Uncertainty, Credit-market Frictions and Corporate Investment Sensitivity to Cash Flow

Transcription

1 Uncertainty, Credit-market Frictions and Corporate Investment Sensitivity to Cash Flow Delong Li The Johns Hopkins University (This is a preliminary draft.) June 2, 2017 Abstract This paper investigates effects of uncertainty and credit-market frictions on corporate investment sensitivity to cash flow (ISC). I present a stochastic heterogeneous-agent model in which a consolidated firm incorporates a group of individual production units that possess different variances of future profitability, while facing occasionally-binding capitalirreversibility and collateral constraints. I show analytically that the interaction among heterogeneous variances, irreversibility, and collateral constraints generates different investment policy across production units as well as distinct unit-level ISC: production units with low individual variances choose to exercise investment having positive ISC because of binding collateral constraints, while those with high variances choose to have zero investment and ISC. I aggregate all unit-level ISC to get the parent firm-level ISC and show that the latter is reduced by the parent firm s uncertainty. In contrast, the degree of being financially constrained has two opposite competing impacts on the ISC. Using an annual panel of more than 10,000 listed firms in the United States over the past 30 years, I find empirical results consistent with my model s predictions. My work thus highlights the importance of uncertainty and credit-market frictions in shaping investment decisions beyond the effects on the investment level - firms could become more or less sensitive to liquidity due to the interactive effect of uncertainty, credit-market frictions and internal liquidity. Given that recessions are typically periods of heightened uncertainty and tighter financial constraints, my finding has implications in evaluating the effectiveness of liquidity-provision policy in downturns. I thank my advisors Greg Duffee, Jon Faust and Jonathan Wright for their advice. I thank Laurence Ball, Robert Barbera, Christopher Carroll, Olivier Jeanne and all the participants of the Johns Hopkins macroeconomic seminar for comments. All mistakes are my own. 1

2 1 Introduction Impacts of uncertainty and credit-market frictions on real economic activities have been popular in the past decade. As Stock and Watson (2012) have documented,.. shocks that produced the recession primarily were associated with financial disruptions and heightened uncertainty. Among others, Gilchrist et al. (2014) show that both uncertainty and credit-market frictions could significantly depress corporate investment level. However, the investment sensitivity to cash flow is to a large extent ignored by the macroeconomic literature, though its close counterpart in the consumer theory the marginal propensity to consume (MPC) attracts great attention. Similar to the MPC, investment sensitivity to cash flow describes firms propensity to invest out of internal funds. It is thus fundamental for understanding the dynamics of corporate investment when the Modigliani-Miller conditions do not hold. 1 In this paper I show that shocks to uncertainty and credit-market frictions not only drive the investment level, but also influence the investment sensitivity to cash flow (ISC). Identifying such impacts is important because it uncovers the interaction among uncertainty, credit-market frictions and internal liquidity. It is thus helpful for understanding the transmission mechanism of uncertainty and financial shocks. Specifically, I show that though both types of shocks depress investment, it could be because firms voluntarily postpone investment decisions until uncertainty resolves or because they are forced to cut investment due to the lack of credit. In the latter case, liquidity-provision stimulus, like the income effect of corporate tax cuts, investment tax credit, or expansionary monetary policy, could contribute to boosting investment, while such policy could be ineffective in the former case. For instance, the Great Recession was not only about liquidity shortage and credit tightening, the market uncertainty also shot up to the historic high. The stimulus policy could then play limited role because firms are reluctant to invest when facing heightened uncertainty this is consistent with what happened in reality, as Stein (2012) documents: firms took advantage of the quantitative easings to borrow cheaply but to buy back stocks, rather than to invest. In summary, since the effectiveness of policy stimulus depends on whether firms investment is responsive enough to liquidity shocks, any study 1 In the world of Modigliani and Miller (1958), real economic decisions do not depend on the firm s capital structure. Thus investment sensitivity to cash flow is zero, as long as the latter is not serial correlated. 2

3 that ignores the interactions among uncertainty, credit-market frictions and internal liquidity is likely to compute misleading estimates of the policy effects. I first present a theoretical model that features heterogeneous uncertainty, capital irreversibility and occasionally-binding collateral constraints. I solve out the firm-level investment sensitivity to cash flow in closed form and show analytically how it is affected by swings in uncertainty and financial shocks that tighten collateral constraints. In the model, each firm incorporates a group of individual production units with a common expected profitability but different variances, with the latter influenced by uncertainty at the firm level. Such heterogeneity in individual variances works with capital irreversibility in generating different investment policy units with low variances are active in making new investment while those with high variances choose to wait having zero investment (called inactive ). This is because when future is uncertain and capital expenditures are irreversible, firms have a tendency to delay investment decisions. This difference in the investment policy generates different ISC. For active units, collateral constraints renders investment sensitive to internal funds (ISC > 0). For inactive units instead, ISC = 0 because investment equals to zero regardless of internal funds. After cross-sectional aggregation, the firm-level ISC equals to the average of all unit-level ISC. Higher uncertainty at the firm level reduces the ISC by shifting the distribution of individual variances to the right and pushing more units to the inactive region they will thus have ISC= 0. A financial shock that tightens collateral constraints in contrast has ambiguous consequences depending on two competing forces: an intensive-margin effect that reduces the positive ISC for the investing units by imposing higher downpayment requirements and an extensive-margin effect that hinders waiting so that more production units will have positive ISC. The net effect depends on the distribution of production units. Moreover, extending the model to include risky borrowing with endogenous default and decreasing returns to scale technology will not affect my model s predictions. I also discuss the robustness with respect to internal competition for resources across different production units within a firm. I test my model predictions about uncertainty, the degree of being financially constrained, and the investment sensitivity to cash flow using firm-level data from the COMPUSTAT North America database. My sample includes an annual panel of around 10,000 listed firms in the United States over the past 30 3

4 years. Specifically, to estimate the ISC I regress individual capital investment on cash flow, with both terms normalized by the size of existing capital stock, while controlling for future profitability (via Tobin s Q and real sales growth) and debt overhang (via net leverage). I further include a measure of uncertainty, the degree of being financially constrained, and their interaction terms with cash flow into the regression to allow the ISC to be state-dependent. To measure uncertainty at the firm level, I use the volatility of enterprise value that equals to stock-return volatility normalized by net leverage. For the degree of being financially constrained, I use the excess bond premium (EBP) in Gilchrist and Zakrajek (2012) to capture the time-series variations, with higher EBP indicating tighter external finance conditions in general, and use firms sizes to gauge the cross-sectional variations, assuming that smaller firms are potentially more constrained. 2 I confirm my model s predictions that corporate investment is less sensitive to cash flow when uncertainty is high, while investment tends to be more sensitive to cash flow for smaller firms or when external finance conditions become tighter (EBP is higher). Results are significant in economic terms. Some existing literature has pointed out that the investment-cash-flow regression suffers from an endogeneity problem caused by measurement errors in Tobins Q. This is because when the latter cannot fully capture future profitability, cash flow may contain information about future profits. If so, it is possible that the positive investment response to cash flow (ISC> 0) simply comes from an omitted-variable bias because investment in theory reacts to changes in expected profits. However, my results are unlikely driven by this alternative explanation because I focus on the change of investment sensitivity upon changing uncertainty rather than the sensitivity per se. Suppose that the ISC is positive because cash flow were informative about future profitability. Then such information should be more relevant to firms investment decisions when uncertainty is high. I should observe investment responds more to cash flow upon high uncertainty, rather than less. Since it is not the case, it rules out this explanation based on omitted-variable bias. To further cope with this issue, I instead estimate the regression adopting the measurement-error-robust estimator using higher-order cumulants in Erickson and Whited (2012), Erickson et al. (2014) and Erickson et al. (2016), treating Tobin s Q as a mis-measured 2 Geanakoplos (2010) and Gilchrist and Zakrajek (2012) provide the theoretical foundation for the use of excess bond premium, while Hennessy and Whited (2007) justifies the use of sizes. See section 4 for more detailed discussions. 4

5 variable. As an alternative methodology, I adopt the Generalized Method of Moments (GMM) in Arellano and Bond (1991) to re-estimate the regression, while treating all firm-level variables as endogenous variables. My results are robust to both methods. In the rest of the paper, in section 2 I will firstly relate my work to the existing literature from several perspectives. I will present the model in section 3 and the empirical work in section 4. Section 5 concludes the paper and points out future research. 2 Related Literature This paper relates to the existing literature of both macroeconomics and corporate finance. Firstly, previous studies in macroeconomics about effects of uncertainty and financial frictions on corporate investment mostly focus on their roles in depressing the investment level. I instead investigate their distinct influences on the investment sensitivity to cash flow, also known as the marginal propensity to invest out of income. The latter has crucial implications in evaluating the effectiveness of stimulus policy. Specifically, uncertainty interacting with capital irreversibility makes firms investment reluctant to liquid-provision stimulus like tax cuts or investment tax credit. This is in line with Dixit and Pindyck (1994), Bloom (2009) and Stein (2012). Stressing this effect of real options, and compare it to the precautionary motives, my paper is also relevant to the discussion about the transmission channels of uncertainty, like in Bloom (2009) and Gilchrist et al. (2014). Secondly in the corporate finance literature, whether the positive investment sensitivity to cash flow (ISC) observed in the data comes from financial frictions or an omitted-variable bias due to informative cash flow is still debatable. Furthermore, even after conceding the role of financial frictions, whether this friction-induced ISC should increase monotonically with the tightness of constraints remain unclear in both theoretical and empirical work. See, among others, Fazzari et al. (1988), Kaplan and Zingales (1997), and Abel (2015) for the debate. In this paper, I provide a structural model in which cash flow is exogenous, thus uninformative mechanically, to explain why investment is sensitive to cash flow with the existence of financial frictions. By solving out the ISC in closed form, I show analytically how this sensitivity changes with individual uncertainty and the degree of firms being 5

6 financially constrained. My work therefore has proper theoretical foundation. On the other hand, I find empirical patterns consistent with my theoretical predictions using the data of U.S. listed firms, wherein I discuss the possible endogeneity carefully including measurement errors in Q that renders cash flow informative causing an omitted-variable bias. Furthermore, I separately identify impacts of individual uncertainty, individual measure of financial constraints, and market-wise investor risk aversion. Evidence in the former shows that not only the degree of being financially constrained, but also the uncertainty a firm faces, could explain the cross-sectional heterogeneity in the ISC. Evidence in the latter instead identifies the time-series variations of the investment sensitivity; it relates to, for example, Chen and Chen (2012) and McLean and Zhao (2014). In the macroeconomic literature, previous studies have showed that financial frictions and uncertainty play an important role in depressing firms investment. Two seminal ways to model financial frictions are Bernanke et al. (1999) ( BGG ) about costly state verifications and Kiyotaki and Moore (1997) about collateral constraints. In these studies, financial frictions reduce investment by making external finance more costly or scarce. On the other hand for uncertainty, a canonical strand of literature focuses on non-convex adjustment costs of capital, like irreversibility, and the generated option value of waiting. Earlier papers like McDonald and Siegel (1986), Dixit and Pindyck (1994), Abel and Eberly (1995), Abel and Eberly (1996) and Bloom (2000) adopt deterministic volatility and more recent papers like Bloom et al. (2007), Bloom (2009) and Bloom et al. (2014) extend the work to stochastic volatility. In both setup, heightened uncertainty depresses investment by inducing firms to delay investment voluntarily when the cost of investment is irreversible. While most of these papers are about the level of investment, Bloom et al. (2007) investigate the sensitivity of investment to productivity shocks and conclude that because of uncertainty corporate investment responds less to the latter. My work is different from theirs because the cash flow in my model is exogenous and orthogonal to productivity shocks. I therefore identify the investment sensitivity to liquidity, rather than productivity shocks. As I will argue later, this is in particular important for isolating the role of financial frictions. Nevertheless, I do hold the same stance as Bloom et al. (2007) that in periods of high uncertainty, because firms are reluctant to respond, certain types of policy stimulus could be less effective. Their work is more about the demand-side stimulus like increasing government spending; my work is more about the liquidity-provision 6

7 policy that boosts firms cash flow, for instances, tax cuts, investment tax credits, or even the income effects of expansionary money policy. 3 A recent strand of literature provides a novel view linking together financial frictions and uncertainty, including Arellano et al. (2012), Caldara et al. (2014), Christiano et al. (2014), and Gilchrist et al. (2014). They adopt the BGG setup of costly state verification to argue that the channel in which uncertainty reduces investment is by making external finance more costly. Specifically, as heightened uncertainty increases firms probability of default and thus the expected dead-weight loss caused by state verification, lenders will require higher compensation, which widens the credit spreads and in turn depresses investment. Caldara et al. (2014) adopts a penalty function approach in a vector auto-regressions framework following Faust (1998) and Uhlig (2005) and argue that conditional on credit spreads, uncertainty have limited depressing effects on investment. By doing so, they claim to reject the real-option effect of uncertainty. In the baseline model of this paper, I model financial frictions by collateral constraints following Kiyotaki and Moore (1997). All borrowings will thus be risk-free and no defaults will happen in equilibrium. Financial shocks are captured by tightening collateral constraints directly. This allows me to separately identify effects of real options and financial frictions. In the model extension, I further include the BGG setup and show the co-existence of realoptions channel and default channel. In the empirical work, I then identify the importance of the former by sub-sample evidence; however, due to the fact that my sample selection biases to big firms that are in general far away from default boundaries, my results tend to under-estimate the role of default channel. Another strand of literature my paper relates to originates from debates on investment sensitivity to cash flow between Fazzari et al. (1988) and Kaplan and Zingales (1997). There has been a huge literature in corporate finance afterwards about this issue; see Hubbard (1997), Tirole (2010), and Roberts and Whited (2012) for detailed surveys. Two main questions asked by this literature are: (1) whether the positive investment sensitivity to cash flow (ISC) observed 3 Dixit and Pindyck (1994) also document in general that uncertainty could make firms less responsive to policy shocks but they only focus on the investment level. Stein (2012) indicates that one possible reason that the quantitative easing by the Federal Reserve may have limited contribution to investment is that firms tend to borrow cheaply and buyback stocks. My paper provides a potential explanation of such behaviour high uncertainty makes firms voluntarily choose not to invest because of the high real-options value. 7

8 in the data comes from financial frictions or informative cash flows; (2) even if the ISC may come from financial frictions, whether this frictions-induced sensitivity increases monotonically with the degree of frictions. One side of the literature, for example Fazzari et al. (1988) (called FHP view for brevity), believes that financial frictions violate the Modigliani and Miller (1958) causing investment sensitive to internal funds (cash flow). Moreover, the sensitivity increases with the degree of being financially constrained for example, FHP find that firms that pay dividends exhibit lower ISC than firms that do not. 4 Lorenzoni et al. (2007) present a simulated model in which the investment sensitivity to cash flow is caused by binding collateral constraints to show that tighter constraints lead to higher ISC. Compared to my work, their model is solved numerically using linear perturbation around the steady states while assuming collateral constraints always bind. It is thus unable to disentangle the two competing impacts of collateral constraints on the ISC. In contrast, another side of the literature, for instance Kaplan and Zingales (1997), doubts both the necessity of financial frictions in generating positive ISC and the monotonicity (called the KZ view). They argue that the positive ISC could present in a frictionless economy when the cash flow is informative about future profitability investment responding positively to higher cash flow is simply reacting to a change of expected profitability. 5 Empirical work from this side points out that the control variable that most studies use for future profitability, the Tobin s Q, contains measurement errors rendering the cash flow informative. 6 In other words, the positive ISC could merely be an omitted-variable (unobserved profitability) bias. They further argue that though financial frictions are sufficient for positive ISC in theory, whether the frictions-induced sensitivity increases or decreases with the degree of frictions is indeterminate, depending on the distribution of firms. In this paper, to investigate roles of financial frictions, I choose the FHP side that investment is sensitive to cash flow because of the violation of Modigliani and Miller (1958). To remove impacts of the information content, I show a structural model in which cash flow is exogenous and orthogonal to future prof- 4 Fazzari et al. (1988) argue that not paying dividends is a good indicator of being financially constrained, because dividends payers can always get more internal funds by reducing their dividends and thus should be regarded as unconstrained firms. Moyen (2004) disagrees with this criterion and uses a model to show that dividends payers could be in fact constrained. 5 For example, see Gomes (2001), Alti (2003), Abel (2015), etc. 6 For example, see Erickson and Whited (2012) and Erickson et al. (2014), etc. 8

9 itability. Using this model, I show analytically that firms ISC is zero when the financial market is frictionless. Moreover, since the ISC in my model could be solved out in closed form, I analyse how it changes with external shocks: not only the degree of financial frictions, but also the individual uncertainty. In the empirical part, I discuss the possible endogeneity caused by measurement errors in Q in three different ways. Neither of them provides evidence that my empirical results are driven by the informative-cash-flow story. I find evidence that smaller firms (that are potentially more financially constrained) tend to exhibit higher ISC, consistent with the FHP view; however, more volatile firms tends to have lower ISC. Such different roles of financial constraints and uncertainty in affecting investment sensitivity highlights the importance of isolating the two types of shocks. If financially-constrained firms are also more volatile, failing to account for uncertainty may pollute the impacts of financial constraints. 7 In addition, my empirical sample contains a long period over the past 30 years. This enables me to identify further the time-series variations in the ISC, though the original debate between FHP and KZ is more about cross-sectional heterogeneity. This relates to Chen and Chen (2012) and McLean and Zhao (2014). 8 The former find that the ISC declines even in the crisis; my work shows that the heightened uncertainty could be a potential driver. The latter instead link the movements of ISC to business cycles; I further distinguish the different roles of heightened uncertainty and financial-market frictions. The closest existing paper is Li et al. (2015) who look into the differential responses of corporate investment and sensitivity in the emerging markets upon U.S. financial and volatility shocks. This paper instead focuses on U.S. firms and provides a theoretical foundation for the empirical work. 3 A Model In this section, I present a theoretical model that features heterogeneous uncertainty, capital irreversibility and occasionally-binding collateral constraints. I solve out firm-level investment sensitivity to cash flow (ISC) in closed form and show analytically how it is affected by swings in uncertainty and the tightness of 7 Baum et al. (2009) also evaluate effects of uncertainty on the ISC; however, they do not model irreversibility and thus cannot discuss the real-options effect. 8 Also see, among others, Allayannis and Mozumdar (2004), Ascioglu et al. (2008), and Ağca and Mozumdar (2008). 9

10 collateral constraints. In the model, each firm incorporates a group of individual production units with constant returns to scale technology and stochastic profitability. Their distribution of profitability shares a common expectation but possesses different variances, with the latter influenced by uncertainty at the firm level. Such heterogeneity in individual variances works with capital irreversibility in generating different investment policy (called real-options effect) units with low variances are active in making new investment while others are not. For active units, furthermore, collateral constraints shape their investment policy by making the latter sensitive to internal funds (ISC > 0). For inactive units, instead, ISC = 0 because investment equals to zero regardless of internal funds. After aggregation, the firm-level ISC equals to the cross-sectional average of all unit-level ISC. The former is thus affected by the combination of uncertainty, the tightness of collateral constraints and distribution of individual production units. I show that higher uncertainty at the firm level could shift the distribution of individual variances to the right. It thus reduces the ISC by pushing more units to the inactive region, while tighter collateral constraints could have ambiguous consequences depending on the competition between an intensive-margin effect and an extensive-margin effect. In the end, I extend the model to include risky borrowing with endogenous default and decreasing returns to scale technology; I also discuss the robustness with respect to internal competition for resources across different production units within a firm. 3.1 Setup Each firm (i) consists of J independent, risk-neutral production units {i j } J j=1 that exist for three periods. They are heterogeneous both ex ante and ex post. At the beginning of period 1, unit (i j ) is given a stochastic cash flow w ij from the parent firm as endowments, and at the same time it learns about the distribution from which it will draw z ij, the profitability of operation that lasts in the following two periods. Specifically, assume all units future profitability is distributed log-normally and subjects to a common mean z i but different variances: log z ij N(log z i σ 2 i j /2, σ 2 i j ). Furthermore, differences in σ ij come from the individual-specific factor loading γ ij to the firm-level uncertainty σ i ; i.e. σ ij = γ ij σ i. An increase in σ i represents a heightened uncertainty shock to the firm that shifts the whole distribution of σ ij to the right. σ i, {γ ij } J j=1 and {w ij } J j=1 are common knowledge to all individuals after they realize. Denote 10

11 the realized cross-sectional distribution of {γ ij } J j=1, {σ i j } J j=1 and {w i j } J j=1 by F i,γ, F i,σ and F i,w, respectively. 9 The model focuses on dynamic capital investment decisions by each production unit that faces both irreversibility and collateral constraints. For the former, once fixed assets are installed, they cannot be uninstalled, unless the whole operation is liquidated at a discount price. By normalizing the purchasing price of fixed assets to 1 per unit, the discount price in liquidation is q < 1 per unit. This discount price is to capture the fact that in reality firms usually cannot buy and sell fixed assets at the same price. 10 Besides, both risk-free savings and borrowings are allowed at a gross interest rate R f per period. Specifically, there exists a two-period bond in period 1 and a one-period bond in period 2 that each production unit could buy or sell. Suppose there is a group of investors who are willing to trade any amount of bonds with production units at R f. Borrowings are risk-free because of collateral requirements as in Kiyotaki and Moore (1997). Both fixed assets and financial assets (i.e. savings) could serve as collaterals while the former has a haircut of (1 θ) between 0 and 1. Hart and Moore (1994) and Geanakoplos (2010) provide a theoretical foundation for the existence of haircut it represents the disagreement between borrowers and lenders in the value of collaterals. This is also a parsimonious way to capture financial shocks, with lower θ indicating a higher haircut, thus tighter external finance conditions; see Lorenzoni et al. (2007), Quadrini (2011) and Cui (2014) for a similar setup. Geanakoplos (2010) further shows that lenders risk aversion could be a source of such financial shocks (leading to a lower θ). In contrast, there is no haircut if financial assets are used as collaterals. For example, an entrepreneur could save at Rf 2 in the first period and borrow against this twoperiod bond in the second period at R f ; effectively, the entrepreneur saves in a one-period bond in period 1 at a rate equals to R2 f R f = R f Here I do not impose any restrictions on the relationship between F i,σ and F i,w. In other words, this modelling setup is neutral to any internal competition for resources across different production units. Specifically, one could imagine that after the realization of F i,σ, the CFO of parent firm could redistribute resources so that F i,w is determined by the former. While modelling such an optimization is beyond the scope of this paper, I discuss how it could qualitatively affect my empirical predictions at the end of this section. 10 This assumption of total, rather than partial, irreversibility is not crucial qualitatively, but it simplifies the algebra. See Caggese (2007) for a similar setup. 11 Therefore, the setup that there is only a two-period bond in period 1 and a one-period bond in period 2 itself does not twist the optimal policy, but the fact that borrowing requires collaterals does. 11

12 Now I start describing the optimization problem for the production unit (i j ) of firm (i). In the following context, I suppress the subscript (i) for brevity; i.e. use (j) to denote (i j ). I further use the subscripts (1j), (2j), (3j) to represent the choice variables in each period 1, 2, 3 or the stock variables at the beginning of each period for the unit (j). In period 1, production unit (j) has an initial capital stock k 1j. There is no production in the first period, so it distributes its exogenous cash flow w j between dividend d 1j and investment in fixed assets, ι 1j. 12 Since future profitability of operation is uncertain and so is the investment return, the unit can also choose not to invest now but to maintain its capital stock at k 1j. In other words, the unit waits (ι 1j = 0) until the uncertainty resolves in the next period. It is also able to save in risk-free bond (b L j < 0) or borrow external debt b L j > 0 limited by a fraction θ < 1 of the liquidation value of fixed assets it chooses to hold. The superscript L stands for long indicating that the saving or borrowing is in two periods. Formally, see equation (1) to (5). Equation (1) and (5) are budget constraint and irreversibility constraint, respectively. Equation (3) is the collateral constraint for a two-period debt; if the unit instead chooses to save, (3) will always hold for the right-hand side is positive. Equation (2) describes the capital accumulation process and equation (4) rules out external equity finance This exogenous-cash-flow setup is to take into account a famous critique in the literature towards investment sensitivity to cash flow, starting from Fazzari et al. (1988) and Kaplan and Zingales (1997). The critique argues that the positive sensitivity of investment to cash flow needs not come from financial frictions but the information content of cash flow about future profitability (e.g. when z j is persistent). Among others, a recent paper by Abel (2015) shows analytically that the ISC could be positive in a frictionless economy when cash flow is serial correlated. In this model, I rule out this alternative explanation by letting cash flow in period 1 be purely exogenous and orthogonal to future cash flow. I also show that under this modelling setup the ISC equals to zero in a frictionless financial market. In the empirical part, I show evidence that support this financial explanation of positive ISC. 13 There is no production in the first period and thus no capital depreciation. I rule out equity finance for simplicity. According to Myers and Majluf (1984), frictions in the equity market are greater than those in the debt market, so excluding equity finance should not qualitatively affect my model s predictions. However, given that smaller firms in reality do depend on equity for external finance, quantitatively my model could over-estimate effects of tighter collateral constraints. 12

13 w j + b L j = d 1j + ι 1j (1) k 2j = k 1j + ι 1j (2) R 2 f b L j θ q(1 δ) 2 k 2j (3) d 1j 0 (4) ι 1j 0 (5) At the beginning of period 2, the profitability of operation z j realizes. All uncertainty resolves in the sense that z j will be carried over to the next period. The unit operates and generates profits z j k 2j but in the form of account receivables it cannot get the proceeds until the last period. This is a technical assumption to mitigate the Oi-Hartman-Abel effect. 14 invest if it did not exhaust its financial resources in period Unit j can choose to In the optimal policy, it will only invest if the realized z j is high enough. Borrowings and savings are allowed in a one-period bond b S j as long as the collateral constraints are satisfied, where the superscript S stands for short-term debt. 16 The remaining resources will be paid out as dividends. Formally, 14 Specifically, without such a setup, if z j is high, the unit will reinvest the proceeds generating a convex marginal-value-of-capital function in terms of zj 2. Such convexity will result in a Jensen s-inequality effect when z j is uncertain: the bigger the variance of z j, the higher the expected marginal value of capital. Therefore uncertainty encourages investment, opposite to the commonly-accepted empirical evidence; e.g. Bloom (2009). The reason is that in a 3-period model under constant returns to scale technology, benefits of investing (and reinvesting) is of order-two of z j, but benefits of waiting to invest is only of at most order-one of z j (because when the firm waits in the first period and invests in the second period, the benefit is in terms of R f z j ). Therefore the rising uncertainty could make investing more beneficial than waiting. This is of similar spirits as Oi (1961), Hartman (1972) and Abel (1983). Such an OHA effect coming from convexity will become smaller when the number of periods get larger and will no longer dominate in a infinite horizon setup, or in a decreasing-returns-toscale setup. In this paper, in order to obtain a closed-form solution, I instead assume that the proceeds are in the form of account receivables and thus cannot be reinvested. Then the convexity of marginal value of capital is reduced. In fact, given the extensive use of trade credit of U.S. firms, this, itself, is not a strong assumption. For example in the COMPUSTAT U.S. sample, the flow of account receivables takes up around 25% of annual cash flow for the median firm, and the number of days sales is around one-year and a half. At the end of this section, I extend the model to incorporate decreasing returns to scale in order to drop this technical assumption. The solution will then be obtained numerically. 15 Financial resources include both internal funds (w j ) and the maximum amount of external funds (collateral borrowing). Since θ < 1, the entrepreneur needs down-payment to buy new fixed assets, then it is straightforward to show that positive investment is possible only if the unit did not exhaust its financial resources in period Equation (6), (9) and (10) together suggest that b S j is non-negative; however, this does not mean the production unit cannot save. In fact, with the two-period savings carried over from period 1, it effectively saves in period 2 if it chooses not to withdraw (borrow against) these amount of savings. 13

14 b S j = d 2j + ι 2j (6) k 3j = (1 δ)k 2j + ι 2j (7) Rf 2 b L j + R f b S j θ q(1 δ)k 3j (8) d 2j 0 (9) ι 2j 0 (10) In period 3, z j is carried over from period 2. Since this is the last period, the production unit operates to get profits but no longer invests or saves; no new debt is allowed as well. The unit also liquidates the total capital stock at a price equals to q per unit. Account receivables and savings, if any, pays back. All proceeds after paying back the debt will be paid out as dividends. Formally, d 3j = z j (k 2j + k 3j ) + q(1 δ)k j3 R 2 f b L j R f b S j (11) d 3j 0 (12) Each production unit maximizes E[d 1j + βd 2j + β 2 d 3j ] where β represents the time preference common to all agents, subject to (1) (12). 3.2 Analytical Solution I this section, I solve the model analytically using backward induction. First of all, I impose Assumption 1 to pin down the solution for the second period (since the last period is trivial, given by (11) and (12)). Assumption 1. βr f < 1 Proposition 1. Under Assumption 1, in period 2 if the realized z j is high enough, the production unit will invest all the financial resources including both internal and external funds; otherwise, it will maintain k 3j = (1 δ)k 2j and consume all the other resources. Formally, If z j ψ 1, k 3j = (1 δ)k 2j 14

15 b S j = 1 R f [ θ q(1 δ) 2 k 2j R 2 f b L j ] d 2j = b S j If z j > ψ 1, where k 3j = (1 δ)k 2j R f b L j 1 θ q(1 δ) R f b S j = 1 R f [ θ q(1 δ)k 3j R 2 f b L j ] d 2j = 0 ψ 1 = 1 1 [1 β q(1 δ) ( β) θ q(1 δ)] β R f The value function is given by: [β(2 δ)z j + (1 δ)(1 βψ 1 )]k 2j R f b L j if z j ψ 1 V 2j (k 2j, b L j, z j ) = [1 δ + βz j + (1 δ)β(zj ψ1) ]k 2j [1 + β(zj ψ1) 1 θ q(1 δ) ]R f b L j R f if z j > ψ 1 1 θ q(1 δ) R f Proof. See Appendix, available from the author upon requests. (13) The production unit distributes limited resources among dividends (d), savings (s) and investment (i). Assumption 1 says that the production unit is impatient enough so that in period 2, it prefers dividends to savings; in short, d s, so the borrowing constraint (8) is binding ( indicates preference). Comparing z j to ψ 1 is in fact comparing [βz j + β q(1 δ) + ( 1 R f β) θ q(1 δ)] to 1. The former represents the benefits of investing, wherein the first term is account receivables and the second term is the liquidation value of the invested fixed assets after depreciation. The last term instead indicates the utility gains from impatience the production unit can borrow and consume some extra amount of funds using this additional amount of fixed assets as collaterals. When this whole term is greater than 1, that is the benefit of consuming dividends, the production unit will choose to invest; i.e. i d s. The amount of investment is proportional to its net worth, [(1 δ)k 2j R f b L j ] because of constant returns to scale. The term, (1 θ q(1 δ) R f ), in the denominator indicates the down-payment requirement: for each fixed asset that is worth 1, only an amount equals to θ q(1 δ) R f is allowed to use external finance; the rest has to be 15

16 paid by internal funds as down payments. On the other hand if the total benefits of investing is smaller than 1, the production unit will choose to pay out (d i and d s) all the liquid assets as dividends and let fixed assets depreciate. 17 Next, to pin down the solution for the first period, I impose Assumption 2. Assumption 2. z > max{ψ 1, ψ 2 } ψ 2 = ψ β [ 1 βr f 1][1 θ q(1 δ) R f ] Proposition 2. Under Assumption 2, in period 1 if the expected marginal value of capital is greater than the expected marginal value of savings, the production unit will invest all resources including both internal and external funds; otherwise, it will save internal funds in the two-period bond, i.e. b L j < 0. Formally, If x kj ( x bj ), k 2j = k 1j b L j = ( w j ) d 1j = 0 If x kj > ( x bj ), k 2j = w j + k 1j 1 θ q(1 δ) 2 R 2 f b L j = θ q(1 δ) 2 Rf 2 k 2j d 1j = 0 where x kj = E[ V 2j(k 2j, b L j, z j) k 2j ] x bj = E[ V 2j(k 2j, b L j, z j) b L ] j Proof. See Appendix, available from the author upon requests. In the above proposition, I use x kj to denote the expected marginal value 17 To simplify the algebra, endogenous exit is assumed away. 16

17 of capital. It is indeed the marginal q in the neoclassical literature. 18 I use x bj to denote the expected marginal value of debt. Since debt is treated as negative savings, ( x bj ) is the expected marginal value of savings. In the later context, I refer it to the real-options value of waiting. Assumption 2 is derived from β( x bj ) > 1 for all j. It is to compare the expected marginal value of savings with the value of dividend payouts. Assumption 2 says that as long as the expected profitability z is high enough, even though the production unit is impatient, it is willing to save rather than to consume dividends (s d). This is because by saving the production unit keeps an option to invest in period 2 and there is a chance to benefit from high returns in production (when z j turns out to be high). In other words, this assumption makes sure that the production unit prefers waiting-to-invest to dividends. Without it, delaying investment (waiting-to-invest) would be impossible because the impatient production unit will just consume everything. Under this assumption, because of the first-order homogeneity of the value function, optimal investment policy is determined by comparing x kj and ( x bj ). If x kj is greater than ( x bj ), the production unit will choose to invest all financial resources in fixed assets and the borrowing constraint is binding (i s d). Again, the amount of new capital equals to the ratio between its net worth w j + k 1j and the down-payment requirement. Otherwise, the production unit will choose to delay investment and save all the resources in the two-period bond (s d and s i). I summarize production unit j s optimal policy in Figure 1 by combining Proposition 1 and 2. In the first period, the production unit faces a good (high z) but uncertain future. It thus chooses between investing now or waiting until the uncertainty resolves later. If it chooses to invest, then it could get higher production in period 2 because of higher capital stocks. Or it could choose to put the money in the risk-free bond and postpone the investment decision until the next period. In period 2, if the realized z j turns out to be high, the production unit will invest. If z j is low, it will not invest but to consume (pay out dividends). By doing this, the production unit foregoes the extra production in period 2 it could get in the case of investing but gain an opportunity to avoid inefficient investment when z j is low. In other words, choosing to wait in period 1 gives the production unit an option, but not an obligation, to invest in period 2; in 18 For example, see Hayashi (1982) and Abel (1983). However, because of financial frictions, it is no longer a sufficient statistics for investment the latter could also depend on the amount of internal funds, w j. 17

18 Figure 1: Policy rules contrast, doing investment in period 1 gives up this option because the installed capital cannot be uninstalled if z j is low keeping capital stocks inefficiently high becomes an obligation. The optimal policy here is thus different from the one in the neoclassical case: the marginal q, i.e. x kj, is compared to ( x bj ), rather than the Hall-Jorgenson users cost of capital. Under uncertainty and irreversibility, the investment threshold for q to surpass is in fact higher than the opportunity cost of funds described by risk-free rate and depreciation (Hall and Jorgenson (1969)). I follow the literature to call this ( x bj ) the option value of waiting. 19 The investment determinant in period 1 is thus the sign of (x kj +x bj ). The next proposition shows that this investment determinant, (x kj + x bj ), decreases with the individual variance σ j. In other words, when the variance is large, the production unit tends to delay investment. This is because it only invests when z j is high. This option makes the marginal value of debt in period 2 a convex function in z j. So larger variance in z j increases ( x bj ) from a Jensen s inequality effect and thus decreases (x kj + x bj ). Formal analysis is in Proposition 3. Proposition 3. (x kj + x bj ), as a function of σ j, is decreasing. Proof. See Appendix, available from the author upon requests. 19 See, among others, Dixit and Pindyck (1994), Abel and Eberly (1995), and Bloom (2009). 18

19 Figure 2: Marginal value functions 19

20 To illustrate Proposition 3 graphically, Figure 2 plots the period 2 s realized marginal value of capital MV K 2j = V 2j (k 2j, b L j, z j)/ k 2j and marginal value of savings ( MV B 2j ) = ( V 2j (k 2j, b L j, z j)/ b L j ) depending on z j under a standard choice of parameters. 20 The realized profitability z j is on the x-axis and MV K 2j is plotted in the line with circular markers. In contrast, ( MV B 2j ) is plot in the solid line without markers. The dashed line indicates ψ 1. It is clear that though both the marginal value of capital and the marginal value of savings are convex in z j, the latter is actually much more convex than the former. Holding constant z, their expectations x kj and ( x bj ) are functions of σ j, depicted in Figure 3 under a same group of parameters with σ j on the x-axis. x kj is plotted in the line with circular markers; ( x bj ) is plot in the solid line without markers, and the dashed line gives their difference (x kj +x bj ). As stated in Proposition 3, the latter decreases with σ j, consistent with the fact that ( MV B 2j ) is more convex in z j than MV K 2j. Because of this monotonicity, the root of x kj + x bj = 0, denoted by σ is unique if it exists. σ is homogeneous across all production units in a particular firm. However different units have their own individual variances σ j leading to different investment policy. Those with low σ j σ will choose to invest in the first period because marginal q, x kj, is higher than real-options value of waiting, ( x bj ). In contrast, those with high σ j > σ will choose to wait. The following proposition guarantees the existence of σ under Assumption Assumption 3. β > (1 δ) 1 R f < (1 δ) + (1 θ q(1 δ) R f ) max{ψ 1, ψ 3 } < z < ψ 4 ψ 3 = [R f (1 δ)](1 βψ1 α ) β[1 + 1 α (1 δ R f )] ψ 4 = [R f (1 δ)] + βψ 1 (1 δ) β[1 + 1 α (1 δ R f )] 20 Specifically, on the annual basis I choose β to be 0.95, R f to be 1.04, q to be 0.9, θ to be 0.7, δ to be 0.1 and z to be 0.3. These parameters only serve as an illustrative example; Proposition 3 does not depend on the specific choice of parameters as long as Assumption 1 and 2 are satisfied. 21 Assumption 3 is actually very weak. It always holds under standard parameterizations like the one that generates Figure 2 and 3. 20

21 Figure 3: Expected marginal value functions 21

22 α = 1 θ q(1 δ) R f Proposition 4. Under Assumption 3, there exists a unique σ such that for any production unit j, if σ j σ, then x kj x bj ; if σ j > σ, then x kj < x bj. Proof. See Appendix, available from the author upon requests. Given that x kj + x bj is continuous and decreases with σ j, the existence of σ comes directly from the intermediate theorem. It requires x kj + x bj to be positive at σ j = 0 and negative at σ j = +, ensured by Assumption 5 when z is in a moderate region. Intuitively, if z is too small, x kj + x bj will always be negative the production unit will always choose to delay investment. If z is too big, x kj + x bj will always be positive and the production unit will always choose to invest immediately. Taking together Assumption 1, 2 and 3, it must be max{ψ 1, ψ 2, ψ 3 } < z < ψ 4. The other two conditions in Assumption 3 are sufficient (not necessary) to make such a choice of z exists, i.e. max{ψ 1, ψ 2, ψ 3 } < ψ 4. If the production unit is too impatient, it will never choose to wait because forgoing near-future profits is just too costly. Moreover, if saving is too profitable, the production unit will always choose to save but never invest. The parameterization I used to generate Figure 2, 3 (and 4 later) satisfies all these assumptions. Specifically, I chose z to be 0.3. This is also close to the median return on capital in my empirical sample (0.27) from COMPUS- TAT U.S. listed firms in the past 30 years. 22 Under this parameterization, σ is around 1.43, lying in the range of firm-level uncertainty I can observe from my sample. Given that unit-level uncertainty could be much bigger than the consolidated firm-level uncertainty due to diversification, the economic significance of this real-options effect could be big in reality. 3.3 Investment Sensitivity to Cash Flow Given optimal investment policy in the first period (Proposition 2), both unit and firm-level investment sensitivity to cash flow (ISC) can be solved out in closed-form. In this section I show that firm-level ISC is in fact the crosssectional average of all unit-level ISC. An uncertainty shock to the firm reduces firm-level ISC by shifting the distribution of unit-level profitability variances to the right (and thus more units will have zero ISC). On the other hand, a financial shock defined by tighter collateral constraints instead has two competing 22 See the next section for details about data. 22

23 impacts it reduces investing units positive ISC (the intensive-margin effect) but at the same time dampens the waiting motives so more production units will have positive ISC (the extensive-margin effect). The net effect of such a financial shock on the firm-level ISC depends on the relative dominance of the two effects. I will bring these predictions to the empirical work in the next section. Lastly, I discuss how internal competition within a firm could potentially change the predictions. To be precise, in theory ISC is defined as the partial derivative of investment ratio (investment normalized by existing capital stock) to cash-flow ratio (cash flow normalized by existing capital stock). The literature usually uses the regression slope of the former on the latter as a counterpart. When firms investment policy is non-linear, this linear regression slope actually captures the average, rather than marginal, ISC. I show in the following context that after aggregation, the firm-level investment in my model is indeed linear in firm-level cash flow. So the ISC in my model is also the marginal propensity to invest. For unit-level investment sensitivity to cash flow, defined as ι1j/k1j w j/k 1j, take Proposition 2, 3 and 4 together: 0 if σ j > σ ISC j = 1 if σ j σ (14) 1 θ q(1 δ) 2 R 2 f To understand it, when individual variance σ j is low, the production unit will have positive investment that is sensitive to internal funds because of the binding collateral constraint only a fraction of the newly-invested fixed assets can be paid by external debt. This is a violation of the Modigliani and Miller (1958) so the amount of cash flow does matter. It is clear that in a frictionless economy where such a borrowing constraint is relaxed by letting θ go to this ISC will become zero even for the investing units. This evidence proves that in my model the positive relationship between investment and cash flow is not due to an omitted-variable bias as in Kaplan and Zingales (1997), Abel (2015) and others. I will further discuss this issue in the empirical part of this paper. In contrast, when the individual variance σ j is high, so is the real-options value of waiting. The production unit optimally chooses to wait and have zero investment regardless of the amount of internal funds, so the investment sensitivity to 23