Nonlinear and time-varying risk premia

Highlights

• There is a significant nonlinear and time-varying contemporary relationship between return and realized volatility, ranging from positive to negative with an up-down-up pattern.

• The relationship is positive when the trading volume is smaller than usual and is negative when the trading volume is large.

• Low and high levels of trading volume have asymmetrical influences on risk premia, with a larger absolute value of risk premia for high levels of trading volume.

• The nonlinear changing autocorrelation of returns is insignificant from zero on normal trading days and most likely different from zero on abnormal trading days.

Abstract

Facing the puzzling risk-return trade-off, this paper proposes a new model for risk premia to capture nonlinear and time-varying features under the influence of trading volume. Using high-frequency data for the US stock market in Wharton Research Data Services' Trade and Quote database, our empirical findings suggest a significant nonlinear and time-varying contemporary relationship between return and realized volatility, ranging from positive to negative with an up-down-up pattern, summarized as follows. First, the contemporary relationship is positive on inactive trading days when the trading volume is smaller than usual, in which case traders may face no new information or event uncertainty. Second, the relationship is significantly negative when the trading volume is large on active trading days, in which case traders may be overconfident and behave in a risk-seeking fashion. Third, the risk premium tends toward zero during extremely abnormal trading days. Finally, low and high levels of trading volume have asymmetrical influences on risk premia, with a larger absolute value of risk premia for high levels of trading volume. Furthermore, the nonlinear changing autocorrelation of returns is insignificant from zero on normal trading days and most likely different from zero on abnormal trading days. These results provide explanations for the conflicts between financial theoretic and empirical studies.

Introduction

Quantifying the relationship between an asset return and its risk is a fundamental but unanswered issue in finance studies despite being the subject of extensive studies for several decades (Badshah, Frijns, Knif, and Tourani-Rad (2016)). On the one hand, under the market efficiency assumption, most traditional capital asset pricing financial theories, such as the intertemporal capital asset pricing model (Merton et al. (1973), hereafter ICAPM), imply that the relationship should be positive as in French, Schwert, and Stambaugh (1987). On the other hand, it has been commonly recognized that the return and volatility of equity are negatively and asymmetrically related; see Bekaert and Wu (2000) and Badshah et al. (2016). These phenomena have been well documented by leverage effects and feedback effects. Other empirical studies contradict these conclusions with uncertain signs of the relationship between return and risk, the so-called risk-return trade-off, as in Glosten, Jagannathan, and Runkle (1993). In this paper, the nonlinear time-varying contemporary relationship between daily returns and realized volatility (hereafter RV) is investigated under the impact of trading volume, which is commonly used as a proxy for levels of news information flow. This nonlinear contemporaneous daily return-volatility relationship cannot be completely characterized by linear simultaneous or asymmetrical relationships, such as the leverage effect and feedback effect, especially at the daily frequency or higher.

Since the seminal work of Merton et al. (1973), which derives a simplified linear and time-invariant partial positive risk-return relationship in the famous ICAPM model, there are many studies to estimate and test this relationship. This risk-return trade-off is so fundamental in financial economics that it could be described as the “first fundamental law of finance”. Unfortunately, empirical studies often yield conflicting results, finding a negative or insignificant return-risk trade-off relationship rather than a positive one, as in Ghysels, Santa-Clara, and Valkanov (2005), Rossi and Timmermann (2015), Liu (2017) and the references therein. Other studies find that the relationship is unstable and time-varying; see, for example, the papers in Nyberg (2012), Kinnunen (2014), and Frazier and Liu (2016) and the references therein. These studies examine the risk-return relations more generally by relaxing the restrictive assumption that these relationships linear and time-invariant, for example, using the Markov-switching specification by Ghysels, Guérin, and Marcellino (2014), considering the asymmetrical relations as responding to the states of an economy and market timing in Wu and Lee (2015), and investigating a variety of possible shapes and potential nonlinearities inherent in return dynamics studied by Frazier and Liu (2016). These mixed and confusing results are usually thought to stem from omitted variable problems or differences of models for return and variance; see Scruggs (1998), Guo and Whitelaw (2006), and Kinnunen (2014), and the references therein. Indeed, after finding that the risk-return relation is considerably time-varying, Brandt and Wang (2010) attributed these conflicts to the limitations of the research design. Most studies assume a constant risk-return relationship over time, which is inconsistent with the understanding that investor preferences change over business cycles. Whitelaw (1994) showed that imposing a constant linear return-risk relation can lead to erroneous inferences because of the unstable risk relation.

Although there are several explanations for and many studies on those above mixed and inclusive results, they are far from sufficient. To the best of our knowledge, the nonlinear and time-varying contemporary return-risk trade-off relationship under the impact of news information flow and endogeneity of volatility is rarely studied in the literature. Facing inclusive results, the asymmetric relationship between return and volatility (risk is usually measured by volatility) has been extensively researched, which are often called leverage effects and feedback effects. The generalized autoregressive conditional heteroskedasticity (GARCH)-type models are often used to study these two effects, such as the models in Brandt and Kang (2004) and Bollerslev, Litvinova, and Tauchen (2006). The GARCH model also describes the linear contemporaneous relationship between return and volatility in its volatility-in-mean equation. However, GARCH-type models are mainly determined by lagged squared returns and lagged variance or other exogenous explanation variables, which are ex ante observable. Harvey (2001) argued that the inclusive result of the relationship between return and volatility not only depends on models but also is affected by exogenous predictors. To avoid relying on exogenous predictors and in a more flexible econometric framework, Brandt and Kang (2004) used a latent vector autoregressive (VAR) model to study contemporaneous and intertemporal relationships between the conditional mean and volatility of stock returns. Actually, this method still does not fully consider endogeneity problems. In our paper, volatility is treated as an endogenous variable directly. At the same time, our motivation also comes from the view that the relevance of the risk-return and autocorrelation can fluctuate with levels of information flow (Kinnunen (2014)). In empirical studies, a first-order autoregressive term is often included in the risk-return specification to account for market inefficiency such as non-synchronous trading (Nelson (1991)) or to test whether the lagged return can help to explain the expected return (Ghysels et al. (2005)). Therefore, the focus in this paper is on the nonlinear and time-varying features of the risk-return trade-off and market efficiency under the influences of levels of information flow, which is believed to be an interest topic and vital in finance research.

Indeed, on the one hand, our focus can be implied by the adaptive markets hypothesis (AMH) of Lo, 2004, which is based on the concept of bounded rationality and evolutionary principles. This hypothesis suggests that market participants adopt satisfactory rather than optimal behaviors through heuristics and an evolutionary process under a permanently changing market environment. Prices reflect both information and the prevailing market ecology. AMH implies that the degree of market efficiency is dynamic and context dependent, and it can change in cyclical fashion with market conditions. The first implication of AMH is that the relation between risk and reward is not stable over time, which means that “the equity risk premium is also time-varying and path-dependent” (Lo (2004)). In this study, it is argued that the level of new information is a key reflection of market conditions. This argument is in line with AMH, which posits that changing market conditions are closely linked to the type and amount of available pricing information and how market participants process and use this information. It seems natural to assume that the survival of market participants and trading strategies depends on the level of new information that should be subsumed in prices.

On the other hand, although economic models usually assume that for positive risk premiums, agents are risk averse or risk neutral, this is not always the case in reality. According to the well-known prospect theory of Kai-Ineman and Tversky (1979), researchers commonly seek risks and tend to overweight outcomes with low probability. Overconfidence is a well-known exception to the rule of risk aversion. The cognitive psychology literature shows that investors are usually overconfident about the precision of their knowledge and behave in an irrational fashion when valuing information. Since the levels of information flow are time varying, it is reasonable to infer that the risk preference of investors is also time varying, which causes risk premiums to change over time under the influence of information flow.

Additionally, trading volume relates to new information. Actually, Easley and O'hara (1992) documented that both the presence and absence of trade may signal the existence of new information. Traders observe and learn from the process of trading. In fact, Jones, Kaul, and Lipson (1994) concluded, “our evidence strongly suggests that the occurrence of transactions per se contains all of the information pertinent to the pricing of securities”. Trading volume is a good proxy for information flows (for details and other introductions, see Section 2.2.3 below).

In other words, it is necessary to relax the assumption of a linear risk-return trade-off and instead consider a nonlinear time-varying relationship under the impact of trading volume. Using high-frequency data from the US stock market, this paper provides new insights into the relationship between return and RV as well as the autocorrelation of returns. Realized volatility has become a common subject of many studies because it is superior and simpler than conventional volatility models such as GARCH and/or stochastic volatility models. RV makes full use of the available intraday information and is less noisy and more informative on the current level of volatility. Various works have explored RV, which can be used in practice; see, for examples, Andersen and Bollerslev (1998), Andersen, Bollerslev, Diebold, and Ebens (2001), Barndorff-Nielsen and Shephard (2002), Andersen, Bollerslev, Diebold, and Labys (2003), and the references therein.

Considering the differences among aggregated market, portfolio and individual stocks, we choose the S&P 500 index (SPX), the SPDR S&P 500 ETF Trust (SPY) and ten large capital companies as our study samples, and then we find strong evidence of a nonlinear and time-varying relationship between return and RV as well as autocorrelation under the impact of trading volume. The relationship can range from positive to negative nonlinearly under the effects of trading volume in a fixed and similar pattern for aggregated market, portfolio and individual stocks.

The main motivation of this study comes from the empirical analysis of the following real example by comparing the model in (Eq. (8)) for the constant regression and the model in (Eq. (9)) for the threshold regression in Section 3. To show our empirical evidence, we plot the estimations of SPX, SPY, Apple Company (AAPL), and Google Company (GOOG) representing the aggregated market, portfolio and individual stocks, respectively, in Fig. 4, Fig. 5, which present the nonlinear and constant (denoted by the green lines) relationships between returns and risk (RV), where the number of trades is used to indicate the trading volume and the proxy for information. Compared with the negative constant risk premium coefficients, these two figures obviously show that the coefficient of the risk-return trade-off is nonlinear and fluctuates with changing trading volume. The detailed results are reported in Section 3.

Our contributions in this paper can be summarized as follows. First, the risk-return relationship is positive on inactive trading days when trading volume is lower than usual. In this case, facing event uncertainty or no news, most investors are risk averse and choose “slow trade” or no trade, requiring a positive premium for risks and liquidity. However, when the trading volume is extremely low, the premium approaches zero, which may indicate the risk-neutral preference of noise traders, who dominate the trading process. As the changes in trading volume increase from negative toward zero, the relationship first increases and then decreases.

Second, the risk-return trade-off decreases to negative values on active trading days when the trading volume is higher than usual. There are three reasons for this phenomenon. First, the increase in trading activities increases the liquidity of stocks, which decreases the risks of inventory. A second reason is the increase in the proportion of informed traders who trade many shares in the direction suggested by their knowledge. Finally, overconfidence leads traders to behave in a risk-seeking fashion, chasing hot stocks and easily overacting. Such traders are likely to be irrational and prefer risk-seeking with negative premiums. However, during extremely active trading days, the relationship trends toward zero after reaching the lowest point because of the different proportions of informed and non-informed participants engaged in speculation and noise trading.

Third, the relationship is approximately zero on normal days or on slightly inactive trading days when the changes in trading volume are approximately zero. During these periods, there is no news, and traders are risk neutral. The main participants are traders who need usual liquidity and risk-neutral noise traders who trade at any time.

Finally, the absolute value of the lowest negative premium is larger than that of the highest positive premium, indicating that the risk-return trade-off is asymmetrically affected by trading volume. In addition, the negative premiums are more significant than positive premiums. This phenomenon is due mainly to multiple effects, such as the increased proportion of informed traders, overconfidence and changes in risk preference during active trading days.

These findings are much more rich and informed than those based on the assumption of a linear relationship between risk and return. Furthermore, we find strong evidence that the autocorrelation or predictability of returns (market inefficiency) is related to trading volume, although there seem to be no significant signs of autocorrelation on normal trading days. The autocorrelation of returns is much more likely to be significantly different from zero on extremely active or inactive trading days, which indicates that the stock market becomes more easily inefficient during abnormal trading days than during normal trading days.

In summary, we find strong and robust evidence that the contemporary relationship between returns and volatilities is nonlinear and time-varying under the impact of changing information flows and the market environment. This relationship has a specific and fixed fluctuating pattern related to trading volume. The autocorrelation or predictability of returns that reflects stock market efficiency is also related to trading volume but with no fixed weaving patterns. Our findings can help explain the inconclusive and mixed results of financial theoretical and empirical studies as well as contradictions among them.

The remainder of this paper is constructed as follows. First, our econometric model and its estimation method are presented in Section 2. Then, we describe the data and report the empirical results in Section 3. Next, some robustness checks for our models are presented in Section 4. Finally, Section 5 concludes the paper.