Can tail risk explain size, book‐to‐market, momentum, and idiosyncratic volatility anomalies?

• DOI: http://doi.org/10.1111/jbfa.12403
• Author: Sofiane Aboura,Y. Eser Arisoy
• Date: 01 October 2019
• Published date: 01 October 2019

DOI: 10.1111/jbfa.12403

Can tail risk explain size, book-to-market,

momentum, and idiosyncratic volatility anomalies?

Sofiane Aboura1Y.Eser Arisoy2

1Université de ParisXIII, Sorbonne Paris Cité, 99

avenueJean-Baptiste Clément, 93430

Villetaneuse, France

2NEOMA Business School, 59 rue Pierre

Taittinger,51100 Reims, France

Correspondence

SofianeAboura, Université de Paris XIII, Sor-

bonneParis Cité, 99 avenue Jean-Baptiste Clé-

ment,93430 Villetaneuse, France.

Email:sofiane.aboura@univ-paris13.fr

Abstract

We examine the impact of tail risk on the return dynamics of size,

book-to-market ratio, momentum and idiosyncratic volatilitysorted

portfolios. Our time-series analyses document significant portfolio

return exposures to aggregate tail risk. In particular, portfolios that

contain small, value, high idiosyncratic volatility and low momentum

stocks exhibitnegative and statistically significant tail risk betas. Our

cross-sectional analyses at the individual stock level suggest that tail

risk helps in explaining the four pricing anomalies, particularly size

and idiosyncratic volatility anomalies.

KEYWORDS

anomalies, idiosyncratic volatility, momentum, size, tail risk, value

JEL CLASSIFICATION

C4, G11, G12

1INTRODUCTION

Tailrisk hedging has gained considerable interest among market participants since the 2008 financial crisis. It is now

widely acknowledged that market returns have much fatter tails than a Normal distribution would generate, and tail

events occur much more frequently than a Normal curve would predict. Therefore, understanding the dynamics of

tail risk and its relation to asset returns is of crucial importance in investment decision making and has implications for

portfolio management.1In this paper,we investigate whether tail risk helps explain four well-documented asset pricing

anomalies – size, value vs. growth, momentum and idiosyncratic volatility.

There are several reasons why tailr isk couldhave important asset pricing implications. First, it is well documented

in the option pricing literature that assets like deep out-of-the money put options can become highly valuable dur-

ing stress periods. For example, Rubinstein (1994) argues that the ‘market’s pricing of indexoptions since the crash

seems to indicate an increasing “crash-o-phobia” among investors’. One would naturally expect a similar sensitivity to

1Keynes (1921) was the first economist to recognize that a decision makerhas to minimize the probability of obtaining an outcome below the mean. In the

same spirit, Roy (1952) explains how the portfolio manager should minimize the chance of disaster based on the ‘safety first’ principle. Similarly,the recent

revivalof the rare-disasters literature, in the wake of the financial crisis, is based on several concepts used in the extreme value theory literature to measure

tailrisk and in particular left tail risk (Embrechts, Klüppelberg, & Mikosch, 1997).

J Bus Fin Acc. 2019;46:1263–1298. wileyonlinelibrary.com/journal/jbfa c

2019 John Wiley & Sons Ltd 1263

1264 ABOURA ANDARISOY

tail risk in the stock market.2For example,Bloom (2009) offers a firm-level tail uncertainty based explanation of tail

risk, in which economic uncertainty affects firms’ investment decisions negatively. According to Bloom (2009), firm-

level tail uncertainty is a channel through which tail risk impacts the equity premium, and the tail exponent, which

measures tail heaviness, is an important determinant of asset prices. As a result, crash-sensitivestocks command a risk

premium.3Second, Bollerslevand Todorov (2011) show that compensation for rare eventsaccounts for a large fraction

ofthe US equity risk premium. Gabaix (2012) reaches similar c onclusions, showing that a time-varying raredisaster risk

framework can explainthe equity premium puzzle.4Huang, Liu, Rhee, and Wu (2012) document a significantly positive

extreme downside risk premium in the cross-section of stock returns, even after controlling for market, size, value,

momentum and liquidity effects. Barro (2005) explains the equity premium puzzle via the ‘peso problem’ and argues

that the selection of no-disaster samples (for instance due to data unavailability) only moderately affects the equity

premium, but the likelihoodof these disasters has major effects on rates of return and the equity premium. Third, Chol-

lete and Lu (2011) suggest that tail risk should induce a monotonic pattern in the cross-section of stock returns in the

sensethat stocks that are more sensitive to tail risk should command higher returns. They argue that such stocks, which

tendto co-move with systemic risk, should be unattractive to risk-averse investors, as they are more difficult to sell dur-

ing stress periods. Hence, they should carry a ‘tail risk premium’.5Similarly, Chabi-Yo, Ruenzi, and Weigert (2018) note

that US stocks that are likely to perform badly during market crashes earn significantly higher averagereturns than

stocks that offer protection against market downturns. Bollerslev,Todorov,and Xu (2015) further suggest that market

fear plays an important role in stock return predictability,since investors demand a special compensation for bearing

tail risk.6

Our paper contributes to the literature in severalways. To the best of our knowledge, this is the first study to exam-

ine the sensitivity of market capitalization (MCAP), book-to-market ratio (B/M), idiosyncratic volatility (IVOL), and

momentum (MOM) sorted portfolios to aggregate tail risk. Given the difficulty in pricing small-value and big-growth

stocks using linear factor models, and the continuing debate over the idiosyncratic volatility puzzle and momentum

crashes, tail risk seems to be a potential candidate as an explanatoryfactor behind these empirically documented pric-

ing puzzles, as it can capture nonlinearities in the left tail of the stock return distribution, which can help improve the

performance of the linear factor models proposed in the literature. Hence, in this paper we revisit size, value, momen-

tum and idiosyncratic volatility anomalies, and study if and how aggregate tail risk relates to the returns on small vs.

big, value vs. growth, high vs. low idiosyncratic volatility, and high vs. low momentum portfolios.7Despite a growing

2See,for example, Yuen (2015) who documents that the option-implied left tail risk of the market portfolio is priced in the cross-section of stock returns, even

aftercontrolling for risk during normal times.

3Storsletten,Telmer and Yaron (2004) and Bloom (2009) further show that firm-leveltail uncertainty is countercyclical.

4See also Tsaiand Wachter (2015), who survey recent disaster risk models that provide explanations for the equity premium puzzle. Building on the seminal

papers of Barro (2006) and Rietz (1988), the authors argue that disaster risk offers a coherent and parsimonious framework explainingasset pricing puzzles,

asthey observe that large drops in financial markets affect prices significantly through the risk premium channel.

5Tailriskis also found to be an important f actor in the cross-section of mutual fund and hedge fund returns. Xiong, Idzorek, and Ibbotson (2014) document that

the tail-risk premium is economically significant in the cross-section of both US equity fund returns and non-US equity fund returns. Using a non-parametric

estimate for hedge funds’ systematic tail risk, Agarwal, Ruenzi, and Weigert (2017) show that tail risk has significant predictivepower inthe cross-section of

equity-orientedhedge fund returns.

6VanOordt and Zhou (2016) further report that stocks with historically high tail betas suffer losses during market crashes. However, they conclude that the

abilityof systematic tail risk to explain the cross-section of expected returns seems limited.

7The literature offers severalexplanations to size, value vs. growth, idiosyncratic volatility and momentum anomalies. For example, studies document that

smalland value stocks are sensitive to risk factors such as aggregate volatility risk (Barinov, 2012), market jump risk (Arisoy,2014; Cremers, Halling, & Wein-

baum, 2014), financial distress risk (Avramov,Chordia, Jostova, & Philipov,2013), and liquidity risk (Asness, Moskowitz, & Pedersen, 2013). Herskovic, Kelly,

Lustig, and Van Nieuwerburg (2016) show that firms’ idiosyncratic volatility obeysa strong factor structure and that shocks to the common idiosyncratic

volatility factor have important implications for asset prices. Hou and Loh (2016) examinea large number of alternative explanations for the idiosyncratic

volatility puzzle and find that many existing explanationsexplain less than 10% of the puzzle. On the other hand, explanations based on investors’ lottery

preferences and market frictions show some promise in explaining the puzzle. Momentum is another established empirical phenomenon. Like size, value

and idiosyncratic volatility,there is much debate and little consensus regarding the explanation driving this premium. Among the potential explanations are

macroeconomic risks and time-varying risk premia (Ahn, Conrad, & Dittmar,2003; Chordia & Shivakumar, 2002), liquditiy risk (Pastor & Stambaugh, 2003;

Sadka, 2006), growth options and opportunities (Berk, Green, & Naik, 1999; Johnson, 2002; Sagi & Seasholes, 2007), investmentsand time-varying riskfac-

tors(Zhang, 2004), and factor structure with other asset classes (Asness et al., 2013).

ABOURA ANDARISOY 1265

literature, it is still not known whether portfolios sorted on different stock characteristicshave different exposures to

aggregate tail risk and, if they do, whether tail risk can help explainseveral stock market anomalies documented in the

literature. Our study is an attempt to fill this gap.

To operationalize our research framework,one needs to come up with an economically plausible measure of tail

risk. Recent studies develop alternative measures of tail risk both at the individual stock level and at the aggregate

level. Bali, Cakici, and Whitelaw (2014) developa stock-level measure of hybrid tail covariance risk (HTCR) motivated

by the underdiversified portfolio holdings of individual investors and bythe assumption that for individual stocks, tail

riskis primarily idiosyncratic. The authors document a significantly positive relationship between hybrid tail covariance

risk and expected stock returns. Kellyand Jiang (2014) develop an aggregate tail risk measure based on extreme value

theory.They use monthly firm-level price crashes to identify common fluctuations in tail risk among individual stocks.

The authors find that their aggregate tail risk measure correlates significantly with tail risk measures extracted from

S&P 500 indexoptions and show that it negatively predicts real economic activity and strongly predicts aggregate mar-

ket returns.8We argue that Kellyand Jiang’s (2014) tail-risk index (TAIL) is a suitable measure to proxy aggregate tail

risk, and hence an ideal tool to study the effect of aggregate tail risk in our time-series analyses at the portfolio level.9

Our choice of Kelly and Jiang’s (2014) aggregate tail risk measure has several reasons. First, TAIL captures extreme

downward movements and time-varying tail risk in the US stock market, and it can be directly estimated from the

cross-section of US stock returns.10 Second, because TAILis estimated using a large cross-section of stocks that expe-

rience individual tail events much more frequently than the aggregate market index,it provides accurate information

about the level of aggregate tail risk.11 Third, Kelly and Jiang (2014) document that their time-varying tail exponentis

highly persistent, strongly predicting future extreme returns of individual stocks. This indicates that the tail risk index

is a potentially important determinant of asset prices. Fourth, TAILis an aggregate measure of tail risk, inferred from

the common tail risk of individual stocks. Since individual stock return tails obey a power law, the tail risk of the mar-

ket portfolio will reflect the tail risk of the individual stocks. Hence, it is an ideal tool to track the evolution of aggre-

gate tail risk. Fifth, TAIL further captures asymmetric downside risk, as it focuses on the extreme component of the

left tail of the distribution. Finally,the sample period that we examine (1963–2013) covers several periods of extreme

market conditions, and is long and representative enough to makestatistical inferences about the effect of aggregate

tail risk on returns of size, book-to-market ratio, idiosyncratic volatility and momentum-sorted portfolios and in the

cross-section.12

Using monthly changes in TAIL (ΔTAIL)as a proxy for aggregate tail risk, our findings at the portfolio level can be

summarized as follows. First, we note that extreme size portfolios (containing smallest and biggest stocks) are more

negatively exposed to aggregate tail risk than mid-cap stocks. In particular,the portfolio of small stocks and the arbi-

trage portfolio that is long in the quintile of smallest stocks and short in the quintile of biggest stocks (Lo-Hi MCAP)

exhibita negative and statistically significant aggregate tail risk exposure, which is much higher in magnitude compared

to any other portfolio. Similarly,we find that the portfolio of value stocks and the arbitrage portfolio that goes long the

8Chapman,Gallmeyer, and Martin (2018) confirm that the Kelly and Jiang (2014) tail risk factor can predict market returns and find that th tail indexexplains,

inparticular, the cross-section of the discount rate component of returns, but not the cash-flow component.

9Thereis a further practical reason for using TAIL for our tests conducted at the portfolio level. Because the tests at the portfolio levelare essentially about the

sensitivity of portfolio returns to aggregate tail risk, we cannot use HTCR,which is by construction a tail risk measure at the individual stock level. However,

we include HTCR in our cross-sectional analyses to test its incremental explanatorypower and we find that HTCR is a strong contender for explaining the

fouranomalies in the cross-section, especially the small size premium. We refer the reader to Section 5 for more details about the cross-sectional explanatory

powerof HTCR and TAIL.

10Most measures of tail risk proposed in the literature are based on statistical metrics that model power laws. Chapman et al. (2018) explain that the Kelly

and Jiang’s (2014) tail risk measure is appropriate for a long-run risks model. Moore et al. (2013) discuss how the downside tail risk of stock returns can be

differentiatedcross-sectionally by including in the analysis not only the tail risk but also the corresponding scale parameter. Andersen et al. (2016) find a clear

distinctionbetween the left tail factor, which predicts the future equity tail risk premium, and the spot variance factor, which predicts the actual future return

variation.Chabi-Yo, Ruenzi, and Weigert (2018) and Weigert(2016) compute a tail risk estimator based on a copulato feature tail dependence.

11In particular,Kelly and Jiang (2014) use daily returns on all NYSE/AMEX/NASDAQ stocks with share codes 10 and 11 from January 1963 to December 2010

toconstruct their aggregate tail risk measure.

12We would like to thank Brian Kelly and Hao Jiang for kindly sharing their aggregate tail risk data.