In this paper, we show that the relation between expected investment and future stock returns (i.e., the expected investment-return relation) is negative and inconsistent with the multiperiod q theory. Further analysis reveals that the expected investment change measure of Hou et al. (2018a) is a poor proxy for future investment because of the mismatch of investment characteristics and the incorrect constraint imposed on the regression.

Introduction

In a multiperiod q theory framework, the first principle of investment implies that the investment return can be expressed as R I i,t+1 = (1 − τt+1) Πi,t+1 Ai,t+1 + 1 2 (1 − τt+1)at+1( Ii,t+1 Ai,t+1 ) 2 + δi,t+1τt+1 + (1 − δi,t+1) [ 1 + (1 − τt+1)at+1( Ii,t+1 Ai,t+1 ) ] 1 + (1 − τt)at( Ii,t Ai,t ) , (1) where RI i,t is the investment return; Πi,t, Ii,t, and Ai,t are the operating profits, investment, and productive assets of firm i in time t; δi,t is the exogenous depreciation rate of capital and is firmspecific and time-varying, τt is the corporate tax rate and is time-varying, and at > 0.3 This equation predicts that, all else equal, stocks with high expected investment should earn higher expected returns than stocks with low expected investment. Hou et al. (2018a,b) argue that decomposing Equation (1) leads to two different components: a “dividend yield”, (۱−τt+1)Πi,t+1/Ai,t+1+(1/2)(1−τt+1)at+1(Ii,t+1/Ai,t+1) 2+δi,t+1τt+1 1+at(Ii,t/Ai,t) , that represents two welldocumented determinants of the expected returns (the profitability and investment-to-assets (I/A) variables) and a “capital gain”, (۱−δi,t+1)[1+(1−τt+1)at+1(Ii,t+1/Ai,t+1)] 1+at(Ii,t/Ai,t) , that represents a third return determinant. They also find that their I/A growth proxy positively relates to future stock returns. In this paper, we focus on the expected investment-return relation, which is directly predicted by the multiperiod q theory, and call attention to the fact that Hou et al.’s decomposition is misleading. By construction, the I/A growth component contains two different anomalies in firm characteristics (the expected and current investment anomalies), where the latter is the same as that in the “dividend yield” component. Therefore, the I/A growth component is not a single but rather a composite anomaly that combines the well-documented asset growth effect of Cooper et al. (2008) with a new expected investment effect.4 In addition, although the results of Hou et al. (2018a,b) appear to be consistent with the multiperiod q theory, they tell us little about whether the documented positive linkage originates from the positive relation between the expected investment and expected returns, as suggested by the q theory, the inverse of the asset growth effect that is difficult to isolate with their proxy, or the mix of both. More importantly, by using the “capital gain” component as a determinant, Hou et al. (2018a,b) simply assume that the relation between expected investment and future stock returns is positive and consistent with the multiperiod q theory, which relies on certain theoretical assumptions that may or may not be valid, whereas the expected investment-return relation remains an open question. As such, we first construct a direct expected investment measure, denoted by Et [I/A], and form an implied expected investment component, denoted by Et [εI/A], by adding current I/A back to the Hou et al.’s (2018a) expected I/A changes (Et [d 1 I/A]), and then conduct both parametric and nonparametric tests to assess their predictive power. Our results reveal that Et [d 1 I/A] positively relates to future returns, which appears to be consistent with the multiperiod q theory. In contrast, we find opposite evidence that both Et [εI/A] and Et [I/A] are negative and statistically significant return predictors. The high-minus-low strategy earns -131 basis points per month on average with a Newey-West t-statistic of -7.977 and a Sharpe ratio of -1.192, which is greater than the average return of -0.817% (t = -5.210; Sharpe ratio = -0.841) in Et [εI/A]. After controlling for the well-known market, size, book-to-market, momentum, profitability, and investment factors discussed by Carhart (1997), Fama and French (1993, 2015), and Hou et al. (2015), we find that the difference between the returns on the decile portfolios with the highest and lowest Et [I/A] (Et [εI/A]) remains negative and highly significant, and the predictability of Et [I/A] is consistently stronger than that of Et [εI/A]. There are two potential reasons why the two expected investment measures and the expected investment change measure produce contrasting conclusions. First, we find that the return spread between the highest and lowest deciles on Et [d 1 I/A] earns positive profits largely because stocks in the lowest (highest) decile are those with high (low) current I/A instead of future I/A, which suggests that Et [d 1 I/A] is a poor proxy for future investment. In contrast, Et [I/A] is an appropriate proxy because it consistently aligns with firms’ future investment. Stocks with high Et [I/A] (decile 10) are those that invest aggressively in the future (0.264), whereas stocks with low Et [I/A] (decile 1) are those that invest conservatively (0.023). Furthermore, we show that the poor linkage between Et [d 1 I/A] and future investment and the weaker predictability of Et [εI/A] are due to the biased parameter estimates, which is constructed based on the incorrect constraint imposed in the regression. Running a regression using Et [d 1 I/A] as the dependent variable is equivalent to running a constrained regression of Et [I/A] and imposing the constraint that the slope of I/A must be one. However, we show that the estimated I/A slopes from the unconstrained regressions are much less than one, which indicates that the constraint imposed by Hou et al. (2018a) is inappropriate. Running a regression by imposing the incorrect restriction can lead to biased parameter estimates (Greene, 2012), hence biasing the fitted dependent variable and the associated Et [εI/A] derived from it.