The Investment CAPM: Part II, Methods

(This blog post is the second of a 4-part sequence based on my working paper fresh from the oven: "q-factors and investment CAPM, which is a solicited, analytical essay on the big-picture of the investment CAPM. Due to its length, I am splitting it into 4 parts on my blog. The link above gives the complete pdf, which also provides detailed references.)


Methods

The investment CAPM literature has taken its key predictions to the data via a variety of approaches, including factor regressions, structural estimation, and quantitative theories.
 
Factor Models

Hou, Xue, and Zhang (2015) propose and test the q-factor model, which implements the investment CAPM via the Fama-French (1993) portfolio approach. The q-factor model says that the expected return of an asset in excess of the riskfree rate is described by its sensitivities to the market factor, a size factor, an investment factor, and a return on equity (Roe) factor. The size, investment, and Roe factors are constructed from 2 by 3 by 3 sorts on market equity, investment-to-assets, and Roe. Empirically, the q-factor model goes a long way toward summarizing the cross section of average stock returns. The model explains many anomalies that bedevil the Fama-French 3-factor model, such as Jegadeesh and Titman’s (1993) momentum (Fama and French, 1996). Most anomalies are just different manifestations of investment and profitability. The data for the q-factors and testing portfolios are available for download at global-q.org.
 
Intuition

On the one hand, sorting on net stock issues, composite issuance, book-to-market and other valuation ratios, as well as long-term reversal is closer to sorting on investment than on profitability. As such, these diverse sorts reflect their common implied sort on investment.

The flow-of-fund constraint of firms says that their uses of funds must equal their sources of funds. As such, all else equal, equity issuers should invest more and have lower costs of capital than nonissuers. In addition, firms use different capital goods in their operating activities, including working capital, physical property, plant, and equipment, and (measured) intangibles. As such, total asset growth is the most comprehensive measure of investment-to-assets, a simple measure that aggregates over investments in heterogeneous capital goods.

The value factor is redundant in the presence of the investment factor. In the investment CAPM, investment increases with marginal q, which in turn equals average q with constant returns to scale. Average q and market-to-book equity are close cousins and are identical twins without debt. As such, value stocks with low valuation ratios should invest less and, all else equal, should earn higher expected returns than growth stocks with high valuation ratios.

High valuation ratios come from a stream of positive shocks on fundamentals, and low valuation ratios a stream of negative shocks on fundamentals. Growth stocks typically have high long-term prior returns, and value stocks low long-term prior returns. As such, long-term reversal also reflects the investment factor. Firms with high long-term prior returns should invest more and have lower costs of capital than firms with low long-term prior returns.

On the other hand, sorting on earnings surprises, short-term prior returns, and financial distress is closer to sorting on profitability than on investment. As such, these diverse sorts reflect their common implied sort on profitability. Intuitively, shocks to earnings are positively correlated with shocks to returns, contemporaneously. Firms with positive earnings shocks experience immediate stock price increases, and firms with negative earnings shocks experience immediate stock price drops. As such, momentum winners should have higher expected profitability and earn higher expected returns than momentum losers.

In addition, less financially distressed firms have higher profitability and, all else equal, should earn higher expected returns than more financially distressed firms. As such, the distress anomaly is just another manifestation of the profitability factor.
 
Subsequent Work

The q-factor model has effectively ended the quarter-century reign of the Fama-French (1993) 3-factor model as the leading model in empirical asset pricing. During the long process, the q-factor model has stimulated a large subsequent literature on factor models.

Fama and French (2015) attempt to fix their 3-factor model by incorporating their own versions of the investment and profitability factors to form a 5-factor model. Fama and French (2018) further add the momentum factor, UMD, to form their 6-factor model.

However, Hou, Mo, Xue, and Zhang (2019a) show that the 4-factor q-model fully subsumes the Fama-French 6-factor model in head-to-head spanning tests. In the 1967-2018 monthly sample, the investment and Roe factors in the q-factor model earn on average 0.38% and 0.55% per month (t-value = 4.59 and 5.44), respectively. Their alphas in the Fama-French 6-factor model are 0.1% and 0.27% (t-value = 2.82 and 4.32), respectively. The Gibbons, Ross, and Shanken (GRS, 1989) test strongly rejects the null hypothesis that the 6-factor model can jointly subsume the investment and Roe factors (p-value = 0.00).

Conversely, HML, CMA (the investment factor), RMW (the profitability factor), and UMD in the Fama-French 6-factor model earn on average 0.32%, 0.3%, 0.28%, and 0.64% per month (t-value = 2.42, 3.29, 2.76, and 3.73), respectively. More important, their alphas in the q-factor regressions are economically small (tiny in many cases) and statistically insignificant: 0.05%, 0.00%, 0.03%, and 0.14% (t-value = 0.49, 0.08, 0.32, and 0.61), respectively. The GRS test cannot reject the null hypothesis that the q-factor model can jointly subsume the HML, CMA, RMW, and UMD factors (p-value = 0.79). In all, despite having two fewer factors, the q-factor model fully subsumes the Fama-French 6-factor model, including UMD.

Stambaugh and Yuan (2017) group 11 anomalies into two clusters based on pairwise cross-sectional correlations. The first cluster, denoted MGMT, contains net stock issues, composite issues, accruals, net operating assets, investment-to-assets, and the change in gross property, plant, and equipment plus the change in inventories scaled by lagged book assets. The second cluster, denoted PERF, includes failure probability, O-score, momentum, gross profitability, and return on assets. The composite scores, MGMT and PERF, are defined as a stock’s equal-weighted rankings across all the variables (realigned to yield positive low-minus-high returns) within a given cluster. Stambaugh and Yuan form their factors from monthly independent 2 by 3 sorts from interacting size with each of the composite scores.

However, Stambaugh and Yuan (2017) deviate from the standard factor construction per Fama and French (1993) in two important ways. First, the NYSE-Amex-NASDAQ breakpoints of 20th and 80th percentiles are used, as opposed to the NYSE breakpoints of 30th and 70th, when sorting on the composite scores. Second, the size factor contains stocks only in the middle portfolios of the composite score sorts, as opposed to stocks from all portfolios. The Stambaugh-Yuan factors are sensitive to their factor construction, and their nonstandard construction exaggerates their factors’ explanatory power. Most important, once replicated via the standard procedure, the MGMT and PERF factors are close to the investment and Roe factors in the q-factor model, with correlations of 0.8 and 0.84, respectively (Hou, Mo, Xue, and Zhang, 2019a).

Hou, Mo, Xue, and Zhang (2019b) perform cross-sectional forecasting regressions of future investment-to-assets changes on the log of Tobin’s q, operating cash flows, and the change in Roe. Independent 2 by 3 sorts on size and expected 1-year-ahead investment-to-assets changes yield an expected growth factor, with an average premium of 0.84% per month (t-value = 10.27) and a q-factor alpha of 0.67% (t-value = 9.75). Hou et al. augment the q-factor model with the expected growth factor to yield the  model. Using a large set of 150 anomalies that are significant with NYSE breakpoints and value-weighted returns compiled in Hou, Xue, and Zhang (2019), Hou et al. conduct a large-scale horse race of latest factor models. The  model is the best performing model that substantially outperforms the Fama-French (2018) 6-factor model. In fact, the q-factor model already compares well with the 6-factor model.

However, unlike investment and profitability, expected growth is unobservable. The performance of the  model depends on its expected growth specification, and crucially, on operating cash flows as a key predictor of future growth. As such, although its underlying intuition is clear, the  model should be interpreted primarily as a tool for dimension reduction.

Structural Estimation

Factor models only explore directional predictions of the investment CAPM. In structural estimation, one takes the model’s key equation directly to the data for econometric estimation and evaluation. Hansen and Singleton (1982) conduct the first such test for the consumption CAPM. Liu, Whited, and Zhang (2009) perform the first structural estimation for the investment CAPM. Although by no means perfect, Liu et al.’s first stab yields much more encouraging results than Hansen and Singleton’s at the consumption CAPM. The baseline investment CAPM with only physical capital manages to explain value and post-earnings-announcement drift separately, albeit not jointly. Liu and Zhang (2014) show that the baseline model can explain Jegadeesh and Titman’s (1993) momentum separately, but not simultaneously with value.

The joint estimation difficulty has been largely resolved by Goncalves, Xue, and Zhang (2019), who introduce working capital into the investment CAPM. With plausible parameter estimates, the two-capital investment CAPM manages to explain the value, momentum, investment, and Roe premiums jointly. Aggregation also plays an important role. Liu, Whited, and Zhang (2009) and Liu and Zhang (2014) construct portfolio-level predicted returns from portfolio-level accounting variables to match with portfolio-level stock returns. In contrast, Goncalves et al. use firm-level accounting variables to construct firm-level predicted returns, which are then aggregated to the portfolio level to match with portfolio-level stock returns.

A surprising insight from Goncalves, Xue, and Zhang (2019) is that value and momentum (as well as investment and Roe) are driven by related, if not identical, mechanisms. Intuitively, current investment and expected investment are locked in a “tug of war” in the investment CAPM equation. When current investment overpowers expected investment, the model predicts the value and investment premiums. When expected investment overpowers current investment, the model predicts the momentum and Roe premiums. The predicted value and investment premiums are long-lived, persisting over 3-5 years after portfolio formation. The predicted momentum and Roe premiums are short-lived, vanishing within 1 year after portfolio formation. The model dynamics are intriguingly consistent with the dynamics in the data.
 
Quantitative Theories

Zhang (2005) constructs the first neoclassical, dynamic investment model for the cross section of returns in the spirit of real business cycles (Kydland and Prescott, 1982; Long and Plosser, 1983). Instead of estimating the first-order conditions formally in structural estimation, quantitative theory studies specify a dynamic model fully, calibrate and simulate it, and compare its implied moments with observed moments in the data. Zhang highlights the role of costly reversibility in explaining the value premium. Intuitively, value firms are burdened with more unproductive capital in bad times, finding it more difficult to downsize so as to yield more cyclical and riskier cash flows and earn higher expected returns than growth firms. In contemporaneous and independent work, Cooper (2006) shows closely related mechanisms at work in a real options model. Also in a related real options model, Carlson, Fisher, and Giammarino (2014) emphasize the role of operating leverage in driving the value premium. The Zhang model, recently labelled by Clementi and Palazzo (2019) as “the standard investment model,” has served as a launching pad for a large subsequent, theoretical literature on the cross section of returns. A full review of this literature is far beyond this analytical essay.

A long-standing controversy in this theoretical literature is that the CAPM alpha of the value premium in Zhang’s (2005) model is economically small, although the average value premium itself matches that observed in the data. Subsequent studies have attempted to explain the failure of the CAPM in explaining the value premium in the post-Compustat sample by breaking the tight link between the stochastic discount factor (SDF) and the market factor with multiple aggregate shocks. Prominent examples include short- and long-run shocks (Ai and Kiku, 2013), investment-specific technological shocks (Kogan and Papanikolaou, 2013), stochastic adjustment costs (Belo, Lin, and Bazdresch, 2014). However, these 2-shock models all fail to explain the long sample evidence from 1926 onward that the CAPM alpha of the value premium is economically small and statistically insignificant.

Bai, Hou, Kung, Li, and Zhang (2019) embed disasters into a general equilibrium model with heterogeneous firms to induce strong nonlinearity in the SDF to explain the CAPM failure. Intuitively, when a disaster hits, value firms are burdened with more unproductive capital, finding it more difficult with costly reversibility to reduce capital than growth firms. As such, value firms are more exposed to the disaster risk than growth firms, giving rise to a high average value premium. However, in a finite sample, in which disasters are not realized, the estimated market beta fails to fully capture the disaster risk embedded in the value premium. Consequently, the CAPM fails to explain the value premium in a finite sample without disasters.

In the general equilibrium model of Bai, Hou, Kung, Li, and Zhang (2019), a nonlinear consumption CAPM holds by construction, yet the standard consumption CAPM fails badly in simulated data from the model. Intuitively, the aggregate consumption growth is a poor proxy for the SDF based on recursive utility. Their correlation in simulated data is close to zero. Surprisingly, the onset of disasters is not associated with particularly low contemporaneous consumption growth, and the onset of recoveries not with particularly high consumption growth.

Intuitively, when a disaster hits, the SDF spikes up immediately because investors anticipate multiple years of high marginal utility (bad times). However, consumption smoothing immediately kicks in, with forward-looking real investment falling drastically to smooth consumption. Consequently, consumption only falls cumulatively over multiple years, making the contemporaneous consumption growth a bad proxy for the SDF. Relatedly, consumption smoothing also explains why the classic CAPM performs better than the standard consumption CAPM. Because stock prices are forward-looking, the market factor is much more correlated with the SDF than the contemporaneous consumption growth.

A more recent controversy concerns the quantitative performance of the standard investment model (Zhang, 2005). Clementi and Palazzo (2019) argue that upon hit by adverse shocks, U.S. public firms have “ample latitude” to divest their unproductive assets. In particular, “each quarter on average 18.2% of firms record negative gross investment (p. 282),” suggesting that “plenty of firms downsize, at all times (p. 287),” and that there exists “no sign of irreversibility (p. 289).” Quantitatively, Clementi and Palazzo argue that for the standard investment model to explain the average value premium, its implied investment rates must be counterfactual, with a tiny fraction of negative rates and a cross-sectional volatility that is an order of magnitude smaller than that in the data.

Bai, Li, Xue, and Zhang (2019) reexamine the evidence of costly reversibility in U.S. public firms. Bai et al. document that the firm-level investment rate distribution is highly skewed to the right, with a small fraction of negative investments, 5.79%, a tiny fraction of inactive investments, 1.46%, and a large fraction of positive investments, 92.75%. The firm-level evidence is even stronger than the prior plant-level evidence in Cooper and Haltiwanger (2006). Sample criteria likely play an important role. While Cooper and Haltiwanger include only relatively large manufacturing plants in continuous operations throughout their 1972-1988 sample, Bai et al. include virtually all Compustat firms in different industries (not just manufacturing), with no restrictions on size or age.

With a careful replication effort, Bai, Li, Xue, and Zhang (2019) trace the differences between their evidence and Clementi and Palazzo’s (2019) to 3 sources. First, both studies measure gross investment rates as net investment rates plus depreciation rates. Both measure net investment rates as the net growth rates of net property, plant, and equipment (PPE) in Compustat. Bai et al. measure depreciation rates as Compustat’s depreciation over net PPE, depreciation rates that are embedded in net PPE. In contrast, Clementi and Palazzo use industry-level geometric depreciation rates estimated by Bureau of Economic Analysis, depreciation rates that are internally incompatible with net PPE in Compustat. Second, Clementi and Palazzo impose sample criteria that are nonstandard in empirical finance, such as removing firm-years with mergers and acquisitions, in which the target’s assets are more than 5% (a low cutoff) of the acquirer’s. Finally, Clementi and Palazzo also engage in a highly questionable research practice by cutting off the right tail of the quarterly investment rate distribution at 0.2.

While Clementi and Palazzo’s (2019) evidence is flawed, their point of matching investment and returns moments jointly in quantitative studies is well taken. Using Simulated Method of Moments (SMM), Bai, Li, Xue, and Zhang (2019) estimate four parameters (the upward and downward adjustment cost parameters, the fixed cost of production, and the conditional volatility of firm-specific productivity) to target seven data moments (the average value premium, the cross-sectional volatility and skewness of individual stock excess returns, the cross-sectional volatility, skewness, and persistence of investment rates, as well as the fraction of negative investment rates). The SMM estimation strongly indicates costly reversibility and operating leverage in U.S. public firms. The downward adjustment cost parameter is estimated to be 508.2 (t-value = 13.39), which is substantially higher than the upward parameter, 0.63 (t-value = 4.6). The fixed cost of production is estimated to be 0.0637 (t-value = 4.24). The model matches the average value premium of 0.43% per month (t-value = 1.97) in the 1962–2018 sample. For investment rates, the cross-sectional volatility is 62% per annum (58.5% in the data) and the fraction of negative investments 5.78% in the model (5.79% in the data). The overidentification test fails to reject the model with the seven moments (p-value = 0.59).