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	economics investment Stock Strategies Part of the the investment sequence Fama-French Three-Factor Model Any discussion of stock strategies should probably start with Eugene Fama - the "father of modern finance" Eugene Fama and winner of the Nobel prize for his "empirical analysis of asset prices" The Prize in Economic Sciences 2013. His most famous contribution is proposing the three levels of market efficiency, but this is closely followed by his extension of the single factor model based on the capital asset pricing model to include two additional observations: small firms and firms with high book-to-market ratios tend to outperform the market even after controlling for market risk ($\beta$) Fama–French three-factor model. . He actually found results much more impressive than just this, namely The Cross-Section of Expected Stock Returns: Though, to give credit where its do, its worth pointing out he acknowledged these individual insights were found by other people The Cross-Section of Expected Stock Returns. After controlling for the book-to-market ratio, leverage no longer correlates with higher return After controlling for market capitalization and the book-to-market ratio, the E/P ratio no longer correlates with higher returns. After controlling for market capitalization, $\beta$ no longer correlates with higher returns. In other words, they find the CAPM model to be completely wrong! This also explains why I found virtually no correlation between $\beta$ and returns when I looked at 100 stocks between 1984 and 2017 Marjanovic. You may think this disproves the CAPM, and I think Fama would agree with you. You may think this disproves the efficient market hypothesis - and there Fama does not: he's pretty adamant that all a paper can do is prove that either the market is not efficient or that the pricing model is wrong, but that it's impossible to prove which is the case. For the skeptical, Fama cites an alternative reason small firms might yield better returns in an efficient market Chan Is there any criticism of the study itself? I haven't seen any besides the obvious: just because the strategy worked in the past (1963 - 1990 in this case) doesn't mean it works now. If this is true, there are two possible reasons: (a) the market itself changed, presumably getting more efficient over time. and (b) the relationship is statistical luck. There is some indication of (a): $\beta$ did predict returns between 1941 and 1965 - but this still goes away after controlling for firm size, suggesting what changed was the relationship between $\beta$ and firm size, not either of them and returns. I decided to do some more recent validation using Vanguard ETFs from the early 200s until today. I found the VSMAX and VXF (indices that tracked small firms) outperformed the wider market by about 0.8% annually, though the t-score was only 0.5. I wanted to make this as idiot-proof and model-agnostic as possible, so here's what I did: I used the VTI, which seeks to track the overall equity market across firm sizes, growth, and value styles. For each index of interest, I found their monthly returns as far back as possible, but no further back than 2001 (when VTI was started). I used the "adjusted close" prices on Yahoo Finance to account for dividend payouts. For each month, I computed the logarithm of the return of the index of interest and the logarithm of the return of VTI and subtracted the two. I report the average of these differences and implied standard error (standard deviation divided by the square root of the number of months). The purpose here is to dramatically reduce the noise, which allows for tighter confidence intervals relative to a truly naive sample mean. Specifically, in my experience, the confidence intervals are roughly half the size (a quarter the variance). I also did a standard estimation of $\alpha$ and $\beta$ and estimated the return by adding $\alpha$ to $\beta$ times the average market return. I got similar results with similar confidence intervals. In my opinion (b) is only plausible for the market capitalization parameter since its p-values ranged from 0.05 to 0.00065. Once you consider all the plausible other parameters that might predict stock trends, this seems plausibly due to chance. The book-to-market ratio's p-values, however, ranged from roughly 1-in-100,000 to roughly 1-in-a-billion, which seems really implausible to be by chance alone. Other Factor Models Carhart four-factor model - the Fama-French Three-Factor Model with an additional momentum factor Carhart four-factor model Fama-French Five Factor Model - the Fama-French Three-Factor Model with two additional factors (robustness of operating profit and investment aggressiveness) Fama–French three-factor model Momentum todo Wikipedia contributors. (2021, July 14). Carhart four-factor model. In Wikipedia, The Free Encyclopedia. Retrieved 21:10, December 1, 2021, from https://en.wikipedia.org/w/index.php?title=Carhart_four-factor_model&oldid=1033568498 Wikipedia contributors. (2021, October 16). Fama–French three-factor model. In Wikipedia, The Free Encyclopedia. Retrieved 10:08, December 1, 2021, from https://en.wikipedia.org/w/index.php?title=Fama%E2%80%93French_three-factor_model&oldid=1050249685 Chan, K. C., & Chen, N. F. (1991). Structural and return characteristics of small and large firms. The journal of finance, 46(4), 1467-1484. https://doi.org/10.1111/j.1540-6261.1991.tb04626.x Wikipedia contributors. (2021, September 14). Eugene Fama. In Wikipedia, The Free Encyclopedia. Retrieved 09:53, December 1, 2021, from https://en.wikipedia.org/w/index.php?title=Eugene_Fama&oldid=1044262637 The Prize in Economic Sciences 2013. (2013). Royal Swedish Academy of Sciences. https://www.nobelprize.org/prizes/economic-sciences/2013/press-release/ Fama, Eugene F.; French, Kenneth R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics. 33 (1): 3–56. https://doi.org/10.1016/0304-405X(93)90023-5. Fama, E. F., & French, K. R. (1992). The Cross-Section of Expected Stock Returns. The Journal of Finance, 47(2), 427–465. https://doi.org/10.1111/j.1540-6261.1992.tb04398.x Fama, E. F., & MacBeth, J. D. (1973). Risk, return, and equilibrium: Empirical tests. Journal of political economy, 81(3), 607-636. https://doi.org/10.1086/260061 Fama, E. F., & French, K. R. (2008). Dissecting anomalies. The Journal of Finance, 63(4), 1653-1678. https://doi.org/10.1111/j.1540-6261.2008.01371.x Lewellen, J. (2014). The cross section of expected stock returns. Forthcoming in Critical Finance Review, Tuck School of Business Working Paper, (2511246). https://dx.doi.org/10.2139/ssrn.2511246 Dreman, D. N., & Berry, M. A. (1995). Overreaction, underreaction, and the low-P/E effect. Financial Analysts Journal, 51(4), 21-30. https://doi.org/10.2469/faj.v51.n4.1917 Hirshleifer, D., & Shumway, T. (2003). Good day sunshine: Stock returns and the weather. The Journal of Finance, 58(3), 1009-1032. https://doi.org/10.1111/1540-6261.00556 Crain, M. A. (2011). A literature review of the size effect. Available at SSRN 1710076. https://dx.doi.org/10.2139/ssrn.1710076 Galariotis, E. C. (2014). Contrarian and momentum trading: a review of the literature. Review of Behavioral Finance. https://doi.org/10.1108/RBF-12-2013-0043 Zaremba, A. (2019). The cross section of country equity returns: A review of empirical literature. Journal of Risk and Financial Management, 12(4), 165. https://doi.org/10.3390/jrfm12040165 Woo, K. Y., Mai, C., McAleer, M., & Wong, W. K. (2020). Review on efficiency and anomalies in stock markets. Economies, 8(1), 20. https://doi.org/10.3390/economies8010020 Alhenawi, Y. (2015). On the interaction between momentum effect and size effect. Review of Financial Economics, 26(1), 36-46. https://doi.org/10.1016/j.rfe.2015.03.005 Astakhov, A., Havranek, T., & Novak, J. (2017). Firm size and stock returns: A meta-analysis (No. 14/2017). IES Working Paper. https://www.econstor.eu/handle/10419/174207 Astakhov, A., Havranek, T., & Novak, J. (2019). Firm size and stock returns: A quantitative survey. Journal of Economic Surveys, 33(5), 1463-1492. https://doi.org/10.1111/joes.12335 Marjanovic. B. (2017). Huge Stock Market Dataset. https://www.kaggle.com/borismarjanovic/price-volume-data-for-all-us-stocks-etfs

economics investment

Stock Strategies

Fama-French Three-Factor Model

Any discussion of stock strategies should probably start with Eugene Fama - the "father of modern finance" Eugene Fama and winner of the Nobel prize for his "empirical analysis of asset prices" The Prize in Economic Sciences 2013.

His most famous contribution is proposing the three levels of market efficiency, but this is closely followed by his extension of the single factor model based on the capital asset pricing model to include two additional observations: small firms and firms with high book-to-market ratios tend to outperform the market even after controlling for market risk ($\beta$) Fama–French three-factor model. . He actually found results much more impressive than just this, namely The Cross-Section of Expected Stock Returns:

After controlling for the book-to-market ratio, leverage no longer correlates with higher return
After controlling for market capitalization and the book-to-market ratio, the E/P ratio no longer correlates with higher returns.
After controlling for market capitalization, $\beta$ no longer correlates with higher returns. In other words, they find the CAPM model to be completely wrong!

You may think this disproves the CAPM, and I think Fama would agree with you. You may think this disproves the efficient market hypothesis - and there Fama does not: he's pretty adamant that all a paper can do is prove that either the market is not efficient or that the pricing model is wrong, but that it's impossible to prove which is the case. For the skeptical, Fama cites an alternative reason small firms might yield better returns in an efficient market Chan

Is there any criticism of the study itself? I haven't seen any besides the obvious: just because the strategy worked in the past (1963 - 1990 in this case) doesn't mean it works now. If this is true, there are two possible reasons: (a) the market itself changed, presumably getting more efficient over time. and (b) the relationship is statistical luck.

There is some indication of (a): $\beta$ did predict returns between 1941 and 1965 - but this still goes away after controlling for firm size, suggesting what changed was the relationship between $\beta$ and firm size, not either of them and returns.

I decided to do some more recent validation using Vanguard ETFs from the early 200s until today. I found the VSMAX and VXF (indices that tracked small firms) outperformed the wider market by about 0.8% annually, though the t-score was only 0.5.

I wanted to make this as idiot-proof and model-agnostic as possible, so here's what I did:

I used the VTI, which seeks to track the overall equity market across firm sizes, growth, and value styles.
For each index of interest, I found their monthly returns as far back as possible, but no further back than 2001 (when VTI was started). I used the "adjusted close" prices on Yahoo Finance to account for dividend payouts.
For each month, I computed the logarithm of the return of the index of interest and the logarithm of the return of VTI and subtracted the two.
I report the average of these differences and implied standard error (standard deviation divided by the square root of the number of months).

The purpose here is to dramatically reduce the noise, which allows for tighter confidence intervals relative to a truly naive sample mean. Specifically, in my experience, the confidence intervals are roughly half the size (a quarter the variance).

I also did a standard estimation of $\alpha$ and $\beta$ and estimated the return by adding $\alpha$ to $\beta$ times the average market return. I got similar results with similar confidence intervals.

In my opinion (b) is only plausible for the market capitalization parameter since its p-values ranged from 0.05 to 0.00065. Once you consider all the plausible other parameters that might predict stock trends, this seems plausibly due to chance. The book-to-market ratio's p-values, however, ranged from roughly 1-in-100,000 to roughly 1-in-a-billion, which seems really implausible to be by chance alone.

Other Factor Models

Carhart four-factor model - the Fama-French Three-Factor Model with an additional momentum factor Carhart four-factor model
Fama-French Five Factor Model - the Fama-French Three-Factor Model with two additional factors (robustness of operating profit and investment aggressiveness) Fama–French three-factor model

Momentum

todo

Wikipedia contributors. (2021, July 14). Carhart four-factor model. In Wikipedia, The Free Encyclopedia. Retrieved 21:10, December 1, 2021, from https://en.wikipedia.org/w/index.php?title=Carhart_four-factor_model&oldid=1033568498 Wikipedia contributors. (2021, October 16). Fama–French three-factor model. In Wikipedia, The Free Encyclopedia. Retrieved 10:08, December 1, 2021, from https://en.wikipedia.org/w/index.php?title=Fama%E2%80%93French_three-factor_model&oldid=1050249685 Chan, K. C., & Chen, N. F. (1991). Structural and return characteristics of small and large firms. The journal of finance, 46(4), 1467-1484. https://doi.org/10.1111/j.1540-6261.1991.tb04626.x Wikipedia contributors. (2021, September 14). Eugene Fama. In Wikipedia, The Free Encyclopedia. Retrieved 09:53, December 1, 2021, from https://en.wikipedia.org/w/index.php?title=Eugene_Fama&oldid=1044262637 The Prize in Economic Sciences 2013. (2013). Royal Swedish Academy of Sciences. https://www.nobelprize.org/prizes/economic-sciences/2013/press-release/ Fama, Eugene F.; French, Kenneth R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics. 33 (1): 3–56. https://doi.org/10.1016/0304-405X(93)90023-5. Fama, E. F., & French, K. R. (1992). The Cross-Section of Expected Stock Returns. The Journal of Finance, 47(2), 427–465. https://doi.org/10.1111/j.1540-6261.1992.tb04398.x Fama, E. F., & MacBeth, J. D. (1973). Risk, return, and equilibrium: Empirical tests. Journal of political economy, 81(3), 607-636. https://doi.org/10.1086/260061 Fama, E. F., & French, K. R. (2008). Dissecting anomalies. The Journal of Finance, 63(4), 1653-1678. https://doi.org/10.1111/j.1540-6261.2008.01371.x Lewellen, J. (2014). The cross section of expected stock returns. Forthcoming in Critical Finance Review, Tuck School of Business Working Paper, (2511246). https://dx.doi.org/10.2139/ssrn.2511246 Dreman, D. N., & Berry, M. A. (1995). Overreaction, underreaction, and the low-P/E effect. Financial Analysts Journal, 51(4), 21-30. https://doi.org/10.2469/faj.v51.n4.1917 Hirshleifer, D., & Shumway, T. (2003). Good day sunshine: Stock returns and the weather. The Journal of Finance, 58(3), 1009-1032. https://doi.org/10.1111/1540-6261.00556 Crain, M. A. (2011). A literature review of the size effect. Available at SSRN 1710076. https://dx.doi.org/10.2139/ssrn.1710076 Galariotis, E. C. (2014). Contrarian and momentum trading: a review of the literature. Review of Behavioral Finance. https://doi.org/10.1108/RBF-12-2013-0043 Zaremba, A. (2019). The cross section of country equity returns: A review of empirical literature. Journal of Risk and Financial Management, 12(4), 165. https://doi.org/10.3390/jrfm12040165 Woo, K. Y., Mai, C., McAleer, M., & Wong, W. K. (2020). Review on efficiency and anomalies in stock markets. Economies, 8(1), 20. https://doi.org/10.3390/economies8010020 Alhenawi, Y. (2015). On the interaction between momentum effect and size effect. Review of Financial Economics, 26(1), 36-46. https://doi.org/10.1016/j.rfe.2015.03.005 Astakhov, A., Havranek, T., & Novak, J. (2017). Firm size and stock returns: A meta-analysis (No. 14/2017). IES Working Paper. https://www.econstor.eu/handle/10419/174207 Astakhov, A., Havranek, T., & Novak, J. (2019). Firm size and stock returns: A quantitative survey. Journal of Economic Surveys, 33(5), 1463-1492. https://doi.org/10.1111/joes.12335 Marjanovic. B. (2017). Huge Stock Market Dataset. https://www.kaggle.com/borismarjanovic/price-volume-data-for-all-us-stocks-etfs