Regularities. December 2006

Transcription

1 Preliminary and incomplete Not for quotation Comments welcome Regularities Laura X. L. Liu School of Business and Management Hong Kong University of Science and Technology Toni M. Whited School of Business University of Wisconsin Madison Lu Zhang Stephen M. Ross School of Business University of Michigan and NBER December 2006 Abstract The q theory of investment implies a purely characteristic-based expected-return model. Under constant return to scale, stock return equals investment return, which is tied directly with firm characteristics. We use a two-period example to show analytically that the investment-return equation is consistent with the relations of average returns with book-to-market, investment-toasset, and earnings surprises. Using GMM, we estimate the q-theoretic expected return model by minimizing the differences between average stock returns in the data and average unlevered investment returns. Our model captures quantitatively the average returns of portfolios sorted on investment-to-asset and on size and book-to-market, including the small-stock value premium. The model is also partially successful in matching the post-earnings-announcement drift and its higher magnitude in small firms. Finance Department, School of Business and Management, Hong Kong University of Science and Technology, Kowloon, Hong Kong, tel: (852) , and fnliu@ust.hk. Department of Finance, School of Business, University of Wisconsin at Madison, 975 University Avenue, Madison WI 53706; tel: (608) , fax: (608) , and twhited@bus.wisc.edu. Finance Department, Stephen M. Ross School of Business, University of Michigan, 701 Tappan Street, E 7605 Bus Ad, Ann Arbor MI ; and NBER, tel: (734) , and zhanglu@bus.umich.edu. For helpful comments, we thank Nick Barberis, David Brown, V. V. Chari, Rick Green, Burton Hollifield, Patrick Kehoe, Narayana Kocherlakota, Leonid Kogan, Owen Lamont, Ellen McGrattan, Antonio Mello, Mark Ready, Bryan Routledge, Martin Schneider, Masako Ueda, and seminar participants at the Federal Reserve Bank of Minneapolis, the Yale School of Management, the University of Wisconsin, Madison, Carnegie-Mellon University, the Society of Economic Dynamics Annual Meetings in 2006, and the 2006 UBC Summer Finance Conference. Some of the theoretical results have previously been circulated in NBER working paper #11322 entitled Anomalies. All remaining errors are our own.

2 1 Introduction The empirical finance literature has documented tantalizing relations between future stock returns and firm characteristics. As surveyed, for example, in Fama (1998) and Schwert (2003), traditional asset pricing models have failed to explain many of these relations, which have therefore been dubbed anomalies. Several prominent studies, such as Shleifer (2000) and Barberis and Thaler (2003), have interpreted this failure as prima facia evidence against the efficient markets hypothesis. We use the neoclassical q theory of investment to provide the microfoundation for time-varying expected returns in the cross section, thus providing a structural framework for understanding these anomalies and for capturing them empirically. As first shown by Cochrane (1991), under constant return to scale, stock returns equal investment returns, which are tied to firm characteristics through the first-order conditions for optimal investment. We use these optimality conditions to show how expected returns vary in the cross section with firm characteristics, corporate policies, and events. In particular, we show that q theory can give rise to the following anomalies. The first is the investment anomaly, that is, the negative correlation between the investment-to-assets ratio and average returns, especially for firms with higher operating income-to-asset ratios. The second is the value anomaly, that is, value stocks (with high book-to-market ratios) earn higher average returns than growth stocks (with low book-to-market ratios), especially for small firms. The third is the post-earnings-announcement drift anomaly, that is, firms with positive earnings surprises earn higher average returns than firms with negative earnings surprises, especially for small firms. The intuition behind the way in which q theory generates these anomalies is most transparent in a simple two-period example. The investment return from time t to t+1 equals the ratio of the marginal profit of investment at t+1 divided by the marginal cost of investment at t. This definition implies two potential explanations for observed asset pricing anomalies. First, optimal investment implied by q theory produces a negative relation between investment and expected returns. The ratio of investment to assets increases with the net present value of capital, and the net present value 2

3 decreases with the cost of capital; that is, the expected return. The investment anomaly occurs because a low cost of capital implies high net present value, which in turn implies high investment. The value anomaly results from the same driving force because investment is an increasing function of marginal q, which is closely linked to the market-to-book ratio. The negative investment-return relation then implies a negative relation between market-to-book and expected returns. Whereas the first explanation operates through the denominator of the investment return, the second operates through the numerator. The marginal product of capital at time t + 1 in the numerator of the investment return equation drives the post-earnings-announcement drift in our model. Under constant return to scale, the marginal product of capital equals the average product of capital, which in turn equals profitability plus the rate of capital depreciation. This link suggests a positive relation between expected profitability and expected returns, all else equal. Because profitability is highly persistent and because earnings surprises and profitability are highly correlated, our model therefore predicts that earnings surprises are positively correlated with expected returns. Intriguingly, our economic explanations of anomalies do not involve risk directly, even though we do not assume over- or under-reaction, as in Daniel, Hirshleifer, and Subrahmanyam (1998). Because we derive expected returns from the optimality conditions of the firm, the stochastic discount factor and its covariances with stock returns do not enter the expected-return determination. Characteristics are sufficient statistics for expected returns, even in efficient markets. This result clearly helps interpret the debate on covariances versus characteristics in empirical finance in, for example, Daniel and Titman (1997) and Davis, Fama, and French (2000). More importantly, we test empirically whether q theory can quantitatively capture asset pricing anomalies. We test a purely characteristic-based, structural expected return model derived from the value maximization problem underlying q theory. To facilitate empirical tests of the theory, we derive new analytical relations between stock and investment returns after incorporating flow operating costs, leverage, and financing costs of external equity. We then use GMM to minimize the differences between the average stock returns observed in the data and the expected stock returns 3

4 implied by the model. Our data comprise several sets of testing portfolios designed to capture cross sectional variations induced by the anomalies. We examine the value anomaly in the Fama- French (1993) 25 size and book-to-market portfolios, the post-earnings-announcement drift in ten portfolios sorted by Standardized Unexpected Earnings (SUE), and the negative relation between expected returns and investment in ten portfolios sorted by the ratio of investment to assets. The q-theoretic mechanisms are empirically important. Average stock returns in the data and model-implied expected returns track each other closely across portfolios sorted by investment-toassets, and across portfolios sorted by size and book-to-market. Specifically, when we apply the benchmark model with only physical adjustment costs to the Fama-French (1993) 25 portfolios (a task comparable to that in Fama and French, 1996, Table I), the average absolute pricing error is only 0.07% per month. The overidentification test fails to reject the null hypothesis that the average pricing error is zero. Further, none of the 25 alphas are significantly different from zero. The small-stock value strategy, in particular, carries an insignificant alpha of 0.04% per month. The model is less successful in matching average returns across the SUE portfolios. When we only use the SUE portfolios in the moment conditions, the benchmark model generates an average difference of 0.81% per month between the returns of the low and high SUE portfolios, but both extreme deciles have significant alphas of 0.21% and 0.36% per month, respectively. The model performs well for the nine double-sorted size and SUE portfolios, in which only three out of nine estimated alphas are significant. More importantly, the model-implied average return spread between the low SUE and high SUE terciles is 0.80% per month in small firms, a difference noticeably higher than the 0.14% per month difference in big firms. Unfortunately, the difference in model-implied average returns between extreme SUE portfolios largely disappears in joint estimation, in which we also ask the model to price the Fama-French (1993) 25 size and book-to-market portfolios, ten investment-to-assets portfolios, and the market portfolio. Our asset pricing tests represent a fundamental departure from traditional tests in the cross section of returns. Derived from optimality conditions, our approach differs from empirically mo- 4

5 tivated factor models, as in Fama and French (1993, 1996). In contrast to tests derived from consumption-based asset pricing (e.g., Hansen and Singleton, 1982; Lettau and Ludvigson, 2001; Bansal, Dittmar, and Lundblad, 2005), we do not use information on preferences or the stochastic discount factor. Our expected return model is based entirely on firm characteristics. In view of the empirical difficulty in estimating risk and risk premia documented in, for example, Fama and French (1997), it is perhaps not surprising that our model outperforms traditional asset pricing models in capturing anomalies. Our theoretical approach is built on Cochrane (1991), who first establishes the link between stock and investment returns. (See also Restoy and Rockinger, 1994). Cochrane (1996) uses the aggregate investment return to price the cross section of returns. Cochrane (1997) also uses the investment-return equation to explain the magnitude of the equity premium. Our contribution consists of employing the q model to address anomalies in the cross section. We also add to studies of the cross section of returns based on intertemporal models of optimal firm behavior, as in Berk, Green, and Naik (1999), Carlson, Fischer, and Giammarino (2004), and Zhang (2005). Our work differs because we do structural estimation on closed-form estimating equations. Our work is also related to the q theory of investment originated by Brainard and Tobin (1968) and Tobin (1969). On a theoretical level the equivalence between stock and investment returns is an algebraic restatement of the equivalence between marginal q and average q, demonstrated in Lucas and Prescott (1971) and Hayashi (1982). Empirically, our work is connected with the investment Euler equation tests designed to understand the investment behavior (e.g., Shapiro, 1986; Whited, 1992). Our extention of this approch consists of restating the investment Euler equation in terms of stock returns and of testing it using data on the cross section of returns. 2 A Two-Period Example We follow Cochrane (1991, 1996) to link expected returns to characteristics using the q theory. 5

6 2.1 The Setup Firms use capital and a vector of costlessly adjustable inputs to produce a perishable output good. Firms choose the levels of these inputs each period to maximize their operating profits, defined as revenues minus the expenditures on these inputs. Taking the operating profits as given, firms then choose optimal investment to maximize their market value. There are only two periods, t and t+1. Firm j starts with capital stock k jt, invests in period t, and produces in both t and t+1. The firm exits at the end of period t+1 with a liquidation value of (1 δ j )k jt+1, in which δ j is the firm-specific rate of capital depreciation. Let π jt =π(k jt,x jt ) denote the operating profit at time t, in which x jt is a vector of exogenous shocks including aggregate and firm-specific productivity shocks. We assume that operating profits exhibit constant return to scale, that is, π(k jt,x jt )=π 1 (k jt,x jt )k jt, in which numerical subscripts denote partial derivatives. The expression π 1 (k jt,x jt ) therefore denotes the marginal product of capital.. The law of motion for capital is k jt+1 = i jt + (1 δ j )k jt, in which i jt denotes capital investment. We use the one-period time-to-build convention: capital goods invested today only become productive at the beginning of next period. Investment incurs quadratic adjustment costs given by θ(i jt,k jt ) = (a/2)(i jt /k jt ) 2 k jt, in which a > 0 is a constant parameter. The function θ (i jt,k jt ) is increasing and convex in i jt, decreasing in k jt, and exhibits constant return to scale, that is, θ(i jt,k jt )=θ 1 (i jt,k jt )i jt + θ 2 (i jt,k jt )k jt. Let m t+1 be the stochastic discount factor from time t to t+1, which is correlated with the aggregate component of x jt+1. Firm j choose i jt to maximize the market value of equity: Payout/external equity at period t {}}{ max π(k jt,x jt ) i jt a [ ] Cash flow at period t+1 2 ijt {}}{ k jt + E t m t+1 π(k jt+1,x jt+1 ) + (1 δ j )k jt+1 {i jt } 2 k jt } {{ } Cum dividend market value of equity at period t+1 (1) The market value contains two parts. The first is net cash flow over period t, which is denoted by π(k jt,x jt ) i jt (a/2)(i jt /k jt ) 2 k jt. Firms use operating profits π(k jt,x jt ) to pay investment costs, 6

7 which include both purchase costs, i jt, and adjustment costs, (a/2)(i jt /k jt ) 2 k jt. If net cash flow is positive, firms distribute it to shareholders (payout). Otherwise, negative net cash flows consist of cash inflows from shareholders (external equity). The second part of the market value contains the discounted value of cash flow from period t+1, given by the sum of operating profits π(k jt+1,x jt+1 ) and the liquidation value of the capital stock at the end of t+1. Taking the partial derivative of equation (1) with respect to i jt yields the first-order condition: Marginal cost of investment at period t {}} [ ]{ ijt 1 + a k jt = E t m t+1 Marginal benefit of investment at period t+1 {}}{ π 1 (k jt+1,x jt+1 ) + (1 δ j ) } {{ } marginal q at period t (2) Equation (2) says that the marginal cost of investment must equal the marginal benefit (commonly dubbed marginal q). To generate one additional unit of capital at the beginning of next period, k jt+1, firms must pay the unit price of capital (normalized to be one) and the marginal adjustment cost, a(i jt /k jt ). The next-period marginal benefit of this additional unit of capital includes the marginal product of capital, π 1 (k jt+1,x jt+1 ), and the liquidation value (net of depreciation), 1 δ j. Discounting this next-period benefit using the pricing kernel m t+1 yields the marginal q at time t. To derive asset pricing implications of the two-period q-theoretic model, we follow Cochrane (1991) and divide both sides of equation (2) by the marginal cost of investment to obtain: E t [m t+1 r I jt+1 ] = 1 (3) in which r I jt+1 denotes the investment return, defined as: r I jt+1 }{{} Investment return from period t to t+1 Marginal benefit of investment at period t+1 {}}{ π 1 (k jt+1,x jt+1 ) + (1 δ j ) 1 + a(i jt /k jt ) }{{} Marginal cost of investment at period t (4) The investment return, rjt+1 I, is therefore the ratio of the marginal benefit of investment at period t+1 divided by the marginal cost of investment at period t. 7

8 Under constant return to scale, stock returns equal investment returns. To derive this result, we note from equation (1) that we can define the ex-dividend equity value at period t, denoted p jt, as: p jt }{{} Ex dividend equity value at period t = E t m t+1 Cash flow at period t+1 {}}{ π(k jt+1,x jt+1 ) + (1 δ j )k jt+1, (5) that is, p jt equals the cum-dividend equity value the maximum in equation (1) minus the net cash flow over period t. We can now define stock returns as follows: rjt+1 S = }{{} Stock return from period t to t+1 Cash flow at period t+1 {}}{ π(k jt+1,x jt+1 ) + (1 δ j )k jt+1, (6) E t [m t+1 [π(k jt+1,x jt+1 ) + (1 δ j )k jt+1 ]] }{{} Ex dividend equity value at period t in which the numerator is the cash flow at period t+1 (the ex-dividend market value of equity is zero). Dividing both the numerator and the denominator of equation (6) by k jt+1 yields: r S jt+1 = π 1 (k jt+1,x jt+1 ) + (1 δ j ) E t [m t+1 [π 1 (k jt+1,x jt+1 ) + (1 δ j )]] = π 1(k jt+1,x jt+1 ) + (1 δ j ) = r I 1 + a(i jt /k jt ) jt+1, (7) in which the second equation follows from the first-order condition given by equation (2). Because of this equivalence, in what follows, we use r jt+1 to denote both stock and investment returns. 2.2 Economic Mechanisms Admittedly, the prediction in equation (7) that stock returns equal investment returns in realizations is restrictive; for example, Cochrane (1991) shows that aggregate investment returns are much less volatile than aggregate stock returns. However, to provide economic intuition on the driving forces of expected returns, we only use the restriction in expectations: E t [r jt+1 ] = }{{} Expected return Expected marginal product of capital {}}{ E t [π 1 (k jt+1,x jt+1 )] + 1 δ j. (8) 1 + a(i jt /k jt ) }{{} Marginal cost of investment In this regard, we follow Cochrane (1997), who shows that average aggregate equity returns are well within the range of plausible parameters for average aggregate investment returns. 8

9 Equation (8) is useful for interpreting anomalies because it ties expected returns directly to characteristics. The equation implies that there are three firm-specific determinants of expected returns: the investment-to-assets ratio, i jt /k jt, the expected marginal product of capital, E t [π 1 (k jt+1,x jt+1 )], and the depreciation rate, δ j. We discuss each below The Value Anomaly Holding constant the expected marginal product of capital and the rate of capital depreciation, equation (8) implies the negative relation between expected returns and investment-to-assets in Cochrane (1991). In the capital budgeting language of Brealey, Myers, and Allen (2006, Chapter 6), investment-to-assets increases with the net present value of capital. All else equal, the net present value decreases with the cost of capital, which is precisely the expected return. Given expected cash flows, high costs of capital imply low net present value, which in turn implies low investment-to-assets ratios. Further, a low cost of capital implies high net present value, which in turn implies high investment-to-assets ratios. Adjustment costs are a crucial ingredient for our model. From equation (8), the partial derivative of expected returns with respect to the investment-to-assets ratio is: 1 E t [r jt+1 ] (i jt /k jt ) = a(e t[π 1 (k jt+1,x jt+1 )] + 1 δ j ) [1 + a(i jt /k jt )] 2 < 0. (9) The negative sign generates the negative relation between investment and expected returns, as illustrated in Figure 1. However, without adjustment costs, a = 0, and expected returns are independent of investment, E t [r jt+1 ]/ (i jt /k jt ) = 0. More importantly, Figure 1 shows that similar to high investment-to-asset firms, growth firms have low expected returns; and similar to low investment-to-asset firms, value firms have high 1 Because of the law of motion for capital, k jt+1 = [i jt/k jt + (1 δ j)]k jt, we should also worry about the effect of investment at period t on the capital at the beginning of period t+1. Let π 11(k jt+1, x jt+1) denote the second order partial derivative of π(k jt+1, x jt+1) with respect to k jt+1. We have: E t[r jt+1] (i jt/k jt) xjt+1)] + 1 δj) Et[π11(kjt+1, xjt+1)kjt] a (Et[π1(kjt+1, xjt+1)] + 1 δj) = a(et[π1(kjt+1, + = [1 + a(i jt/k jt)] a(i jt/k jt) [1 + a(i jt/k jt)] 2 The term involving π 11(k jt+1, x jt+1) drops out because π 11(k jt+1, x jt+1) = 0 (constant returns to scale). 9

10 expected returns. This pattern follows directly from a one-to-one mapping between investment-toassets and market-to-book. Equations (2) and (5) imply that [ ] ijt q jt = 1 + a }{{} k jt Marginal q = Average Q {}}{ p jt k jt+1 Q jt, (10) that is, marginal q = average Q. It follows that growth firms invest more and earn lower expected returns, and that value firms invest less and earn higher expected returns. Formally, equation (10) implies that (i jt /k jt )/ Q jt = 1/a > 0. The value anomaly follows because the chain rule of partial derivative implies that E t [r jt+1 ]/ Q jt = [ E t [rjt+1 S ]/ (i jt/k jt )][ (i jt /k jt )/ Q jt ] < 0. To show that the value anomaly is stronger in small firms, we need a few more lines of algebra. The main driving force is the convexity of the investment-return relation. From equation (9): 2 E t [r jt+1 ] (i jt /k jt ) 2 = 2a2 [E t [π 1 (k jt+1,x jt+1 )] + 1 δ j ] [1 + a(i jt /k jt )] 3 > 0. (11) We also need (i jt /k jt )/ p jt > 0. However, equation (10) implies that p jt = [1+a(i jt /k jt )](i jt /k jt + 1 δ)k jt and p jt / (i jt /k jt ) = q jt k jt +ak jt+1 > 0. The interpretation of p jt / (i jt /k jt ) > 0 is that high q firms invest more than low q firms (e.g., Fama and French 1995; Xing 2006). 2 The stronger value anomaly in small firms says that E t [r jt+1 ]/ Q jt / p jt < 0. In our model: E t [r jt+1 ] Q jt / p jt = 2 E t [r jt+1 ] Q jt p jt = 1 2 E t [r jt+1 ] (i jt /k jt ) a (i jt /k jt ) 2 < 0, (12) p jt in which the second equality follows from the chain rule and the inequality follows from the convexity of the investment-return relation and p jt / (i jt /k jt ) > 0 (high q firms invest more than low q firms). 2 p jt/ (i jt/k jt) > 0 does not imply that big firms invest more than small firms. Although firm size means market capitalization in empirical finance (e.g., Fama and French 1993), firm size means physical size in macroeconomics and in our model. For example, Evans (1987) and Hall (1987) show that small firms invest more and grow faster than big firms, using the logarithm of employment as the measure of firm size. The measure corresponds to log(k jt) in our model. More important, from equation (2). i jt/k jt is independent of k jt in our model (constant return to scale). 10

11 2.2.2 Post-Earnings-Announcement Drift The relation between earnings and average returns arises naturally in our neoclassical model. Intuitively, the marginal product of capital in the numerator of the expected-return equation (8) is closely related to earnings. Accordingly, expected returns increase with earnings. The earnings-return relation persists even without capital adjustment costs. In this sense, the earnings-return relation is more fundamental than the investment-return relation (and the valuereturn relation). Specifically, when a = 0, equation (8) reduces to: E t [r jt+1 ] = }{{} Expected return Expected marginal product of capital {}}{ E t [π 1 (k jt+1,x jt+1 )] + (1 δ j ) (13) i.e., expected (net) returns equal the expected marginal product of capital minus the depreciation rate. Earnings equals operating cash flows minus the capital depreciation (the only accruals in our model). 3 Let e jt denotes earnings, then: e jt }{{} Earnings Operating cash flows {}}{ π(k jt,x jt ) δ j k jt }{{} Capital depreciation (14) We can now link expected returns to expected accounting profitability. Because constant return to scale, equation (13) becomes: E t [r jt+1 ] = }{{} Expected return Average product of capital { [ }} ]{ πjt+1 E t k jt+1 i.e., expected returns equal expected profitability. [ ] ejt δ j = E t k jt+1 }{{} Expected profitability + 1 (15) To link expected returns further to earnings surprises, we note that profitability is highly per- 3 We assume implicitly that accruals are only used to mitigate the accounting timing and matching problems that deviate operating cash flows from earnings (e.g., Dechow 1994). In particular, we do not model earnings management. 11

12 sistent. 4 In particular, we assume that profitability follows an autoregressive process: e jt+1 [ ejt = ē(1 ρ k e ) + ρ e jt+1 k jt }{{} Expected profitability ] + Earnings surprise {}}{ ε e jt+1 (16) in which ē and 0 < ρ e < 1 are the long-run average and the persistence of profitability, respectively. ε e jt+1 denotes the earnings surprise. Combining equations (15) and (16) yields: E t [r jt+1 ] }{{} Expected return Profitability {[ }} ]{ ejt = ē(1 ρ e ) + ρ e k jt }{{} Expected profitability + 1 (17) Expected returns thus increase with the current-period profitability. This insight only depends on the persistence in profitability, not the specific functional form in equation (16). Plugging the one-period-lagged equation (16) into equation (17) yields: E t [r jt+1 ] }{{} Expected return [ ] Earnings surprise = ē(1 ρ e )(1 + ρ e ) + ρ 2 ejt 1 {}}{ e + ρ k e ε e t jt 1 }{{} Expected profitability + 1 (18) Expected returns therefore have a positive loading, ρ e, on earnings surprises, ε e t. The model thus predicts that the magnitude of the post-earnings-announcement drift should increase with the average persistence in firm-level profitability in the sample. But equation (18) predicts that the loading of expected returns on earnings surprises is constant across different market capitalization groups. To generate the stronger post-earnings-announcement drift in small firms, we need to introduce adjustment costs. We go back to equation (8) for this purpose. Combining equations (8) and (14), we have: E t [r jt+1 ] }{{} Expected return = Expected profitability {}}{ E t [e jt+1 /k jt+1 ] a(i jt /k jt ) }{{} Marginal cost of investment (19) 4 There is much evidence on the persistence in profitability at the portfolio level and at the firm level (e.g., Fama and French 1995, 2000). For example, Fama and French (2006) document that the current-period profitability is the strongest predictor of profitability one to three years ahead. 12

13 Therefore, controlling for the denominator that is equivalent to market-to-book, 1 + a(i jt /k jt ) = q jt = Q jt, expected returns increase with expected profitability. Empirically, Haugen and Baker (1996) and Fama and French (2006) show that, controlling for market valuation ratios, firms with high expected profitability earn higher average returns than firms with low expected profitability. More important, equation (19) implies that the loading of expected returns on the expected profitability equals 1/Q jt = k jt+1 /p jt that is inversely related to the market capitalization, p jt. 5 This prediction is consistent with Cohen, Gompers, and Vuolteenaho s (2002) evidence that the relation between expected profitability and future returns is stronger in small firms. Moreover, combining equations (14) and (8), we have: E t [r jt+1 ] }{{} Expected return = Expected profitability [ ] ejt 1 {}}{ ē(1 ρ e )(1 + ρ e ) + ρ 2 e k jt a(i jt /k jt ) }{{} Marginal cost of investment + ρ e ε e t + 1 }{{} Earnings surprise (20) Therefore, controlling for market-to-book, expected returns increase with earnings surprises, and the loading of expected returns on earnings surprises equals ρ e k jt+1 /p jt, which is inversely related to the market capitalization, p jt. This prediction is consistent with Bernard s (1993) evidence that the magnitude of the post-earnings-announcement drift is stronger in small firms The Depreciation-Return Relation Equation (8) shows that, all else equal, firms that have higher rates of capital depreciation earn lower expected returns than firms that have lower rates of capital depreciation. Our prediction on the depreciation-return relation is consistent with the theoretical results of Tuzel (2005). Using a two-sector model, Tuzel shows that firms with more structure earn higher average returns than firms with more equipment. The reason is that structure depreciates more slowly than equipment. 5 Fama and French (1992) document that the average cross-sectional correlation between book-to-market and market capitalization is

14 2.3 Characteristics-Based Asset Pricing As a fundamental departure from traditional asset pricing theory, our q-theoretic model determines expected returns entirely from firm characteristics via the first principle of investment without any information about the pricing kernel. The reason is that, m t+1, and its covariances with returns do not enter the expected-return equation (8). This insight carries over to the dynamic setting (see Proposition 1 below). Characteristics are sufficient statistics for expected returns, and investmentbased asset pricing can be developed independently from consumption-based asset pricing. 6 This practice only means that the effect of m t+1 is indirect, not irrelevant. If m t+1 is a constant, m, then equation (26) implies that the expected return E t [r jt+1 ] = 1/m, a constant uncorrelated with firm characteristics. If the correlation between m t+1 and x jt+1 is zero, i.e., the operating profits of firms are unaffected by aggregate shocks, then equation (3) implies that E t [r jt+1 ]= r ft, in which r ft 1/E t [m t+1 ] is the risk free rate. In this case, there is no cross-sectional variation in expected returns, and our results only provide time-series correlations between the risk free rate and firm characteristics. Because we study expected returns instead of expected excess returns, we do not need to specify m t+1 necessary for determining the risk free rate, and we do not need to restrict the correlation between m t+1 and x jt+1 as necessary for determining expected excess returns. More important, the characteristic-based approach is internally consistent with the traditional risk-based approach. From equation (3) and equivalence between stock and investment returns, E t [m t+1 r jt+1 ] = 1. Following Cochrane (2001, p. 19), we rewrite this equation as the beta-pricing form, E t [r jt+1 ] = r ft +β jt λ mt, in which β jt Cov t [r jt+1,m t+1 ]/Var t [m t+1 ] is the amount of risk, and λ mt Var t [m t+1 ]/E t [m t+1 ] is the price of risk, but from E t [r I jt+1 ] = E t[r S jt+1 ] = r ft +β jt λ mt, we have β jt = E t [r I jt+1 ] r ft/λ mt, which provides a one-to-one mapping between covariances and characteristics. However, apart from this mechanical link, risk only plays a secondary role in our characteristic-based determination of expected returns. 6 This approach is foretold by Rubinstein (2001, p. 23): For the most part, financial economists take the stochastic process of stock prices, the value of the firms, or dividend payments as primitive. However, to explain some anomalies, we may need to look deeper into the guts of corporate decision making to derive what the processes are. 14

15 3 The Dynamic Model We first describe the model with all equity financing. Then we add more realistic ingredients such as leverage and costly external equity. 3.1 All Equity Financing Firms use capital and a vector of costlessly adjustable inputs to produce a homogeneous output. The levels of these inputs are chosen each period to maximize operating profits, defined as revenues minus the expenditures on these inputs. Taking the operating profits as given, firms then choose optimal investment to maximize their market value. Investment involves physical costs of adjustment. Let π jt π(k jt,x jt ) denote the maximized operating profits of firm j at time t, in which k jt is the capital stock and x jt is a vector of exogenous shocks such as aggregate and firm-specific shocks to the production technology and the prices of costlessly adjusted inputs and industry- and firm-specific shocks to the demand for firm output. π(k jt,x jt ) exhibits constant return to scale. Firms that stay in production each period must incur a flow operating cost proportional to capital stock, ck jt, in which the parameter c>0 is a constant common to all firms. End-of-period capital equals investment plus beginning-of-period capital net of depreciation: k jt+1 = i jt + (1 δ jt )k jt (21) We assume that capital depreciates at an exogenous proportional rate of δ jt. This rate is firmspecific and time-varying because later we use data on capital depreciation divided by capital stock to measure the rate of depreciation in our tests. When firms invest, they incur purchase/sales costs and convex costs of physical adjustment. Purchase/sales costs are incurred when firms buy or sell uninstalled capital. When firms disinvest, the sales costs are negative, representing revenues. Convex costs of physical adjustment are nonnegative costs that are zero when i jt = 0. These costs are continuous, strictly convex in i jt, non-increasing in k jt, and differentiable with respect to i jt and k jt everywhere; and the second-order 15

16 partial derivative of the convex-cost function with respect to k jt is nonnegative. We let φ(i jt,k jt ) denote the total cost of investment that represents the sum of purchase/sale costs and convex costs of physical adjustment, and we refer to it as the augmented adjustment-cost function. φ(i jt,k jt ) satisfies φ 1 0,φ 2 0, and φ 11 >0, in which the subscript i denotes the first-order partial derivative with respect to the i th argument, and multiple subscripts denote high-order derivatives. Let q jt be the present-value multiplier associated with equation (21). We can formulate the market value of firm j, denoted v(k jt,x jt ), as follows: v(k jt,x jt ) = [ max E t {i jt+s, k jt+1+s } s=0 s=0 m t+s (π(k jt+s,x jt+s ) ck jt+s φ(i jt+s,k jt+s ) q jt+s [k jt+s+1 (1 δ jt+s )k jt+s i jt+s ]) ] (22) The first-order conditions with respect to i jt and k jt+1 are, respectively, q jt = φ 1 (i jt,k jt ) (23) q jt = E t [m t+1 [π 1 (k jt+1,x jt+1 ) c φ 2 (i jt+1,k jt+1 ) + (1 δ jt+1 )q jt+1 ]] (24) Combining equations (23) and (24) yields: φ 1 (i jt,k jt ) = E t [m t+1 [π 1 (k jt+1,x jt+1 ) c φ 2 (i jt+1,k jt+1 ) + (1 δ jt+1 )φ 1 (i jt+1,k jt+1 )]] (25) Dividing both sides by φ 1 (i jt,k jt ) yields: E t [m t+1 r I jt+1 ] = 1 (26) in which r I t+1 denotes the investment return, defined as: r I jt+1 π 1(k jt+1,x jt+1 ) c φ 2 (i jt+1,k jt+1 ) + (1 δ jt+1 )φ 1 (i jt+1,k jt+1 ) φ 1 (i jt,k jt ) (27) Intuitively, equation (27) says that the investment return is the ratio of the marginal benefit of investment at time t+1 divided by the marginal cost of investment at time t. The denominator, φ 1 (i jt,k jt ), is the marginal cost of investment. By optimality, it equals marginal q jt the shadow value of capital, or the expected present value of the marginal profits from investing in capital 16

17 goods. In the numerator of equation (27), π 1 (k jt+1,x jt+1 ) c is the extra operating profits, net of the flow operating costs generated by the extra capital at t+1; the effect of extra capital on the augmented adjustment cost is captured by φ 2 (i jt+1,k jt+1 ); and (1 δ jt+1 )φ 1 (i jt+1,k jt+1 ) is the expected present value of marginal profits evaluated at time t+1, net of depreciation. Proposition 1 (Cochrane 1991) Define the ex-dividend firm value, p jt, as p jt p(k jt,k jt+1,x jt ) = v(k jt,x jt ) π(k jt,x jt ) + ck jt + φ(i jt,k jt ) and stock return as: r S jt+1 p jt+1 + π(k jt+1,x jt+1 ) ck jt+1 φ(i jt+1,k jt+1 ) p jt (28) Under constant return to scale, we have p jt = q jt k jt+1 and r S jt+1 = ri jt+1. Proof. See Appendix A. The result is first derived by Cochrane (1991) (see also Restoy and Rockinger 1994). The equivalence between stock and investment returns is an algebraic restatement of the equivalence between marginal q and average Q (e.g., Hayashi 1982). 3.2 Leverage The benchmark model assumes that all firms are entirely equity financed. If firms finance investment using both equity and debt, the investment return is the leverage-weighted average of equity return and corporate bond return. 7 For simplicity, we follow Hennessy and Whited (2005) and model only one-period debt. At the beginning of period t, firm j can choose to issue a certain amount of one-period debt, denoted b jt+1, that must be repaid at the beginning of next period. Negative b jt+1 represents cash holdings. The interest rate on b jt is r(x jt ), and is a function of the exogenous state variable, x jt, and can be stochastic. r(x jt ) can also be firm-specific because x jt contains both aggregate and firm-specific shocks. 7 This result was first established by Gomes, Yaron, and Zhang (2006) but not reported in the published paper. 17

18 We can now formulate the cum-dividend market value of equity as: v(k jt,b jt,x jt ) = max {i jt+s,k jt+s+1,b jt+s+1 } s=0 E t s=0 m t+s[π(k jt+s,x jt+s ) ck jt+s φ(i jt+s,k jt+s ) + b jt+s+1 r (x jt+s ) b jt+s q jt+s (k jt+s+1 (1 δ jt+s )k jt+s i jt+s )] The optimality conditions with respect to i jt, k jt+1, and b jt+1 are, respectively: (29) q jt = φ 1 (i jt,k jt ) (30) q jt = E t [m t+1 [π 1 (k jt+1,x jt+1 ) c φ 2 (i jt+1,k jt+1 ) + (1 δ jt+1 )q jt+1 ]] (31) 1 = E t [m t+1 r (x jt+1 )] (32) It follows that E t [m t+1 rjt+1 I ]=1, and E t[m t+1 rjt+1 B ]=1, in which the investment return is defined as: r I jt+1 π 1(k jt+1,x jt+1 ) c φ 2 (i jt+1,k jt+1 ) + (1 δ jt+1 )φ 1 (i jt+1,k jt+1 ) φ 1 (i jt,k jt ) (33) and the corporate bond return is defined as: r B jt+1 r (x jt+1 ) (34) Proposition 2 (Gomes, Yaron, and Zhang 2006) Define the ex-dividend equity value as: p(k jt,b jt,x jt ) v(k jt,b jt,x jt ) π(k jt,x jt ) + ck jt + φ(i jt,k jt ) b jt+1 + r (x jt )b jt and r S jt+1 [p jt+1 + π(k jt+1,x jt+1 ) ck jt+1 φ(i jt+1,k jt+1 ) + b jt+2 r(x jt+1 )b jt+1 ]/p jt as stock returns. Under constant return to scale, q jt k jt+1 = p(k jt,b jt,x jt ) + b jt+1 (35) More important, the investment return is the leverage-weighted average of stock and bond returns: r I jt+1 = ν jt r B jt+1 + (1 ν jt )r S jt+1 (36) in which ν jt is the market leverage ratio: ν jt b jt+1 /[p(k jt,b jt,x jt ) + b jt+1 ]. 18

19 Proof. See Appendix A. Equation (36) says that the investment return is the leverage-weighted average of stock returns and bond returns. Although we make some simplifying assumptions to derive this equation, it is likely to hold under more general conditions given its well-known application in capital budgeting (e.g., Grinblatt and Titman 2001, p. 381). 3.3 Costly External Equity The benchmark framework assumes that firms can finance capital investment using external equity costlessly. In reality, issuing equity is often costly (e.g., Smith 1977; Altinkilic and Hansen 2000; Hennessy and Whited 2006). To capture the equity financing costs, we let ψ(o jt,k jt ) denote the financing-cost function of issuing equity, in which o jt is the amount of financing, o jt [φ(i jt,k jt ) + ck jt π(k jt,x jt )]1 o jt (37) and in which 1 o jt 1 {φ(i jt,k jt )+ck jt π(k jt,x jt ) 0} is the indicator function that takes the value of one if the firm uses outside equity and zero otherwise. We assume that the financing-cost function is increasing, ψ 1 >0, convex, ψ 11 >0, exhibits economies of scale, ψ 2 0 and constant return to scale: ψ(o jt,k jt ) = ψ 1 (o jt,k jt )o jt + ψ 2 (o jt,k jt )k jt (38) v(k jt,x jt ) = We can now formulate the cum-dividend market value of equity as: max {i jt+s, k jt+s+1 } s=0 E t [ s=0 m t+s[π(k jt+s,x jt+s ) ck jt+s φ(i jt+s,k jt+s ) ψ(o jt+s,k jt+s ) q jt+s [k jt+s+1 (1 δ jt+s )k jt+s i jt+s ]] ] (39) The optimality conditions with respect to i jt and k jt+1 are, respectively: q jt = φ 1 (i jt,k jt )[1 + ψ 1 (o jt,k jt )1 o jt] [ [ (π 1 (k jt+1,x jt+1 ) c φ q jt = E t m 2 (i jt+1,k jt+1 ))(1 + ψ 1 (o jt+1,k jt+1 )1 o jt+1 ) t+1 ψ 2 (o jt+1,k jt+1 )1 o jt+1 + (1 δ jt+1)q jt+1 ]] 19

20 Combining the two equations yields E t [m t+1 rjt+1 I ]=1, in which the investment return is: r I jt+1 = [ (π 1 (k jt+1,x jt+1 ) c φ 2 (i jt+1,k jt+1 ))(1 + ψ 1 (o jt+1,k jt+1 )1 o jt+1 ) ψ 2 (o jt+1,k jt+1 )1 o t+1 + (1 δ jt+1)φ 1 (i jt+1,k jt+1 )(1 + ψ 1 (o jt+1,k jt+1 )1 o jt+1 ) φ 1 (i jt,k jt )[1 + ψ 1 (o jt,k jt )1 o jt ] (40) ] The investment return in equation (40) is still the ratio of the marginal benefits of investment evaluated at period t + 1 divided by the marginal costs of investment at period t. Increasing one unit of capital entails marginal purchase/sales and physical adjustment costs that sum up to φ 1 (i jt,k jt ). If this investment is partially financed by external equity, its marginal financing cost is then ψ 1 (o jt,k jt ) o jt / i jt = ψ 1 (o jt,k jt )φ 1 (i jt,k jt ). Adding all three parts of the marginal cost yields the denominator in equation (40). The numerator of equation (40) contains three terms. The interpretation of the first term, π 1 (k jt+1,x jt+1 ) c φ 2 (i jt+1,k jt+1 ), is the same as that in the benchmark model. If the firm issues external equity at t+1, then the marginal effect of the extra unit of capital on the amount of financing costs is ψ 1 (o jt+1,k jt+1 )1 o jt+1 o jt+1/ k jt+1 = [π 1 (k jt+1,x jt+1 ) c φ 2 (i jt+1,k jt+1 )]ψ 1 (o jt+1,k jt+1 )1 o jt+1. The extra unit of capital also lowers financing costs because of economies of scale. This benefit is captured by ψ 2 (o jt+1,k jt+1 )1 o jt+1. At the end of period t+1, the firm is left with 1 δ jt+1 units of capital net of depreciation. This capital is worth marginal q evaluated at time t+1, which equals the marginal costs of investment at that time. Proposition 3 Define the ex-dividend market value of equity, p jt, as p jt p(k jt,k jt+1,x jt ) = v(k jt,x jt ) π(k jt,x jt ) + ck jt + φ(i jt,k jt ) + ψ(o jt,k jt ) and stock return as r S jt+1 [p jt+1 + π(k jt+1,x jt+1 ) ck jt+1 φ(i jt+1,k jt+1 ) ψ(o jt+1,k jt+1 )]/p jt. If the operating-profit, the adjustment-cost, and the financing-cost functions all exhibit constant return to scale, then p jt = q jt k jt+1 and r S jt+1 =ri jt+1, in which ri jt+1 is given by equation (40). Proof. See Appendix A. It is tempting to incorporate time-to-build into the model. Time-to-build says that multiple periods are required to build new capital projects, instead of the one-period convention embedded 20

21 in the standard capital accumulation equation (21). Theoretically, several studies have demonstrated the importance of time-to-build in driving business cycle fluctuations (e.g., Kydland and Prescott 1982). Empirically, Lamont (2000) shows that investment plans can predict excess stock returns better than actual aggregate investment, probably because of investment lags. However, the analytical link between the stock and investment returns breaks down under time-to-build. The reason is that the investment return measures the trade-off between the marginal benefits and the marginal costs of new investment projects. In contrast, the stock return is the return to the entire firm that derives its market value not only from the new but also from the old incomplete projects. Detailed derivations are not reported but are available upon request. 4 Empirical Methodology Our chief goal in this section is to evaluate how well the economic mechanisms developed in Section 2 can quantitatively explain the anomalies in the data. Section 4.1 outlines the design of the benchmark test, and Section 4.2 discusses its various perturbations. 4.1 The Benchmark Specification Proposition 2 shows that the investment return is the leverage-weighted average of the stock return and the corporate bond return. We unlever the investment return in equation (36) as: r S jt+1 = ri jt+1 ν jtr B jt+1 1 ν jt. (41) Because anomalies primarily concern first moments, we therefore test the ex-ante restriction implied by equation (41): expected stock returns equal expected unlevered investment returns, that is, E t [r S jt+1 ] = E t [ r I jt+1 ν jt r B jt+1 1 ν jt We test this restriction using GMM on the following moment conditions: E [[ r S jt+1 ri jt+1 ν jtr B jt+1 1 ν jt ] ] Z t ]. (42) = 0, (43) 21

22 in which denotes the Kronecker product, and Z t is a vector of portfolio-specific and aggregate instrumental variables known at time t. Our test differs from investment-based asset pricing tests in Cochrane (1991, 1996), Balvers and Huang (2006), Gomes, Yaron, and Zhang (2006), and Whited and Wu (2006). Cochrane (1991, 1996) and Gomes et al. (2006) parameterize the pricing kernel as a linear function of aggregate investment returns constructed from macroeconomic variables. Similarly, Balvers and Huang parameterize the pricing kernel as a linear function of the Solow residual. However, firm characteristics are absent in these papers. Whited and Wu (2006) test equation (26) by assuming m t+1 is a linear combination of the Fama-French (1993) factors and by constructing firm-level investment returns. However, stock returns are absent. Our test has its roots in Cochrane (1991), who compares the time series properties of aggregate stock and investment returns. We add more ingredients into the q-theoretic framework, and use it to understand the anomalies in the cross section of returns. Although based on the same investment Euler equation, our test differs from the Euler equation tests in the investment literature (e.g., Shapiro, 1986; Whited, 1992; Love, 2003). These tests assume a constant pricing kernel, m t+1, which in turn implies that all stocks earn the risk-free rate ex ante. We do not include the quantity moment condition from equation (25) into our set of return moment conditions. Doing so requires us to make strong parametric assumptions about m t Testing Portfolios We use GMM to implement the test given by equation (46) on portfolios sorted on anomaly-related characteristics. We use portfolio-level data two reasons. First, as pointed out in Whited (1998), simple versions of investment Euler equations are almost always strongly rejected at the firm level. The reason is that real investment is lumpy at the firm level, especially in small firms. To capture lumpy investment, we can incorporate fixed costs, but we lose the differentiability of the adjustment-cost function, φ(i jt,k jt ), at the point where the investment, i jt, equals zero. Second, and more importantly, asset pricing anomalies are often documented at the portfolio level, so it is 22

23 natural to conduct our tests using portfolios. We use 55 testing portfolios: the Fama-French 25 size and book-to-market portfolios; ten portfolios sorted on the investment-to-assets ratio; ten portfolios sorted on Standardized Unexpected Earnings, or SUE; nine portfolios sorted on size and SUE; and the aggregate stock-market portfolio. Our choice of testing portfolios captures a wide array of anomalies. We include book-to-market and SUE portfolios because the value anomaly and post-earnings-announcement drift are arguably two of the most celebrated anomalies (e.g., Fama and French 1992, 1993; Bernard and Thomas 1989, 1990). We include the investment-to-assets portfolios because the q-theory explanation of the value anomaly works primarily though capital investment. We also include the aggregate market portfolio to facilitate comparison with Cochrane (1991), who essentially tests the implication that the aggregate stock return equals the aggregate investment return Bond Yields Because firm-level corporate bond data are rather limited, and because few or none of the firms in several portfolios have corporate bond ratings, we use the Baa-rated bond yield as rjt+1 B in equation (43) for all portfolios. Although simplistic, this strategy avoids the use of firm-level bond return data that have a sample size much smaller than that for firm-level stock return data. Moreover, asset pricing anomalies are mostly documented for equity returns. Assuming no cross-sectional variations in average bond returns across anomaly-related portfolios therefore serves as a natural first cut Instrumental Variables Our list of instrumental variables includes a vector of ones and three portfolio-specific variables: investment-to-assets, sales-to-capital, and book-to-market. To construct portfolio-level investmentto-assets ratios, we divide the sum of investments for all the firms in the portfolio by the sum of their capital stocks. We use similar procedures to construct all portfolio-level characteristics. We also use in the list of instrumental variables a vector of macroeconomic variables that include the dividend yield, the default premium, the term premium, and the short-term interest rate. 23

24 These conditioning variables are commonly used to predict future stock market returns (e.g., Fama and French 1989; Ferson and Harvey 1991) Functional Forms We follow the empirical investment Euler equation literature in specifying the marginal product of capital, π 1 (k jt,x jt ), and the adjustment-cost function, φ(i jt,k jt ). We first need to relate the unobservable marginal product of capital to observables. As shown in Love (2003), if firms have a Cobb-Douglas production function with constant returns to scale, the marginal product of capital of portfolio j is given by: π 1 (k jt,x jt ) = κ y jt k jt, (44) in which y jt denotes sales, and κ denotes the capital share. Equation (44) assumes that the shocks to the operating profits, x jt, are reflected in the realizations of sales. To parameterize the adjustment-cost function, we follow Whited (1998) and Whited and Wu (2006) and use a flexible functional form that is linearly homogeneous but allows for nonlinearity in the marginal adjustment-cost function: N φ ( ) φ(i jt,k jt ) = i jt + 1 n n a ijt n k jt, (45) k jt n=2 in which a n,n=2,...,n φ are coefficients to be estimated, and N φ is a truncation parameter that sets the highest power of i jt /k jt in the expansion. If N φ = 2, then equation (45) reduces to the standard quadratic adjustment-cost function. To determine N φ, we follow Whited and Wu (2006) and use the test developed by Newey and West (1987). First, we choose a high starting value for N φ and estimate the model. Next, using the same optimal weighting matrix, we estimate a sequence of restricted models for progressively lower values of N φ, in which the corresponding coefficient, a Nφ +1, is set to zero. The final value for N φ is then the highest one for which the exclusion restriction on the parameter a Nφ +1 is not 24