Nonlinear Factor Returns in the US Equity Market

Abstract

We examine nonlinear return-to-characteristic relationships for five equity market factors: value, momentum, small size, low beta, and profitability. Our study employs monthly returns and characteristics for the largest one thousand US stocks from 1964 to 2023 with a focus on average active returns over the last 20 years. Beyond simplicity in modeling the return-generating process, we find no reason to assume a linear relationship between characteristics and security returns. Allowance for nonlinearity leads to increases in information ratios for factor portfolios neutralized with respect to nonlinear exposure to the other factors.

Keywords:

• equity market factors
• low beta
• momentum
• nonlinear return analysis
• profitability
• quantitative portfolio management
• small size
• value factor

PL Credits: 2.0:

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 The factor portfolios specified by the regression coefficients in Equation (1) have a one–standard deviation exposure to the factor of interest. The portfolios are commonly known as 120/20 long-short where the amount of shorted security capitalization varies from about 10 to 30 percent.

2 Panel regressions that simultaneously include all 20*12*1,000 = 240,000 observations using market capitalization weights have similar results for average active returns with larger t statistics. However, the estimated coefficients do not equate to identifiable portfolios and the panel regressions do not provide a time-series active risk parameter.

3 The security return model $r_i={\alpha }_i+{\beta }_i{r}_M$ and portfolio active weights of $w_{\mathit{\text{Mi}}}{s}_i$ give the portfolio’s alpha as ${\alpha }_P=r_P-{\beta }_P{r}_M$ where ${\beta }_P=({\sum }_{i=1}^N{w}_{\mathit{\text{Mi}}}{s}_i{\beta }_i)-1$ is the portfolio’s active beta.

4 We use four rather than two standard errors (i.e., a t statistic of 2.0) to emphasize larger correlation coefficients because there are many separately calculated numbers: 10 in Table 2 and 105 in Table 4.

5 As in Figure 3, the nonlinear fitted return curves from the monthly weighted Fama–Macbeth regressions do not cross at zero due to squared characteristic scores that are adjusted to have a mean of zero. In Figures 5 and 6, the fitted return at zero is subtracted along the entire score spectrum each month so the average active return and risk are exactly zero at the origin.

6 The weighted cross-sectional characteristic distributions vary by factor and month but have approximately normal distributions, with total capitalization of about 15 percent in the tail categories and 35 percent in the middle categories. The pure earnings yield and pure momentum characteristics tend to be thin-tailed compared to normal, with about 10 percent and 40 percent in the tail versus middle categories, respectively.

7 The direct formula for R-squared in observationally weighed cross-sectional regressions is $1-{\sum }_{i=1}^N{w}_i{e}_i^2/{\sum }_{i=1}^N{w}_i{r}_i^2$ where $e_i$ are the residual returns and $w_i$ are the weights. Adjusted R-squared is often employed in multivariate regressions to account for the number of independent variables. As a practical matter, the change in degrees of freedom using 15 rather than 5 regressors has little impact because of the large sample size, N = 1,000, an Effective N due to observational weighting of about 100 to 200 a month.

8 LOESS allows for a nonpolynomial shape in the regression curves that replace the $c_k$ in Equation (1). LOESS is computationally difficult to implement in our weighted-observation large-sample context and less known in financial markets research.

Additional information

Notes on contributors

Roger Clarke

Roger Clarke, PhD, is the former President of Ensign Peak Advisors, Salt Lake City.

Harindra de Silva

Harindra de Silva, PhD, CFA, is a Portfolio Manager at Allspring Global Investments, Los Angeles.

Steven Thorley

Steven Thorley, PhD, CFA, is an Emeritus Professor of Finance at the Marriott School of Management.