They first look at the returns for each of these factors by putting them into quintiles and comparing returns across the 1971 - 2013 period in the US, excluding micro-cap companies which are found to be most sensitive to the factors in prior literature. They find all the factors to exhibit monotonic returns with changes in the quintiles in favor of the known factor relations.
They then split the quintiles into quintiles of the remaining factors into 5x5 portfolios to show the sensitivity of certain factors within the primary factors. They find that smaller companies tend to be more sensitive to size, profitability, and the investment factors, with momentum generally being unaffected by size (but this is consistent with prior studies that show momentum to increase only in the smallest of companies, which are excluded from this study). They then find that growth companies tend to be larger, more profitable, make higher investments in assets, and have higher momentum. Next, they find that less profitable companies tend to be smaller, categorized as value, make less investments, and to have no effect on momentum. Next, they find that companies who make smaller investments in assets seem to be smaller, be considered value, have lower profitability, and have no effect on momentum. Finally, high momentum companies tend to be characterized as growth and be irrespective of other factors.
Next the authors double-sorted the portfolios into 5x5 portfolios to see what are the returns of different levels of the factors in combinations. They also calculate a monotonic relationship to see which factors are likely to explain monotonic return increases with changes in those factors.
In relation to size, the authors find that the smallest companies are the only ones who show a monotonic increase in returns with increases to the value factor; none of the company sizes result in monotonic changes in returns with increases to the profitability factor; only the smallest companies show a monotonic increase in returns with increases to the investment factor; and all but one of the company size quintiles show a monotonic increase in returns with increases to the momentum factor.
In relation to value, the authors find that the top 3 quintiles of value show monotonic increases in returns with decreases in company size; none of the levels of book-to-price result in monotonic changes in returns with increases to the profitability factor, only companies in the top quintile of value show monotonic increases in returns with decreases in investment; and only the growth companies show monotonic increases in returns with increases in momentum.
In relation to profitability, there are monotonic relationships in several of the quintiles of the other factors; however, there does not seem to be a relationship where more or less profitable companies exhibit consistently higher or lower returns with increases or decreases in the other factors. In general, the returns of more or less profitable companies are not predictable in combination with other factors. However, companies with low profitability and high investment tend to perform the worst; while companies with low profitability and high momentum tend to perform the best. This contrast is evidence that level of profitability may not have an effect on returns when combining with other factors whose influence is more prominent.
In relation to investment, there are monotonic relationships with a few of the levels of company size and momentum; however, the changes in level of the other factors does not seem to have a monotonic relationship when controlling for level of investment. The highest return comes from high momentum, low investment companies as well as the low investment and small size companies; the smallest returns come from the high investment, low momentum companies.
In relation to momentum, the authors find the top 3 quintiles of momentum show monotonic increases in returns with decreases in company size; only the past losers tend to monotonically increase in return with increases to the value factor; and the changes in level of the other factors does not seem to have a monotonic relationship when controlling for the level of momentum. The highest returning portfolio seems to be the high momentum, with low investment, profitability, book-to-price ratios, or company size. The least profitable portfolio seems to be those with low momentum, combined with high investment or low book-to-price ratio.
Next, the authors pull the corner portfolios from the previously discussed 5x5 sort to explore the risk adjusted returns (i.e., sharpe ratios, volatility, SKAD) of long-only, short-only, and zero-cost portfolios with each factor combination. They find that the long-only portfolio outperforms the short-only portfolio in all cases except one (i.e., the ROA/BP portfolio). They find that the highest sharpe ratio of the zero-cost portfolio comes from the BP/MOM portfolio, which is in line with prior studies that find combining value companies with momentum companies creates good diversification effects and therefore good risk-adjusted returns. This portfolio also has the second-highest long-only return (i.e., after BP/ROA). Portfolios that include the investment factor also tend to be the ones with the highest sharpe ratios.
Finally, the authors form Fama-McBeth regressions with each of the factors to see how much of the portfolios' returns are explained by the factors. When considering only two factors at a time, they find that a 1 standard deviation increase in the book-to-price ratio and past 6-month return can produce a 4.1% increase in return to the portfolio; and a 1 standard deviation decrease in the past year percentage change in assets and increase in past 6-month returns can produce a 4.3% increase in return to the portfolio. When all factors are regressed at once, they find a 1 standard deviation decrease in the past year percentage change in assets and increase in past 6-month returns can produce a 4.2% increase in return to the portfolio; the value factor was not statistically significant in the all-factor regression. The authors suggest the likely reason the other factors are not prominent is because they are mostly emphasized in micro-cap companies which are excluded from this study. The investment and momentum factors are significant in all combinations with other factors.
Karell, V., & Yeomans, J. S. (2018). Anomaly Interactions and the Cross-Section of Stock Returns. Fuzzy Economic Review, 23(1), 33–61.
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