The MAX Effect and Firm Characteristics: Size, Value, and Momentum

We analyze the relationship between the MAX effect and other well-known firm characteristics and risk factors, such as size, value, and momentum.

The MAX effect, while a effective anomaly in its own right, does not exist in a vacuum. It is intertwined with other well-documented firm characteristics and risk factors that have been shown to explain a significant portion of the cross-section of stock returns. Understanding these relationships is important for isolating the true source of the MAX effect and for developing robust trading strategies. In this article, we will explore the interplay between the MAX effect and three of the most prominent factors: size, value, and momentum.

The MAX Effect and the Size Effect

The size effect, first documented by Banz (1981), refers to the empirical finding that small-cap stocks tend to outperform large-cap stocks over the long run. When we examine the characteristics of high MAX stocks, we find that they are predominantly small-cap stocks. This is not surprising, as smaller, less liquid stocks are more likely to experience the kind of extreme price movements that lead to high MAX values.

This overlap between the MAX effect and the size effect raises an important question: Is the MAX effect simply a manifestation of the size effect? The research by Bali, Cakici, and Whitelaw (2011) addresses this issue directly. They perform bivariate sorts, first sorting stocks by size and then by MAX, and find that the MAX effect persists within each size quintile. In other words, even among the largest stocks, those with the highest MAX values still tend to underperform those with the lowest MAX values. This suggests that the MAX effect is a distinct phenomenon and not merely a proxy for the size effect.

The MAX Effect and the Value Effect

The value effect, popularized by Fama and French (1992), is the tendency for stocks with low book-to-market ratios (growth stocks) to underperform stocks with high book-to-market ratios (value stocks). When we look at the relationship between the MAX effect and the value effect, the picture is less clear. Some studies find that high MAX stocks have a slight value tilt, while others find no significant relationship.

To disentangle the two effects, we can again turn to bivariate sorts. By sorting stocks first by their book-to-market ratio and then by MAX, we can examine the performance of the MAX effect within different value and growth portfolios. The evidence suggests that the MAX effect is present across all book-to-market quintiles, indicating that it is not subsumed by the value effect.

The MAX Effect and Momentum

The momentum effect, identified by Jegadeesh and Titman (1993), is the tendency for stocks that have performed well in the past to continue to perform well in the future, and for stocks that have performed poorly to continue to perform poorly. The relationship between the MAX effect and momentum is complex. On the one hand, high MAX stocks often have high returns in the portfolio formation month, which could be interpreted as a short-term momentum signal. On the other hand, the MAX effect is a negative predictor of future returns, which is the opposite of the momentum effect.

To control for the influence of momentum, we can use the Fama-French-Carhart four-factor model, which includes a momentum factor (MOM). The formula for the four-factor model is as follows:

R_i,t - R_f,t = α_i + β_i,mkt(R_m,t - R_f,t) + β_i,smb(SMB_t) + β_i,hml(HML_t) + β_i,mom(MOM_t) + ε_i,t

R_i,t - R_f,t = α_i + β_i,mkt(R_m,t - R_f,t) + β_i,smb(SMB_t) + β_i,hml(HML_t) + β_i,mom(MOM_t) + ε_i,t

Where:

• R_i,t is the return of stock i in month t.
• R_f,t is the risk-free rate in month t.
• α_i is the alpha of the stock.
• R_m,t is the return of the market in month t.
• SMB_t is the return of the size factor in month t.
• HML_t is the return of the value factor in month t.
• MOM_t is the return of the momentum factor in month t.
• β are the factor loadings.
• ε_i,t is the error term.

When we regress the returns of MAX-sorted portfolios on the four factors, we find that the high MAX portfolio has a large and statistically significant negative alpha. This indicates that the MAX effect is not explained by the size, value, or momentum factors.

Actionable Example: Controlling for Firm Characteristics

A quantitative analyst who wants to build a strategy based on the MAX effect must carefully control for other firm characteristics to ensure that the strategy is capturing the intended anomaly. Here is a practical approach:

1. Factor-Neutral Portfolio: Construct a portfolio that is neutral to the size, value, and momentum factors. This can be achieved by using a portfolio optimization process that constrains the portfolio's factor loadings to be close to zero.
2. Residual MAX: Calculate the residual MAX for each stock by regressing its MAX value on its size, book-to-market ratio, and momentum. The residual from this regression represents the portion of the MAX value that is not explained by these other factors.
3. Trading Strategy: Build a trading strategy based on the residual MAX. This could involve going long on stocks with the lowest residual MAX and short on stocks with the highest residual MAX.

By taking these steps, the analyst can be more confident that the strategy is truly exploiting the MAX effect and not just inadvertently loading up on other known risk factors.

In conclusion, while the MAX effect is related to other firm characteristics, it appears to be a distinct and robust anomaly. Its persistence after controlling for size, value, and momentum suggests that it is driven by a unique set of behavioral biases and market frictions that are not captured by traditional asset pricing models.

References

[1] Bali, T. G., Cakici, N., & Whitelaw, R. F. (2011). Maxing out: Stocks as lotteries and the cross-section of expected returns. Journal of Financial Economics, 99(2), 427-446.

[2] Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56.

[3] Carhart, M. M. (1997). On persistence in mutual fund performance. The Journal of Finance, 52(1), 57-82.