THE REACTIVE BETA MODEL

• Author: Denis Grebenkov,Sofiane Aboura,Sebastien Valeyre
• Date: 01 March 2019
• Published date: 01 March 2019
• DOI: http://doi.org/10.1111/jfir.12176

THE REACTIVE BETA MODEL

Sebastien Valeyre

John Locke Investments, University of Paris XIII

Sofiane Aboura

CNRS - Ecole Polytechnique

Denis Grebenkov

University of Paris XIII

Abstract

We present a reactive beta model that accounts for the leverage effect and beta elasticity.

For this purpose, we derive a correlation metric for the leverage effect to identify the

relation between the market beta and volatility changes. An empirical test based on the

most popular market-neutral strategies is run from 2000 to 2015 with exhaustive data

sets, including 600 U.S. stocks and 600 European stocks. Our findings confirm the

ability of the reactive beta model to remove an important part of the bias from the beta

estimation and from most popular market-neutral strategies. To examine the robustness

of the reactive beta measurement, we conduct Monte Carlo simulations over seven

market scenarios against five alternative methods. The results confirm that the reactive

model significantly reduces the bias overall when financial markets are stressed.

JEL Classification: C5, G01, G11, G12, G32

I. Introduction

Finding an appropriate measure of market betas is of paramount importance for many

financial applications, including market-neutral hedge fund managers who target a near-

zero beta. Contrary to common belief, perfect beta-neutral strategies are difficult to

achieve in practice, as the mortgage crisis in 2008 exemplified, when most market-

neutral funds remained correlated with stock markets and experienced considerable

unexpected losses. This exposure to the stock index (Banz 1981; Fama and French 1992,

1993; Carhart 1997; Ang et al. 2006) is even stronger during down market conditions

(Moreira and Muir 2017; Agarwal and Naik 2004; Bussi

ere, Hoerova, and Klaus 2015).

In such a period of market stress, hedge funds may even add no value (Asness, Krail, and

Liew 2001).

In this article, we derive a stock market beta measure that we implement to test

the quality of hedging for four popular strategies in the hedge funds industry. The first

and most important strategy captures the low-beta anomaly (Black 1972; Black, Jensen,

and Scholes 1972; Haugen and Heins 1975; Haugen and Baker 1991; Ang et al. 2006;

Baker, Bradley, and Taliaferro 2013; Frazzini and Pedersen 2014; Hong and Sraer 2016)

that defies conventional wisdom on the risk and reward trade-off predicted by the capital

The Journal of Financial Research Vol. XLII, No. 1 Pages 71–113 Spring 2019

DOI: 10.1111/jfir.12176

71

© 2019 The Southern Finance Association and the Southwestern Finance Association

asset pricing model (CAPM) (Sharpe 1964). According to this anomaly, high-beta stocks

underperform low-beta stocks. Similarly, stocks with high idiosyncratic volatility earn

lower returns than stocks with low idiosyncratic volatility (Malkiel and Xu 1997; Goyal

and Santa-Clara 2003; Ang et al. 2006, 2009). The related strategy consists of shorting

high-beta stocks and buying low-beta stocks. The second important strategy captures

the size effect (Banz 1981; Reinganum 1981; Fama and French 1992), in which stocks of

small firms tend to earn higher returns, on average, than stocks of larger firms. The related

strategy consists of buying stocks with small market capitalization and shorting those

with high market capitalization. The third strategy captures the momentum effect

(Jegadeesh and Titman 1993; Carhart 1997; Grinblatt and Moskowitz 2004; Fama and

French 2012), where past winners tend to continue to show high performance. This

strategy consists of buying the past year’s winning stocks and shorting the past year’s

losing stocks. The fourth strategy captures the short-term reversal effect (Jegadeesh

1990), where past winners in the last month tend to show low performance. This strategy

consists of buying the past month’s losing stocks and shorting the past month’s winning

stocks, which would be highly profitable if there were no transaction cost and no market

impact. Testing the quality of the hedge of the strategies is equivalent to assessing the

quality of the beta measurements, which is difficult to realize directly as the true beta is

not known.

The implementation of all these strategies requires a reliable estimation of the

betas to maintain the hedge. Ordinary least squares (OLS) estimation remains the most

frequently employed method, even though it is impaired in the presence of outliers,

especially from small companies (Fama and French 2008), illiquid companies (Amihud

2002; Acharya and Pedersen 2005; Ang, Shtauber, and Tetlock 2013), and business

cycles (Ferson and Harvey 1999). In these circumstances, the OLS beta estimator might

be inconsistent. To overcome these limitations, our approach consists of renormalizing

the returns to make them closer to Gaussian and thus to make the OLS estimator more

consistent. In addition, many papers report that betas are time varying (Blume 1971;

Fabozzi and Francis 1978; Jagannathan and Wang 1996; Fama and French 1997;

Bollerslev, Engle, and Wooldridge 1988; Lettau and Ludvigson 2001; Lewellen and

Nagel 2006; Ang and Chen 2007; Engle 2016). This can lead to measurement errors that

could create serious bias in the cross-sectional asset pricing test (Shanken 1992; Chan

and Lakonishok 1992; Meng, Hu, and Bai 2011; Bali, Engle, and Tang 2017). In fact,

firms’stock betas do change over time for several reasons. The firm’s assets tend to vary

over time via acquiring or replacing new businesses, which makes them more diversified.

The betas also change for firms that change in dimension to be safer or riskier. For

instance, financial leverage may increase when firms become larger, as they can issue

more debt. Moreover, firms with higher leverage are exposed to a more unstable beta

(Galai and Masulis 1976; DeJong and Collins 1985). One way to account for the time

dependence of betas is to consider regime changes when the return history used in the

beta estimation is long enough. Surprisingly, only one paper (Chen, Zhang, and Wu

2005) suggests a solution to capture the time dependence and discusses regime changes

for the beta using a multiple structural change methodology. The study shows that the

risk related to beta regime changes is rewarded by higher returns. Another approach is to

examine the correlation dynamics. Francis (1979) finds that “the correlation with the

72 The Journal of Financial Research

market is the primary cause of changing betas . . . the standard deviations of individual

assets are fairly stable”(p. 989). This finding calls for special attention to the correlation

dynamics addressed in our article but are apparently insufficiently investigated in other

works.

Despite the extensive literature on this issue, little attention has been paid to the

link between the leverage effect

1

and the beta. The leverage effect is defined as the

negative correlation between the securities’returns and their volatility changes. This

correlation induces residual correlations between the stock overperformances and beta

changes. In fact, earlier studies have heavily focused on the role of the leverage effect on

volatility (Black 1976; Christie 1982; Campbell and Hentchel 1992; Bekaert and Wu

2000; Bouchaud, Matacz, and Potters 2001; Valeyre et al. 2013). Surprisingly, despite its

theoretical and empirical underpinnings, the leverage effect has not been considered so

far in beta modeling, while it is a measure of risk. We aim to close this gap.

Our article starts by investigating the role of the leverage effect in the correlation

measure by extending the reactive volatility model (Valeyre et al. 2013), which

efficiently tracks the implied volatility by capturing both the retarded effect induced by

the specific risk and the panic effect, which occurs whenever the systematic risk becomes

the dominant factor. This allows us to set up a reactive beta model incorporating three

independent components, all of which contribute to a reduction in the hedging bias. First,

we take into account the leverage effect on beta, where the beta of underperforming

stocks tends to increase. Second, we consider a leverage effect on correlation, in which a

stock index decline induces an increase in correlations. Third, we model the relation

between the relative volatility (defined as the ratio of the stock’s volatility to the index’s

volatility) and the beta. When the relative volatility increases, the beta increases as well.

All three independent components contribute to a reduction in the biases in the naive

regression estimation of the beta and therefore considerably improve hedging strategies.

The main contribution of this article is the formulation of a reactive beta model.

The economic intuition behind the reactive beta model is the derivation of a suitable beta

measure allowing market beta estimation with reduced bias and a smaller standard

deviation. The model is coined “reactive”because the beta measurement is adjusted as

soon as prices move. An empirical test is performed based on an exhaustive data set that

includes the 600 largest American stocks and the 600 largest European stocks from 2000

to 2015, which includes several business cycles. This test validates the superiority of the

reactive beta model over conventional methods.

We further examine the robustness of the reactive beta measurement using

Monte Carlo simulation against five alternative methods (OLS, minimum absolute

deviation (MAD), trimean quantile regression [TRM], dynamic conditional correlation

1

Note that we are not dealing with the restricted definition of the “leveraged beta”that comes from the degree

of leverage in the firm’s capital structure. Notice that the market beta may be nonlinearly related to the market

return, which could lead to spurious inference in beta measurement (DeBondt and Thaler 1987), whereas the

leverage effect could be a major explanation of such nonlinearity. For example, Garlappi and Yan (2011) relate

leverage to default probability, Daniel, Jagannathan, and Kim (2012) relate the financial leverage to the operating

leverage, Choi (2013) relates leverage to economic conditions, Mitchell and Pulvino (2001) relate leverage and

volatility managed portfolios, and Liu, Stambaugh, and Yuan (2018) relateleverage to the beta-idiosyncratic

volatility relation. In this context, the time-variation effect in conditional beta adds on this bias (Boguth et al. 2011).

The Reactive Beta Model 73