Estimating the Equity Tail Risk Premium

Unrealistic Optimism and Asymmetry in the Pricing of Equity Tail Risk K. Victor Chow Jingrui Li and Ben Sopranzetti Ma...

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Unrealistic Optimism and Asymmetry in the Pricing of Equity Tail Risk

K. Victor Chow Jingrui Li and Ben Sopranzetti

May 27, 2018

Corresponding author: Ben Sopranzetti, [email protected] K. Victor Chow, Distinguished Professor of Global Business & Finance, West Virginia University; Jingrui Li, Finance doctoral student, West Virginia University; Ben Sopranzetti, Professor of Finance, Rutgers Business School - Newark and New Brunswick. We thank Kose John, Sophia Zhengzi Li, Bingxin Li, Aurelio Vasquez, George J. Jiang, Alexander Kurov, Diego Amaya, Dan Weaver, Daniela Osterrieder, Richard McGee (MFA2018 discussant) and seminar participants at West Virginia University for helpful comments and feedback. 1

Unrealistic Optimism and Asymmetry in the Pricing of Equity Tail Risk ABSTRACT Unrealistic Optimism is the psychological phenomenon whereby human beings are unreasonably optimistic about the likelihood and impact of tail events; further, they incorporate information in a selective way in order to reinforce this optimistic bias. Unrealistic Optimism results in new information about extreme positive events being processed quickly and efficiently, but new information about extreme negative events being processed with a lag, and when it is finally processed, it is done so with an optimistic bias which prolongs the amount of time that it takes for the information to be fully and correctly incorporated into prices. Moreover, the more remote the likelihood of the tail event and the larger its potential impact, the greater is the severity of the optimistic bias. To test these predictions, this paper introduces a novel methodology to directly calculate the tail risk premium for individual stocks, and then uses this measure to examine its characteristics in the cross section of individual stock returns. The empirical results are robust to a variety of specifications. Positive tail risk seems to be efficiently priced, i.e., the existence of a premium for bearing positive tail risk today has no predictive power for future returns. However, the same cannot be said of the pricing of negative tail risk, where the existence of a premium for bearing negative tail risk today is significantly associated with lower returns in the future. Moreover, the greater the premium for bearing negative tail risk today, the more negative and persistent are the future returns.

Keywords: tail risk, asymmetry, cross-section of stock returns, return prediction, behavioral finance, investor psychology, empirical asset pricing. JEL Classification: G11, G12, G17


Unrealistic Optimism and Asymmetry in the Pricing of Equity Tail Risk


Introduction Harris and Guten (1979) and Weinstein (1980, 1982, 1984, 1987, and 1989) document the

existence of Unrealistic Optimism, the phenomenon whereby human beings are unreasonably optimistic about the likelihood of extreme events that could potentially occur to them. Humans believe, and their memory processes systematically reinforce this belief, that negative (positive) tail events have a lower (higher) probability of occurring to themselves than occurring to others. Consequently, when people calculate the expected impact of extreme tail events, the results are exaggerated and optimistically biased. In addition, the more extreme the event, the greater is the exaggeration of reality. Sharot, Guitart-Masip and Korn (2011), Sharot, Korn and Dolan (2011) and Sharot, et al. (2012a) provide evidence that the human memory process actually reinforces the distortions associated with Unrealistic Optimism. People update their priors more frequently when presented with favorable information than they do with negative information, i.e., they selectively incorporate information to support their optimistically biased outlook. In the economics literature, Van den Steen (2004) formally models the behavior of diverse, rational agents and finds that these agents (1) are likely to be overoptimistic about their chances of success; (2) tend to blame failure on external factors but believe that success is a direct result of their own actions; (3) have the erroneous belief that they will achieve better outcomes than will others, and (4) tend to overestimate both the precision of their forecasts and the extent to which they have control over their own outcomes. Given the extensive evidence on the existance and pervasiveness of Unrealistic Optimism, an investigation into the application of this phenomenon to asset pricing,


and specifically on the pricing of information related to extreme events, is natural and could potentially be very interesting. Compensation for extreme, tail event risk is formally referred to as a “tail risk premium.” Bollerslev, Todorov and Xu (2015) shows that a majority of the predictability in the variance risk premium is attributed to this premium for bearing jump tail risk, and that, specifically, it is negative tail risk and not positive tail risk that seems to be priced1. The impact of tail events on returns is well documented at the aggregate market level, but not so much is known about its impact at the individual stock level. One reason why is that out of the money call and put options are required to determine the tail risk in the risk neutral probability space. Although out of the money options are prevalent for an index such as the S&P500, they either do not exist or are illiquid for most stocks. For this reason, to date no paper has directly examined both the impact of tail risk and its premium on the cross section of individual stock returns. Given that the return distribution for individual stocks will likely exhibit a greater proclivity for extreme events than the return distribution for a diversified market portfolio (where extreme negative events in some securities might be tempered by extreme positive events in others) one would expect that tail risk should play a more prominent role in the returns for individual stocks than it would for a market portfolio. Consequently, to empirically examine the impact of Unrealistic Optimism on asset prices, a exploration of the impact of tail risk on the returns of individual stocks, rather than a diversified portfolio, is warranted. Kelly and Jiang (2014) is the first to examine tail risk in the cross section of individual stock returns. This important paper employs an aggregate measure of time-varying tail risk that


Variance risk premium of market index possesses return predictive power was first documented in Bollerslev, Tauchen and Zhou (2009).


relies on panel estimation from the cross section of stock returns. It then measures a stock’s sensitivity to this measure of tail risk by sorting portfolios into quintiles based on tail betaexposure, and documents that the lowest tail beta quintile is associated with the lowest future returns, while the highest tail beta quintile is associated with the highest future returns. Although Kelly and Jiang provides strong evidence that tail risk is priced, it does not directly calculate the tail risk premium nor does it examine any asymmetry in the way positive and negative tail are priced. The current paper differentiates itself from Kelly and Jiang (2014) in three critical ways. First, rather than using an aggregate measure of tail risk and indirectly examining the sensitivity of a stock’s return to this aggregate measure, the current paper directly calculates the tail risk premium for individual stocks and examines how this premium varies across the cross-section of stock returns. Specifically, the paper introduces a novel, non-parametric approach to directly determine the tail risk premium, that avoids the need for the use of liquid out of the money stock options (which don’t exist for most stocks). The second contribution is that this new approach allows tail risk to be deconstructed into both positive and negative components, where negative tail risk requires a positive return premium (investors demand a higher return today than otherwise expected for bearing negative tail risk) and positive tail risk yields a negative premium (investors are willing to accept a lower return today when there is a chance for extreme positive events). Third, almost all of the extreme jumps are concentrated in the first and tenth deciles; consequently, an analysis of deciles, and even percentiles, rather than quintiles is necessary if researchers are to better understand how extreme jump tail risk is priced. The results in this paper provide an interesting new perspective on the differential pricing of information related to negative and positive tail risk that is consistent with the predictions of 5

Unrealistic Optimism. Bollerslev, Todorov and Xu (2015) and Kelly and Jiang (2014) find evidence of pricing for negative tail risk, but neither fully examines the extent to which positive tail risk is priced. The current paper finds that both positive and negative tail risk are priced in the cross section of individual stock returns. Consistent with the behaviors predicted by Unrealistic Optimism, the evidence further reveals that information about positive tail risk and negative tail risk seem to be processed by the market differentially. Positive tail risk is efficiently priced at the monthly level; that is, the existence of a premium for bearing positive tail risk today holds no predictive power for future monthly returns. This implies that information about positive tail risk is quickly incorporated into prices. Interestingly, however, information about negative tail risk seems not to be as quickly priced; there seems to be a lag before it is fully processed and incorporated into prices. In other words, the existence of a positive premium for bearing negative tail risk today implies that the market will require a positive premium again for bearing this risk in the following month. Overall the results in this paper indicate that - at the individual stock level investors are inefficient at pricing negative tail events but relatively efficient at pricing positive tail events. In addition, this paper presents evidence that the size of the tail risk premium matters in predicting future returns. The more positive the current tail risk premium (that is, the greater the concerns about a big negative jump), the greater and longer lasting is its predictive power for future returns. This paper is organized as follows; Section II contains the literature review; Section III demonstrates individual stock level tail risk premium estimation and the data. Section IV presents the psychological framework for tail risk perception. Section V contains the tail risk premium cross-sectional pricing characteristics and cross-sectional return tests. Section VI includes robustness checks. Finally, Section VII concludes. 6


Literature Review


Tail risk In addition to the papers mentioned in the introduction, several recent papers on tail risk

are related to this paper. Chabi-Yo, Ruenzi and Weigert (2015) finds that investors are crashaverse; that is, they receive positive compensation for holding crash-sensitive stocks through the measure of “lower tail dependence” from individual stock price distributions. Bali, Cakici and Whitelaw (2014) introduces a hybrid tail risk measure that measures stock return tail covariance risk. The measure is based on the basic form of lower partial moments, and documents a significant positive premium captured in the cross section. Almeida et al. (2017) adopts the Almeida and Garcia (2016) methodology to risk neutralize the cross section of stock returns in order to construct a nonparametric tail risk measure. It finds that the risk-neutral tail risk measure possesses short-term (three to six month ahead) predictive power. The findings indicate that investors are downside tail risk averse and require compensation for bearing this risk, resulting in a positive risk premium in the cross section. Traditional finance theory assumes a normal distribution of asset returns, for which mean and variance together are sufficient to characterize the entire return distribution. The capital asset pricing model (Sharpe (1964), Lintner (1965) and Mossin (1966)) predicts that market volatility is a determinant in market equity premium. Contrary to this notion, Ang, et al. (2006) examines whether aggregate volatility innovation is priced in cross-section of stock returns, and conclude that high sensitivity stocks have subsequent low average returns and vice versa. Given this controversy, it is natural to ask is whether other return distributional characteristics are also priced in the cross section. Chang, Christofferson and Jacobs (2013) shows that the cross-section of stock returns has substantial exposure to higher moments. Cremers, Halling and Weinbaum (2015) finds 7

that although both jumps and volatility are priced in cross section, jumps seem to have larger impact on returns than does volatility. Lu and Murray (2017) constructs a proxy for bear-market risk and finds it to be negatively priced; that is, stocks with a high sensitivity to bear-market risk are found to underperform their low-sensitivity counterparts. Since jumps and higher order moments are directly related to distributional asset tails, this leads to one to conjecture that tail risk carries a negative premium in the cross section. B.

Unrealistic Optimism For almost 50 years, starting with McGuire (1960), Roth (1975), and Robertson (1977),

Unrealistic Optimism has been studied in psychology literature. Human beings have an optimistic bias about their personal risk; specifically, they perceive their own future as more optimistic compared to others. They rate negative future events as less likely to happen to themselves than to the average person and positive events as more likely to happen to themselves than to the average person (e.g., Burger and Burns (1988), Campbell et al. (2007), Harris and Guten (1979), Harris and Middleton (1994), Kirscht, et al. (1966), Lek and Bishop (1995), Otten and Pligt (1996), Perloff and Fetzer (1986), Regan, Snyder, and Kassin (1995), Weinstein (1980, 1982, 1984, 1987), Weinstein and Klein (1995), and Whitley and Hern (1991)). Recently, a flurry of research (Chowdhury, et al. (2014), Garrett and Sharot (2014), Korn, et al. (2014), Kuzmanovic, et al. (2015, 2016), Moutsiana, et al. (2013), Sharot, et al. (2012b), and Sharot, et al. (2011)) has extended the understanding of unrealistic optimism by investigating how this optimistic bias might be maintained. These studies find that participants were selectively incorporating new information in order to maintain an optimistic outlook. Specifically, Sharot, et al. (2011) and Sharot, et al. (2012a) find that humans form beliefs asymmetrically and that there exists a striking asymmetry in human belief-updating process, whereby people update their beliefs 8

more in response to information that is better than expected compared to information that is worse than expected. In addition, Moutsiana, et al. (2013) shows that humans possess a natural tendency to discount bad news while incorporating good news into beliefs and that this bias decreases with age. It is this interesting asymmetry in the way that positive and negative extreme information is incorporated that is the motivation for the current paper.


Calculation of the Tail Risk Premium in the Cross Section of Individual

Stock Returns A.

Methodology This section discusses the construction of the tail risk premium associated with jumps in

returns for individual stocks. The methodology is an innovation on the well-established notion Bollerslev, Todorov and Xu (2015), Carr and Wu (2009) and others - that the jump tail risk premium can be calculated as the difference between the expectation of the tail variation in the physical probability space (ℙ-space) and its counterpart in the risk-neutral probability space (ℚspace). To this end, we define the infinite-order polynomial variation of log returns, which captures not only the second-order (quadratic) variation (see Carr and Wu, 2009), but also the higher-order (third-order and up) variations, which Jiang and Oomen (2008) has shown to be associated with jumps in stock returns. We denote the simple return #$ = &'

ln .&


&' (&')* &')*

and logarithmic return +$ =

/ over a period from 1 − 1 to 1 . Formally, based on Merton (1976)’s jump diffusion

process, the realized infinite-order polynomial variation (ℙ4) for individual asset 5 at time t can be expressed as follows: 9


1 1 1 A D G = 2 × ( × + + × + + ⋯ + × +6,$ ) 6,$ 6,$ = 2