Low‐Risk Anomalies?

• Author: PAUL SCHNEIDER,CHRISTIAN WAGNER,JOSEF ZECHNER
• Date: 01 October 2020
• Published date: 01 October 2020
• DOI: http://doi.org/10.1111/jofi.12910

THE JOURNAL OF FINANCE •VOL. LXXV, NO. 5 •OCTOBER 2020

Low-Risk Anomalies?

PAUL SCHNEIDER, CHRISTIAN WAGNER, and JOSEF ZECHNER∗

ABSTRACT

This paper shows that low-risk anomalies in the capital asset pricing model and

in traditional factor models arise when investors require compensation for coskew-

ness risk. Empirically, we find that option-implied ex ante skewness is strongly

related to ex post residual coskewness, which allows us to construct coskewness

factor-mimicking portfolios. Controlling for skewness renders the alphas of betting-

against-beta and betting-against-volatility insignificant. We also show that the re-

turns of beta- and volatility-sorted portfolios are driven largely by a single principal

component, which in turn is explained largely by skewness.

∗Paul Schneider is at USI Lugano and SFI. Christian Wagner and Josef Zechner are at WU

Vienna University of Business and Economics. This paper received the 2015 Jack Treynor Prize

sponsored by the Q-Group (The Institute for Quantitative Research in Finance). We are grate-

ful to Kevin Aretz; Turan Bali; Nick Baltas; Michael Brennan; John Campbell; Mikhail Chernov;

Peter Christoffersen; Mathijs Cosemans; Andrea Gamba; Patrick Gagliardini; Christopher Hen-

nessy; Christopher Hrdlicka; Leonid Kogan; Loriano Mancini; Miriam Marra; Ian Martin; Yoshio

Nozawa; Lasse Pedersen; Paulo Rodrigues; Ivan Shaliastovich; Christian Schlag; Fabio Trojani;

Rossen Valkanov; Pietro Veronesi; Arne Westerkamp; Paul Whelan; Liuren Wu; and participants

at the American Finance Association Meetings 2017 (Chicago), the European Finance Association

Meetings 2016 (Oslo), the UBS Quantitative Investment Conference 2016 (London), the Spring

Seminar of the Q Group 2016 (Washington, D.C.), the Annual Conference on Advances in the

Analysis of Hedge Fund Strategies 2015 (London), the IFSID Conference on Derivatives 2015

(Montreal), the SAFE Asset Pricing Workshop 2015 (Frankfurt); and seminar participants at Cass

Business School, Dauphine, Copenhagen Business School, Hong Kong University of Science and

Technology, Imperial College, Singapore Management University,Stockholm School of Economics,

University of Geneva, University of Toronto (Rotman), Warwick Business School, and WU Vi-

enna for helpful comments. Paul Schneider acknowledges support from the Swiss National Science

Foundation grant “Model-Free Asset Pricing.” Christian Wagner acknowledges support from the

Center for Financial Frictions (FRIC), grant no. DNRF102. We are especially indebted to Stefan

Nagel and Kenneth Singleton (the Editors), two anonymous referees, and an anonymous associate

editor for their extensive comments that have greatly helped to improve the paper. The authors

alone are responsible for any errors and for the views expressed in the paper. We have read The

Journal of Finance disclosure policy and have no conflicts of interest to disclose.

Correspondence: Josef Zechner,Department of Finance, Accounting and Statistics, Institute for

Finance, Banking and Insurance, Vienna University of Economics and Business, Welthandelsplatz

1, A-1020 Vienna, Austria; e-mail: josef.zechner@wu.ac.at.

This is an open access article under the terms of the Creative Commons Attribution-Non

Commercial-NoDerivs License, which permits use and distribution in any medium, provided the

original work is properly cited, the use is non-commercial and no modifications or adaptations are

made.

DOI: 10.1111/jofi.12910

© 2020 The Authors. The Journal of Finance published by Wiley Periodicals LLC on behalf of

American Finance Association

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2674 The Journal of Finance®

EMPIRICAL FINDINGS THAT LOW-BETA stocks outperform high-beta stocks

and that (idiosyncratic) volatility negatively predicts equity returns have

spurred a large literature on “low-risk anomalies” (LRAs; e.g. Haugen and

Heins (1975), Ang et al. (2006), Baker, Bradley, and Wurgler (2011), Frazzini

and Pedersen (2014)). In this paper, we show that the returns to trading such

LRAs can be explained by the skewness of equity returns, which is ignored by

standard measures of market and idiosyncratic risk.

Our theoretical analysis starts with a stochastic discount factor (SDF) that

is quadratic in the market excess return (e.g., Kraus and Litzenberger (1976),

Harvey and Siddique (2000)). This SDF implies that investors demand com-

pensation for covariance risk as in the capital asset pricing model (CAPM),

and that investors accept lower (demand higher) expected returns on assets

with positive (negative) coskewness, as expressed by the covariance between

the asset and squared market excess returns. If excess returns are determined

in this way, then some trading strategies will generate alphas and thus appear

as anomalies in the CAPM. Such alphas, however, merely reflect compensation

for coskewness risk ignored by the CAPM. We show that these alphas are in-

deed driven by the correlation between CAPM residual returns and squared

market returns.

Next, we demonstrate the link between coskewness and LRA alphas in

a Merton-type model of firm risk, where (co)skewness arises endogenously

from leverage and stochastic asset volatility. In this model, we simulate three

worlds. First, we simulate a world with a standard CAPM pricing kernel. In

this case, we find no LRAs. Second, we simulate a skew-aware world in which

the pricing kernel also depends on the squared market return. In this case,

we find that LRAs appear and are driven by coskewness. Third, we simulate a

world in which all moments higher than skewness are also taken into account.

We find that these higher moments contribute much less toward explaining

LRAs. Taken together, the simulation results suggest that (co)skewness is the

main determinant of LRAs. We also use the model to demonstrate that there

is a direct link between a firm’s ex ante skewness, its realized coskewness,

and its alpha. This motivates our empirical approach of using equity option-

implied ex ante skewness to construct coskewness factor-mimicking portfolios

in our study of LRAs.

We establish our main empirical results for a cross-section of approximately

5,000 U.S. firms over the period January 1996 to August 2014. This sample

covers all CRSP firms for which data on common stock and equity options are

available. To comprehensively capture asymmetries in the return distribution,

we compute three measures of ex ante skewness from portfolios of out-of-the-

money (OTM) options: upper skewness from OTM call options, which covers

the right part of the distribution, lower skewness from a portfolio that is short

OTM put options, which by definition is negative, to account for the left part

of the distribution, and total skewness, which is the sum of upper and lower

skewness. Thus, total skewness becomes more negative, as the cost of put op-

tions relative to call options rises, that is, as the premium that investors are

willing to pay for protection against downside risk rises.

Low Risk Anomalies? 2675

Our empirical analysis starts by showing that in the data, ex ante skew-

ness is related to residual coskewness and alphas in the same way as in our

simulated skew-aware world: the more extreme a firm’s ex ante skewness, the

higher its residual coskewness and the lower its CAPM alpha. The results are

virtually unchanged when we compute alphas and residual coskewness rel-

ative to the Fama-French three-factor model (FF3, Fama and French (1993)).

When we additionally control for momentum (FF4, Carhart (1997)), the results

become quantitatively less pronounced but the qualitative patterns remain the

same for lower and upper ex ante skewness. These findings suggest that ex

ante skewness is linked to residual coskewness and alphas in a way that is

consistent with skew-aware asset pricing, but is not captured empirically by

standard risk factors.

Having established that ex ante skewness is a forward-looking proxy for

residual coskewness, we study the main prediction of our model, namely, that

controlling for skewness should eliminate positive alphas and negative resid-

ual coskewness of beta- and volatility-related LRAs. To do so, we construct

coskewness factor-mimicking portfolios from decile portfolios sorted by mea-

sures of firms’ ex ante skewness. Using several alternative specifications of

skewness factors, we find that LRA alphas decrease substantially and become

statistically insignificant when we control for skewness. For instance, using

our most flexible specification, we find for betting-against-beta (BaB) that the

CAPM alpha drops from 125 to 33 basis points per month, the FF3 alpha from

109 to 21 basis points, and the FF4 alpha from 73 to 21 basis points. For all

anomalies, the reduction in alphas is in lockstep with a reduction in the strate-

gies’ negative coskewness. These results suggest that controlling for ex ante

skewness does indeed render alphas insignificant because it captures coskew-

ness risk. This is confirmed in cross-sectional regressions of alphas on resid-

ual coskewness betas for 80 CAPM beta- and volatility-sorted portfolios. With-

out controlling for skewness, the regression R2s are 73%, 73%, and 48% using

CAPM, FF3, and FF4 alphas and residual coskewness, respectively. When we

control for skewness, the R2s drop to 22% for the CAPM-based regression and

to less than 4% for the FF3- and FF4-based regressions.

Given the skew-aware SDF, the different anomalies that prior literature

has established as mostly unrelated asset pricing puzzles, should all be ex-

posed to a common factor. To explore this conjecture, we proceed in three steps.

First, using principal component analyses (PCAs) of anomaly returns, we show

that the anomalies based on CAPM betas, idiosyncratic volatility, and option-

implied variance do indeed have a common determinant. We find that the first

principal component (PC) explains more than 90% of the variation in anomaly

excess returns and more than 70% of the variation in FF4 residual returns.

Second, we show that the first PC is related to the returns of skewness factors.

When we regress the first PC on skew factor returns, we find R2sofupto95%

for excess returns and up to 80% for FF4 residual returns. Furthermore, we

show that the skew exposures of beta- and volatility-sorted portfolios mono-

tonically decrease in beta and volatility. These results provide strong evidence

that LRAs have a common determinant related to (co)skewness.