sectional
Failure probabilityCampbell, Hilscher, and Szilagyi2008Journal of FinanceCross-Sectional
Earnings announcement returnChan, Jegadeesh, and Lakonishok1996Journal of FinanceCross-Sectional
Earnings forecast revisionsChan, Jegadeesh, and Lakonishok1996Journal of FinanceCross-Sectional
Advertising ExpenseChan, Lakonishok, and Sougiannis2001Journal of FinanceCross-Sectional
R&D over market capChan, Lakonishok, and Sougiannis2001Journal of FinanceCross-Sectional
Asset GrowthCooper, Gulen and Schill2008Journal of FinanceCross-Sectional
Intangible returnDaniel and Titman2006Journal of FinanceCross-Sectional
Share issuance (5 year)Daniel and Titman2006Journal of FinanceCross-Sectional
Momentum-ReversalDe Bondt and Thaler1985Journal of FinanceCross-Sectional
Long-run reversalDe Bondt and Thaler1985Journal of FinanceCross-Sectional
Exchange SwitchDharan Ikenberry1995Journal of FinanceCross-Sectional
Credit Rating DowngradeDichev Piotroski2001Journal of FinanceCross-Sectional
EPS Forecast DispersionDiether et al.2002Journal of FinanceCross-Sectional
Unexpected R&D increaseEberhart et al.2004Journal of FinanceCross-Sectional
Organizational CapitalEisfeldt and Papanikolaou2013Journal of FinanceCross-Sectional
Pension Funding StatusFranzoni and Martin2006Journal of FinanceCross-Sectional
52 week highGeorge and Hwang2004Journal of FinanceCross-Sectional
TangibilityHahn and Lee2009Journal of FinanceCross-Sectional
Industry concentration (Herfindahl)Hou and Robinson2006Journal of FinanceCross-Sectional
Momentum (12 month)Jegadeesh and Titman1993Journal of FinanceCross-Sectional
Momentum (6 month)Jegadeesh and Titman1993Journal of FinanceCross-Sectional
Change in recommendationJegadeesh et al.2004Journal of FinanceCross-Sectional
Short term reversalJegedeesh1989Journal of FinanceCross-Sectional
Long-term EPS forecastLa Porta1996Journal of FinanceCross-Sectional
Cash flow to marketLakonishok, Scheifer, and Vishny1994Journal of FinanceCross-Sectional
Revenue Growth RankLakonishok, Scheifer, and Vishny1994Journal of FinanceCross-Sectional
Momentum and VolumeLee Swaminathan2000Journal of FinanceCross-Sectional
Public Seasoned Equity OfferingsLoughran Ritter1995Journal of FinanceCross-Sectional
Dividend InitiationMichaely et al.1995Journal of FinanceCross-Sectional
Dividend OmissionMichaely et al.1995Journal of FinanceCross-Sectional
Institutional ownership interactions with anomaliesNagel2005Journal of FinanceCross-Sectional
Dividend YieldNaranjo et al.1998Journal of FinanceCross-Sectional
Share issuance (1 year)Pontiff and Woodgate2008Journal of FinanceCross-Sectional
Initial Public OfferingsRitter1991Journal of FinanceCross-Sectional
Firm Age - MomentumZhang2004Journal of FinanceCross-Sectional
Book to marketStattman1980The Chicago MBACross-Sectional
Bid-ask spreadAmihud and Mendelsohn1986Journal of Financial EconomicsCross-Sectional
Institutional Ownership for stocks with high short interestAsquith, Pathak, and Ritter2005Journal of Financial EconomicsCross-Sectional
Cash-based operating profitabilityBall, Gerakos, Linnainmaa, and Nikolaev2016Journal of Financial EconomicsCross-Sectional
SizeBanz1981Journal of Financial EconomicsCross-Sectional
Market leverageBhandari1988Journal of Financial EconomicsCross-Sectional
Past trading volumeBrennan, Chordia, and Subrahmanyam1998Journal of Financial EconomicsCross-Sectional
Breadth of ownershipChen Hong Stein2002Journal of Financial EconomicsCross-Sectional
Turnover volatilityChordia, Subrahmanyam, and Anshuman2001Journal of Financial EconomicsCross-Sectional
Volume VarianceChordia, Subrahmanyam, and Anshuman2001Journal of Financial EconomicsCross-Sectional
Conglomerate returnCohen and Lou2012Journal of Financial EconomicsCross-Sectional
SpinoffsCusatis et al.1993Journal of Financial EconomicsCross-Sectional
Short InterestDechow, Hutton, Meulbroek, and Sloan2001Journal of Financial EconomicsCross-Sectional
O ScoreDichev1998Journal of Financial EconomicsCross-Sectional
Altman Z-ScoreDichev1998Journal of Financial EconomicsCross-Sectional
operating profits / book equityFama and French2006Journal of Financial EconomicsCross-Sectional
Industry MomentumGrinblatt Moskowitz1999Journal of Financial EconomicsCross-Sectional
DividendsHartzmark Salomon2013Journal of Financial EconomicsCross-Sectional
net income / book equityHaugen and Baker1996Journal of Financial EconomicsCross-Sectional
Cash-flow varianceHaugen and Baker1996Journal of Financial EconomicsCross-Sectional
Volume to market equityHaugen and Baker1996Journal of Financial EconomicsCross-Sectional
Volume TrendHaugen and Baker1996Journal of Financial EconomicsCross-Sectional
Return SeasonalityHeston and Sadka2008Journal of Financial EconomicsCross-Sectional
Sin Stock (selection criteria)Hong Kacperczyk2009Journal of Financial EconomicsCross-Sectional
Share repurchasesIkenberry et al.1995Journal of Financial EconomicsCross-Sectional
Revenue SurpriseJegadeesh and Livnat2006Journal of Financial EconomicsCross-Sectional
Option Volume relative to recent averageJohnson So2012Journal of Financial EconomicsCross-Sectional
Option Volume to Stock VolumeJohnson So2012Journal of Financial EconomicsCross-Sectional
Days with zero tradesLiu2006Journal of Financial EconomicsCross-Sectional
Intermediate MomentumNovy-Marx2012Journal of Financial EconomicsCross-Sectional
gross profits / total assetsNovy-Marx2013Journal of Financial EconomicsCross-Sectional
Cash to assetsPalazzo2012Journal of Financial EconomicsCross-Sectional
Debt IssuanceSpiess Affleck-Graves1999Journal of Financial EconomicsCross-Sectional
Slope of smileYan2011Journal of Financial EconomicsCross-Sectional
Amihud's illiquidityAmihud2002Journal of Financial MarketsCross-Sectional
Share VolumeDatar, Naik, and Radcliffe1998Journal of Financial MarketsCross-Sectional
Enterprise MultipleLoughran and Wellman2011Journal of Financial and Quantitative AnalysisCross-Sectional
Efficient frontier indexNguyen Swanson2009Journal of Financial and Quantitative AnalysisCross-Sectional
InvestmentTitman, Wei, and Xie2004Journal of Financial and Quantitative AnalysisCross-Sectional
Convertible debt indicatorValta2016Journal of Financial and Quantitative AnalysisCross-Sectional
Volatility smirkXing Zhang Zhao2010Journal of Financial and Quantitative AnalysisCross-Sectional
Sustainable GrowthLockwood Prombutr2010Journal of Financial ResearchCross-Sectional
Momentum and LT ReversalChan and Kot2006Journal of Investment ManagementCross-Sectional
Employment growthBelo, Lin, and Bazdresch2014Journal of Political EconomyCross-Sectional
CAPM beta squaredFama and MacBeth1973Journal of Political EconomyCross-Sectional
Number of consecutive earnings increasesLoh Warachka2012Management ScienceCross-Sectional
Governance IndexGompers, Ishii and Metrick2003Quarterly Journal of EconomicsCross-Sectional
Change in Forecast and AccrualBarth and Hutton2004Review of Accounting StudiesCross-Sectional
Excluded ExpensesDoyle et al.2003Review of Accounting StudiesCross-Sectional
Mohanram G-scoreMohanram2005Review of Accounting StudiesCross-Sectional
Order backlogRajgopal, Shevlin and Venkatachalam2003Review of Accounting StudiesCross-Sectional
Inventory GrowthThomas and Zhang2002Review of Accounting StudiesCross-Sectional
Operating LeverageNovy-Marx2010Review of FinanceCross-Sectional
Decline in Analyst CoverageScherbina2008Review of FinanceCross-Sectional
Earnings surprise of big firmsHou2007Review of Financial StudiesCross-Sectional
Industry return of big firmsHou2007Review of Financial StudiesCross-Sectional
Price delayHou and Moskowitz2005Review of Financial StudiesCross-Sectional
Tail risk betaKelly and Jiang2014Review of Financial StudiesCross-Sectional
Kaplan Zingales indexLamont, Polk, and Saa-Requejo2001Review of Financial StudiesCross-Sectional
Growth in advertising expensesLou2014Review of Financial StudiesCross-Sectional
Composite debt issuanceLyandres, Sun and Zhang2008Review of Financial StudiesCross-Sectional
Real estate holdingsTuzel2010Review of Financial StudiesCross-Sectional
Book-to-market and accrualsBartov and Kim2004Review of Quantitative Finance and AccountingCross-Sectional
Weekend EffectSmirlock and Starks1986Journal of Financial EconomicsTime-Series
January EffectKeims1985Journal of Financial EconomicsTime-Series
Turn of the Month EffectAgrawal and Tandon1994Journal of International Money and FinanceTime-Series

References
  1. Schwert, G. William (2003). "Anomalies and Market Efficiency" (PDF). Handbook of Economics and Finance. doi:10.1016/S1574-0102(03)01024-0.
  2. Kent, Daniel; Hirshleifer, David (Fall 2015). "Overconfident Investors, Predictable Returns, and Excessive Trading". Journal of Economic Perspectives.
  3. Barberis, Nicholas (2018). "Psychology-based Models of Asset Prices and Trading Volume" (PDF). NBER Working Paper. WIDER Working Paper. 2018. doi:10.35188/UNU-WIDER/2018/444-5. ISBN 978-92-9256-444-5.
  4. McLean, David; Pontiff, Jeffrey (February 2016). "Does Academic Research Destroy Return Predictability?". The Journal of Finance. 61 (1): 5. doi:10.1111/jofi.12365.
  5. Novy-Marx, Robert; Velikov, Mihail (2015). "A taxonomy of anomalies and their trading costs". The Review of Financial Studies.
  6. Fama, Eugene (1970). "Efficient Capital Markets: A Review of Theory and Empirical Work". Journal of Finance. 25 (2): 383–417. doi:10.2307/2325486. JSTOR 2325486.
  7. Harvey, Campbell R. (January 2016). "... and the Cross-Section of Expected Returns". The Review of Financial Studies. doi:10.1093/rfs/hhv059.
  8. Cochrane, John (2017). "Macro-Finance". Review of Finance. 21 (3): 945–985. doi:10.1093/rof/rfx010.
  9. Campbell, John Y. (July 1988). "Stock Prices, Earnings, and Expected Dividends" (PDF). The Journal of Finance. 43 (3): 661–676. doi:10.1111/j.1540-6261.1988.tb04598.x. JSTOR 2328190.
  10. Banz, Rolf W. (March 1981). "The relationship between return and market value of common stocks". Journal of Financial Economics. 9: 3–18. doi:10.1016/0304-405X(81)90018-0.
  11. Lakonishok, Josef; Shleifer, Andrei; Vishny, Robert W. (December 1994). "Contrarian Investment, Extrapolation, and Risk" (PDF). Journal of Finance. 49 (5): 1541–1578. doi:10.1111/j.1540-6261.1994.tb04772.x. S2CID 55404532.
  12. Jagannathan, Ravi; Wang, Zhenyu (March 1995). "The Conditional CAPM and the Cross‐Section of Expected Returns". Journal of Finance.
  13. Fama, Eugene; French, Kenneth (1993). "Common risk factors in the return on stocks and bonds". Journal of Financial Economics. 33: 3–56. doi:10.1016/0304-405X(93)90023-5.
  14. Savov, Alexi (2011). "Asset pricing with garbage". The Journal of Finance. 66: 177–201. doi:10.1111/j.1540-6261.2010.01629.x.
  15. Stambaugh, Robert; Yuan, Yu (2016). "Mispricing Factors". Review of Financial Studies.
  16. Carhart, Mark (1997). "On persistence in mutual fund performance". Journal of Finance. 52: 57–82. doi:10.1111/j.1540-6261.1997.tb03808.x.
  17. Fama, Eugene; French, Kenneth (2015). "A five-factor asset pricing model". Journal of Financial Economics. 116: 1–22. doi:10.1016/j.jfineco.2014.10.010.
  18. Ball, Ray; Kothari, S.P.; Shanken, Jay (1995). "Problems in measuring portfolio performance An application to contrarian investment strategies". Journal of Financial Economics. 38: 79–107. doi:10.1016/0304-405X(94)00806-C.
  19. Stoll, Hans; Whaley, Robert (1983). "Transaction costs and the small firm effect". Journal of Financial Economics. 12: 57–79. doi:10.1016/0304-405X(83)90027-2.
  20. Jensen, Michael; Bennington, George (1970). "Random walks and technical theories: Some additional evidence". Journal of Finance. 25 (2): 469–482. doi:10.1111/j.1540-6261.1970.tb00671.x.
  21. Sullivan, Ryan; Timmermann, Allan; White, Halbert (2001). "Dangers of data mining: The case of calendar effects in stock returns". Journal of Econometrics. 105: 249–286. doi:10.1016/S0304-4076(01)00077-X.
  22. Astakhov, Anton; Havranek, Tomas; Novak, Jiri (2019). "Firm Size and Stock Returns: A Quantitative Survey". Journal of Economic Surveys. 33 (5): 1463–1492. doi:10.1111/joes.12335. S2CID 201355673.
  23. Harvey, Campbell; Liu, Yan; Zhu, Heqing (2016). "... and the cross-section of expected returns". The Review of Financial Studies. doi:10.1093/rfs/hhv059.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.