The Inventory Growth Spread

Transcription

1 Frederico Belo University of Minnesota Xiaoji Lin London School of Economics and Political Science Previous studies show that firms with low inventory growth outperform firms with high inventory growth in the cross-section of publicly traded firms. In addition, inventory investment is volatile and procyclical, and inventory-to-sales is persistent and countercyclical. We embed an inventory holding motive into the investment-based asset pricing framework by modeling inventory as a factor of production with convex and nonconvex adjustment costs. The augmented model simultaneously matches the large inventory growth spread in the data, as well as the time-series properties of the firm-level capital investment, inventory investment, and inventory-to-sales. Our conditional single-factor model also implies that traditional unconditional factor models such as the CAPM should fail to explain the inventory growth spread, although not with the same large pricing errors observed in the data. (JEL E22, E23, E44, G12) Inventory investment is highly volatile and positively correlated with the business cycle. This observation has led macroeconomists to examine inventory investment as an important driver in the propagation and the amplification of shocks in the economy. At the same time, the asset pricing literature documents substantial variation in risk premiums over the business cycle, both in the time series and in the cross-section. The previous facts are naturally linked if inventory investment, like physical capital investment, responds to changes in risk premiums, a measure of firms cost of capital. Thus, understanding the economic mechanism behind these facts seems important to understand both the business cycle itself and the variation in risk premiums over the business cycle, which are central questions in macroeconomics and finance. This article provides an empirical and theoretical analysis of the link between inventory investment and risk premiums in the cross-section of publicly We thank Robert Goldstein, François Gourio, Erica Li (CEPR discussant), Selale Tuzel, Pietro Veronesi (the editor), and Lu Zhang for helpful comments and suggestions. We are especially grateful to the anonymous referee for constructive comments that have significantly improved the exposition of the article. We also thank seminar participants at CEPR/Studienzentrum Gerzensee European Summer Symposium in Financial Markets, the Econometric Society World Congress, the London School of Economics and Political Science, and the University of Minnesota. Xiaoji Lin is thankful for the research support of Financial Market Group (FMG) and STICERD Research Grant. All errors are our own. Send correspondence to Frederico Belo, Department of Finance, University of Minnesota, th Avenue South, Office 3-137, Minneapolis, MN 55455; telephone: (612) [email protected]. c The Author Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please [email protected]. doi: /rfs/hhr069 Advance Access publication August 25, 2011

2 traded firms. On the asset price side, as first documented in Thomas and Zhang (2002), there is a large inventory growth spread: Firms with low inventory growth rates outperform firms with high inventory growth rates by a value between 6.6% (value-weighted) and 10.7% (equal-weighted) per annum. The inventory growth spread is pervasive: It shows up in both small and large firms, although it is larger among small firms. In addition, we show that the standard CAPM and, to a lesser extent, the Fama and French (1993) three-factor model cannot explain the size of the inventory growth spread in the data. On the real quantity side, extending previous work by Khan and Thomas (2007a) to the firm level, we document that the inventory investment rate is volatile and strongly procyclical, and the inventory-to-sales ratio is smooth, persistent, and countercyclical. To understand the empirical facts, we incorporate an inventory holding motive into the investment-based asset pricing framework (e.g., Cochrane 1991; Zhang 2005; Cooper 2006). This framework has been successfully used to explain several asset pricing facts and thus provides a natural framework for understanding the economic determinants of the inventory growth spread. The macroeconomics literature has proposed several alternative mechanisms to rationalize positive holdings of the inventory stock, typically viewed as a zero return asset. Here, following Ramey (1989), we specify inventories as a factor of production. Thus, like the physical capital stock, in this specification the inventory stock provides a flow of services to the firm. This approach can be motivated in several ways. For example, cars in the showroom are necessary to generate sales. More generally, the existence of setup costs in the production process makes it optimal for firms to accumulate half-completed items and finish them in a batch job. This procedure reduces the number of periods during which machines must be set up that is, periods in which workers are idle and thus unproductive. In addition to modeling inventories as a factor of production, we consider a flexible representation of inventory adjustment costs that includes both convex and nonconvex costs. Adjustment costs have been shown to be important to match several asset pricing facts both in the cross-section and at the aggregate level, and we investigate here the importance of these costs in the context of an investment-based model with inventory holdings. 1 In the model, firms make physical capital investment and inventory investment decisions to maximize the value of the firm for shareholders. Optimal capital and inventory investment determine firms dividends and market value, thus establishing an endogenous link between the inventory investment rate and firms risk. Through simulations, we then investigate if the model can endogenously generate a sizable inventory growth spread, and simultaneously 1 Examples include Zhang (2005) and Li, Livdan, and Zhang (2009) in the cross-section, and Cochrane (1991) and Jermann (1998) at the aggregate level. 279

3 The Review of Financial Studies / v 25 n match the properties of inventory investment and inventory holdings observed in the data. The main results from the model can be summarized as follows. On the real quantity side, we show that the model overall matches the business cycle facts well. The model is calibrated to match the time-series properties (volatility and autocorrelation) of the physical capital and inventory investment rates. The model then endogenously matches key properties of the inventoryto-sales ratio. The inventory-to-sales ratio is countercyclical, because sales are highly procyclical and co-move more with aggregate shocks than the inventory stock. In addition, the inventory-to-sales ratio is smooth and highly persistent. As emphasized in Ramey and West (1999), the high persistence of the inventory-to-sales ratio is a key stylized fact of inventories, and thus should be taken seriously in the evaluation of any candidate model of inventory behavior. On the asset pricing side, the model is calibrated to match aggregate asset pricing moments (aggregate Sharpe ratio and properties of the risk-free rate) as well as the value premium. Our analysis then shows that the model endogenously generates a sizeable inventory growth spread. For value-weighted portfolios, the inventory growth spread is 4.6% in the model versus 6.6% in the data. In addition, the model matches the pattern of the inventory growth spread across firms with different sizes observed in the data: The inventory growth spread is higher among small firms than across large firms. When we compute the model-implied levered returns, we show that the difference between the inventory growth spreads generated by the model and the data is not statistically significant. The investment-based model is less successful at replicating the failure of the CAPM across the inventory growth portfolios. Even though the model generates a pattern of pricing errors that is qualitatively consistent with the data, the size of the pricing errors is too small. In the data, the mean absolute pricing errors for the CAPM are 5.3% per annum across equal-weighted portfolios, and 2% per annum across value-weighted portfolios, while the corresponding pricing errors in the model are only 0.6% and 0.5% per annum. This result contrasts with the findings in Gomes, Kogan, and Zhang (2003) and Li, Livdan, and Zhang (2009), who show that a production economy with one aggregate systematic shock, as we have here, can replicate the failures of the CAPM. Our analysis is different because we test the CAPM using the Fama French portfolio approach, whereas the previous studies use Fama and MacBeth (1973) cross-sectional regressions. This difference is important. By performing the asset pricing tests at the portfolio level, the effect of measurement errors in the betas is significantly reduced in our approach. In addition, the sorting procedure based on a variable (inventory growth) that is correlated with the firm s true conditional beta generates portfolio betas that are significantly more stable over time than the firm-level betas. As a result, the portfolio s unconditional betas are on average closer to its true conditional betas, thus helping 280

4 improve the performance of unconditional asset pricing models at the portfolio level. To understand the economic mechanism driving the results in the model, we investigate the sensitivity of the results across alternative specifications of the model. Our analysis shows that a combination of both convex and nonconvex inventory adjustment costs is essential for the good fit of the investmentbased model on both quantities and asset prices. Without inventory adjustment costs, the inventory growth spread generated by the model is tiny, the inventory investment rate too volatile, and the inventory-to-sales ratio not persistent enough relative to the data. Intuitively, without inventory adjustment costs, firms take advantage of the costlessly adjustable inventory input (a storage technology) to smooth their dividends. In turn, this mechanism destroys the large risk dispersion generated by the standard one-capital-good investment-based model, despite the existence of physical capital adjustment costs. The negative relationship between inventory growth and future stock returns generated by the model follows naturally from standard q-theory. The mechanism is analogous to standard explanations of the well-documented negative relationship between physical capital investment and future stock returns. Since the inventory stock is a capital input that provides a flow of service over time, inventory investment responds to changes in the cost of capital (risk premium). Naturally, firms increase their inventory stock when their cost of capital is low, generating the observed negative relationship between inventory growth and future stock returns. Our contribution is thus quantitative: We show that the investment-based model augmented with inventory holdings can quantitatively match the magnitude of the link between inventory growth and future stock returns observed in the data. Taken together, the results in this article provide support for modeling inventories as a factor of production, as well as for the existence of nonconvex inventory adjustment costs, consistent with the results in Khan and Thomas (2007b). In addition, the ability of the model to simultaneously match the large inventory growth spread observed in the data and key business cycle moments in a setup with rational expectations and profit maximizing firms suggests that the observed inventory growth spread is consistent with a riskbased interpretation. Firms with low inventory investment rates have a high cost of capital (risk), which explains the high average returns of these firms in the data. The work most closely related to ours is that of Jones and Tuzel (2011), who show that risk premiums are strongly negatively related to future inventory growth at the aggregate, industry, and firm levels. In addition, they show that the effect is stronger for firms in industries that produce durables rather than nondurables, exhibit greater cyclicality in sales, and require longer lead times for new orders. We instead focus on evaluating quantitatively the ability of an investment-based asset pricing model to explain the inventory growth spread as 281

5 The Review of Financial Studies / v 25 n well as the time-series properties of inventory. Our work is also related to the large macroeconomic literature on aggregate inventory behavior and business cycles. 2 Our article contributes to this strand of literature by examining the asset pricing implications of existent macroeconomic models as well as the importance of nonconvex costs for matching asset pricing facts. Finally, our work contributes to the investment-based asset pricing literature with multiple and heterogeneous capital goods by showing the importance of inventories for explaining cross-sectional asset pricing moments. 3 The article proceeds as follows. Section 1 documents the business cycle properties of inventory investment and its related asset pricing facts in the cross-section. Section 2 presents the investment-based model with inventory holdings. Section 3 calibrates the model and reports the cross-sectional moments generated from the simulation of the model, which we compare with the real data. Section 4 provides a detailed analysis of the economic mechanisms driving the results in the model. Section 5 concludes. An online appendix with additional analysis and robustness checks is available from the author s Web pages. 1. Facts In this section, we document the business cycle properties of inventory investment and its related asset pricing facts in the cross-section. To characterize the business cycle facts, we document the summary statistics of firm-level inventory investment, physical capital investment, sales growth, and inventoryto-sales ratio. To characterize the asset pricing facts, we construct portfolios sorted on the firm-level inventory growth rate, as well as portfolios two-waysorted on size and inventory growth rate. We then investigate the characteristics of the portfolio s average stock returns and perform standard asset pricing tests. The analysis of the business cycle facts in this section complements the analysis in Khan and Thomas (2007a) for the aggregate U.S. economy (see their Table 1). The analysis of the asset pricing facts complements the analysis in Thomas and Zhang (2002); Lyandres, Sun, and Zhang (2008); and Jones and Tuzel (2011). 4 2 Our approach of modeling inventories as a factor of production follows that of Ramey (1989) and has also been used in Kydland and Prescott (1982), Christiano (1988), Jones and Tuzel (2011), and several other studies. Blinder and Maccini (1991) and Ramey and West (1999) emphasize the production-smoothing motive for holding inventories, while Kahn (1987, 1992) and Bils and Kahn (2000) focus on the stockout avoidance motive for the firm to hold inventory. More recently, Fisher and Hornstein (2000) and Khan and Thomas (2007b) stress the importance of (S,s) rules implied by nonconvex costs for explaining inventory investment over business cycles. 3 For example, Bazdresch, Belo, and Lin (2010), Lin (forthcoming), and Tuzel (2010) also study economies with multi-capital inputs (labor, R&D capital, and real estate capital, respectively). 4 Because inventory is part of accruals, the asset pricing results reported here are also related to the large literature on the accruals anomaly initiated with Sloan (1996). 282

6 The results in this section provide the facts that we use to evaluate the performance of the investment-based asset pricing model with inventory holdings that we present in Section 2 below. 1.1 Data Monthly stock returns are from the Center for Research in Security Prices (CRSP), and accounting information is from the CRSP/Compustat Merged Annual Industrial Files. The sample is from July 1965 to December We exclude from the sample any firm-year observation for which total assets or the capital stock are either zero or negative. In addition, as standard, we omit firms whose primary standard industry classification (SIC) is between 4900 and 4999 (regulated firms) or between 6000 and 6999 (financial firms). Following Fama and French (1993), we require each firm to have at least two years of data in Compustat before it is included in the sample. The data for the three Fama French factors (small-minus-big [SMB], high-minus-low [HML], and market) are from Kenneth French s Web page. The key variable for the empirical work is the firm-level inventory investment rate. We construct this variable as follows. Total inventory stock (N t ), which includes raw materials, finished goods, and work-in-progress, is given by Compustat data item INVT. We transform this variable into real terms by dividing it by the consumer price index. Net inventory investment is given by the change in the stock of total inventories in year t from year t 1 (N t N t 1 ). The inventory investment rate is then given by the ratio of the change in the stock of total inventories to the beginning of the period stock of inventories (HN t = (N t N t 1 )/N t 1 ). Thus, our inventory investment rate is effectively the real net growth rate of total inventories of the firm. Firms that do not report inventory holdings are excluded from the sample because the theory in this article does not apply to those firms. In addition, we also keep track of several other firm-level variables. To construct the physical capital investment rate (IK), we measure firm-level capital investment (I t ) by data item CAPX (capital expenditures) minus SPPE (sales of property, plant, and equipment). The physical capital stock (K t ) is given by data item PPENT (net property plant and equipment). The physical capital investment rate is then given by the ratio of physical capital investment to the beginning of the period capital stock (IK t = I t /K t 1 ). Firms sales are given by data item SALE. Real sales growth (SG) rate is thus measured by the ratio of the change in the sales from year t to year t 1 to the sales in year t 1, deflated by the consumer price index. The inventory-to-sales ratio (NS) is measured as the ratio of total inventories in year t divided by sales in year t. Market equity (Size) is price times shares outstanding at the end of December of t, from CRSP. BM is the book-equity-to-market-equity ratio, in which book equity is computed as in Fama and French (1993). Physical-capital-to-market-equity ratio (K/ME) is physical-capital stock divided by market-equity. Leverage (Lev) 283

7 The Review of Financial Studies / v 25 n Table 1 Summary statistics Percentile Correlations Mean S.D. AC1 10th 90th HN SG NS IK HN SG NS This table reports the mean, standard deviation (S.D.), autocorrelation (AC1), the 10th and 90th percentiles, and the pairwise correlation of the following variables: (1) firm-level physical capital investment rate (IK), measured by Compustat data items CAPX minus SPPE divided by PPENT; (2) net real inventory growth rate (HN), measured by the net growth rate in Compustat data item INVT deflated by the CPI; (3) net real sales growth rate (SG), measured by the net growth rate in Compustat data item SALE deflated by the CPI; and (4) inventory-to-sales ratio (NS), measured by the ratio of Compustat data items INVT and SALE. To decrease the influence of outliers, the firm-level data are winsorized at the top and bottom 1%. The data are annual, and the sample is July 1965 to December is book liabilities (given by total assets, Compustat data item AT, minus book value of equity) in year t divided by the market value of the firm (market equity plus total assets minus book value of equity). 1.2 Business cycle facts Table 1 reports summary statistics of the firm-level physical capital investment rate (IK), inventory growth rate (HN), inventory-to-sales ratio (NS), and real sales growth (SG) variables. All variables are reported at annual frequency. The firm-level inventory and physical capital investment rates are very volatile (standard deviation of 33% and 27% per annum, respectively), are positively correlated (26%), and have low autocorrelation ( 7% and 17%, respectively). In addition, both investment rates are procyclical, as measured by the positive correlation with sales growth (43% for inventory growth and 27% for physical investment). The inventory-to-sales ratio is on average 17% per annum, and it is smooth, with a standard deviation of 7%. In addition, it is persistent, with an autocorrelation of 50%, and is countercyclical (correlation with sales growth of 10%). Overall, these summary statistics are roughly consistent with those reported in Khan and Thomas (2007a) for the U.S. economy at the aggregate level. 1.3 Asset pricing facts One-way-sorted inventory growth portfolios. We construct ten oneway-sorted inventory growth portfolios as follows. Following Fama and French (1993), in June of year t, we first sort the universe of common stocks into ten portfolios based on the deciles of the cross-sectional distribution of the inventory growth rate at the end of year t 1. Once the portfolios are formed, their value- and equal-weighted returns are tracked from July of year t to June of year t + 1. The procedure is repeated in June of year t

8 Panel A in Table 2 reports the average equal- and value-weighted excess stock returns (r S, in excess of the risk-free rate) of the ten one-way-sorted inventory growth portfolios, and Panel B reports average portfolio characteristics. Consistent with the results in Thomas and Zhang (2002), both the equaland value-weighted average excess returns of these portfolios are decreasing in the inventory growth rate. The Patton and Timmermann (2010) monotonic relationship test strongly rejects the hypothesis that the average returns of these portfolios are all equal against the hypothesis that they are decreasing in the level of inventory growth, with a p-value of 0.1% for equal-weighted portfolios and 2.2% for value-weighted portfolios. The spread in the average excess returns of these portfolios is large. For equal-weighted portfolios, firms with low inventory growth rates outperform firms with high inventory growth rates (L-H) by 10.7% per annum, and this value is more than 6.6 standard errors from zero. For value-weighted portfolios, the spread is 6.6% per annum and is 3.2 standard errors from zero. We label this fact as the inventory growth spread. Panel B in Table 2 also reports the time-series average of median portfoliolevel characteristics of the inventory growth portfolios. The characteristics of these portfolios reveal that the book-to-market ratio (BM) is significantly negatively correlated with the average inventory growth rate. This fact is consistent with standard q-theory since, in general, the book-to-market ratio can be shown to be a decreasing function of firms investment rate. The average (log) size characteristic across portfolios exhibits an inverted U-shape, with the portfolios in the middle having slightly larger firms. Finally, firm leverage (Lev) and inventory growth rates are slightly negatively correlated Two-way-sorted inventory growth and size portfolios. A wellestablished fact in the empirical asset pricing literature is the observation that the link between stock return predictability and firm characteristics in the cross-section tends to be stronger among small or tiny firms. In fact, many of the so-called asset pricing anomalies are robust only across small firms and do not exist among large firms (for several examples and a survey of the literature, see Fama and French 2008). If this is also the case for the link between inventory growth and stock returns, then the economic relevance of this link is debatable because small/tiny firms tend to be less liquid and thus more difficult to trade. In addition, the universe of small firms represents a disproportionately small fraction of the overall total stock market value. To investigate if the link between inventory growth and stock returns is a robust feature of the overall economy, we form nine portfolios two-way-sorted on size (market capitalization) and inventory growth rates. At the end of June of year t, the universe of common stocks is allocated to groups based on its size and inventory growth rate at the end of year t 1. The nine portfolios are constructed as the intersection of three inventory growth and three size portfolios (the breakpoints are the 33 th and 66 th percentiles of the corresponding sorting 285

9 The Review of Financial Studies / v 25 n Table 2 One-way-sorted inventory growth portfolios Panel A: Returns Low High L-H MAE Equal-Weighted Portfolios r S [t] α [t] α FF [t] Value-Weighted Portfolios r S [t] α [t] α FF [t] Panel B: Accounting Variables HN IK Size BM K/ME Lev This table reports the average excess returns and alphas of ten portfolios sorted on inventory growth rate. Panel A reports the statistics for the following variables: r S is the average annualized ( 1200) excess stock return; α is the intercept from the monthly CAPM regression in annual percentage; and α FF is the intercept from the monthly Fama French (1993) three-factor regression in annual percentage. t are heteroskedastic and autocorrelation-consistent t-statistics. L-H stands for the low-minus-high inventory growth portfolio, and the average returns of this portfolio is the inventory growth spread. MAE is the mean absolute pricing error. Panel B reports the time-series averages of median characteristics of the ten inventory growth portfolios. HN is inventory growth rate, IK is the physical capital investment rate, Size is the log market capitalization, BM is the book-equity-to-market-equity ratio, K/ME is the physical-capital-to-market-equity ratio, and Lev is the leverage ratio. The sample is from July 1965 to December

10 variable). Once the portfolios are formed, their value- and equal-weighted returns are tracked from July of year t to June of year t + 1. The procedure is repeated in June of year t + 1. The results reported in column L-H of Table 3 show that the inventory growth spread is a robust feature of the overall economy, although it is larger among small firms. The inventory growth spread is statistically significant across small, medium, and large firms for both equal- and value-weighted portfolios. Across small firms, the inventory growth spread is 7.1% and 5.8% for equal- and value-weighted portfolios, respectively. Across large firms, the inventory growth spread is 3.3% and 3.8% for equal- and value-weighted portfolios, respectively. Although the inventory growth spread for large firms is smaller than that for small firms, the spreads are all more than 2.9 standard errors from zero for both equal- and value-weighted portfolios Asset pricing tests. We also investigate if the spread in the average returns across the inventory growth portfolios is explained by exposure to standard risk factors, as captured by the capital asset pricing model (CAPM) and the Fama and French (1993) three-factor model. To test the CAPM, we run monthly time-series regressions of the excess returns of each portfolio on a constant and the excess returns of the market portfolio (Market). To test the Fama-French three-factor model, we augment the previous CAPM regressions with the SMB and HML factors. Panel A in Table 2 reports the intercepts (alphas) for both the CAPM (α) and the Fama French three-factor model (α F F ) on the ten portfolios one-way-sorted on inventory growth. The CAPM cannot explain the pattern of the returns of these portfolios. The CAPM alphas are large and in general statistically different from zero, especially for equal-weighted portfolios. The CAPM mean absolute pricing errors are 5.3% per annum for equal-weighted portfolios and 2.0% per annum for value-weighted portfolios. The Fama-French three-factor model is more successful than the CAPM at explaining the average returns of these portfolios, but the magnitude of the alphas is still large and in general statistically significant, especially across equal-weighted portfolios. The Fama French mean absolute pricing errors are 2.5% per annum for equal-weighted portfolios and 1.6% per annum for valueweighted portfolios. The analysis of the results for the nine portfolios twoway-sorted on inventory growth and size reported in Table 3 is qualitatively similar to the analysis for the one-way-sorted inventory growth portfolios, and so its analysis is omitted here. To sum up, the results in this section document an economically large and statistically significant inventory growth spread: Firms with low inventory growth significantly outperform firms with high inventory growth. This inventory growth spread is pervasive across the economy: It shows up in small, medium, and large firms, although it is larger among small firms. Finally, the 287

11 The Review of Financial Studies / v 25 n Table 3 Two-way-sorted inventory growth and size portfolios Inventory Growth Inventory Growth Inventory Growth Low Mid High L-H Low Mid High L-H Low Mid High L-H MAE Equal-Weighted Portfolios r S α α FF CAPM FF Small Mid Big t ( r S) t(α) t ( α FF) Small Mid Big Value-Weighted Portfolios r S α α FF CAPM FF Small Mid Big FF) t(r S ) t(α) t ( α FF) Small Mid Big This table reports the summary statistics of nine portfolios two-way-sorted on inventory growth rate and firm size. r S is the average annualized ( 1200) excess stock return. α is the intercept from the monthly CAPM regression in annual percentage. α FF is the intercept from the monthly Fama French (1993) three-factor regression in annual percentage. t are heteroskedastic and autocorrelation-consistent t-statistics. Low, Mid, and High stand for the sorting on inventory growth rates, and Small, Mid, and Big stand for the sorting on size (market capitalization). L-H stands for the low-minus-high inventory growth portfolio, and the average returns of this portfolio is the inventory growth spread. MAE is the mean absolute pricing error. The sample is monthly from July 1965 to December

12 inventory growth spread is not explained by standard asset pricing models such as the CAPM and, to a lesser extent, the Fama French three-factor model An Investment-based Model with Inventory Holdings In this section, we specify an investment-based asset pricing model with inventory holdings that we use to understand the empirical evidence presented in the previous section. By studying the producers optimal production decisions, the model establishes an endogenous link between the firm s inventory growth rates and the firm s level of risk and expected stock return. 2.1 Economic environment The economy is composed of a large number of firms that produce a homogeneous good. Firms are competitive and take as given the market-determined stochastic discount factor M t,t+1, used to value the cash flows arriving in period t + 1. The existence of a strictly positive stochastic discount factor is guaranteed by a well-known existence theorem if there are no arbitrage opportunities in the market (see, for example, Cochrane 2001, Chapter 4.2) Technology. We focus on the optimal production decision problem of one firm in the economy (we suppress any firm-specific subscript to save on notation). The firm uses capital inputs K t and inventory inputs N t to produce output Y t, according to the following constant elasticity of substitution (CES) technology: Y t = e x t +z t [ α k K ρ t ] θ + (1 α k )Nt ρ ρ, (1) where α k > 0 controls the relative weight of the two inputs in the production process, 0 < θ 1 is the degree of returns to scale, and the parameter ρ determines the elasticity of substitution (ES) between physical capital and the inventory stock, defined as ES = (1 + ρ) 1. In the limit, when ρ 0 the CES aggregator collapses to the standard Cobb-Douglas case, when ρ 1 the two inputs are perfect substitutes, and when ρ the two inputs are perfect complements (Leontief). x t is aggregate productivity, and z t is firm s specific productivity. Aggregate productivity follows the process x t+1 = ˉx(1 ρ x ) + ρ x x t + σ x ε x t+1, (2) 5 In the online appendix for this article, we show that inventory growth retains its strong predictive power in cross-sectional regressions that include several firm-level stock-return predictors such as the physical capital investment rate as well as size, book-to-market ratio, momentum, asset growth, and sales growth. In addition, we show that the inventory growth spread is mostly a within-industry effect, not a cross-industry effect. 289

13 The Review of Financial Studies / v 25 n where εt+1 x is an independently and identically distributed (i.i.d.) standard normal shock. Firm-specific productivity follows the process z t+1 = ρ z z t + σ z ε z t+1, (3) where ε z t+1 is an i.i.d standard normal shock that is uncorrelated across all firms in the economy, and εt+1 x is independent of εz t+1 for each firm. In the model, the aggregate productivity shock is the driving force of economic fluctuations and systematic risk, and the firm-specific productivity shock is the driving force of firm heterogeneity. In every period t, the capital stock K t depreciates at rate δ k and is increased (or decreased) by gross investment I t. The law of motion of the capital stock is given by K t+1 = (1 δ k )K t + I t, 0 < δ k < 1. (4) Similarly, the firm s inventory stock N t depreciates at rate δ n and is increased (or decreased) by gross inventory investment H t. The law of motion of the inventory stock is given by N t+1 = (1 δ n )N t + H t, 0 < δ n < 1. (5) Following Bils and Kahn (2000) and Ramey and West (1999), net sales are measured as Sales t = Y t H t ; that is, net sales are specified by total output minus the gross investment in the inventory stock Adjustment costs and operating fixed costs. The production activity of the firm is subject to three different types of costs: physical capital adjustment costs, inventory adjustment costs, and operating fixed costs. Capital adjustment costs are specified by the following adjustment cost function ( ) 2 a + K t + c+ k It 2 K Kt t if I t > 0 K adj t 0 if I t = 0 a K t + c k 2 ( It K t ) 2 Kt if I t < 0, where c k +, c k, a+, a > 0 are constants. This specification includes both nonconvex adjustment costs, captured by the first term a +, K t, as well as convex ( ) 2 adjustment costs, captured by the last term c+, k It 2 K Kt t. In addition, we allow these costs to be asymmetric, as in Zhang (2005), to capture the fact that (6) 290

14 cutting the capital stock may be more costly than expanding it. The nonconvex costs capture the costs of adjusting capital that are independent of the size of the investment (e.g., Abel and Eberly 2002). To scale these costs across firms with different sizes, we make the size of the nonconvex cost to be proportional to the firm s size of capital stock. As standard from the q-theory of investment literature, the nonconvex and convex capital adjustment costs include planning and installation costs, learning the use of new equipment, or the fact that production is temporarily interrupted. Firms also incur in inventory adjustment costs, which are specified by the following functional form: ( ) 2 b + N t + c+ n Ht 2 N Nt t if H t > 0 Nadj t = 0 if H t = 0 (7) b N t + c n 2 ( Ht N t ) 2 Nt if H t < 0, where c + n, c n, b+, b > 0 are constants. For symmetry, this specification is similar to the specification of capital adjustment costs. Naturally, because the two capital inputs are different, the adjustment cost parameters will be different in the two specifications as well. The convex component of inventory adjustment cost in Equation (7) follows from Gomes, Kogan, and Yogo (2009) and, apart from its asymmetries, it is the standard specification used in standard q-theory of investment. The nonconvex adjustment cost specification for inventories was first proposed in Scarf (1960) and captures a fixed cost incurred when the firm adjusts its stock of inventories. This fixed cost includes, for example, the number of labor hours that the firm hires to undertake inventory investment, irrespective of the size of the investment. Using firm-level U.S. data, McCarthy and Zakrajsek (2000) provide evidence that the inventory adjustment is nonlinear and asymmetric. These features are both captured here by the asymmetry and nonconvexity of the adjustment cost function. We discuss the magnitude of the adjustment cost parameters in the calibration section below. Finally, the firm also incurs in operating fixed costs of production that are independent of firm size, which are captured by a positive parameter f. The positive fixed cost captures the existence of fixed outside opportunity costs for some scarce resources, such as managerial labor used by the firms Stochastic discount factor. Following Zhang (2005), we directly specify the stochastic discount factor without explicitly modeling the consumer s problem. The stochastic discount factor is given by log M t,t+1 = log β + γ t (x t x t+1 ) (8) γ t = γ 0 + γ 1 (x t ˉx), (9) 291

15 The Review of Financial Studies / v 25 n where M t,t+1 denotes the stochastic discount factor from time t to t + 1. The parameters {β, γ 0, γ 1 } are constants satisfying 1 > β > 0, γ 0 > 0 and γ 1 < 0. According to this specification, the risk-free rate (R f,t ) and the maximum Sharpe ratio (S R t ) in the economy are given by 1 R f,t = [ ] = 1 E t Mt,t+1 β e γ t (1 ρ x )(x t ˉx) 1 2 γ t 2σ x 2 (10) S R t = σ [ ] t Mt,t+1 [ ] = e γ t 2σ x 2 1. E t Mt,t+1 (11) Equation (8) can be motivated as a reduced-form representation of the intertemporal marginal rate of substitution for a fictitious representative consumer or the equilibrium marginal rate of transformation, as in Belo (2010). According to Equation (9), γ t is time varying and decreases in the demeaned aggregate productivity shock x t ˉx to capture the well-documented countercyclical price of risk with γ 1 < 0. The precise economic mechanism driving the countercyclical price of risk can be, for example, time-varying risk aversion, as in Campbell and Cochrane (1999). 2.2 Firm value, risk, and return All firms in the economy are assumed to be all-equity financed, so we define D t = Sales t I t Nadj t K adj t f (12) to be the dividends distributed by the firm to the shareholders. Negative dividends are interpreted as equity issuance. Define the vector of state variables as s t = (K t, N t, x t, z t ) and let V cum (s t ) be the cum-dividend market value of the firm in period t. The firm makes capital investment I t and inventory investment H t decisions to maximize its cum-dividend market value by solving the problem V cum (s t ) = max I t+ j,h t+ j, j=0.. E t M t,t+ j D t+ j, (13) subject to the capital and inventory accumulation Equations (4) and (5) and the flow of funds constraint (12) for all dates t. The operator E t [.] represents the expectation over all states of nature given all the information available at time t. In the model, risk and expected stock returns are determined endogenously along with the firm s optimal production decisions. To make the link explicit, j=0 292

16 we can evaluate the value function in Equation (13) at the optimum, V cum [ (s t ) = D t + E t Mt,t+1 V cum (s t+1 ) ] (14) [ 1 = E t Mt,t+1 Rt+1 s ] (15) E t [ R s t+1 ] = R f t R f t Cov t [ R s t+1, M t,t+1 ], (16) where Equation (14) is the Bellman equation for the value function, the Euler Equation (15) follows from the standard formula for stock return R s t+1 = V cum (s t+1 )/ [ V cum (s t ) D t ], and Equation (16) follows from simple algebra using Equation (15) and R f t = E t [ Mt,t+1 ] 1. According to Equation (16), firms whose stock returns have a high negative covariance with the stochastic discount factor (i.e., provide low returns when the marginal utility of consumption is high) are risky, and thus the average stock returns of these firms must be high in equilibrium to compensate investors for bearing the risk of holding these assets. 3. Model Implications for the Cross-section This section reports our main findings. We investigate the ability of the investment-based model to match the cross-sectional empirical facts reported in Section 1. To generate the model s implied cross-sectional moments for asset prices and quantities, we calibrate the model and simulate 200 artificial panels, each of which has 3,600 firms and 480 monthly observations. We then replicate the empirical procedures on the artificial data simulated by the model and report the cross-sample average results. All the endogenous variables in the model, including the firm s physical capital and inventory investment rates, risk, and expected returns, are functions of the state variables. Because the functional forms are not available analytically, we solve for these functions numerically. Appendix A1 provides a description of the solution algorithm and the numerical implementation of the model. 3.1 Calibration The model is solved at monthly frequency. Because all the quantity variables in the data are available only at the annual frequency, we aggregate the monthly quantity variables to the annual frequency and we calibrate the model to match selected annual moments as closely as possible. Table 4 reports the set of parameter values used to solve the model. The first set of parameters specifies the technology of the firm. The second set of parameters describes the exogenous stochastic processes that the firm faces, including the aggregate and idiosyncratic productivity shock, and the stochastic discount factor. The choice of the parameter values is based on the parameter values reported in previous studies whenever possible, or by matching known 293

17 The Review of Financial Studies / v 25 n Table 4 Parameter Values Parameter Symbol Value Technology Weight of physical capital in the production function α k 0.78 Returns to scale θ 0.70 Elasticity of substitution between capital and inventory ρ 0.5 Rate of depreciation for capital δ k 0.01 Rate of depreciation for inventory δ n 0.02 Convex parameter in capital adjustment cost c k + /c k 3/30 Convex parameter in inventory adjustment cost c n + /cn 6.5/65 Nonconvex parameter in capital adjustment cost a + /a 0.05/0.12 Nonconvex parameter in inventory adjustment cost b + /b 0.20/0.07 Operating fixed cost f Stochastic Processes Persistence coefficient of aggregate productivity ρ x /3 Conditional volatility of aggregate productivity σ x 0.007/3 Persistence coefficient of firm-specific productivity ρ z 0.97 Conditional volatility of firm-specific productivity σ z 0.10 Time-preference coefficient β Constant price of risk γ 0 50 Time-varying price of risk γ This table presents the calibrated parameter values of the baseline investment-based model. aggregate asset pricing facts, as well as key firm-level moments reported in Table 1. Firm s technology. We set the returns to scale in the production function (1) to be θ = 0.7, roughly in the lower end of the estimates in Basu and Fernald (1997) and Burnside, Eichenbaum, and Rebelo (1995). The share of capital in the production function is set to be α k = 0.78 and the elasticity of substitution between capital and inventory stock to be ρ = 0.5 to match the inventoryto-sales ratio of 17%. The capital depreciation rate δ k is set as 1% per month as in Zhang (2005). The depreciation rate of inventory is set at δ n = 2% per month, following Jones and Tuzel (2011), who argue that the depreciation rate for inventory is higher than that of physical capital. The empirical evidence on inventory adjustment costs is scarce. Although intuition suggests that inventory adjustment costs are likely to be low, the empirical evidence is mixed. Chirinko (1993) estimates inventory adjustment costs to be small. However, Chirinko s analysis takes the data (e.g., firms sales and inventory investment) as given, and thus it ignores the importance of adjustment costs in a setup in which the data are endogenously determined, as we have here. Ramey and Vine (2006) document that a typical plant in the U.S. automobile industry closes for about 4% of the total working weeks in a typical year due to inventory adjustments. Similarly, Hall (1996) reports average plant shutdowns of about 8.5 weeks each year for inventory adjustments. Although these costs have been interpreted as evidence of fixed setup costs of production 294

18 (e.g., the fixed cost of opening a plant for a week), they also show that inventory adjustments are associated with substantial losses due to interruptions in the production process for relatively long periods. Since the inventory adjustment cost parameters determine the dynamics of inventory investment, we calibrate these parameters to match as closely as possible the volatility and autocorrelation of the firm s level inventory growth rate. Convex adjustment costs are important in determining the volatilities, while nonconvex adjustment costs are crucial to match the autocorrelation. We set the convex inventory adjustment costs to be c n + = 6.5, c k = 65 and the nonconvex inventory adjustment costs to be b + = 0.20, b = The asymmetry in the adjustment cost function is consistent with the empirical evidence in McCarthy and Zakrajsek (2000), who show that firm-level inventory adjustment is nonlinear and asymmetric. Similarly, we calibrate the physical capital adjustment costs to match as closely as possible the firm-level volatility and autocorrelation of the physical capital investment rate. This exercise leads to the following combinations of parameters: The convex capital adjustment costs are set to c k + = 3, ck = 30, and the nonconvex capital adjustment costs are set to be a + = 0.05, a = The asymmetry between upward and downward costs is set to be ck /c+ k = 10, consistent with Zhang (2005). We set the operating fixed cost f to match as closely as possible the value premium in the data (5.46% per annum), defined as the difference in the average value-weighted returns of the high-minus-low decile portfolio sorted on book-to-market ratio, and at the same time generate a reasonable firm-level physical-capital-to-market-equity ratio (K/ME) of Stochastic processes. We set the persistence of the aggregate productivity shock at ρ x = /3 and its conditional volatility at σ x = 0.007/3, which roughly corresponds to the quarterly estimates in Cooley and Prescott (1995). The long-run average level of aggregate productivity, ˉx, is a scaling variable. We set ˉx = 2.1, which implies the average long-run physical capital in the economy at one-half. To calibrate the persistence parameter ρ z and the conditional volatility parameter σ z of the firm-specific productivity shock, we follow Zhang (2005) and restrict these two parameters using their implications on the degree of dispersion in the cross-sectional distribution of firms stock return volatilities. Thus, we set ρ z = 0.97 and σ z = 0.10, implying an average annual volatility of individual stock returns of 34%, approximately the value of 32% reported in Vuolteenaho (2001). Following Zhang (2005), we pin down the three parameters governing the stochastic discount factor, β, γ 0, and γ 1 in Equations (8) and (9), by matching 6 We target the physical-capital-to-market-equity ratio instead of the more standard book-equity-to-market-equity ratio because in the theoretical model we can measure the stock of physical capital but not the book value of equity (since we have two capital inputs). 295

19 The Review of Financial Studies / v 25 n three aggregate return moments: the average real interest rate, the volatility of the real interest rate, and the average Sharpe ratio in the U.S. economy (approximately 0.4). This procedure yields β = 0.994, γ 0 = 50, and γ 1 = The previous parameter values specify the main calibration of the model, which we define as the baseline model (specification 1). To help understand the economic determinants of inventory investment and risk in the model, we also consider six additional alternative specifications (specifications 2 to 6), which we analyze in Section Matching business cycle facts Panel A of Table 5 reports averages of selected firm-level moments (real quantities) of the firm-level physical capital investment rate, inventory growth, inventory-to-sales ratio, and sales growth across simulated data from the model. The table reports the results for the baseline model (specification 1) as well as across the alternative specifications of the model (specifications 2 to 7). In this section, we focus the analysis on the results for the baseline model. Overall, the baseline model successfully matches several firm-level quantity moments. The calibration of the model targets the mean inventory-to-sales ratio, as well as the volatility and autocorrelation of the physical capital investment rate and inventory growth rate. The calibration of the model matches these moments well, although the autocorrelation of inventory growth is slightly higher in the model, 7% in the data versus 11% in the model. 7 More interestingly, the model also matches quantity moments not used in the calibration. The moments for firm-level sales growth are close to the data, although the model slightly overshoots its volatility, 23% in the data versus 40% in the model, as well as its autocorrelation, 6% in the data versus 21% in the model. The implied correlations generated by the model are also qualitatively consistent with the data: Both physical capital investment and inventory growth are procyclical (positively correlated with sales growth), and the inventory-to-sales ratio is countercyclical, because sales are procyclical while the inventory stock is much smoother. However, the magnitude of these correlations is considerably higher than in the data, a result that can be partially explained by the absence of measurement error in the simulated data. These high correlations are also expected given the very simple stochastic structure of the model in which economic fluctuations are driven by one aggregate shock. Finally, and most importantly, the model also successfully matches the high autocorrelation of the inventory-to-sales ratio, 50% in the data versus 54% in the model. As emphasized by Ramey and West (1999), the high autocorrelation and the countercyclicality of the inventory-to-sales ratio is a key stylized fact 7 We do not report the mean of physical investment rate because it is pinned down by, and hence is equal to, the physical capital depreciation rate. Similarly, the mean inventory growth and sales growth are pinned down by the aggregate growth rate in the economy. Because there is no growth in the model, these values are approximately zero. 296

20 Table 5 Data versus model-implied moments across alternative calibrations Panel A: Real Quantities Avg S.D. Autocorrelation Correlation Spec. NS σ (IK) σ (HN) σ (NS) σ (SG) AC(IK) AC(HN) AC(NS) AC(SG) ρ(ik,hn) ρ(ik,sg) ρ(hn,sg) ρ(ns,sg) Data Baseline model Alternative specification without nonconvex inventory adj. costs (b + = b = 0) Alternative specification without convex inventory adj. costs (c n + = c n = 0) Alternative specification without inventory adj. costs (b + = b = c n + = c n = 0) Alternative specification without fixed operating costs ( f = 0) Alternative specification with low K and N elasticity of substitution (ρ = 0.7) Alternative specification with high K and N elasticity of substitution (ρ = 0.05) (continued) 297

21 The Review of Financial Studies / v 25 n Table 5 Data versus model-implied moments across alternative calibrations (cont.) Panel B: Asset Prices Inventory Growth Spread Sharpe Value Equal-weighted Value-weighted Spec. Ratio R f σ ( ) R f K/ME Premium All Small Mid Big All Small Mid Big Data Baseline model Alternative specification without nonconvex inventory adj. costs (b + = b = 0) Alternative specification without convex inventory adj. costs (c n + = c n = 0) Alternative specification without inventory adj. costs (b + = b = c n + = c n = 0) Alternative specification without fixed operating costs ( f = 0) Alternative specification with low K and N Elasticity of Substitution (ρ = 0.7) Alternative specification with high K and N Elasticity of Substitution (ρ = 0.05) This table presents selected moments in the data and implied by the simulation of the model under alternative specifications (Specifications 1 to 7). The reported statistics are averages from 200 samples of simulated data, each with 3,600 firms and 480 monthly observations. Panel A reports firm-level moments real quantities. It reports the mean of the inventory-to-sales ratio (Avg NS) as well as the standard deviation, denoted by σ (.), the autocorrelation, denoted by AC(.), and the correlation, denoted by ρ(x, y), of the firm-level investment rate (IK), net inventory growth rate (HN), inventory-to-(net) sales ratio (NS), and real sales growth (SG). Panel B reports aggregate and cross-sectional asset pricing moments: the aggregate Sharpe ratio, the mean and the standard deviation of the risk-free rate, the physical-capital-to-market-equity ratio (K/ME), the value-weighted value premium, as well as the equal- and value-weighted inventory growth spread across the ten one-way-sorted inventory growth portfolios (All), as well as across the portfolios two-way-sorted on inventory growth and size (Small, Mid, and Big). 298

22 of inventories and thus must be taken seriously by any candidate theoretical model of inventory behavior. 3.3 Matching asset pricing facts Panel B of Table 5 reports the averages of aggregate and selected crosssectional asset pricing moments in the data simulated from the model. The calibration of the baseline model targets key aggregate asset pricing moments, as well as the value premium. Despite the nonlinearities in the model, the model matches these moments well. The model generates a large Sharpe ratio (0.41) and a low and smooth risk-free rate (mean of 2% and volatility around 3%). The level of asset prices (firm value) in the model is also close to the data: The mean physical capital to market equity ratio (K/ME) is 0.41 in the data, and 0.32 for the model. Finally, the model matches the value premium. In our data sample, the value premium is 5.46% (value-weighted), and in the model is 5.58% One-way-sorted portfolios. Table 6 reports the average equal- and value-weighted excess returns and asset pricing test results of the ten one-waysorted inventory growth portfolios (Panel A), as well as of the nine portfolios two-way-sorted on size and inventory growth (Panel B). We replicate the construction of these portfolios and the asset pricing tests on the simulated data following the procedure used in the real data, as described in Section 1.3. The model generates a positive inventory growth spread: Firms with low inventory growth outperform firms with high inventory growth. Focusing on the returns across the ten one-way-sorted inventory growth portfolios, Panel A shows that the model generates an inventory growth spread that is more than three standard errors from zero, and is economically large. For equal-weighted portfolios, the inventory growth spread is 5.56% in the model versus 10.68% in the data. The model is considerably more successful on value-weighted portfolios. Here, the inventory growth spread is 4.61% in the model versus 6.62% in the data, a difference of only two percentage points. The magnitude of the spreads generated by the model is slightly lower than those observed in the data, which can be partially explained by the absence of debt (leverage) in our model (in the data reported in Panel B of Table 2, firms with low inventory growth rates have slightly higher leverage ratios than firms with high inventory growth rates). To test if the difference between the model s implied inventory growth spreads is significantly different from those observed in the data, Figure 1 plots the histogram of the levered inventory growth spread based on the two hundred simulated panels. We levered the inventory growth spread in the model to make it directly comparable to the data. 8 Interestingly, the inventory growth spread in 8 We compute the model-implied levered return as R e t = R a t + Lev (R a t R f t ), where R a is the return of the all-equity firm in the model, R f is the risk-free rate, and Lev is the portfolio-level average leverage ratio from Panel B in Table

23 The Review of Financial Studies / v 25 n Table 6 Inventory growth portfolios on simulated data Panel A: One-Way-Sorted Inventory Growth Portfolios Low High L-H MAE Equal-Weighted Portfolios r S [t] α [t] α FF [t] Value-Weighted Portfolios r S [t] α [t] α FF [t] (continued) 300

24 Table 6 Inventory growth portfolios on simulated data (cont.) Panel B: Two-Way-Sorted on Size and Inventory Growth Portfolios Inventory Growth Inventory Growth Inventory Growth Low Mid High L-H Low Mid High L-H Low Mid High L-H MAE Equal-Weighted Portfolios r S α α FF CAPM FF Small Mid Big t ( FF) r S) t(α) t ( α FF) Small Mid Big Value-Weighted Portfolios r S α α FF CAPM FF Small Mid Big t ( FF) r S) t(α) t ( α FF) Small Mid Big This table reports the average excess returns, alphas, and corresponding t-statistics of ten one-way-sorted inventory growth portfolios (Panel A) and nine two-way-sorted inventory growth rate and size portfolios (Panel B), constructed from data simulated by the baseline investment-based model. r S is the average annualized ( 1200) excess stock return. α is the intercept from the monthly CAPM regression in annual percentage. α FF is the intercept from the monthly Fama and French (1993) three-factor regression in annual percentage. t are heteroskedastic and autocorrelation-consistent t-statistics. Low, Mid, and High stand for the sorting on inventory growth rates, and Small, Mid, and Big stand for the sorting on size (market capitalization). L-H stands for the low-minus-high inventory growth portfolio, and the average return of this portfolio is the inventory growth spread. MAE is the mean absolute pricing error. The reported statistics are averages from 200 samples of simulated data, each with 3,600 firms and 480 monthly observations. 301

25 The Review of Financial Studies / v 25 n Figure 1 Histogram of the levered inventory growth spread across simulations The figure shows the histogram of the equal-weighted (top panels) and value-weighted (bottom panels) levered inventory growth spread across ten one-way-sorted inventory growth portfolios (left panels) as well across small firms (middle panels) and big firms (right panels) generated by the baseline investment-based model with inventory holdings. The histograms are based on data simulated by the model across 200 samples, each with 3,600 firms and 480 monthly observations. The arrow in each panel shows the corresponding inventory growth spread in the real data reported in Panels A and B of Table 2. the data is well inside the distribution of the levered inventory growth spread generated by the model across all firms, for both equal- and value-weighted portfolios (top and bottom left panels) Two-way-sorted inventory growth and size portfolios. The average returns of the nine portfolios two way-sorted on size and inventory growth generated by the model are also large and overall consistent with the pattern observed in the data. In particular, for both equal- and value-weighted portfolios, the inventory growth spread is larger among small firms than across large firms (see L-H column). However, the model predicts the inventory growth spread for medium-sized firms to be smaller than the spread for both small and large firms, a pattern that is not consistent with the data. Importantly, we cannot reject the hypothesis that the inventory growth spread in the model across firms with different sizes is equal to the corresponding inventory growth spreads in the data. As reported in Figure 1, the inventory growth spread across small and large firms observed in the data is inside the distribution of the levered inventory growth spread generated by the model (top and bottom middle and right panels) Asset pricing tests. Finally, we investigate if the model can replicate the failure of the unconditional CAPM and, to a lesser extent, of the Fama and French (1993) three-factor model in explaining the inventory growth spread. 302

26 Table 6 shows that the unconditional CAPM in the simulated data performs significantly better than in the real data. Panel A shows that the model generates a pattern of abnormal returns across the ten one-way-sorted inventory growth portfolios that is qualitatively consistent with the data: Firms with low inventory growth rates have higher abnormal returns than firms with high inventory growth rates. For equal-weighted portfolios, the CAPM alpha of the low-minus-high spread portfolio is economically large, 1.9% per annum, and this value is more than 3.5 standard errors from zero. For value-weighted portfolios, the CAPM alpha of the spread portfolio is 1.7% per annum, and this value is also more than 3.5 standard errors from zero. However, the sizes of the CAPM alphas in the model are considerably lower than those observed in the data. The mean CAPM absolute pricing errors in the model are 0.6% and 0.5% per annum for equal-weighted and value-weighted portfolios, respectively, whereas in the data the corresponding values are 5.3% and 2.0%. The analysis of the results for the Fama French three-factor model in the simulated data, as well as the asset pricing test results across the portfolios two-way-sorted on size and inventory growth (Panel B), is qualitatively similar to the analysis of the unconditional CAPM across the ten portfolios one-way-sorted on inventory growth, and so its discussion is omitted here. The inability of the investment-based model to quantitatively replicate the failure of the CAPM is perhaps surprising in light of the results reported in Gomes, Kogan, and Zhang (2003, Table 7) and Li, Livdan, and Zhang (2009, Table 3). These studies examine a production economy with only one aggregate shock, as we have here, and conclude that their theoretical models are consistent with the failure of the CAPM. Specifically, using firm-level Fama and MacBeth (1973) cross-sectional regressions, they show that betas estimated using standard rolling regressions (as in Fama and French 2002) based on past return data are negatively correlated with firms future stock returns. In addition, standard firm characteristics such as size and book-to-market ratio have predictive power for stock returns even controlling for the firm s estimated beta. The asset pricing tests reported here are different because we test the CAPM using the Fama French portfolio approach, whereas the previous studies use Fama MacBeth cross-sectional regressions. This difference is important. Consistent with the previous studies, our model replicates the failure of the CAPM using firm-level cross-sectional regressions as well (results reported in the online appendix). As emphasized in the previous studies, the standard empirical procedures used to estimate the unobserved time-varying firm-level betas are subject to large measurement errors, which in turn helps explain the weak performance of the CAPM both in the data and in the model. By performing the asset pricing tests at the portfolio level, the effect of measurement errors in the betas is significantly reduced in our approach. 303

27 The Review of Financial Studies / v 25 n In addition, the sorting procedure based on a variable (inventory growth) that is correlated with the firm s true conditional beta generates portfolio betas that are significantly more stable over time than the firm-level betas. 9 As a result, the portfolio s unconditional betas are on average closer to its true conditional betas, thus helping improve the performance of unconditional asset pricing models at the portfolio level. To replicate the failure of the CAPM at the portfolio level is likely to require changes in the setup of the investment-based model. We do not pursue these changes here because our focus is on the study of the realized inventory growth spread. A possible reason for the counterfactually good fit of the CAPM at the portfolio level in the model is the assumption of a single source of aggregate risk. The better empirical performance of multifactor models such as the Fama and French (1993) three-factor model or the Chen, Novy-Marx, and Zhang (2011) model, however, suggests that the single source of aggregate risk might not be a good assumption. Thus, adding additional sources of aggregate risk in the investment-based model (for example, investment-specific shocks along the lines of Kogan and Papanikolaou 2010) is one promising direction for improving the fit of the investment-based model on asset pricing tests. 4. Inspecting the Mechanism The previous analysis shows that the baseline investment-based model with inventory holdings generates a large inventory growth spread in the data and at the same time matches key firm-level quantity and aggregate asset pricing moments. In this section, we study the alternative specifications of the model (specifications 2 to 7) to understand the economic forces driving these results. Table 5 reports the fit of each alternative specification on both asset pricing and real quantity moments in the cross-section. In each additional specification, we vary one set of parameters at a time while keeping the other parameters equal to the calibration of the baseline model. Thus, the results from each alternative specification can be interpreted as a standard comparative static exercise. In specifications 2 to 4, we shut down nonconvex (b n + = b n = 0) or convex (b n + = b n = 0) inventory adjustment costs, both separately and jointly. In specification 5, we set the operating fixed cost to zero ( f = 0), and in specifications 6 and 7, we consider a low (ES = 0.59) and high (ES = 1.05) value of the elasticity of substitution between physical capital and inventory inputs. We do not investigate alternative specifications of the physical capital adjustment cost function because our main focus is on inventory. To help in the interpretation of the results and in understanding the fundamental determinants of inventory investment, Figure 2 plots the policy function of the gross inventory investment (i.e., the level of total inventory 9 In the simulated data, the average cross-sectional correlation between firms inventory growth rate and conditional beta is 50%. In addition, as reported in the online appendix, the standard deviation of the estimated rolling beta is 7.65 at the firm level but only 0.88 at the portfolio level. 304

28 Figure 2 Inventory investment policy function This figure plots the policy functions of gross inventory investment H(K t, N t, x t, z t ) against the current inventory stock across four alternative specifications of the investment-based model with inventory holdings: Panel A is the baseline model with both convex and nonconvex inventory adjustment costs; Panel B is a specification with only nonconvex inventory adjustment costs; Panel C is a specification with only convex inventory adjustment costs; and Panel D is a specification without any inventory adjustment costs. In the plots, we fix the aggregate productivity x t and capital K t at their respective long-run average levels of ˉx and ˉK. The inventory stock is normalized to be between zero and one. Each of these panels has two curves corresponding to the low firm-level productivity z t (solid line) and high productivity z t (dashed line). investment) as a function of the current inventory stock implied by the baseline model as well as by selected alternative specifications. In addition, to understand the determinants of firm s risk, Figure 3 plots the firm s conditional beta β(k t, N t, x t, z t ) against the firm s current inventory stock in the baseline model and its alternative specifications. The conditional beta is given by β t Cov t [ R s t+1, M t,t+1 ] V art [ Mt,t+1 ] 1, which follows from Equation (16). 10 [ 10 Specifically, we can write Equation (16) in the standard expected return-beta form as E t R t+1 s ] = R f t + β t λ t, in which β t Cov t [R t+1 s ], M [ ] 1 t,t+1 V ar t Mt,t+1 is the quantity of risk of the asset, and λt R f t [ ] V ar t Mt,t+1 is the price of risk. 305

29 The Review of Financial Studies / v 25 n Figure 3 Fundamental determinants of risk This figure plots the firm s conditional beta β(k t, B t, x t, z t ), given by β t Cov t [R t+1 s ], M t,t+1 V ar t [ Mt,t+1 ] 1, which follows from Equation (16), against the current level of inventory stock. Each panel reports the conditional beta for the baseline model as well as for an alternative specification of the model: Panel A is the baseline model across two different levels of productivity (high and low); Panel B is a model with only nonconvex inventory adjustment costs; Panel C is a model with only convex inventory adjustment costs; Panel D is a model without inventory adjustment costs; Panel E is a model with no fixed operating costs; and Panel F is a model with two different levels of physical capital and inventory elasticity of substitution (ES) (high and low). In Panels B to F, we fix the aggregate productivity x t, firm-level productivity z t, and physical capital K t stock at their respective long-run average levels of ˉx, ˉz, and ˉK. The inventory stock is normalized to be between zero and one. 4.1 The role of nonconvex and convex inventory adjustment costs To investigate the importance of inventory adjustment costs for inventory investment dynamics and risk, we examine the results from the alternative specifications 2 to 4. Figure 2 shows that, all else equal, more productive firms have higher levels of inventory investment. In the baseline model (Panel A), for relatively low levels of inventory stocks, inventory investment is increasing in the current level of the inventory stock, but at a diminishing rate. This diminishing rate is due to convex inventory adjustment costs that penalize the firm from making large and swift changes in the inventory stock. Importantly, firms inventory investment and disinvestment decisions are associated with inventory stock thresholds levels due to nonconvex adjustment costs. These thresholds are the levels of inventory stock at which the fixed 306

30 costs of adjusting inventory are equal to the difference between the marginal benefits and marginal convex costs of making the adjustment. The region inside the thresholds is the inaction region where firms find it optimal not to make any changes in the inventory stock. The baseline model s implied optimal inventory investment policy is close to an (s, S) type rule, which leads to lumpiness in inventory investment. In the model with only nonconvex inventory adjustment costs (Panel B), the firm makes full inventory adjustments to the optimal frictionless level except across a small range of current inventory stock (again, the inaction region) in which the fixed costs of inventory adjustment are too large relative to the benefits from adjusting the inventory stock. In the model with only convex adjustment costs (Panel C), firms continuously adjust their inventory stock (except at the point of zero inventory investment), but at a diminishing rate up to a certain level, which gives rise to the concave policy function. Finally, without any inventory adjustment costs (Panel D), firms always adjust to the optimal inventory stock level. The qualitative properties of the inventory investment policy function help explain the poor fit of the alternative specifications 2 to 4 reported in Table 5. Clearly, both nonconvex and convex inventory adjustment costs are crucial for the model to match the data, both on the quantity side and on the asset pricing side. In terms of real quantities (Panel A), specification 2 shows that by removing nonconvex inventory adjustment costs, the model counterfactually generates an investment growth rate that is too persistent (50% here versus 11% in the baseline model and 7% in the data). Consistent with the analysis in Panel C of Figure 2, with only convex inventory adjustment costs, firms spread the inventory investment over time due to increasing marginal adjustment costs. In turn, this makes the inventory investment considerably more persistent than in the data. The results for specification 3 show that by removing convex adjustment costs, the model generates an investment growth rate that is too volatile (77% here versus 30% in the baseline model and 33% in the data). Consistent with the analysis in Panel B of Figure 2, with only nonconvex inventory adjustment costs, inventory investment is lumpy, with occasional infrequent but very large inventory investments. In turn, this makes the unconditional volatility of inventory investment too high relative to the data. Specification 4 shows that this unreasonably large volatility of inventory investment remains the main problem in a specification of the model in which both convex and nonconvex adjustment costs are eliminated. Turning to the analysis of the effects of inventory adjustment costs on asset prices, Panels A to D of Figure 3 plot the firm s conditional beta in the baseline model and across the alternative specifications of the inventory adjustment cost function considered here. According to Panel A, all else equal, a firm s risk is decreasing in the firm s productivity and decreases with the inventory stock. Because inventory investment is increasing in productivity, this fact 307

31 The Review of Financial Studies / v 25 n helps explain the negative correlation between firms, inventory growth rates and future returns observed in the data. Importantly, eliminating either convex (Panel B), nonconvex (Panel C), or both (Panel D) significantly reduces the firm s risk relative to the baseline model. In addition, without any inventory adjustment costs, the firm s conditional beta is flat as a function of the inventory stock. The positive relationship between the size of adjustment costs and the firm s risk is well known (e.g., Jermann 1998; Zhang 2005). In production economies, the firm s risk is inversely related to its flexibility in using investment to mitigate the effect of shocks on its dividend stream. The more flexible a firm is in this regard, the less risky it is. The size of the adjustment costs controls the firm s ability to smooth its dividends, and hence controls its flexibility. Thus, here, the lower the inventory adjustment costs a firm faces, the more flexible it is in adjusting its stock of inventories, and thus the less risky the firm is. Consistent with the analysis of the firm s conditional beta in Figure 3, Panel B of Table 5 shows that eliminating the inventory adjustment costs substantially reduces the return s spreads, and thus deteriorates the fit of the model on the asset pricing dimension as well. Specification 2 shows that by removing nonconvex adjustment costs, the model generates a tiny value premium (0.63% here versus 5.58% in the baseline model and 5.35% in the data) and reduces the inventory growth spread of both equal- and value-weighted portfolios to about half the size of the spreads in the baseline model. Specification 3 shows that removing convex adjustment costs has a smaller effect on spreads than removing nonconvex costs, but the model-implied spreads are still lower than those generated by the baseline model. Finally, when both convex and nonconvex adjustment costs are removed in specification 4, the model-implied spreads are too small, especially the value premium (1.17% here versus 5.58% in the baseline model and 5.35% in the data). The previous analysis makes an important point about the endogenous determination of risk in investment-based models with multi-capital inputs. Even though the model has physical capital adjustment costs, which in a one-capitalgood investment-based model can endogenously generate a sizeable risk dispersion (e.g., the value premium in Zhang 2005), the addition of a costlessly adjustable capital input (inventory) destroys the large risk dispersion generated by the model. Intuitively, without inventory adjustment costs, firms take advantage of the costlessly adjustable inventory input (a storage technology) to smooth their dividends. This extra flexibility endogenously reduces overall risk in the economy to the point that the model loses its ability to match the size of the stock return spreads observed in the data. 4.2 The role of operating fixed costs To examine the importance of operating fixed costs, we examine the results from the alternative specification 5, which sets the operating fixed cost to 308

32 zero. On the real quantity side, Panel A of Table 5 shows that shutting down operating fixed costs has a negligible impact on firm quantity moments. In fact, the real quantity moments generated by this alternative specification of the model are almost indistinguishable from those generated by the baseline model. Because operating fixed costs are essentially sunk costs, these costs have a negligible impact on the inventory investment and physical investment policy functions. On the asset pricing side, however, shutting down operating fixed costs has a strong negative effect on the ability of the baseline model to match the data. Panel E of Figure 3 shows that eliminating fixed operating costs reduces significantly the firm s risk relative to the baseline model. This result is consistent with that of Carlson, Fisher, and Giammarino (2004), who argue that operating leverage increases risk: When a firm is hit with negative shocks, its operating profits fall relative to the fixed costs. As a result, cash flows are more sensitive to aggregate shocks (see Li, Livdan, and Zhang 2009 for a similar analysis in the context of an investment-based model with one capital input). Panel B of Table 5 reports the quantitative effect of eliminating the operating fixed costs on the model s fit. Without operating fixed costs, the value premium is tiny (1% here versus 5.58% in the baseline model and 5.35% in the data) and the inventory growth spread for both equal- and value-weighted portfolios is too low, approximately half the spread in the baseline model. In addition, the inventory growth spread is slightly larger across larger firms than across small firms, in sharp contrast with the data. This suggests that operating fixed costs are especially important for the risk dispersion across small firms, consistent with the analysis in Li, Livdan, and Zhang (2009). Finally, without operating fixed costs, the model generates firm values that are too large, reflected by the low average physical-capital-to-market-equity ratio (18% here versus 32% in the baseline model and 41% in the data). Taken together, these results show that without operating fixed costs, the overall risk in the economy is reduced, which generates small stock return spreads as well as firm-level market values that are too high due to low discount rates. 4.3 The role of physical capital and inventory elasticity of substitution Finally, to examine the importance of the elasticity of substitution (ES) between capital and inventory for the good fit of the investment-based model, we examine the results from the alternative specifications 6 and 7. In the baseline model, ES = (1 + ρ) 1 = Here, in specification 6, we consider a low elasticity ρ = 0.7 (ES = 0.59), and in specification 7, we consider a high elasticity ρ = 0.05 (ES = 1.05), a value that is close to the standard Cobb-Douglas production function (ES = 1). The real quantity moments reported in Panel A of Table 5 show that the ES parameter has a first-order effect in determining the average inventory-to-sales 309

33 The Review of Financial Studies / v 25 n ratio: When the ES is low, firms hold too much inventory (average inventoryto-sales ratio of 21%), and when the ES is high, firms hold too few inventories (average inventory-to-sales ratio of 10%), relative to the data (average of 17%). In addition, when the ES is high, the inventory growth volatility is higher. The most interesting analysis is the effect of the ES on firms risk. Panel F of Figure 3 shows that, all else equal, firms risk decreases with the elasticity of substitution between physical capital and inventory stock. This result is intuitive. When the ES between the two inputs is high (i.e., capital and inventory are more substitutable), firms are more flexible because they can use relatively more of the capital input with lower adjustment costs to smooth the impact of shocks on dividends. As a result, firms overall risk is lower when the ES is high. Consistent with this analysis, Panel B of Table 5 shows that, all else equal, the magnitude of the inventory growth spread is negatively related with the size of the ES: Relative to the baseline model, the inventory growth spread is higher with a low ES, and the spread is lower with a high ES. 5. Conclusion We incorporate an inventory holding motive into the investment-based asset pricing framework by modeling inventory as a factor of production subject to convex and nonconvex adjustment costs. The model replicates the large inventory growth spread observed in the data, as well as the firm-level properties of physical capital investment, inventory investment, and inventory-to-sales ratio. Our conditional single-factor model also implies that traditional factor models such as the CAPM and, to a lesser extent, the Fama and French (1993) three-factor model should fail to explain the inventory growth spread, although not with the same large pricing errors observed in the data. This result suggests that introducing additional sources of aggregate risk in the standard investment-based model may be necessary to capture the size of the CAPM violation observed in the data. Our results have implications for both the asset pricing and the macroeconomics literatures. For asset pricing, our results, based on a model with rational expectations and firm value maximization, suggest that the large inventory growth spread observed in the data is in principle consistent with a risk-based interpretation. Firms with low inventory investment rates have a high cost of capital (risk), which explains the high average returns of these firms in the data. For the macroeconomics literature, our results show that time-varying risk is an important determinant of inventory investment. Given the importance of inventory investment in business cycle fluctuations, our results suggest that incorporating time-varying risk premiums in current macroeconomic models of inventory behavior is important for an accurate understanding of the inventory investment dynamics over the business cycle and how inventory investment propagates and amplifies the effect of shocks in the economy. 310

34 A1. Numerical Algorithm To solve the model numerically, we use the value function iteration procedure to solve the firm s maximization problem. The value function and the optimal decision rule are solved on a grid in a discrete state space. We use a multi-grid algorithm in which the maximum number of points is 50 in each dimension. In each iteration, we specify a grid of points for capital and inventory, respectively, with upper bounds ˉk and ˉn that are large enough to be nonbinding. The grids for capital and inventory stocks are constructed recursively, following McGrattan (1999) that is, k i = k i 1 + c k1 exp(c k2 (i 2)), where i = 1,..., 50 is the index of grids points and c k1 and c k2 are two constants chosen to provide the desired number of grid points and two upper bounds ˉk and ˉn, given two prespecified lower bounds ḵ and ṉ. The advantage of this recursive construction is that more grid points are assigned around ˉk and ˉn, where the value function has most of its curvature. The state variables x and z have continuous support in the theoretical model, but they have to be transformed into discrete state space for the numerical implementation. We use a three-state Markov process for the x and z processes. The popular method of Tauchen and Hussey (1991) does not work well when the persistence level is above 0.9. Because both the aggregate and idiosyncratic productivity processes are highly persistent, we use the method described in Rouwenhorst (1995) for a quadrature of the Gaussian shocks. In all cases, the results are robust to finer grids as well. Once the discrete state space is available, the conditional expectation can be carried out simply as a matrix multiplication. Cubic interpolation is used extensively to obtain optimal investment and inventory investment that do not lie directly on the grid points. Finally, we use a simple discrete, global search routine in maximizing the firm s problem. References Abel, A., and J. C. Eberly Investment and q with Fixed Costs: An Empirical Analysis. Working Paper, University of Pennsylvania. Basu, S., and J. G. Fernald Returns to Scale in U.S. Production: Estimates and Implications. Journal of Political Economy 105: Bazdresch, S., F. Belo, and X. Lin Labor Hiring, Investment, and Stock Return Predictability in the Cross-section. Working Paper, University of Minnesota and London School of Economics and Political Science. Belo, F Production-based Measures of Risk for Asset Pricing. Journal of Monetary Economics 57: Bils, M., and J. A. Kahn What Inventory Behavior Tells Us About Business Cycles. American Economic Review 90: Blinder, A. S., and L. J. Maccini Taking Stock: A Critical Assessment of Recent Research on Inventories. Journal of Economic Perspectives 5: Burnside, C., M. Eichenbaum, and S. Rebelo Capital Utilization and Returns to Scale. NBER Macroeconomics Annual 10: Campbell, J., and J. Cochrane By Force of Habit: A Consumption-based Explanation of Aggregate Stock Market Behavior. Journal of Political Economy 107: Carlson, M., A. Fisher, and R. Giammarino Corporate Investment and Asset Price Dynamics: Implications for the Cross-section of Returns. Journal of Finance 59: Chen, L., R. Novy-Marx, and L. Zhang An Alternative Three-factor Model. Working Paper, Ohio State University. Chirinko, R. S Multiple Capital Inputs, Q, and Investment Spending. Journal of Economic Dynamics and Control 17: Christiano, L Why Does Inventory Investment Fluctuate So Much? Journal of Monetary Economics 21:

35 The Review of Financial Studies / v 25 n Cochrane, J. H Investment-based Asset Pricing and the Link Between Stock Returns and Economic Fluctuations. Journal of Finance 46: Asset Pricing. Princeton, NJ: Princeton University Press. Cooley, T. F., and E. C. Prescott Economic Growth and Business Cycles. In Thomas F. Cooley (ed.), Frontiers of Business Cycle Research. Princeton NJ: Princeton University Press. Cooper, I Asset Pricing Implications of Nonconvex Adjustment Costs and Irreversibility of Investment. Journal of Finance 61: Fama, E. F., and K. R. French The Cross-section of Expected Stock Returns. Journal of Finance 47: Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics 33: Dissecting Anomalies. Journal of Finance 63: Fama, E. F., and J. D. MacBeth Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy 81: Fisher, J., and A. Hornstein (S,s) Inventory Policies in General Equilibrium. Review of Economic Studies 67: Gomes, J. F., L. Kogan, and M. Yogo Durability of Output and Expected Stock Returns. Journal of Political Economy 117: Gomes, J. F., L. Kogan, and L. Zhang Equilibrium Cross-section of Returns. Journal of Political Economy 111: Hall, G Nonconvex Costs and Capacity Utilization: A Study of Production and Inventories at Automobile Assembly Plans. Working Paper, Federal Reserve Bank of Chicago. Jermann, U Asset Pricing in Production Economies. Journal of Monetary Economics 41: Jones, C., and S. Tuzel Inventory Investment and the Cost of Capital. Working Paper, University of Southern California. Kahn, J. A Inventories and the Volatility of Production. American Economic Review 77: Why Is Production More Volatile Than Sales? Theory and Evidence on the Stockout Avoidance. Quarterly Journal of Economics 107: Khan, A., and J. Thomas. 2007a. Explaining Inventories: A Business Cycle Assessment of the Stockout Avoidance and (S,s) Motives. Macroeconomic Dynamics 11: b. Inventories and the Business Cycle: An Equilibrium Analysis of (S,s) Policies. American Economic Review 97: Kogan, L., and D. Papanikolaou Growth Opportunities and Technology Shocks. American Economic Review, Papers and Proceedings 100: Kydland, F., and E. Prescott Time to Build and Aggregate Fluctuations. Econometrica 50: Li, E. X. N., D. Livdan, and L. Zhang Anomalies. Review of Financial Studies 22: Lin, X. Forthcoming. Endogenous Technological Progress and the Cross-section of Stock Returns. Journal of Financial Economics. Liu, L. X., T. M. Whited, and L. Zhang Investment-based Expected Stock Returns. Journal of Political Economy 117: Lyandres, E., L. Sun, and L. Zhang The New Issues Puzzle: Testing the Investment-based Explanation. Review of Financial Studies 21:

36 McCarthy, J., and E. Zakrajsek Microeconomic Inventory Adjustment: Evidence from U.S. Firm-level Data. Working Paper, Federal Reserve Bank of New York. McGrattan, E. R Application of Weighted Residual Methods to Dynamic Economic Models. In Ramon Marimon and Andrew Scott (eds.), Computational Methods for the Study of Dynamic Economies. Oxford, UK: Oxford University Press. Patton, A. J., and A. Timmermann Monotonicity in Asset Returns: New Tests with Applications to the Term Structure, the CAPM, and Portfolio Sorts. Journal of Financial Economics 98: Ramey, V. A Inventories as Factors of Production and Economic Fluctuations. American Economic Review 79: Ramey, V. A., and D. Vine Declining Volatility in the U.S. Automobile Industry. American Economic Review 96: Ramey, V. A., and K. D. West Inventories. In M. Woodford and J. Taylor (eds.) Handbook of Macroeconomics, pp Amsterdam: Elsevier Science. Rouwenhorst, K. G Asset Pricing Implications of Equilibrium Business Cycle Models. In Thomas Cooley (ed.), Frontiers of Business Cycle Research. Princeton, NJ: Princeton University Press. Scarf, H The Optimality of (S,s) Policies in the Dynamic Inventory Problem. In K. J. Arrow, S. Karlin, and H. Scarf (eds.), Mathematical Methods in the Social Sciences. Stanford, CA: Stanford University Press. Sloan, R. G Do Stock Prices Fully Reflect Information in Accruals and Cash Flows About Future Earnings? Accounting Review 71: Tauchen, J., and R. Hussey Quadrature-based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models. Econometrica 59: Thomas, J. K., and H. Zhang Inventory Changes and Future Returns. Review of Accounting Studies 7: Tuzel, S Corporate Real Estate Holdings and the Cross-section of Stock Returns. Review of Financial Studies 23: Vuolteenaho, T What Drives Firm-level Stock Returns? Journal of Finance 57: Whited, T Problems with Identifying Adjustment Costs from Regressions of Investment on QQ. Economics Letters 46: Wu, J., L. Zhang, and F. Zhang The q-theory Approach to Understanding the Accrual Anomaly. Journal of Accounting Research 48: Zhang, L The Value Premium. Journal of Finance 60: