Which of the following methods does not consider the investment’s profitability?

The investment CAPM predicts that a firm’s expected return is increasing in its expected profitability and decreasing in its investment rate (Hou, Xue, Zhang, 2015, Zhang, 2017). Empirical tests of the investment CAPM exploit the cross-sectional variation in investment and profitability and find strong support for expected return variation in those dimensions. For instance, an empirical q-factor model motivated by the investment CAPM largely subsumes other recently proposed factor premiums (see Hou et al., 2019).

We show that the cross-sectional relation between investment and profitability is crucial for interpreting empirical tests of the investment CAPM. To see this, note that the static version of the investment CAPM per Hou et al. (2015) predicts a triangular relationship between a firm’s investment rate, expected profitability, and expected returns. Expected returns are decreasing in investment holding profitability constant and increasing in profitability holding investment constant. These conditional predictions suggest that unconditional sorts may not be capable of detecting investment and profitability premiums. In particular, when investment and profitability are highly positively correlated, the cross-sectional variation in investment is primarily driven by the variation in expected profitability instead of discount rates. As a result, even though investment negatively predicts returns within the profitability buckets, unconditional sorts on investment may not identify this predictability. In sum, the investment CAPM suggests an economic reason for weak unconditional investment and profitability premiums when the cross-sectional investment-profitability correlation is high. In contrast, conditional sorts that orthogonalize investment and profitability, as in the empirical design of Hou et al. (2015)’s q-factors, can mitigate the impact of the correlation and detect the predicted premiums.

To test our predictions, we construct two samples of U.S. firms with significantly positive and negative cross-sectional relations between investment (I/A) and profitability (Roe). We classify industries featuring high I/A-Roe correlation as the High-Corr sample and industries featuring low I/A-Roe correlation as the Low-Corr sample.

We first examine unconditional investment and profitability premiums in these samples. To do so, we compute portfolio returns based on 2-by-3 size-I/A and size-Roe sorts separately within the High-Corr and Low-Corr samples. Consistent with the predictions from the investment CAPM, investment and profitability factors constructed using these double-sorted portfolios earn significantly lower average returns in the High-Corr sample compared to their full sample counterparts and the benchmark q-factors of Hou et al. (2015). The investment premium in the High-Corr sample is insignificant and less than one fifth of that for the benchmark q-factors, and the profitability premium, while statistically significant, is about three fifth of that for the q-factors. In contrast, in the Low-Corr sample, both premiums are statistically significant and indistinguishable from their full sample counterparts and the benchmark q-factors. The difference in premiums between High-Corr and Low-Corr is striking in that the fundamental investment and profitability spreads do not differ between these two samples.

Next, we investigate whether conditional sorts, which we view as the correct method to test the investment CAPM, mitigate the impact of the high investment-profitability correlation on factor premiums. We apply the empirical design of Hou et al. (2015)’s q-factors and conduct 2-by-3-by-3 independent sorts on size, I/A, and Roe. We confirm that both investment and profitability factors based on triple sorts earn significant average returns in the High-Corr sample. Both premiums are no longer statistically different from the benchmark q-factors that use the full sample. This is consistent with our prediction that triple sorts help detect premiums by properly addressing the correlation between investment and profitability.

We repeat our cross-industry analysis using the investment (CMA) and profitability (RMW) factors of Fama and French (2015). Specifically, we form High-Corr and Low-Corr samples using Fama and French (2015)’s investment (INV) and profitability (OP) measures. We observe that CMA and RMW, which are constructed based on 2-by-3 sorts on size-INV and size-OP, both earn insignificant average returns in the High-Corr sample. Meanwhile, factors based on triple sorts on size, INV, and OP earn significant average returns. This finding has important implications for constructing asset pricing factors. In particular, note that Fama and French (2015) construct CMA and RMW based on double sorts instead of triple sorts. Hence, insignificant CMA and RMW factors, such as those in our High-Corr sample, are not sufficient to reject the importance of investment and profitability for expected returns.

After understanding the impact of the investment-profitability correlation on premiums in the static investment CAPM, we extend our analysis to the dynamic setting in Hou et al. (2021). The dynamic investment CAPM predicts that expected returns are also increasing in expected investment growth. Following our prior argument, a highly positive correlation between investment and expected growth can also weaken the unconditional relation between these characteristics and expected returns. Hou et al. (2021) construct an expected growth factor based on predictions of investment-to-asset changes and identify Ball et al. (2016)’s cash-based operating profitability (Cop) as the most important determinant. Intuitively, Cop can provide a summary statistic of firms’ intangible investments and financial flexibility that are both related to expected growth. Consequently, we use Cop as a proxy for expected growth and repeat our tests in two samples featuring highly positive and negative correlations between I/A and Cop. Consistent with the dynamic investment CAPM, factors based on 2-by-3 sorted size-I/A and size-Cop portfolios earn significantly lower returns compared to their full sample counterparts in the High-CorrCop sample. The premiums in the Low-CorrCop sample are highly significant. Triple sorts mitigate the impact of the positive I/A-Cop correlation, resulting in significant I/A and Cop factors that are no longer distinguishable from their full sample counterparts.

Next, we ask whether a full orthogonalization of investment, profitability, and expected growth further mitigates the impact of the investment-profitability correlation on premiums. We conduct quadruple sorts (2-by-3-by-3-by-3) on size, I/A, Roe, and Cop within the High-Corr and Low-Corr samples based on the I/A-Roe correlation. We find that all three premiums earn significant average returns within both High-Corr and Low-Corr samples, and five out of six premiums pass the high t> 3 hurdle per Harvey et al. (2016).

Finally, we leverage insights from our cross-industry tests to address an ongoing concern about the out-of-sample robustness of investment and profitability premiums. Linnainmaa and Roberts (2018) and Wahal (2019) show that both CMA and RMW factors of Fama and French (2015) earn insignificant average returns in the pre-Compustat era.1 We use historical data from Wahal (2019) and reveal that Fama and French (2015)’s investment (INV) and operating profitability (OP) are highly positively correlated in the cross-section during the early decades and negatively correlated in the recent decades. We then construct CMA and RMW using triple sorts on size, INV, and OP. Once again, we find that investment and profitability premiums become significant once the correlation between INV and OP is properly addressed.

Our evidence suggests that the insignificance of CMA and RMW in the out-of-sample test moving back in time does not necessarily imply that the in-sample premiums are the result of data snooping. In fact, because of the highly positive correlation between investment and profitability in the early decades, the insignificant unconditional premiums and the significant conditional investment and profitability premiums are fully consistent with the investment CAPM.

Our paper contributes to the literature studying the empirical implementations of the investment CAPM. Hou et al. (2015) propose the q-factor model guided by the static investment CAPM which is derived from the first principal of investment. The q-factor model includes investment and profitability factors, in addition to market and size factors.2 Hou et al. (2021) extend the framework to a dynamic setting and introduce the q5-factor model by further including an expected growth factor. These factor models subsume other previously proposed factors, such as Fama and French (2015)’s CMA and RMW (see Hou et al., 2019), and help explain prominent quantitative security analysis strategies motivated by Graham and Dodd (1934) (see Hou et al., 2022). We uncover a nuanced insight from the investment CAPM that the cross-sectional investment-profitability relation affects the significance of investment and profitability premiums. Our findings highlight that investment and profitability factors based on triple sorts, that is, the q-factors, offer a proper test for the investment CAPM. Factors based on double sorts, such as Fama and French (2015)’s CMA and RMW, may have insignificant average returns when the investment-profitability correlation is high even though the investment CAPM is valid.

Our work also speaks to the economic interpretation of out-of-sample test results of cross-sectional premiums. A recent debate centers on whether in-sample premiums remain significant out of sample (e.g., Mclean, Pontiff, 2016, Fama, French, 2017, Linnainmaa, Roberts, 2018, Wahal, 2019, and Jensen et al., 2021). Poor out-of-sample performance of investment and profitability factors, according to traditional inference, indicates that the in-sample premiums are a statistical artifact and discourages researchers from exploring the economic channels behind the premiums. We demonstrate that the investment CAPM, that is the theoretical foundation underlying the investment and profitability factors, helps us interpret the out-of-sample test results economically. In particular, the absence of significant CMA and RMW factors in certain samples does not reject investment and profitability’s relation to expected returns. In fact, the investment CAPM predicts weak unconditional investment and profitability premiums when the fundamental investment-profitability relation is positive but strong premiums otherwise. Therefore, the insignificance of the premiums in high-correlation samples is just as consistent with the investment CAPM as their significance in other samples. In sum, our work highlights that the economic conditions (e.g., the investment-profitability correlation) under which an out-of-sample test is performed should be inspected, because they can affect whether or not we expect to find significant out-of-sample results.

The paper is organized as follows. Sections 2 and 3 present the results demonstrating the importance of the investment-profitability correlation channel in the investment CAPM in static and dynamic settings, respectively. Section 4 discusses the historical evolution of the investment-profitability correlation, and its implications for factors from the perspective of the investment CAPM. Section 5 concludes.

Conceptual framework

In this section, we discuss the correlation between investment and profitability and what it implies for the cross-section of expected returns. Consider a static, two-period version of the investment CAPM as in Hou et al. (2015). The first principle for investment implies that the conditional expected equity return of firm j, E0[Rj,1] , is given byE0[Rj,1]=E 0[Πj,1]1+a(Ij,0/Aj,0),where E0[Πj,1] is expected profitability, Ij,0/Aj,0 is the investment rate, and a>0 is a capital adjustment cost

The dynamic investment CAPM

Our analysis so far highlights a key message that the investment CAPM predicts conditional return premiums because expected returns, investment, and expected profitability form a triangular relation in the static investment CAPM. In this section, we examine whether the insight of conditional return predictability extends to the dynamic investment CAPM model.

Historical investment and profitability premiums

Using hand-collected pre-Compustat accounting data, recent literature has questioned the robustness of various asset pricing factors. In particular, Linnainmaa and Roberts (2018) construct historical investment and profitability factors following a similar methodology as in Fama and French (2015). They find that the historical CMA and RMW earn low and insignificant average returns suggesting that the investment and profitability premiums in the Compustat sample are an artifact of data snooping.

Conclusion

In this paper, we uncover two new insights from the triangular cross-sectional relation between investment, expected profitability, and expected returns predicted by the investment CAPM. First, a high cross-sectional investment-profitability correlation weakens the unconditional relation between investment and expected returns and between profitability and expected returns. We find strong empirical support for this prediction in industries featuring a high cross-sectional

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