Does aggregate uncertainty explain size and value anomalies?

ABSTRACT

This paper examines the impact of aggregate uncertainty on return dynamics of size and book-to-market ratio sorted portfolios. Using VVIX as a proxy for aggregate uncertainty, and controlling for market risk, volatility risk, correlation risk and the variance risk premium, we document significant portfolio return exposures to aggregate uncertainty. In particular, portfolios that contain small and value stocks have significant and negative uncertainty betas, whereas portfolios of large and growth stocks exhibit positive and significant uncertainty betas. Using a quasi-natural experimental setting around the financial crisis, we confirm the differential sensitivity of small versus big and value versus growth portfolios to aggregate uncertainty. We posit that due to their negative uncertainty betas, uncertainty-averse investors demand extra compensation to hold small and value stocks. Our results offer an uncertainty-based explanation to size and value anomalies.

KEYWORDS:

• Uncertainty
• vol-of-vol
• VVIX
• size
• value

JEL CLASSIFICATION:

• G01
• G10
• G11

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ While the expected-utility and rational-expectations framework is still the dominant paradigm in economics, critics point to empirical and experimental counter-examples that are inconsistent with rational optimizing agents and economic equilibrium (Kahneman, Slovic, and Tversky 1982; Thaler 1993).

² See Epstein and Schneider (2010) and Guidolin and Rinaldi (2013) for a detailed review of literature.

³ See Segal (1987, 1990), Klibanoff, Marinacci and Mukerji (2005), Nau (2006), Ergin and Gul (2009), Seo (2009) and Neilson (2010) for studies which establish the link between second-order risk and uncertainty aversion.

⁴ Hansen, Sargent and Tallarini (1999), Chen and Epstein (2002), Anderson, Hansen and Sargent (2003), Uppal and Wang (2003), Kogan and Wang (2003), Maenhout (2004), 2006), Liu, Pan and Wang (2005), Cao, Wang and Zhang (2005), Hansen and Sargent (2007) and Anderson, Ghysels and Juergens (2009) for studies that theoretically motivate why and how uncertainty affects investors’ optimal decision-making and asset prices.

⁵ According to Knight (1921), risk is associated with situations when decision makers can assign probabilities to the range of outcomes associated with the actions available to them. In contrast, uncertainty is associated with situations when there is insufficient information to assign numerical probabilities to outcomes necessitating the use of estimates rather than assigned probabilities.

⁶ VRP is defined as the difference between implied volatility and RV. See Bollerslev, Gibson and Zhou (2011), Drechsler and Yaron (2011), Bekaert, Hoerova and Duca (2013) and Bali and Zhou (2015) for follow-up studies that use VRP in different settings.

⁷ The measure proposed in Bollerslev, Tauchen and Zhou (2009) assumes that the conditional variance of stock market returns is a martingale; however, Bekaert and Hoerova (2014) show that this assumption is not supported by the data, leading to potentially biased variance premia. Nevertheless, we perform additional robustness tests to clearly distinguish between our measure and VRP and include VRP in all the specifications.

⁸ See Lettau and Ludvigson (2001), Petkova and Zhang (2005) and Gulen, Xing and Zhang (2011) for the link between economic conditions, business cycles and the value premium.

⁹ The motivation for using an option-implied measure of vol-of-vol is also related to the well-established strand of literature in option pricing with stochastic volatility. It is now common in option pricing models to assume stochastic volatility for the dynamics of the underlying asset. For example, Bakshi, Cao and Chen (1997) document that option pricing models which incorporate stochastic volatility perform better in terms of internal consistency, yield lower out-of-sample pricing errors, and most notably perform better in hedging. Buraschi and Jiltsov (2007) argue that stochastic volatility in option pricing models can be rationalized by the presence of heterogeneous agents who are exposed to model uncertainty and have different beliefs regarding expected returns. Drechsler and Yaron (2011) draw a link between uncertainty and investors’ demand for compensation against stochastic volatility. Using volatility-of-volatility implied by the cross section of VIX options (VVIX), Park (2013) shows that the model-free risk-neutral VVIX index has forecasting power for future tail risk in hedge fund returns. Huang and Shaliastovich (2014) show that volatility-of-volatility risk (measured by VVIX) is priced in the cross section of option returns. Therefore, an option-implied volatility of market volatility is a much direct and clean measure of vol-of-vol which has sound theoretical underpinnings.

¹⁰ VVIX is calculated from the price of a portfolio of liquid at- and out-of-the-money VIX options. The portfolio can be traded to manage the volatility risk of VIX exposures and to obtain the risk premium between the expected and RV of VIX forward prices. These properties make VVIX a plausible measure to capture uncertainty about expected volatility of the market portfolio.

¹¹ The VIX index is a volatility index comprised of S&P 500 index options, with the price of each option reflecting the market’s expectation of future volatility. The components of the VIX calculation are near- and next-term put and call options with more than 23 days and less than 37 days to expiration. For more details on VIX and its calculation methodology, please see https://www.cboe.com/micro/vix/vixwhite.pdf.

¹² It is well documented that during periods of financial stress and adverse economic conditions VIX tends to rise, indicating market’s expectation of financial turmoil, and as economic conditions recover and investor fear subsides, VIX index goes down. For example, Durand, Lim and Zumwalt (2011) document that the market risk premium and the value premium are sensitive to changes in the VIX.

¹³ Our study is based on daily observations of VVIX in order to gain as much data as possible during a relatively short sample period and to increase the power of statistical tests.

¹⁴ http://www.cboe.com/publish/scheduledtask/MktData/datahouse/VIXcurrent.csv.

¹⁵ http://www.cboe.com/publish/scheduledtask/mktdata/datahouse/VVIXtimeseries.csv.

¹⁶ http://www.cboe.com/micro/impliedcorrelation/implied-correlation-hist.csv.

¹⁷ We would like to thank Gerd Heber, Asger Lunde, Neil Shephard and Kevin Sheppard for making the data publicly available.

¹⁸ Following the literature, VRP is defined as the difference between risk-neutral expected volatility and RV, that is, IV-RV.

¹⁹ For CORR time-series data, we use CBOE ticker symbol JCJ.

²⁰ The test portfolios are constructed at the end of each June using NYSE breakpoints. Ten size-sorted portfolios are constructed using June value of market equity. Ten BE/ME-sorted portfolios are formed based on BE/ME for June of year t which is calculated as the book equity for the last fiscal year end in t − 1 divided by ME for December of t − 1. 5 × 5 portfolios based on size and BE/ME are constructed at the end of each June using the intersections of 5 portfolios formed on size and 5 portfolios formed on BE/ME. The portfolios for July of year t to June of t + 1 include all NYSE, AMEX and NASDAQ stocks for which we have market equity data for June of t.

²¹ http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data-library.html.

²² See Edelman et al. (2012) who mark March 2009 as the end of subprime crisis.

²³ We confirm this finding by using the Augmented Dickey–Fuller (ADF) test with a time trend and four lags. We cannot reject the null of a unit root for VIX with ADF test-statistic of −3.224, and 5% critical value of −3.410. On the other hand, ADF test-statistic for VVIX is −6.124, which rejects the existence of unit root; thus, stationarity cannot be rejected for VVIX.

²⁴ See Copeland and Copeland (1999), Moise (2007), and Arisoy (2014) who document the relationship between aggregate volatility risk and returns of size-sorted portfolios.

²⁵ Our definition of sub-periods is based on Edelman et al. (2012), who identifies March 2009 as a structural break point associated with the end of credit crisis. Our results are robust to alternative sub-periods ending at December 2008, January 2009 and February 2009.

²⁶ For the sake of brevity, we report results using 10 size and 10 B/M portfolios only. Results with 25 portfolios using 5 size and 5 B/M quintiles are available upon request.