Predictable asset price dynamics, risk-return tradeoff, and investor behavior

Abstract

Using a testable Slutsky equation derived from a formal utility maximization model of portfolio choice under uncertainty, we examine whether the momentum component in daily returns is induced by the interaction of the intertemporal risk-return tradeoff and investor tendency to correct prior mispricing. We find that a substantial portion of short-horizon momentum is generated by the asymmetric intertemporal risk-return tradeoff that the positive risk-return relation is strengthened conditional on a prior negative market return but is attenuated conditional on a prior positive market return. With the observation of a highly positive correlation between the trading signals and price change dummies, we further explore the link between technical trading profits and the two pricing factors. Our empirical results provide strong evidence that the profits associated with technical strategies come from exploiting the same momentum component induced by the interaction of the risk-return relation and investor adjustment behavior. We conclude that technical trading profits are the result of rational pricing factors and therefore not evidence of market inefficiency.

Appendices

Appendix A: Momentum portfolio

The capital market equilibrium condition with the law of one price of risk implies that the intertemporal risk-return relation applies not only to the market portfolio but also to other portfolios. In particular, the Intertemporal Capital Asset Pricing Model (ICAPM) implies that the predicted slope from regressing a portfolio’s expected returns on the conditional covariance with the market portfolio measures the representative investor’s relative risk aversion, and hence a portfolio’s expected return can be predicted as a linear function of its conditional covariance with the market portfolio return. We examine whether the significant asymmetric intertemporal risk-return relation observed for the equal- and value-weighted market portfolios is also present for the distinctive returns on a momentum portfolio. In particular, we examine whether the momentum components embedded in the return on a momentum portfolio are explained by the asymmetric ICAPM along with the price adjustment process.

We employ daily return data on the momentum factor from the Fama–French-Carhart four-factor model over the period from November 1926 to December 2017. The momentum factor is constructed as the average return on the two high past return portfolios (large and small stocks) minus the average return on the two low past return portfolios. Thus, the momentum factor represents a zero-net investment strategy that holds long high past return portfolios and shorts low past return portfolios while maintaining approximate neutrality to market risk and the size effect. To examine our conjecture, we estimate the following asymmetric ICAPM specifications.

$$ {\text{Model 1 }}\left( {{\text{A}} - {\text{M1}}} \right):r_{i,t + 1} = \mu + \left( {\delta_{P} Pd_{t + 1} + \delta_{N} Nd_{t + 1} } \right)\hat{\sigma }_{im,t + 1} + \varepsilon_{i,t + 1} $$

(A1)

$$ {\text{Model 2 }}\left( {{\text{A}} - {\text{M2}}} \right):r_{i,t + 1} = \mu + \left( {\delta_{P} Pd_{t + 1} + \delta_{N} Nd_{t + 1} } \right)\hat{\sigma }_{im,t + 1} + \mathop \sum \nolimits_{j = 1}^{5} \phi_{j} r_{i,t - j} + \varepsilon_{i,t + 1} $$

(A2)

$$ {\text{Model 3 }}\left( {{\text{A}} - {\text{M3}}} \right):r_{i,t + 1} = \left( {\mu_{P} Pd_{t + 1} + \mu_{N} Nd_{t + 1} } \right) + \left( {\delta_{P} Pd_{t + 1} + \delta_{N} Nd_{t + 1} } \right)\hat{\sigma }_{im,t + 1} + \mathop \sum \nolimits_{j = 1}^{5} \phi_{j} r_{i,t - j} + \varepsilon_{i,t + 1} , $$

(A3)

where ${r}_{i,t+1}$ is daily realized return of the momentum factor. ${\widehat{\sigma }}_{im,t+1}$ is the conditional covariance of the momentum factor return with the market return and is estimated from the bivariate diagonal vech GARCH (1,1) model with the maximum likelihood method under the multivariate student’s t-distribution assumption. ${\phi }_{j}$ is the jth order return autocorrelation coefficient to capture the price adjustment process resulting from investor behavior to correct the prior mispricing. ${Pd}_{t+1}$ (${Nd}_{t+1}$) is the dummy variable to represent good (bad) market news and capture prior d-day consecutive positive (negative) price changes in the market portfolio, where d = 1, 2, and 3. The RRA parameter of the momentum portfolio is measured by ${\delta }_{P}$ (${\delta }_{N}$) under prior d-day consecutive positive (negative) price changes in the market portfolio. If our conjecture is correct, the RRA parameter under prior positive returns should be non-positive while the RRA parameter under prior negative returns should be strongly positive, i.e., ${\delta }_{P}\le 0$ and ${\delta }_{N}>0$.

The estimation results are presented in Appendix Table A1 with the Newey-West (1987) adjusted t-statistics reported in parentheses. Column (1) of the table shows ${\mu }_{P}>0$ and ${\mu }_{N}<0$, indicating that the average daily realized return is significantly positive after a positive return but is significantly negative after a negative return. This positive (negative) average realized return following a positive (negative) return is associated with the often-documented component in the return on the momentum strategy. Columns (2) and (3) of the table show the estimation result of the simple ICAPM with and without the AR(5) process, ${r}_{i,t+1}=\mu +\delta {\widehat{\sigma }}_{im,t+1}+{\varepsilon }_{i,t+1}$ and ${r}_{i,t+1}=\mu +\delta {\widehat{\sigma }}_{im,t+1}+{\sum }_{j=1}^{5}{\phi }_{j}{r}_{i,t-j}+{\varepsilon }_{i,t+1}$, respectively, where δ measures the constant RRA parameter in the simple ICAPM. The estimated values of δ are all positive and are statistically significant at the 1% level for all three sample periods. The results indicate that the intertemporal risk-return relation estimated by the constant RRA parameter of the ICAPM is economically and statistically positive for the momentum portfolio. This significant positive ICAPM parameter is consistent with the estimates of Bali (2008), who finds a strong positive ICAPM parameter for the various portfolios. In the other columns in the table, we examine whether the positive ICAPM parameter associated with the momentum portfolio is also distorted conditional on good and bad market news.

The results in Appendix Table A1 indicate that our main results concerning the asymmetric ICAPM parameter obtained from the equal- and value-weighted market portfolios are also present for the momentum portfolio. First, the estimation results of A-M2 and A-M3 show that the asymmetry in the intertemporal risk-return relation documented in the two market portfolios is still significant in the momentum portfolio for all three values of d. Within all three sample periods, the estimated RRA parameter of the ICAPM is significantly positive at the 5% level under a prior negative market return but insignificant under a prior positive market return, implying that good (bad) market news weakens (strengthens) the positive risk-return relation. Second, compared to the results of A-M1, the results of A-M2 and A-M3 indicate that the inclusion of the price adjustment terms significantly decreases (increases) the magnitude of ${\delta }_{P}$ (${\delta }_{N}$) for all three sample periods. The result verifies that ignoring the price adjustment process in the estimation of the ICAPM leads to model misspecification and induces an upward (downward) bias in estimates of the RRA parameter conditional on good (bad) market news. Third, the magnitude of the asymmetric ICAPM parameter estimates is greater using the dummies that indicate prior 2- and 3-day consecutive price changes, again verifying that the distortion of the positive ICAPM parameter is greater when conditioning on 2- and 3-day consecutive price changes. Fourth, the values of $\phi (1)$ are all positive and statistically significant at the 1% level for all three sample periods, indicating that the price adjustment process resulting from investor behavior to correct prior mispricing leads to return persistence.

Table 8 Momentum portfolio

Appendix B Momentum Crash

In this section, we explore the possibility that our main results documenting an asymmetric intertemporal risk-return relation along with the importance of investor adjustment behavior might also partly explain momentum crashes. Despite the significant profitability associated with a high mean return and high Sharpe ratio, the momentum strategy is known to display large negative skewness and excess kurtosis caused by large, but infrequent negative returns. These so-called momentum crashes tend to occur after down markets are followed by a rebound and a high level of ex-ante market volatility, coupled with an abrupt rise in contemporaneous market returns. Several studies attribute momentum crashes to the time-varying beta of a momentum strategy. Kothari and Shanken (1992) were the first to document the time variation in betas of the returns on sorted portfolios. Grundy and Martin (2001) also show that momentum portfolios have time-varying exposure to the market factor by construction. Using these features of momentum crashes, Daniel and Moskowitz (2016) develop an optimal dynamic momentum strategy, which significantly outperforms the standard static momentum strategy.

Our empirical results for the full period reported in Table 3 show that the estimated values of the RRA parameter conditional on severely bad market news (i.e., d = 3) are much greater than that of the RRA parameter conditional on a typical bad market news (d = 1), indicating that such severe bad market news causes a considerable increase in the market risk premium, inducing a high level of market volatility. We extend the estimation to the case with even more extreme consecutive bad market news, i.e., d = 4, 5, 6 to test whether extreme consecutive bad market news continues to cause a dramatic increase in the market risk premium. We estimate the original Model 1B (M1B) for the two market portfolios with ${h}_{m,t}^{2}$ as the conditional market volatility and the asymmetric ICAPM (A-M3) for the momentum portfolio. The estimation results are reported in Appendix Table B1, which shows that the estimated values of ${\delta }_{N}$ are much greater than estimates of the same parameter reported in Table 3. The average estimated values of ${\delta }_{N}$ for d = 4, 5, 6 are 24.222 and 14.017 for the equal- and value-weighted market portfolio, respectively, while the estimates of ${\delta }_{N}$ for d = 2 in Table 3 are only 12.607 and 8.853 for the equal- and value-weighted portfolios. The estimation results of A-M3 for the momentum portfolio show the same result; the average estimated value of ${\delta }_{N}$ for d = 4, 5, 6 is 20.860 while the estimated value of ${\delta }_{N}$ for d = 2 in Appendix Table B1 is 9.741. The estimation results imply the extreme market environments with consecutive market losses cause a dramatic increase in both the market and the individual risk premiums, which in turn reduces the market price of portfolios considerably. The results suggest that momentum crashes can be attributed not only to the negative net beta of the strategy when markets rise, but also to a severe drop in stock prices resulting from a dramatic increase in risk premiums during the period of high market stress.

See Tables 9, 10, 11.

Table 9 Momentum Crash

Table 10 Technical trading profits on residuals of momentum regression

Table 11 Technical trading profits from using residuals of Model 1A