The ${\mathit{q}}^5$ model and its consistency with the intertemporal CAPM

Highlights

• We test the consistency of the $q^5$ model of Hou, Mo, Xue, Zhang, 2019, Hou, Mo, Xue, Zhang, 2020 with the Intertemporal CAPM (ICAPM).

• Adding the expected growth factor subsumes the pricing power of the profitability factor.

• The profitability factor carries an insignificant covariance risk price.

• The expected growth factor is not a valid risk factor under the ICAPM.

• The new $q^5$ model cannot be justified as an empirical application of the ICAPM.

Abstract

In this paper, we test the consistency of the $q^5$ model of Hou et al. (2019, 2020) with Merton’s (1973) intertemporal capital asset pricing model (ICAPM) framework. We find that all but one factors in the $q^5$ model carry significantly positive covariance risk prices. The profitability factor, however, has little explanatory power for the cross-section of expected returns. The time-series tests show that the investment factor predicts a significant decline in stock market volatility, thereby being consistent with its positive price of covariance risk and satisfying the sign restrictions associated with the ICAPM. Importantly, the expected growth factor that is found to be helpful in describing cross-sectional average returns fails to predict future investment opportunities with the correct sign, which indicates that it is not a valid risk factor under the ICAPM. Overall, the ICAPM cannot be used as a theoretical background for the $q^5$ model.

Introduction

Assessing whether there is a common theoretical background that legitimates factor models has been one of the major topics in the finance literature. Motivated by the two-period $q$-theory of investment, Hou et al. (2015) propose a $q$-factor model that contains a market factor (Mkt), a size factor ($r_{\text{ME}}$), a profitability factor ($r_{\text{ROE}}$), and an investment factor ($r_{\text{I/A}}$). Cooper and Maio (2019) provide supportive evidence that both the profitability and investment factors in the $q$-factor model of Hou et al. (2015) and the five-factor model of Fama and French (2015) are consistent with Merton’s (1973) ICAPM framework, the important pillar of rational asset pricing. Liu and Zhang (2014) extend the two-period $q$-theory and show that the new multiperiod $q$-theory of investment captures the momentum effect. George et al. (2018) and Lin (2020) also find that the multiperiod $q$-theory helps explain the 52-week high anomaly of George and Hwang (2004) and the idiosyncratic momentum anomaly of Blitz et al. (2011). In a more recent study, Hou et al. (2019) augment an expected growth factor ($r_{\text{Eg}}$), which is motivated by the multiperiod $q$-theory of investment, to their $q$-factor model to form a $q^5$ model and show that this new model consistently outperforms a series of competing models in the literature by using the factor spanning test of Fama, French, 2015, Fama, French, 2016. Hou et al. (2020) further find that the $q^5$ model is the overall best performer in explaining 150 significant anomalies.

Although the multiperiod $q$-theory of investment provides a benchmark for explaining the cross-section of stock returns, it is silent regarding the driving force of the expected profitability, investment, and expected investment-to-assets growth effects on average stock returns.¹ In this paper, we directly test whether the characteristics considered by the multiperiod $q$-theory of investment can serve as proxies for covariances with the risk factors that investors require a premium to hold. That is, we investigate whether the $q^5$ model of Hou, Mo, Xue, Zhang, 2019, Hou, Mo, Xue, Zhang, 2020 can fulfil the sign restrictions that a multifactor model must satisfy to be justifiable by Merton’s (1973) ICAPM theory.

Maio and Santa-Clara (2012) show that for a given factor to be justified as an empirical application of the ICAPM, its cross-sectional return predictability must be in line with the time-series predictability of the opportunity set for the state variable associated with that factor; it must carry a significant price of covariance risk, and the state variable associated with this factor must predict at least one dimension of future investment opportunities (the excess market return and/or stock market volatility) with the correct sign. In other words, if a state variable positively (negatively) predicts expected excess market return at the aggregate level, its innovation should carry a positive (negative) price of covariance risk in the cross-section. Similarly, if a state variable displays positive (negative) predictability for stock market variance in the time-series, its innovation should carry a price of covariance risk with an opposite sign in the cross-section. The intuition is as follows: an asset that covaries positively with (the innovation in) the state variable also positively predicts future investment opportunities (an increase in future aggregate market return and/or a decrease in future market volatility). In this sense, such an asset cannot provide a hedge for reinvestment risk (adverse changes in future investment opportunities) because it would be less profitable when the aggregate market return is expected to be lower or market volatility is expected to be higher. Thus, a risk-averse rational investor will require compensation to hold the asset, implying a positive price of covariance risk for the non-market “hedging” factor.

To estimate the prices of covariance risk for the $q^5$ factors from Hou, Mo, Xue, Zhang, 2019, Hou, Mo, Xue, Zhang, 2020, we employ the ordinary least squares (OLS) and generalized least squares (GLS) cross-sectional regression (CSR) methodologies proposed by Kan et al. (2013) with robust $t$-values that account for estimation error in the covariances and potential model misspecification. As in Cooper and Maio (2019), we use 80 equity portfolios as test assets, which are deciles sorted on size, book-to-market, momentum, return on equity, operating profitability, asset growth, accruals, and net share issues. We find that the $q^5$ model cannot serve as a perfect model because the null of $H_0:\text{GLS}{R}^2=1$ can be rejected with power, which implies that the $q^5$ model is potentially misspecified. This finding collaborates the results of Lewellen et al. (2010), Kleibergen and Zhan (2015), and Gospodinov and Robotti (2020), among others, that the GLS $R^2$ statistic, not its OLS version, can be applied to judge whether a model/factor is mean-variance efficient. We also reject the null that the $q^5$ model has no pricing power for cross-sectional stock returns ($H_0:R^2=0$) at the conventional level. In addition, all but one $q^5$ factors carry significant positive prices of covariance risk using the GLS estimation, which suggests that these factors have explanatory power for the cross-section of expected returns. The covariance risk price estimates of the market factor (the coefficient of relative risk aversion (RRA)) remain in an economically plausible range, satisfying one of the ICAPM criteria from Maio and Santa-Clara (2012). Although the profitability factor earns a reliably positive risk premium, its price of covariance risk is found to be statistically insignificant, which indicates that adding the expected growth factor to the $q$-factor model of Hou et al. (2015) subsumes its pricing power and that the profitability factor fails to be a candidate for the ICAPM state variable. In this sense, our results coincide with Cochrane (2005), Kan et al. (2013), Feng et al. (2020), and Gospodinov and Robotti (2020), among others, who argue that only the prices of covariance risk allow us to identify important factors in the cross-section unless the factors in a given model are orthogonal.

To ensure that our findings are not driven by other well-known ICAPM factors, we reestimate the $q^5$ model by using the innovations in the traditional ICAPM state variables in the literature as risk factors, including the term spread (TERM), default spread (DEF), log dividend yield (DY), one-month T-bill rate (TB), and value spread (VS) (e.g., Campbell and Vuolteenaho, 2004; Hahn, Lee, 2006, Petkova, 2006; Cooper and Maio, 2019). Similar to our findings that only include the $q^5$ factors, the results show that controlling for the traditional ICAPM factors has little impact on the covariance risk price estimates of the $q^5$ factors; except for the profitability factor, all $q^5$ factors carry significantly positive GLS covariance risk price estimates, which suggests that these factors incorporate useful information about the cross-section of expected returns above and beyond that of the traditional ICAPM factors. Our findings provide evidence supporting Gospodinov and Robotti (2020), who argue that Kan et al.’s (2013) GLS CSR methodology with misspecification-robust $t$-value is robust to not only potential model misspecification but also possible lack of identification. Our results are quantitatively similar when we use the approach proposed by Gospodinov et al. (2014), which extends the distance metric introduced by Hansen and Jagannathan (HJ, 1997) to allow for model misspecification and identification failure.

We then investigate whether the equity state variables associated with the $q^5$ factors of Hou, Mo, Xue, Zhang, 2019, Hou, Mo, Xue, Zhang, 2020 are predictive of future investment opportunities. As in Maio and Santa-Clara (2012), the $q^5$ state variables are constructed using the past 60-month rolling sum on the $q^5$ factors. To address the concerns regarding the Stambaugh (1999) small-sample bias and overlapping return and variance observations, we employ the Newey and West (1987) asymptotic $t$-value and a wild bootstrap procedure that accounts for the regressors’ time-series properties. Empirically, we find that the state variable associated with the investment factor predicts a significant decrease in stock market volatility, which complements Cooper and Maio (2019), who show that the investment factor of Hou et al. (2015) correlates with stock market volatility rather than the excess market return. However, in contrast to Cooper and Maio (2019), the profitability factor displays marginal return predictability only at the 24-month horizon, conditional on other state variables. More importantly, although the state variable associated with the expected growth factor exhibits no predictive power for the excess market return at any forecasting horizons, it significantly predicts future stock market volatility with the same sign as its covariance risk price estimates. In this sense, the expected growth factor, which is found to be important in spanning the cross-section of stock returns, cannot satisfy the sign restrictions underlying the ICAPM. We also test if the predictive ability of the $q^5$ state variables for future investment opportunities is subsumed by traditional ICAPM state variables. Consistent with the benchmark evidence, we find significantly negative (positive) stock variance predictability of the investment (expected growth) state variable, which suggests that these two state variables contain useful information that cannot be captured by the alternative ICAPM state variables.

In the last part of the paper, we discuss the role of the profitability factor in the $q^5$ model of Hou, Mo, Xue, Zhang, 2019, Hou, Mo, Xue, Zhang, 2020, as it carries a statistically insignificant price of covariance risk in the CSR analysis. We first assess whether the profitability factor is redundant in the $q^5$ model using the factor spanning test, as suggested by Fama, French, 2015, Fama, French, 2016. Barillas and Shanken (2017) prove that, for traded factors, as in our case, test assets are irrelevant and that evaluating the pricing performance of a factor in a given model based on the Sharpe criterion is equivalent to testing whether that factor can be fully spanned by its exposures to the remaining factors in the model. The spanning test confirms the insignificant covariance risk price estimates of the profitability factor; the intercept in the regressions of the profitability factor on other $q^5$ factors are positive albeit insignificant at the standard level. The mean-standard deviation (MSTD) frontier analysis from Gospodinov and Robotti (2020) also yields similar resutls. One potential reason why the profitability factor adds nothing to the description of U.S. average returns is that the expected investment-to-asset changes used to form the expected growth factor are constructed with two profitability-related measures – the cash-based operating profitability of Ball et al. (2016) and the change in return on equity. Novy-Marx (2015) shows that the profitability factor in the $q$-factor model can be fully priced by the factor formed on the change in return on equity. Hou et al. (2020) also find that their expected growth factor is highly correlated with the factor formed on cash-based operating profitability, with a correlation coefficient of 0.710. Hence, the redundancy of the profitability factor in the $q^5$ model can be largely attributed to the strong pricing power of these two measures in the cross-section.

This paper relates to the work of Cooper and Maio (2019), which extends the analysis in Maio and Santa-Clara (2012) for investigating the consistency of the profitability and investment factors in several popular asset pricing models with the ICAPM. Our paper differs along the following dimensions. First, we investigate the ICAPM consistency of the new $q^5$ model of Hou, Mo, Xue, Zhang, 2019, Hou, Mo, Xue, Zhang, 2020, which augments an expected growth factor to Hou et al.’s (2015) $q$-factor model. Cooper and Maio (2019) argue that the $q$-factor model shares the same theoretical background as the Novy-Marx (2013) four-factor model and the Fama and French (2015) five-factor model. However, we find that the expected growth factor captures the pricing power of the profitability factor in the cross-section and that its associated state variable fails to predict future investment opportunities with the correct sign. In this sense, the $q^5$ model cannot be justified as an empirical application of the ICAPM. Second, we follow Merton’s (1973) ICAPM and focus on covariance risk prices, rather than risk premia, by using Kan et al.’s (2013) approach. Our findings echo previous studies and confirm that only the covariance risk prices can speak to the ability of factors to span asset prices (See Cochrane, 2005, Feng, Giglio, Xiu, 2020, Gospodinov and Robotti, 2020, Kan, Robotti, Shanken, 2013). Third, in addition to the wild bootstrap procedure, we also implement Kostakis et al.’s (2015) IVX and Johnson’s (2019) weighted least squares with ex ante return variance (WLS-EV) estimation methodologies in the predictability analysis. Forth, we test the predictability of the equity state variables across business cycles. We find evidence of return and variance predictability in both economic expansion and recession periods, which stands in stark contrast to previous studies that return predictability concentrates on economic recessions alone (Rapach, Strauss, Zhou, 2010, Huang, Jiang, Tu, Zhou, 2015, Cujean, Hasler, 2017, Golez, Koudijs, 2018). In sum, our results largely complement and extend the results provided in Cooper and Maio (2019). Our paper is also related to the recent researches that assess the pricing performance of the $q^5$ model for cross-sectional returns (e.g., Hou, Mo, Xue, Zhang, 2019, Hou, Mo, Xue, Zhang, 2019, Hou, Mo, Xue, Zhang, 2020). Our paper differs from this strand of literature by testing whether the $q^5$ model can be validated as a correct ICAPM application rather than evaluating its pricing ability in the cross-section. We should emphasize that our remarks are empirical in nature and the objective of this paper is not to discredit Hou et al.’s results. In fact, Hou et al.’s conclusions are informative and their $q^5$ model represents a significant step forward in understanding the driving forces of return anomalies and is now the new benchmark for asset pricing studies. Our objective is to point out that the $q^5$ model cannot share the same ICAPM background with the $q$-factor model. Moreover, we show that the profitability factor is redundant in this new model.

The rest of the paper is organized as follows. In Section 2, we present the ICAPM specification for the $q^5$ model and describe the data. In Sections 3 and 4, we analyze whether the covariance risk prices for the $q^5$ factors from cross-sectional asset pricing tests are consistent with their time-series predictability of future investment opportunities. Section 5 presents further evidence for the pricing performance of the profitability factor. Finally, in Section 6, we conclude.