Skip to main content
SpringerLink
Log in

Tail risk, beta anomaly, and demand for lottery: what explains cross-sectional variations in equity returns?

  • Published:
Empirical Economics Aims and scope Submit manuscript

Abstract

This study investigates the tail-risk and returns relationship in an emerging equity market—India. We observe both the beta and the tail risk anomalies at the univariate portfolio level. However, after controlling for the lottery effects (MAX and idiosyncratic volatility), the relationship between the systematic tail risk measures and the expected returns turns positive for some of the proxies of tail risk, while the relationship between beta and expected returns becomes flat. The lottery effect explains the tail risk anomaly reported in the literate. These results suggest that investors care for probabilities of extreme changes in stock prices but do not care much about moderate variations in stock returns. Emerging markets have a significant presence of individual investors, who consider investing in the stock market as an opportunity to gamble and earn lottery-like payoffs, which results in an overvaluation of lottery stocks. Since these stocks also have high systematic risk and a high probability of extreme negative returns, we observe that the expected returns are negatively correlated with beta and left tail risk measures before controlling for the lottery effects. These results are robust after controlling for other relevant variables such as the size of the firm, book-to-market ratio, momentum, retail ownership, and illiquidity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

We have all the data related to our study and we are ready to provide all the data and the research codes used in the study whenever will be asked by the journal for the verification of our results produced in the manuscript.

Notes

  1. see, for example, Harris et al. (2019), Chabi-Yo et al. (2018), Bollerslev et al. (2015), Kelly and Jiang (2014), Bali et al. (2014), Xiong et al. (2014), Huang et al. (2012).

  2. In some studies, no premium was observed for stocks with higher tail risk (e. g., Van Oordt and Zhou 2016).

  3. Earlier studies using this approach for estimating tail-beta include Post and Versijp (2007) and Bali et al (2014).

  4. Individual investors generally keep concentrated portfolio of a few stocks, but at the same time they also invest in well diversified portfolios, like mutual funds.

  5. The distribution has four parameters (location parameter \(\xi ,\) scale parameter \(\omega ,\) degree-of-freedom parameter, \(\nu\) and skewness parameter \(\alpha .\) In multivariate case of \(d\)-dimension \(\xi\), \(\nu\) and \(\nu\) are vectors of length\(d\), while \(\omega\) is a \(d\times d\) positive-definite square matrix. the Azzalini and Capitanio (2014, Chapter 6) give detail description of the distribution and the ‘sn package’ (maintained by Adelchi Azzalini) provides implementation of this distribution in Yoshiba (2018) provides detailed procedure for estimating skew-t copula and its tail dependence.

  6. We are using an unbalanced panel data as many stocks were listed after year 2000 and many other stocks were delisted during the sample period. The actual number of total sample stocks vary each month depending on availability of data. Each decile portfolio for the month contains nearest integer number of stocks dividing the number of available sample stocks by ten. Any excess or shortfall in number of stocks is adjusted in last portfolio.

  7. The value weighted average excess return is 9.98 percent per annum. The market indices (Nifty and Sensex) have also earned positive risk premium of around 3.12 percent per annum. While, the equal weighted average excess returns of the sample stocks is negative. This implies that majority of stocks have given returns lower than the risk-free rate and positive market risk premium is attributed to a small number of better performing large stocks.

  8. Max and IVOL are more correlated with overall measure of tail risk (i.e., CVaR), rather than the measures of systematic tail risk. It implies that the high correlation between CVaR and measures of lottery effect is due to idiosyncratic part of tail risk and stocks with lottery like features also have high idiosyncratic tail risk.

  9. However, the insignificant or the negative size premiums have been observed in different markets across the world in recent empirical studies and existence of size premium is being debated (see Van Dijk 2011; Astakhov et al. 2019, for a survey).

  10. Ownership percentage of retail investors explains the inverted size premium more consistently. Retail investors have higher ownership in small stocks and speculative trading activities of these investors are likely to cause overvaluation of these stocks. We will elaborate on this in the next section.

  11. Similar results were observed from Chinese market (Hai et al. 2020; Wan 2018). The literature suggest that the IVOL effect may also be caused by factors like heterogeneous belief, short sale constraints, information asymmetry, short-run return reversal and asymmetric limits to arbitrage (see Hou and Loh 2016, Footnote 1). He and Xue (2022) classify these factors in two groups—The first group focuses on some proxies for the lottery preferences of investors, including skewness and expected idiosyncratic skewness coskewness beta and idiosyncratic coskewness beta, and maximum daily return. The second group focuses on various forms of market frictions, such as order imbalances, return reversals, illiquidity and the effect of arbitrage costs.

  12. Recent literature has focused on how individuals allocate their funds among and within the several investment layers. See for example, Das et al. (2010), Baptista (2012), Parker (2021).

  13. Simultaneous investment in lottery and insurance is known as Friedman-Savage Puzzle in the literature.

  14. This trend is also observed in other emerging markets Badarinza. (2016). In developed market on the other hand, most of the investment is channelized through financial intermediaries (institutional investors) and household investors are shifting towards passive investing Anadu et al.(2020).

  15. According to RBI data only 4 percent of household financial investments are in stocks and bonds in 2018–19, while 37 percent is in the form of bank deposits, 19 percent in provident funds and other retirement saving schemes and 10 percent in government sponsored saving schemes. (Reserve Bank of India, Handbook of Statistics on Indian Economy, Table 12).

  16. Barberis and Huang (2008) argue that the investors with cumulative prospect theory utility overweight tiny probabilities of large gains.

  17. While individual investors are mainly considered responsible for lottery-effect but studies shows that the institutional investors also invest in lottery stocks for example Akbas and Genc (2020) and Agarwal et al. (2021) document preference of mutual funds for lottery stocks.

References

  • Agarwal V, Jiang L, Wen Q (2021) Why do mutual funds hold lottery stocks? J Financ Quan Ana (in press)

  • Akbas F, Genc E (2020) Do mutual fund investors overweight the probability of extreme payoffs in the return distribution? J Financ Quan Anal 55(1):223–261

    Article  Google Scholar 

  • Amihud Y (2002) Illiquidity and stock returns: cross-section and time-series effects. J Financ Mark 5(1):31–56

    Article  Google Scholar 

  • Anadu K, Kruttli M, McCabe P, Osambela E (2020) The shift from active to passive investing: risks to financial stability? Finance Ana J 76(4):23–39

    Article  Google Scholar 

  • Ang A, Chen J, Xing Y (2006a) Downside risk. Rev Financ Stud 19(4):1191–1239

    Article  Google Scholar 

  • Ang A, Hodrick RJ, Xing Y, Zhang X (2006b) The cross-section of volatility and expected returns. J Financ 61:259–299

    Article  Google Scholar 

  • Annaert J, De Ceuster M, Verstegen K (2013) Are extreme returns priced in the stock market? European evidence. J Bank Financ 37(9):3401–3411

    Article  Google Scholar 

  • Arellano-Valle RB (2010) On the information matrix of the multivariate skew-t model. Metron 68(3):371–386

    Article  Google Scholar 

  • Astakhov A, Havranek T, Novak J (2019) Firm size and stock returns: a quantitative survey. J Econ Surveys 33(5):1463–1492

    Article  Google Scholar 

  • Atilgan Y, Bali TG, Demirtas KO, Gunaydin AD (2020) Left-tail momentum: underreaction to bad news, costly arbitrage, and equity returns. J Financ Econ 135(3):725–753

    Article  Google Scholar 

  • Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J R Statis Soc Statis Method 65(2):367–389

    Article  Google Scholar 

  • Azzalini A, Capitanio A (2014) The skew-normal and related families. Camb Uni Press, IMS Mono series

    Google Scholar 

  • Badarinza C, Campbell JY, Ramadorai T (2016) International comparative household finance. Ann Rev Econ 8:111–144

    Article  Google Scholar 

  • Badarinza C, Vimal B, and Tarun R (2017) The Indian household finance landscape. India Policy Forum 2016-17. New Delhi: National Coun Appl Econ Res. Available at http://dx.doi.org/https://doi.org/10.2139/ssrn.2797680

  • Bali TG, Cakici N, Whitelaw RF (2011) Maxing out: stocks as lotteries and the cross-section of expected returns. J Financ Econ 99(2):427–446

    Article  Google Scholar 

  • Bali TG, Cakici N, Whitelaw RF (2014) Hybrid tail risk and expected stock returns: when does the tail wag the dog? Rev Asset Pric Stud 4(2):206–246

    Article  Google Scholar 

  • Bali TG, Brown SJ, Murray S, Tang Y (2017) A lottery-demand-based explanation of the beta anomaly. J Financ Quan Ana 52(6):2369–2397

    Article  Google Scholar 

  • Bali TG, Hirshleifer DA, Peng L, and Tang Y (2018) Attention, social interaction, and investor attraction to lottery stocks. In: 9th Miami Behavioral Finance Conference. Available at SSRN: https://ssrn.com/abstract=3343769

  • Baptista AM (2012) Portfolio selection with mental accounts and background risk. J Bank Financ 36(4):968–980

    Article  Google Scholar 

  • Barberis N, Huang M (2008) Stocks as lotteries: The implications of probability weighting for security prices. Amer Econ Rev 98(5):2066–2100

    Article  Google Scholar 

  • Bawa VS, Lindenberg EB (1977) Capital market equilibrium in a mean-lower partial moment framework. J Financ Econ 5(2):189–200

    Article  Google Scholar 

  • Bi J, Zhu Y (2020) Value at risk, cross-sectional returns, and the role of investor sentiment. J Emp Financ 56:1–18

    Article  Google Scholar 

  • Black F, Michael C J, and Myron S (1972) The capital asset pricing model: some empirical tests. In: Jensen MC (ed) Studies in the theory of capital markets, New York: Praeger, 1972.

  • Bollerslev T, Todorov V (2011) Tails, fears, and risk premia. J Finance 66(6):2165–2211

    Article  Google Scholar 

  • Bollerslev T, Todorov V, Xu L (2015) Tail risk premia and return predictability. J Financ Econ 118(1):113–134

    Article  Google Scholar 

  • Campbell JY, Ramadorai T, Ranish B (2019) Do the rich get richer in the stock market? Evidence from India. Ameri Econ Rev 1(2):225–240

    Google Scholar 

  • Carhart MM (1997) On persistence in mutual fund performance. J Financ 52(1):57–82

    Article  Google Scholar 

  • Chabi-Yo F, Ruenzi S, Weigert F (2018) Crash sensitivity and the cross section of expected stock returns. J Financ Quan Anal 53(3):1059–1100

    Article  Google Scholar 

  • Chichernea DC, Kassa H, Slezak SL (2019) Lottery preferences and the idiosyncratic volatility puzzle. Euro Financ Man 25(3):655–683

    Article  Google Scholar 

  • Das S, Markowitz H, Scheid J, Statman M (2010) Portfolio optimization with mental accounts. J Financ Quan Ana 45(2):311–334

    Article  Google Scholar 

  • DiTraglia FJ, Gerlach JR (2013) Portfolio selection: an extreme value approach. J Bank Financ 37(2):305–323

    Article  Google Scholar 

  • Fama EF, French KR (1993) Common risk factors in the returns on stocks and bonds. J Financ Econ 33(1):3–56

    Article  Google Scholar 

  • Fama EF, French KR (1997) Industry costs of equity. J Financ Econ 43(2):153–193

    Article  Google Scholar 

  • Fama EF, French KR (2005) Financing decisions: who issues stock? J Financ Econ 76(3):549–582

    Article  Google Scholar 

  • Fama EF, French KR (2015) A five-factor asset pricing model. J Financ Econ 116(1):1–22

    Article  Google Scholar 

  • Fama EF, MacBeth JD (1973) Risk, return, and equilibrium: empirical tests. J Polit Econ 81(3):607–636

    Article  Google Scholar 

  • Frazzini A, Pedersen LH (2014) Betting against beta. J Financ Econ 111(1):1–25

    Article  Google Scholar 

  • Gao X, Lin TC (2015) Do individual investors treat trading as a fun and exciting gambling activity? Evidence from repeated natural experiments. Rev Financ Stud 28(7):2128–2166

    Article  Google Scholar 

  • Gao GP, Lu X, Song Z (2019) Tail risk concerns everywhere. Man Science 65(7):3111–3130

    Article  Google Scholar 

  • Grinblatt M, Keloharju M (2009) Sensation seeking, overconfidence, and trading activity. J Finance 64(2):549–578

    Article  Google Scholar 

  • Han B, Kumar A (2013) Speculative retail trading and asset prices. J Financ Quan Ana 48(2):377–404

    Article  Google Scholar 

  • Harlow WV, Rao RK (1989) Asset pricing in a generalized mean-lower partial moment framework: theory and evidence. J Financ Quan Ana 24(3):285–311

    Article  Google Scholar 

  • Harris RD, Nguyen LH, Stoja E (2019) Systematic extreme downside risk. J Inter Financ Mar Inst Money 61:128–142

    Article  Google Scholar 

  • He Z, Xue W (2022) Idiosyncratic volatility puzzle exists at the country level. North Amer J Econ Financ 62:101765

    Article  Google Scholar 

  • Hou K, Loh RK (2016) Have we solved the idiosyncratic volatility puzzle? J Financ Econ 121(1):167–194

    Article  Google Scholar 

  • Huang W, Liu Q, Rhee SG, Wu F (2012) Extreme downside risk and expected stock returns. J Bank Financ 36(5):1492–1502

    Article  Google Scholar 

  • Jahankhani A (1976) EV and ES capital asset pricing models: some empirical tests. J Financ Quant Anal 11:513–528

    Article  Google Scholar 

  • Joe H (2006) Discussion of copulas: tales and facts. Extremes 9(1):37–41

    Article  Google Scholar 

  • Joe H (2015) Dependence modeling with copulas. CRC Press

    Google Scholar 

  • Kelly B, Jiang H (2014) Tail risk and asset prices. Rev Financ Stud 27(10):2841–2871

    Article  Google Scholar 

  • Kollo T, Pettere G (2010) Parameter estimation and application of the multivariate skew t-copula. In: copula theory and its applications. Springer, Berlin, Heidelberg 289–298

  • Kotz S, Nadarajah S, (2004) Multivariate t-distributions and their applications. Camb Uni Press.

  • Kumar A (2009) Who gambles in the stock market? J Financ 64(4):1889–1933

    Article  Google Scholar 

  • Lewellen J, Nagel S (2006) The conditional CAPM does not explain asset-pricing anomalies. J Financ Econ 82(2):289–314

    Article  Google Scholar 

  • Lin TC, Liu X (2018) Skewness, individual investor preference, and the cross-section of stock returns. Rev Financ 22(5):1841–1876

    Google Scholar 

  • Long H, Jiang Y, Zhu Y (2018) Idiosyncratic tail risk and expected stock returns: evidence from the Chinese stock markets. Finan Res Lett 24:129–136

    Article  Google Scholar 

  • McNeil AJ, Frey R, and Embrechts P (2015) Quantitative risk management: concepts, techniques, and tools. Prince Uni Press, revised ed

  • Newey W, West KD (1987) A simple, positive definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55:703–708

    Article  Google Scholar 

  • Oehler A, Horn M (2020) Behavioral portfolio theory revisited: lessons learned from the field. Account Financ

  • Parker FJ (2021) A goals-based theory of utility. J Behav Finance 22(1):10–25

    Article  Google Scholar 

  • Post T, Versijp P (2007) Multivariate tests for stochastic dominance efficiency of a given portfolio. J Financ Quant Anal 42(2):489–515

    Article  Google Scholar 

  • Roy AD (1952) Safety first and the holding of assets. Econometrica 20(3):431–449

    Article  Google Scholar 

  • Shefrin H, Statman M (2000) Behavioral portfolio theory. J Financ Quant Anal 35(2):127–151

    Article  Google Scholar 

  • Shefrin HM, Thaler RH (1988) The behavioral life-cycle hypothesis. Econ Inq 26(4):609–643

    Article  Google Scholar 

  • Sklar M (1959) Fonctions de repartition an dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8:229–231

    Google Scholar 

  • Van Dijk MA (2011) Is size dead? A review of the size effect in equity returns. J Bank Finance 35(12):3263–3274

    Article  Google Scholar 

  • Van Oordt MR, Zhou C (2016) Systematic tail risk. J Financ Quant Anal 51(2):685–705

    Article  Google Scholar 

  • Van Hai H, Park JW, Tsai PC, Eom C (2020) Lottery mindset, mispricing and idiosyncratic volatility puzzle: evidence from the Chinese stock market. North Amer J Econ Fin 54:101266

    Article  Google Scholar 

  • Wan X (2018) Is the idiosyncratic volatility anomaly driven by the MAX or MIN effect? Evidence from the Chinese stock market. Inter Rev Econ Fin 53:1–15

    Article  Google Scholar 

  • Xiong JX, Idzorek TM, Ibbotson RG (2014) Volatility versus tail risk: which one is compensated in equity funds? J Port Man 40(2):112–121

    Article  Google Scholar 

  • Yoshiba T (2018) Maximum likelihood estimation of skew-t copulas with its applications to stock returns. J Stat Compu Simu 88(13):2489–2506

    Article  Google Scholar 

  • Zhen F, Ruan X, Zhang JE (2020) Left-tail risk in China. Pac Basin Financ J 63:101391

    Article  Google Scholar 

Download references

Funding

No funding was received for conducting this study. The authors have no relevant financial or non-financial interests to disclose.

Author information

Authors and Affiliations

  1. Birla Institute of Technology and Science Pilani, Dubai Campus, Dubai, United Arab Emirates

    Asgar Ali

  2. Indian Institute of Management Kashipur, Udham Singh Nagar, Uttarakhand, 244713, India

    K. N. Badhani

Authors
  1. Asgar Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. N. Badhani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Asgar Ali.

Ethics declarations

Conflicts of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Ethics approval

This manuscript complies with all the ethical standards.

Informed consent

All authors are agreeing with the content of manuscript, and all give explicit consent to submit this manuscript to the publication in "Empirical Economics" Journal.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ali, A., Badhani, K.N. Tail risk, beta anomaly, and demand for lottery: what explains cross-sectional variations in equity returns?. Empir Econ 65, 775–804 (2023). https://doi.org/10.1007/s00181-022-02355-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00181-022-02355-w

Keywords

JEL Classification

Search

Navigation