Does the Investment-based Model Explain Expected Returns? Evidence from Euler Equations

Transcription

1 Does the Investment-based Model Explain Expected Returns? Evidence from Euler Equations Stefanos Delikouras Robert F. Dittmar October 30, 2015 Abstract We investigate empirical implications of the investment based pricing model for the stochastic discount factor that prices equity returns, investment returns, and both equity and investment returns simultaneously. Our methodology is based on the equivalence of the return on investment and equity implied by the investment based model and the existence of stochastic discount factors in the linear span of returns. We find that the minimum variance stochastic discount factor that satisfies the Euler equation for equity returns cannot satisfy the Euler equation for investment returns, in contradiction of the predictions of the investment model. We consider the implications for this result for recent factor models. Our results suggest that joint restrictions on the optimality of investment and consumption pose stringent conditions for candidate stochastic discount factors. keywords: asset pricing, value premium, investment, q-theory, consumption JEL classification: G12 The authors would like to thank Erica Xuenan Li, Chen Xue, and Lu Zhang for helpful comments and discussions. All errors are the responsibility of the authors. Department of Finance, School of Business Administration, University of Miami, Coral Gables, FL 33124, Department of Finance, Stephen Ross School of Business, University of Michigan, Ann Arbor, MI 48109,

2 1 Introduction The structural investment-based model of asset pricing introduced in Cochrane (1991) has been instrumental in shaping our understanding of the relation between firms investment decisions and the expected returns on their equity. Zhang (2005) shows that firms with high book-to-market ratios can have low book-to-market ratios if their ability to reduce capital stock in economic downturns is costly. More recently, Hou, Xue, and Zhang (2015) propose a four-factor return model inspired by investment based pricing. Their motivation for the model is that firms return on investment is a function of their operating efficiency, measured as the ratio of output to capital, and their investment intensity, measured as the ratio of investment to capital. The authors use these insights to motivate the use of return factors based on portfolios of firms sorted on their return on equity and investment to assets ratio. Their evidence suggests that the model fares quite well in explaining expected returns for a large set of portfolios. While this framework has offered considerable insight into why firm characteristics might be related to expected returns, empirical tests of the mechanisms behind the model have been more limited. Cochrane (1991) emphasizes that when firms invest optimally relative to the stochastic discount factor that satisfies the Euler equation for equity returns, returns on investment will be equal to the returns on equity. A testable hypothesis of this condition is that regressing investment returns and equity returns on the same variables should produce the same coefficients. In an expanded framework, Liu, Whited, and Zhang (2009) show that the return on levered investment should be equal to the return on equity, and suggest that a testable restriction of this restriction is the equality of the means of investment and equity returns. The authors find statistical, but more limited economic support of this condition. We explore restrictions implied by the investment based model that we believe are especially interesting, and related to those explored in Cochrane (1991) and Liu, Whited, and Zhang (2009). An implication of the equality of the return on investment and the return on equity is that projecting the stochastic discount factor onto the span of investment returns or equity returns should produce the same coefficients. Put differently, a projected, and therefore minimum variance, stochastic discount factor that is a linear combination of equity returns should satisfy the Euler condition for investment returns. As noted in Cochrane, satisfaction of this condition suggests that firm investment is optimal given equilibrium in the financial markets. Our analysis represents a direct test of this hypothesis. These tests are of particular economic interest because the investment model does not inform us as to the object that drives pricing, the stochastic discount factor. Rather, it states that, conditional on a stochastic discount factor, investment should be optimal, and that investment will produce a return that is a function of firm characteristics. Firms with higher ratios of output to capital, for 1

3 example, have higher returns not just because their profitability happens to be higher, but rather because these firms output to capital covaries less with the stochastic discount factor than firms with low ratios of output to capital. Our empirical investigations are designed to explicitly examine whether this is true, and therefore whether factors that are designed to maximize variation along these characteristics are consistent with the implications of investment-based pricing. Our results indicate that it is quite challenging for a stochastic discount factor to simultaneously satisfy Euler equations for both equity and investment returns. Investment returns tend to covary positively with the stochastic discount factor in the linear span of equity returns, suggesting that investment pays off in bad states of the world, that is when the stochastic discount factor is high. As a result, it is difficult to generate a risk premium on investment similar to the risk premium on equity, which tends to pay off in bad states of the world, that is when the stochastic discount factor is low. Our tests strongly reject the equivalence of stochastic discount factors that price investment and equity returns, and suggest that the restriction of equality of investment and equity returns, as implied by the investment model, does not hold. These findings are of particular interest in light of factor models that seem to describe expected equity returns well. Hou, Xue, and Zhang (2015) propose a four-factor model that is inspired by the investment model, and Fama and French (2015) propose a similar model that is motivated by clean surplus accounting. Both models fare well in describing equity return variation. However, our earlier results suggest that simply because a model fares well in pricing equity returns does not mean it will also satisfy the Euler equation for investment, the key implication of the investment based model. Our empirical results support this conjecture. Neither the Fama and French (2015) nor the Hou, Xue, and Zhang (2015) models fare well at pricing the joint set of equity and investment returns, although the five-factor model fares better based on typical measures of model fit. Both models tend to generate a positive covariation of both equity and investment returns with the stochastic discount factor due to the dominance of profitability factors in the determination of this stochastic discount factor. This result is puzzling as it suggests that equities tend to have high returns when the profitability factor is low. Several earlier papers empirically investigate the implications of investment-euler equations for cross-sectional variation in returns. Our approach is closely related to Cochrane (1996) and Gomes, Yaron, and Zhang (2006), who investigate investment-euler equations implications for expected equity returns. Cochrane (1996), uses investment returns as factors, and investigates the ability of a stochastic discount factor that is a linear function of investment returns to price a set of size-sorted portfolios. He finds that the model performs about as well as the CAPM or the Chen, Roll, and Ross (1986) factor model in explaining cross-sectional variation in returns on these portfolios. Gomes, Yaron, and Zhang (2006) pursue a similar exercise in investigating the role financial frictions play in explaining cross-sectional variation in returns. Our approach differs significantly from theirs in 2

4 that we construct our investment returns from firm characteristics, following Liu et al. (2009), rather than aggregate macroeconomic data. Our focus is also explicitly on the role that optimal investment plays in understanding the value premium across firms. The remainder of the paper is organized as follows. In Section 2, we discuss the theoretical links between firms investment decisions and expected returns and their empirical implications. In Section 3, we discuss our empirical approach to testing these implications. Section 4 discusses the empirical results, and concluding comments are presented in Section 5. 2 Production-Based Pricing and the Cross-Section of Returns 2.1 Firm Investment Decisions and Expected Equity Returns Cochrane (1991) provides one of the first explicit links between firms investment policies and equity returns. He notes that, following Jonathan E. Ingersoll (1988) and Hansen and Richard (1987), prices of contingent claims on real assets will satisfy an Euler equation, [ E t Mt+1 Rt+1] S = 1, (1) where Rt+1 S represents the gross return to a contingent claim, which we superscript S for stock, as our main interest in this paper is the firm s equity return. The variable M t+1 is a stochastic discount factor, which in this framework captures the information in the returns to a set of Arrow- Debreu securities that return one dollar in a particular state of the economy and zero in all others. Equation (1) is common to equilibrium models of asset pricing in which a consumer optimizes her consumption and portfolio choice given a distribution of payoffs to the contingent claims on the real assets in the economy. Producers in the economy will have access to the same set of Arrow-Debreu securities, but a different objective function. The producer s objective is to choose an investment policy to maximize the value of its firm s contingent claim price. Cochrane (1991) shows that this optimal investment decision will satisfy another Euler equation, [ E t Mt+1 Rt+1] I = 1, (2) where Rt+1 I is the gross return to the firm s investment. A large body of literature has investigated the implications of (2) for asset pricing under different assumptions about the stochastic discount factor, the production function that determines the return on investment, and constraints that affect the return on the firm s investment. 3

5 There are at least three ways to interpret the Euler equation as discussed by Cochrane (1991). First, equation (1) represents a hyperplane in which all contingent claim prices must lie to prevent arbitrage. Since the producer has access to the contingent claims market, he will adjust investment until there is no arbitrage between the investment return and portfolios of the contingent claims. Second, if markets are complete, there will always be a unique contingent claims return that is proportional to the investment return. Again, to prevent arbitrage, the producer will adjust investment to make these returns equal. Finally, he notes that equation (2) implies that firms will adjust investment until the M t+1 that satisfies the Euler equation for contingent claims also satisfies the Euler equation for investment returns. The principal empirical implication of these conditions, as noted in Cochrane (1991), is that the return on investment and the return on investing in a share of equity are identical, R I t+1 = R S t+1 t. (3) The implicit assumption underlying this statement is that equity represents the entire claim to the firm s assets. In the presence of debt claims, Liu, Whited, and Zhang (2009) note that R IL t+1 = R S t+1 t, (4) where Rt+1 IL is the return on a levered claim to investment. In this context, the return on investment will be identical to the weighted average cost of capital. Both sets of authors note that it is not realistic to take the equality condition literally due to specification and measurement errors. Therefore, the authors instead exploit expectational conditions of the restrictions. Liu, Whited, and Zhang (2009) exploit condition (4), and note that it implies E [ Rt+1 IL ] [ (R ) IL 2 ] E t+1 = E [ Rt+1 S ] [ = E (R t+1 ) 2], that is the expected levered investment return should equal the expected equity return and the variances of each return should be the same. The authors find that, in their sample, neither condition can be statistically rejected in the data. Further, when confronted only with the restrictions on means, absolute pricing errors on portfolios are reasonably small. However, when both the mean and volatility restrictions are imposed, pricing errors are larger. The authors conclude that their results suggests that firms align their investment policies with costs of capital and that this alignment can explain many of the perceived anomalies detected in empirical studies of equity returns. An alternative implication, exploited in Cochrane (1991), is that condition (3) implies that regressing stock and investment returns on any variables should produce the same coefficients. 4

6 That is in regressions of (de-meaned) variables, we obtain ˆbI = Cov ( X t+j, Rt+1 I ) V ar (X t+j ) R I t+1 = b I X t+j + e I t+1 R S t+1 = b S X t+j + e S t+1, = Cov ( X t+j, Rt+1 S ) = V ar (X t+j ) ˆb S for any j. His empirical findings suggest that in using returns on investment and equity to forecast various aggregate variables, that forecasts from both sets of returns are the same. Cochrane (1991) also investigates whether regressing a candidate variable on investment and equity returns produces the same coefficients. That is, X t+j = d I R I t+1 + u I t+1 X t+j = d S R S t+1 + u S t+1, yields point estimates ˆd I = Cov ( X t+j, Rt+1 I ) V ar ( ) Rt+1 I = Cov ( X t+j, Rt+1 S ) V ar ( ) Rt+1 S = ˆd S, assuming V ar ( R S t+1) = V ar ( R I t+1 ), which is implied by condition (3). His results suggest that one cannot reject equality in coefficients of forecasting regressions where returns on equity and investment are used to forecast future GNP growth. Our empirical work exploits a version of this last condition, where we utilize a vector of investment and equity returns. We note that equation (3) implies that for N-vectors of returns R I t+1 and R S t+1, E [ X t+jr I t+1] = E [ Xt+j R S t+1]. (5) This condition is similar to the regression conditions explored in Cochrane (1991), as it implies E [X t+j ] E [ R I t+1] + Cov ( Xt+j, R I t+1) = E [Xt+j ] E [ R S t+1] + Cov ( Xt+j, R S t+1). From Liu, Whited, and Zhang (2009) and Cochrane (1991), we know that the expectation of the return on investment should be equal to the expectation of the return on equity and that the covariances of each return with a given variable X t+j should be the same. In addition to being a testable restriction implied by the investment based model, the restriction has interesting economic implications that we discuss below. 5

7 2.2 Implications of the Investment Model for the Stochastic Discount Factor One possible choice of a random variable to examine in equation (5) is the stochastic discount factor, M t+1. This choice is of particular economic interest, as it yields equality of the Euler equations for equity returns and investment returns, which is implied by equilibrium in the investment-based model. We investigate whether these conditions hold in this section. factor. Unfortunately, there is not universal agreement as to the identification of the stochastic discount However, Hansen and Jagannathan (1991) suggest an approach for identifying a set of stochastic discount factors that satisfy Euler equations and are consistent with the framework explored in Cochrane (1991). The authors suggest projecting the stochastic discount factor onto the linear span of returns, M t+1 = c + d ( R t+1 R ) + ξ t+1, (6) where c is a pre-specified mean of the stochastic discount factor. The authors note that for a choice of parameters [ (Rt+1 ˆd = E R ) ( Rt+1 R )] 1 [ (Rt+1 E R ) ] (Mt+1 c). When sample moments converge to population moments, the Euler equation implies ˆd = Σ 1 ( 1 c R ), (7) where Σ is the sample covariance matrix of returns. Since the projection is orthogonal to the error, the resulting stochastic discount factor is minimum variance given its mean. As stated in the previous section, an implication of the investment based model is that if one regresses a random variable on the returns on investment, one should obtain the same regression coefficients that one gets regressing that random variable on the returns on equity. Thus, if we regress the stochastic discount factor on both sets of returns, M t+1 = c + ( d I) ( R I t+1 R I) + ξ I t+1 M t+1 = c + ( d S) ( R S t+1 R S) + ξ S t+1, the resulting coefficients, d I and d S should be equal. A different way of stating this is that if one takes the coefficients on the equity returns, since the coefficients define a stochastic discount factor that satisfies the Euler equation for contingent claims, that [( E c + ( d S) ( R S t+1 R S)) ] R I t+1 = 1. (8) 6

8 Similarly, the Euler equation for equity returns should hold using the stochastic discount factor in the linear span of investment returns. An appealing economic feature of this restriction is that it is fully in accordance of the interpretation incochrane (1991) of the Euler equation for investment. That is, firms will adjust investment until the stochastic discount factor that satisfies the Euler equation for contingent claims also satisfies the Euler equation for investment. Thus, equation (8) represents a null hypothesis for the investment-based pricing model. We discuss a framework for testing this null hypothesis, and its complement that a stochastic discount factor that satisfies the Euler equation for investment returns satisfies the Euler equation for investment returns. 3 Empirical Methodology 3.1 Stochastic Discount Factors for Investment and Equity Returns Our empirical approach is based in extensions of the insights of Hansen and Jagannathan (1991) to generate testable restrictions on stochastic discount factors that price a set of assets. Specifically, we follow DeSantis (1995), Bekaert and Urias (1996), and Chen and Knez (1996) in implementing a testing methodology. Bekaert and Urias (1996) outline an approach that asks if the returns on a set of assets, R J t+1 spans the mean-variance space, or equivalently law of one price stochastic discount factor space implied by the original set of assets joined with an additional set of asset returns, R K t+1. The authors purpose in outlining this methodology is to assess whether adding a set of assets to the original set provides diversification benefits. Chen and Knez (1996) investigate the performance of mutual funds by asking whether the Euler equation for the mutual fund return can be satisfied by a stochastic discount factor in the linear span of a set of basis asset returns. The more straightforward approach is that of Chen and Knez (1996). The authors note that Hansen and Jagannathan (1991) suggests that there are some coefficients, d, such that for an M-dimensional set of basis assets, E [ R J t+1m t+1 ] = E [ R J t+1 R J t+1d ] = 1 M. (9) The authors suggest two testing approaches in this framework. The first is to analytically calculate the solution d = E [ R J 1 t+1 t+1] RJ 1 and calculating a vector of pricing errors for the N assets R K t+1, e T = 1 T t [( ) ] R J t+1ˆd R K t+1 1 N, (10) and conducting a Wald test of the hypothesis that the errors, e T, are jointly zero. Alternatively, 7

9 one can specify moment conditions h T = 1 T t ( R J t+1ˆd ( R J t+1ˆd ) R J t+1 1 M ) R K t+1 1 N, (11) and estimate parameters d via GMM. The test of overidentifying restrictions with N degrees of freedom represents a test that the pricing errors are jointly zero. A limitation of the Chen and Knez (1996) approach, noted in Dahlquist and Söderlind (1999) and Farnsworth, Ferson, Jackson, and Todd (2002) is that the approach does not constrain the mean of the stochastic discount factor. Consequently, one can obtain a stochastic discount factor estimate with a mean greater than zero, implying a negative risk-free rate. One solution to this problem is to include the risk-free asset in the set of basis assets, R J t+1. Alternatively, one can pursue the approach in Bekaert and Urias (1996), which asks the question of whether the basis assets span the joint space of the basis assets and the returns of interest, R K t+1. This approach also implicitly asks whether any discount factor in the linear span of R J t+1 prices the assets RK t+1. The Bekaert and Urias (1996) procedure suggests forming sample moment conditions for the Euler equations for the joint set of returns R t+1 = {R J t+1 ; RK t+1 }, ( h T (d 1, d 2 ) = 1 ( R t+1 c1 + d ( 1 Rt+1 R )) ) 1 M+N ( T R t+1 c2 + d ( 2 Rt+1 R )) 1 M+N t where d k = {d J k ; dk k } for k = 1, 2. The constants c 1 and c 2 are two arbitrary choices for the mean of the stochastic discount factor. Similar to the insights in Black (1972) that any efficient portfolio can be represented as the combination of two arbitrary efficient portfolios, any minimum variance stochastic discount factor can be represented as the combination of two arbitrary minimum variance stochastic discount factors with means c 1 and c 2. The parameters d 1 and d 2 can be estimated via GMM; without restrictions, the system is exactly identified. Bekaert and Urias (1996) suggest two test statistics associated with estimation of parameters in equation (12). The first, a Wald statistic, leaves all parameters unrestricted, with parameters estimated using the exactly identified GMM system. (12) The test statistic is a Wald test of the hypothesis that the parameters {d K 1 ; dk 2 } are jointly equal to zero. Failure to reject this hypothesis implies that information in the returns R J t+1 alone are sufficient to satisfy the Euler equations given means c 1 and c 2. Put differently, the Euler equation for returns R K t+1 is satisfied by stochastic discount factors in the linear span of R J t+1 for means c 1 and c 2, and therefore for any arbitrary c k. Replacing J with equity returns and K with investment returns, this test represents a test that the Euler equation for investment returns is satisfied by a stochastic discount factor in the linear span 8

10 of equity returns, our null hypothesis, equation (8). The second test statistic explicitly restricts d K 1 = d K 2 = 0. Because the test imposes this restriction, the authors refer to it is a likelihood ratio test. In practice, imposing these restrictions result in overidentifying restrictions in the GMM estimation. Consequently, the GMM test of overidentifying restrictions represents a test of whether stochastic discount factors in the linear span of R J t+1 with means c 1 and c 2 satisfy the Euler equation for asset R K t+1. Again, replacing J and K with equity and investment returns allows us to test equation (8), our null hypothesis for the investment-based model. 3.2 Measuring the Return on Investment Thus far, we have discussed the return on investment as if it were an observable quantity. In reality, this return is the outcome of the firm s choice of investment given an unobserved production technology and law of motion for capital accumulation. As a result, one must posit a production technology and costs of adjusting capital. Hence, any results that we obtain are conditioned on our choice of the parametric form of this technology and costs. A failure of the implications of the investment model may be a result of our misspecification of these functions. We proceed, however, to use a production technology and adjustment costs that has been used widely in the literature. Specifically, we utilize a simplified version of the model explored in Liu, Whited, and Zhang (2009). In this model, a firm generates consumption goods by deploying capital, K t to generate output, Y t via a constant return to scale production technology, Π (K t+1, Z t+1 ), where Z t+1 is a productivity shock. Investing in new capital, I t, results in convex adjustment costs, ( It ) 2, Φ (I t, K t ) = a 2 K t where a > 0 is the adjustment cost parameter. Capital accumulates according to the law of motion K t+1 = K t (1 δ t ) + I t, where δ t is the depreciation rate. Finally, firms distribute profits as dividends D t, D t = Π (K t, X t ) Φ (I t, K t ) I t. The firm chooses the level of investment in order to maximize the value of cum-dividend equity, defined as the expected discounted value of current and future dividends. Discounting is achieved by a stochastic discount factor, M t+1, determined by consumers optimal consumption and portfolio decisions. 9

11 In this framework, the authors show that optimal investment satisfies an Euler equation as in equation (2), E [ M t+1 R I t+1] = 1, where, in the absence of debt, R S t+1 = R I t+1 = ( ) 2 ( ) α Y t+1 K t+1 + a It+1 2 K t+1 + (1 δt+1 ) 1 + a I t+1 K t+1. (13) 1 + a It K t The parameter α represents the share of capital in the firm s production function and Y t+1 is the firm s output at time t. Given parameter estimates for a and α, and measurement of output, capital, and investment, one can use equation (13) to compute returns on firms investment. Liu, Whited, and Zhang (2009) explicitly model the impact of debt financing and taxes on firms investment decisions. Incorporating these considerations results in an altered return on investment, which incorporates the benefits of tax shields from debt and depreciation. In this context, the return on equity is equal to the levered investment return, R S t+1 = RI t+1 ω tr B t+1 1 ω t, where R B t+1 is the after-tax debt return and ω t is the market leverage of the firm. In principle, this framework is more realistic, as most publicly traded firms have some degree of long term debt. In practice, obtaining estimates of the after-tax bond return for firms is quite difficult, as noted in Liu, Whited, and Zhang (2009). We opt to consider the unlevered investment return for this reason. Our decision can be motivated by the fact that the stochastic discount factor that satisfies the Euler equation for equity returns should also satisfy the Euler equation for both investment and bond returns. However, for robustness, we investigate results of our tests for samples of firms with no long term debt reported in Compustat. Equation (13) provides a specification for investment returns conditional on parameters a and α, which must be estimated. Liu, Whited, and Zhang (2009) suggest estimating these parameters using moment conditions implied by equation (13), e T = 1 T t ( R S t R I ) t,. (14) For a set of assets N = 2, the system is exactly identified. When N > 2, it is overidentified, and the authors use these overidentifying restrictions as a test of the implications of the investmentbased model. We follow their approach and use single-stage GMM to estimate the parameters a and α. We note, however, that rejection of the test of overidentifying restrictions suggests at least 10

12 three possibilities. First, the rejection could indicate that the investment-based model fails because E [ Rt+1] I [ ] E R S t+1. Second, the rejection could indicate that the functional form of the return on investment is misspecified. Third, and related, a rejection could be a rejection of a cross-sectionally constant share of capital in production, α, and adjustment cost, a. We consider these possibilities in our empirical results below. 3.3 Data and Summary Statistics We utilize a set of portfolios in our empirical tests proposed in Hou, Xue, and Zhang (2015). The authors propose a four-factor model that is inspired by a simplified version of the model for investment returns discussed in the previous section. In that model, the return on investment is driven by the ratios of output to capital and investment to capital. The authors suggest that the functional form in the resulting simplified version of equation (13) implies that expected returns should be increasing in profitability and decreasing in investment intensity. 1 This motivates the construction of portfolios sorted on the dimensions of return on equity and the ratio of investment to assets. These portfolios also form the basis of our empirical tests. We sort all firms in the annual CRSP/Compustat merged database into terciles on the basis of their ratio of investment to assets. We follow Hou, Xue, and Zhang (2015) and define investment to assets as the change in total assets, Compustat item AT, divided by beginning of period assets. Following the authors procedure, and consistent with Fama and French (1993), we assume that financial statement information becomes available to investors no sooner than six months after the end of the fiscal year. Therefore, investment to asset ratios in calendar year t are matched to CRSP returns from July of year t + 1 through June of year t + 2. All firms in the quarterly CRSP/Compustat merged database are sorted into terciles on the basis of their return on equity (ROE), defined as the ratio of income before extraordinary items, Compustat item IBQ, to book value of equity at the beginning of the fiscal quarter. As in Hou, Xue, and Zhang (2015), we define book value of equity following Davis, Fama, and French (2000), shareholder s equity plus balance sheet deferred taxes and investment tax credits, less preferred stock. 2 We treat the return on equity as observed on the quarterly earnings reporting date, Compustat item RDQ. Returns on equity are matched to equity returns from CRSP for the month following the month of the earnings reporting date. Finally, we require that earnings be reported 1 Their two period model results in investment returns of the form R I t+1 = Π t a (I t/k t), where Π t+1 is the return on equity. 2 For a detailed description of the construction of these variables, refer to Hou, Xue, and Zhang (2015). 11

13 for the fiscal quarter that is no more than six months from portfolio formation, consistent with the authors approach. We then form value-weighted portfolios for the intersections of the terciles of investment to assets and return on equity. Return on investment is a function of output, Y t+1, capital, K t+1, investment, I t+1, and depreciation, δ t+1. Following Liu, Whited, and Zhang (2009), we measure output as total net sales, Compustat item SALEQ and depreciation as the ratio of depreciation and amortization expense, item DPQ, to beginning of period capital. Liu, Whited, and Zhang (2009) and Hou, Xue, and Zhang (2015) suggest different measures of capital. The former authors use gross property, plant and equipment, whereas the latter use total assets. Since total assets data are available for more firms, we follow Hou, Xue, and Zhang (2015) and measure capital, K t+1, as beginning of period total assets. Similarly, investment is measured as the change in total assets over the quarter. In the quarterly Compustat data, total asset information is not broadly available until As a result, our final sample spans the period January, 1975 through December, We utilize quarterly data largely due to issues regarding econometric inference and sample size. Data to compute the return on equity is not generally available until 1972, which would yield only 43 annual observations. With nine portfolios and 43 annual observations, statistical power for estimation of the various GMM systems described above would be very low. However, seasonalities may impact the measurement of the quantities of interest in the return on investment from quarterly data. In our empirical tests, we consider two alternatives: we sum data over the past four quarters and repeat the tests using annual data. The timing of variables used to measure the return on investment is an issue discussed at length in Liu, Whited, and Zhang (2009). The authors propose matching financial statement information from December of year t through December of year t + 1 to returns on portfolios held from June of year t + 1 to June of year t + 2. Their rationale is that the market may be unaware of financial statement information for year t until June of year t + 1, similar to the rationale used in constructing the book-to-market ratio as a basis for portfolio formation. However, the economic link between returns on equity and investment link the return on equity from year t to year t + 1 to investment decisions over year t to year t + 1. We choose to match quarterly financial information contemporarily with returns on equity portfolios, except as the financial information contributes to identifying terciles for portfolio formation. In order to ensure that this implicit foresight on the part of the equity market does not drive our results, we also repeat our tests matching quarterly data entering the return on investment to returns one quarter in the future. Summary statistics for the variables are presented in Table 1. As suggested by Fama and French (1995), we compute ratios by separately aggregating the numerator and denominator to the portfolio level, and then dividing. That is, the ratio represents the sum across firms of the 12

14 numerator variable divided by the sum across firms of the denominator variable. Average returns exhibit large differences across investment and profitability terciles. In each of the three ROE terciles, average returns decrease monotonically across investment terciles, and in each of the three investment terciles, returns increase monotonically across ROE terciles. These results are consistent with the hypotheses and empirical findings of Hou, Xue, and Zhang (2015). Averaging across investment terciles, firms in the top profitability tercile earn an average return of 4.25% per quarter, compared to 1.19% for the low profitability tercile, for a profitability premium of 3.06% per quarter. Averaging across profitability terciles, low investment firms earn an average return of 3.61% per quarter, compared to 2.24% per quarter for high investment firms, for an investment premium of 1.37% per quarter. Average ratios are presented in the remaining panels of the table. The ratio of sales to capital, or return on assets, is reported in Panel B. Return on assets increases monotonically across return on equity terciles, as one might expect. The association of return on assets with investment to assets, however, is modest at best. The pattern is not monotonic across any of the return on equity terciles, although there is a tendency for return on assets for the lowest investment tercile to be higher than that of the highest investment tercile. The magnitude of the return on assets is much lower than that reported in Liu, Whited, and Zhang (2009). The reason for this difference is due to the use of quarterly sales, which is a quarterly flow number, compared to assets, an annual stock number. We report the ratio of investment to capital in Panel C. By construction, the ratio of investment to capital increases across the investment terciles. Interestingly, however, the ratio also increases monotonically across profitability terciles. This pattern suggests that more profitable firms tend to invest relatively intensively, but that it is not the case that all investment-intensive firms are relatively profitable. Another point of interest is that the average ratio of investment to capital for the low profitability and low investment portfolio is negative. This result is surprising on face value, as one would not expect a firm to be divesting on average. However, it is instead indicative of turnover in the portfolio; by holding a portolio of low investment, low profitability firms, one is holding firms that on average are divesting in that quarter. 4 Estimation Results 4.1 Returns to Investment We first estimate the parameters of the investment return, a and α, representing the cost of adjustment of new capital and the share of capital in the production function, respectively. We follow Liu, 13

15 Whited, and Zhang (2009) in estimating the parameters using GMM and the moment conditions in equation (14) using the identity matrix in single-stage GMM estimation and calculating standard errors using Newey-West with five lags. Our results suggest that inference about standard errors is not particularly sensitive to the number of lags in the Newey- West estimate. Results of the estimation are presented in Table 2. Point estimates and model specification tests are presented in Panel A. The point estimate for the share of capital in the production function, α, is 0.17 (SE=0.03). This estimate is similar in magnitude to that estimated in Liu, Whited, and Zhang (2009) using the corporate investment portfolio sort of Titman, Wei, and Xie (2004). The point estimate for the adjustment cost parameter, a, is 3.09 (SE=1.04), which is smaller in magnitude than the point estimates in Liu, Whited, and Zhang (2009) due to our use of quarterly data. The benefits of the use of quarterly data are reflected, however, in the precision of the parameter estimates, both of which are well over 2 standard errors from zero. Finally, the test of moment restrictions suggests that equality of means cannot be rejected; the J-statistic suggests that the hypothesis that the errors are jointly zero cannot be rejected at the 29% probability level. While the J-statistic cannot reject the joint equality of means of investment and equity returns, some individual errors are large. In Panel B of Table 2, we present average investment returns, moment condition errors, and associated t-statistics. Means are matched fairly well for the medium profitability, high investment and high profitability, high investment portfolios, with absolute values of pricing errors of less than 20 basis points each. Overall performance is somewhat poorer, however, with mean absolute pricing errors of 1.09% per quarter. This performance is driven in particular by the low profitability firms, which exhibit errors of -1.90% (t=-2.04), 1.24% (t=2.49), and -3.12% (t=-2.05) per quarter for the low, medium, and high investment portfolios respectively. As discussed above, one can interpret these results in several ways. From a statistical standpoint, one cannot reject equality of means of the return on equity and the return on investment, an implication of the investment-based pricing model explored in Liu, Whited, and Zhang (2009). Despite the lack of statistical evidence, the economic magnitude of several of the pricing errors seems large. This may indicate failure of the investment-based model, mis-specification of the production function, or rejection of constant capital share and adjustment costs across the nine investment and profitability portfolios. We investigate the first possibility in greater detail in the next section and the final possibility further below. 4.2 SDFs Implied in Equity and Investment Returns We utilize the investment returns implied by the parameter estimates in the previous section to test restrictions implied by equation (8), that a stochastic discount factor in the linear span of equity returns satisfies the Euler equation for investment, and that a stochastic discount factor in 14

16 the linear span of investment returns satisfies the Euler equation for equity returns. We test these hypotheses as discussed in Section 3.1 using three test statistics. The first is the overidentifying restriction test from Chen and Knez (1996), tested using GMM and the moment conditions in equation (11). The second and third are the Wald and likelihood ratio tests from Bekaert and Urias (1996) using the moment conditions in equation (12). The results of these tests are presented in Table 3. The answers appear unequivocal; according to all three test statistics, the null hypotheses are rejected. First, the Chen and Knez (1996) test indicates that the minimum variance stochastic discount factor in the linear span of equity returns cannot satisfy the Euler equation for equity returns, indicated by the GMM J-statistic of (p-value=0.000). Similarly, the minimum variance stochastic discount factor in the linear span of investment returns fails to satisfy the Euler equation for equity returns. Results of the more flexible tests of Bekaert and Urias (1996) are similar. The likelihood ratio tests, which are direct complements to the Chen and Knez (1996) test, reject the restrictions of zero coefficients on either the investment or the equity returns in the stochastic discount factor when pricing both sets of assets at below the 1% critical level. Similarly, the Wald tests of whether the unrestricted coefficients are zero rejects the null hypothesis at below the 1% critical value in both specifications. We interpret these results as a failure for the investment-based pricing model, since they suggest that projecting the stochastic discount factor on investment returns produces different coefficients than projecting on equity returns. A visual interpretation of the test statistics is presented in Figure 1, where we plot the Hansen and Jagannathan (1991) bounds on the volatility of the stochastic discount factor implied by equity returns, investment returns, and the combined set of investment of equity returns. The figure demonstrates that the lowest bounds are for the investment returns. The usual interpretation of this relatively low bound is that it represents a low threshold for a candidate stochastic discount factor; a stochastic discount factor that prices investment returns needs to have less volatility for a given mean than a stochastic discount factor that prices equity returns. However, although the bounds for equity returns is higher, it is not as high as the combined bounds for equity and investment returns. Thus, even if a candidate stochastic discount factor generates enough volatility to price equity returns, it will not necessarily be sufficiently volatile to price both equity and investment returns. While such a stochastic discount factor exists, it does not seem to satisfy the condition that the equity return and investment return are equivalent. In Panel B of Table 3, we present slope coefficients obtained by regressing investment returns and equity returns on a minimum variance stochastic discount factor implied by the returns on equity, and a complementary set of results by regressing returns on the minimum variance stochastic discount factor implied by the returns on investment. We choose the stochastic discount factor with a mean E [M t+1 ] = 0.988, since this is the mean of the stochastic discount factor implied by annual 15

17 returns on 30-day Treasury Bills from CRSP over our sample period. Hansen and Jagannathan (1991) show that this stochastic discount factor corresponds to the tangency portfolio on the efficient frontier with the maximum Sharpe ratio. The Euler equations for equity and investment imply [ E R I,S t+1 ( ) ] Cov R I,S t+1, M t+1 R f =. (15) E [M t+1 ] That is, risk premia for investment and equity returns should be negatively proportional to these slope coefficients. The first set of columns represents point estimates of regressions of equity and investment returns on the minimum variance stochastic discount factor implied by investment returns. A first issue that arises is that the slope coefficients from regressing equity returns on the investment stochastic discount factor yields uniformly positive coefficients. This implies that equities have relatively high payoffs when the stochastic discount factor is high, and low when the stochastic discount factor is low. This is counterintuitive, as we generally think of equities as assets that pay off when marginal utility is low and provide poor insurance against bad states of the world. There is an association between the slope coefficients and average returns on equities; the two exhibit correlation of 65%. However, the association is positive, rather than negative as implied by the Euler equation. By construction, the stochastic discount factor in the linear span of investment returns satisfies the Euler equation for investment. Consequently, the slope coefficients are perfectly negatively correlated with average returns. In the final column, we present t-statistics for the equality of the slope coefficients. These t-statistics represent a test of the condition suggested in Cochrane (1991), that when regressing returns on investment and equity on other variables, one should obtain the same point estimates. The table shows that these hypotheses are strongly rejected. Only for the low profitability, moderate investment portfolio can one not reject equality of the coefficients. The second set of columns provide the complement to the first; returns on equity and investment are regressed on the minimum variance stochastic discount factor in the linear span of equity returns. Results are similar to those for regressing returns on the investment stochastic discount factor. Average returns on equity are perfectly negatively, and the coefficients are nearly uniformly negative. Some of the investment return coefficients are positive and some are negative. However, the relation between mean investment returns and slope coefficients is positive, with a correlation of 88%. We reject the hypothesis of equality of the slope coefficients for roughly half of the portfolios under consideration. In order to get a bit more insight into the results in this section, we regress the minimum variance stochastic discount factor in the linear span of equity returns with mean equal to the inverse of the gross risk free return on the components of the return on investment. The return 16

18 on investment is positively associated with the contemporaneous ratios of output and investment to capital and negatively associated with the lagged ratio of investment to capital. As emphasized in equation (15), an asset earns a higher risk premium if its returns covary more negatively (less positively) with the stochastic discount factor. Thus, the investment-based pricing model suggests that the return premium for investment should be associated with greater negative, or less positive, covariance of the ratio of contemporaneous output and investment to capital with the stochastic discount factor, and less negative, or more positive covariance of the ratio of lagged investment to capital with the stochastic discount factor. Results of these regressions are presented in Table 4. The table shows that, as we would expect, there is a negative relation between contemporaneous output to capital and the equity stochastic discount factor across all nine portfolios. We can interpret this result as saying that output to capital tends to be high in states that are good news for returns and low in states that are bad news for returns. Six of these coefficients are statistically significant at the 10% critical level. While the coefficients are negative, it is not clear that they align well with average investment returns, however. The most negative point estimate is for the first quintile ROE, second quintile IA portfolio, which has the lowest average investment return of the nine portfolios. In fact, the correlation between the coefficients and average investment returns is somewhat positive, with a correlation coefficient of approximately 50%. This correlation suggests that portfolios of firms who have relatively high output in bad times, but relatively low output in good times, earn higher investment returns. Both the contemporaneous and lagged ratios of investment to output exhibit positive coefficients, although fewer of these coefficients are statistically significantly different than zero. Only the three low investment portfolios have coefficients that are at least marginally statistically distinguishable from zero. The preponderance of positive coefficients suggest that firms in general tend to invest more intensively when the stochastic discount factor is high, or alternatively less intensively when the stochastic discount factor is low. Like the output to capital ratio, there is a positive correlation between the coefficients and average investment returns of approximately 47%. This relation suggests that firms that invest more intensively when the stochastic discount factor is high or less intensively when the stochastic discount factor is low earn higher average investment returns. To some extent, this result seems consistent with the mechanism in Zhang (2005), who suggests that firms with low valuation ratios face extreme adverse costs of reducing investment in bad states of the world, and thus invest more conservatively in good states of the world. In general, the results in this section suggest that the implications of the investment model are violated. In particular, since the investment model predicts that equity returns will be equal to investment returns, we expect that a projection of the stochastic discount factor onto the two sets of assets will yield the same coefficients. This restriction appears to be strongly rejected. At 17

19 least some of the reason for this rejection appears to be the fact that the investment-based model predicts that firms with high ratios of output to capital will have high average returns. Based on the Euler equation, this should be a result of the fact that the output of firms with high ratios of output to capital have higher output when the stochastic discount factor is low than firms with low ratios of output to capital. The data suggest a tendency to the opposite; firms whose output is higher when the stochastic discount factor is low tend to have lower average returns on investment. 4.3 Cross-Sectional Variation in Production Parameters We consider the possibility that different firms face different production technologies where these differences manifest themselves in different adjustment cost and investment share parameters. Since the set of firms that we analyze are distinguished on two dimensions, return on equity and investment to assets, we believe it is sensible to consider firms that have similar profitability or similar investment policies as having more similar production technology parameters. We find that means of equity and investment returns are better matched when we group together firms in the same profitability terciles than when we examine firms in the same investment to asset terciles. Consequently, we repeat our analysis estimating parameters using moment conditions (14) separately for each of the three investment to asset portfolios within profitability terciles. With nine moment conditions and six parameters, the system now has three overidentifying restrictions. Point estimates are presented in Table 5. As shown, the flexibility in estimating parameters comes at the cost of precision of estimates, particularly of the adjustment cost parameters. The point estimates suggest that low profitability firms have relatively low investment share in their production function with a point estimate of approximately α = 0.10, which is over two standard errors from zero. The higher return on equity portfolios have investment shares of approximately α = 0.20, which are also precisely estimated. None of the adjustment cost parameters can be statistically distinguished from zero. The point estimates suggest that both low and high profitability firms have relatively low costs of adjustment, while moderately profitable firms face relatively high costs of adjustment. However, due to the imprecision of the estimates, we are reluctant to draw any strong conclusions from these parameters. The benefit of the flexibility of parameter estimation appears through the magnitude of the moment condition errors. As shown in the Table, the mean absolute error falls from 109 basis points per quarter in Table 2 to 35 basis points per quarter. The test of overidentifying restrictions fails to reject the null that the means of the return on investment are equal to the means of the returns on equity. Average returns on investment and pricing errors suggest that the point estimates better replicate the patterns observed in equity returns. Low profitability firms have low investment returns, and there is a modest tendency for high investment firms to have lower investment returns 18

20 than low investment firms. Differences in means range from 4 basis points for the first ROE tercile, second IA tercile portfolio to 66 basis points for the first ROE, third IA tercile portfolio. While still non-negligible, this difference is substantially smaller than the 312 basis points exhibited in Table 2. We next examine tests of the equality of the minimum variance stochastic discount factors implied by investment and equity returns. Results of these tests are shown in Table 6. Unfortunately, flexibility in the parameter estimates do not seem to generate stochastic discount factors that are similar across equity and investment returns. As in the case where the parameters are held constant across assets, all three tests reject the null hypothesis that the stochastic discount factor in the linear span of equity returns satisfies the Euler equation for the return on investment and vice versa. Thus, while flexibility in parameters improves the ability of the framework to match first moments of equity returns to first moments of investment returns, it does not seem to suggest that the coefficients projecting the stochastic discount factor onto investment and equity returns are the same. As above, we get some additional insight into these results by plotting the minimum standard deviation bounds of Hansen and Jagannathan (1991), shown in Figure??. The figure shows that, relative to Figure??, the bounds on the minimum variance stochastic discount factor in the linear span of investment returns is shifted sharply upward, and is much more acute. This result arises because the cross-sectional variation in mean investment returns when we allow for flexibility in production function parameters is much higher than when we force these parameters to be constant across firms. In fact, the bounds is pushed up sufficiently that it is almost universally higher than the bounds for stochastic discount factors in the linear span of equity returns. However, the bounds in the linear span of both sets of returns is higher still. Thus, the joint set of investment returns and equity returns remains a steep hurdle for a stochastic discount factor that satisfies the Euler equation for both investment returns and equity returns. Finally, as in the case with constant parameters, we depict slope coefficients from regressing returns on investment and equity on minimum variance stochastic discount factors in Panel B of Table 3. As before, slope coefficients from regressing equity returns on the investment stochastic discount factor are uniformly positive. However, there is now a negative relation between the slope coefficients and the average equity returns on the portfolio, with a correlation of -67%. However, we continue to strongly reject the null hypothesis that the sloe coefficients are equivalent. In the complementary case, slope coefficients from regressing investment returns on the minimum variance equity stochastic discount factor exhibit virtually no correlation with mean investment returns, with a correlation coefficient of -15%. The results in this section suggest that allowing for greater flexibility in the parameterization of 19