Debt Refinancing and Equity Returns

Transcription

1 Debt Refinancing and Equity Returns Nils Friewald Florian Nagler Christian Wagner This Version: January 16, 2018 We are grateful to Jennie Bai, Jules van Binsbergen, Harjoat Bhamra, John Y. Campbell, Gian Luca Clementi, Paolo Colla, Thomas Dangl, Christian Riis Flor, Lorenzo Garlappi, Thomas Gehrig, Nicola Gennaioli, Li He, Christopher Hennessy, Søren Hvidkjær, Alexandre Jeanneret, Michael Kisser, Christian Laux, Nathalie Moyen, Guillermo Ordonez, Lasse Pedersen, Thomas Kjaer Poulsen, Julien Sauvagnat, Roberto Steri, Alexander Wagner, Ramona Westermann, Toni M. Whited, Josef Zechner, and participants at the 2017 European Winter Finance Summit (EWFS), the Western Finance Association (WFA) Meetings 2016, the European Finance Association (EFA) Meetings 2016, the SDU Finance Workshop 2016, the 2016 Meeting of the Swiss Society for Financial Market Research (SGF), the 2015 VGSF conference, as well as seminar participants at Aarhus University, Bocconi University, Copenhagen Business School, the Helsinki Finance Seminar, the Lisbon Finance Seminar (Católica/Nova), the Norwegian School of Economics (NHH), and the University of Lugano for providing helpful comments and suggestions. Christian Wagner acknowledges support from the Center for Financial Frictions (FRIC), grant no. DNRF102. Norwegian School of Economics and CEPR. Bocconi University and IGIER. Copenhagen Business School.

2 Debt Refinancing and Equity Returns Abstract We show that the mixed evidence on how financial leverage affects stock returns can be reconciled by accounting for firms debt maturity structures. In our model, firms jointly optimize leverage and debt maturity by balancing benefits and rollover risk of short-term relative to long-term debt. Shareholders require returns that increase with, both, leverage and debt refinancing intensity. Book-to-market and size are related to leverage and refinancing intensity but refinancing intensities convey generic return-relevant information. Empirically, stock returns increase with leverage and refinancing intensity. The information in debt refinancing intensities is not subsumed by book-to-market, size, or other firm characteristics. JEL Classification: G12, G32, G33. Keywords: Equity returns, optimal capital structure, leverage, debt refinancing, rollover risk, book-to-market, size.

3 I. Introduction Empirical studies report mixed evidence on how a firm s financial leverage affects the expected return on its equity. Our paper complements these previous findings by showing, theoretically and empirically, that firms expected equity returns increase with leverage when controlling for the immediacy of debt refinancing. In the model, the firm optimizes its capital structure by jointly choosing the amount of debt as well as the maturity structure of debt, by balancing benefits and refinancing risk of short-term compared to long-term debt. The firm has to refinance debt according to its maturity structure and shareholders commit to cover potential shortfalls arising from the rollover of maturing debt. For shareholders to accept this commitment, expected equity returns have to increase with the firm s leverage and with the firm s debt refinancing intensity, which measures the immediacy of debt refinancing needs. Consequently, the firms leverage alone is not sufficient to gauge the effect of debt-related risks on expected equity returns. Moreover, our framework provides a novel perspective on the link of book-to-market and size to leverage as well as their relation to expected equity returns. First, book-to-market generally increases whereas size generally decreases with, both, leverage and refinancing intensity. Second, consistent with extant empirical evidence, expected equity returns increase with book-to-market and decrease with size. Third, book-to-market and size capture leverage effects on stocks but do not subsume the return-relevant information conveyed by debt refinancing intensities. Specifically, we show that the firm s debt refinancing intensity is positively related to stock returns because it is informative about the (unobservable) tradeoff arising from the relative benefits and refinancing risks of short-term compared to long-term debt. Our empirical analysis confirms that debt refinancing intensities have explanatory power for the cross-section of equity returns beyond leverage, size, book-to-market, and other firm characteristics. To model the interaction between leverage, debt refinancing, and equity returns, we use the model of Leland (1998) and embed insights from the recent bond literature on rollover risk (e.g., He and Xiong, 2012; He and Milbradt, 2014). The firm optimizes its capital structure by jointly choosing the amount of debt (i.e. its leverage) as well as the underlying debt maturity structure. More specifically, the firm decides optimally on how much debt to raise by issuing 1

4 a short-term bond and how much to raise through a long-term bond. Along this maturity dimension, the firm faces a trade-off: On the one hand, short-term debt offers a benefit relative to long-term debt, e.g., it is cheaper in the sense that the fixed issuance costs are lower. On the other hand, increasing the fraction of debt issued through the short-term bond exposes the firm to debt refinancing risk. Given this trade-off, the model implies that firms with comparably low cash flow risk choose higher leverage and longer debt maturities, whereas firms with higher cash flow risk choose lower leverage and shorter debt maturities. These leverage/debt maturity patterns are in line with empirical evidence reported by, e.g., Barclay and Smith (1995), Stohs and Mauer (1996) and Custódio et al. (2013). When we also allow for cross-sectional variation in benefits of short- relative to long-term debt, firms with higher benefits optimally choose higher amounts of short-term debt than firms with lower benefits, which in turn reduces their overall capacity for leverage due to their increased refinancing risk. Hence, the firm s optimal leverage depends on the riskiness of its cash flows as well as its relative benefits of short-term debt. In our discussion of model-implied return expectations and firm characteristics, we first focus on the role of cash flow riskiness and later also account for variation in debt benefits. When firms only differ in their cash flow riskiness, the model implies a one-to-one mapping between leverage, refinancing intensity, and equity returns at the time the firm chooses its initial capital structure. Subsequently, the firm s leverage can change, in response to fluctuations in the market value of equity driven by cash flow realizations, but the firm s refinancing intensity remains unchanged. As a consequence, leverage and refinancing intensity now convey complimentary information about expected returns, with the former informing about the firm s current capital structure and the latter about the debt rollover policy the firm is committed to. For a given refinancing intensity, expected stock returns increase with leverage. Similarly, for a given leverage, expected stock returns increase with refinancing intensity. Hence, neither leverage nor the refinancing intensity alone is sufficient to understand the impact of a firm s debt related risks on equity returns: A firm with high leverage and low debt refinancing intensity may have the same expected return as a firm with low leverage but high debt refinancing intensity. Put differently, the model implies that, in the cross-section of firms, expected equity returns increase with leverage only when controlling for firms refinancing intensities. But we also show that in this setup, without cross-sectional variation in debt benefits, book-to-market 2

5 and size subsume all debt-related information for expected equity returns, much in the spirit of Fama and French (1992). In other words, when the optimal capital structure only depends on cash flow risk, the joint distribution of book-to-market and size embeds the same information as leverage and refinancing intensity. Once we allow for cross-sectional variation in the relative benefits of short- compared to long-term debt, book-to-market and size do not subsume the return-relevant information conveyed by debt refinancing intensities. The intuition is that two firms can have the same leverage, size, and book-to-market today and have started from the same initial leverage but nonetheless are committed to different debt rollover policies. In this case, the refinancing intensity provides complementary information about the firms tradeoff between short- and longterm debt, which has led to the same optimal leverage choice based on different combinations of cash flow risk and debt benefits. More specifically, our model implies that book-to-market and size jointly capture leverage effects in stock returns but that, even after controlling for these characteristics, equity returns increase with debt refinancing intensities. For the empirical analysis, we merge the CRSP- and COMPUSTAT-databases to obtain a sample of approximately 1.4 million firm-month observations across more than 12,000 different firms over the period from 1972 to Our empirical results generally provide strong support for the model predictions. First, we use portfolio double-sorts to show that the data matches the general predictions of our model. In a cross-section of 25 portfolios, obtained from 5 5 double sorts on refinancing intensities and leverage, we find that average equity excess returns increase with leverage for a given refinancing intensity and vice versa. Using the same 25 portfolios, we show that the data also supports the model predictions with respect to bookto-market ratios and firm size: Firms with high (low) leverage and high (low) refinancing intensities have high (low) book-to-market ratios and are small (big) in size. Accordingly, we find that size and book-to-market effects in stock returns are related to, both, leverage and debt refinancing intensities. We then explore the role of debt refinancing for equity returns more rigorously in Fama- MacBeth regressions on the individual firm level. In regressions of stock returns on leverage and refinancing intensities the coefficient estimates on both are significantly positive, even when including beta. Once we add size and book-to-market to these regressions, the coefficient on 3

6 leverage becomes insignificant (as in Fama and French, 1992) but the coefficient for the debt refinancing intensity remains significantly positive. Our results are robust to changes in the regression specification, leverage definitions, and controlling for other firm characteristics such as cash holdings, profitability, past returns, and proxies for financial constraints. Finally, we study the empirical relation of book-to-market and size to leverage and refinancing intensity in more depth. The idea is to provide evidence on the extent to which crosssectional variation in book-to-market and size can be related to leverage and debt refinancing, and on the extent to which value- and size-premiums reflect debt related information. We find that, both, leverage and refinancing intensity have explanatory power for cross-sectional variation in book-to-market and size, where the statistical link appears stronger for refinancing intensities. Moreover, we show that high-minus-low portfolio factor returns based on leverage and refinancing intensity explain around 33% and 24% of the variation in corresponding valueand size-factor returns, respectively. In line with the predictions of our model, these results suggest that book-to-market and size effects in stock returns can to a sizable extent indeed be attributed to debt-related risks. Overall, our results suggest that shareholders indeed care about firms debt maturity choices and that debt rollover risk is an important determinant of their return expectations. Taking the role of debt refinancing into account reconciles previous, mixed evidence on the relation between leverage and stock returns: equity returns increase with leverage when controlling for firms debt refinancing risk. Unlike leverage effects on stock returns, the return-relevant information embedded in debt refinancing intensities cannot be subsumed by book-to-market, size, or other firm characteristics that have been shown to predict returns. The intuition is that the debt refinancing intensity is the only characteristic that allows the shareholder to assess the tradeoff a firm faces in its capital structure decision that optimally balances the benefits and rollover risks of short-term relative to long-term debt. Related literature. Our paper is related to the literature that elaborates on the relation between leverage and stock returns. Gomes and Schmid (2010) argue that the mixed empirical evidence on whether this relation is positive, negative, or whether there is no significant relation at all may be a result of previous papers not accurately accounting for the complexity of the link between a firm s financial leverage and the return on its equity (see, e.g., Bhandari, 1988; 4

7 Fama and French, 1992; Penman et al., 2007; George and Hwang, 2010). Specifically, they argue that the link between leverage and stock returns depends on a firm s investment opportunities. We explore the relation between leverage and stock returns from a different angle which does not require to model the firm s investment policies but emphasizes the role of a firm s debt maturity profile and refinancing risk. Several empirical studies provide evidence that firms issue debt with dispersed maturity dates and that firms choices of leverage and debt maturity profile depend on their risk attributes. Diamond (1991) models the debt maturity choice as a signal for private information about the issuers quality. Guedes and Opler (1996) document an inverse U-shaped pattern between firms debt maturities and credit ratings. Choi et al. (2017) argue that firms spread out their debt maturity dates over time in order to avoid lumpiness in the aggregate issuance amount of debt. More specifically, the optimal capital structure implications of our model that firms with comparably low (high) cash flow risk choose higher (lower) levels of leverage with longer (shorter) debt maturities are consistent with empirical evidence provided by, for instance, Barclay and Smith (1995), Stohs and Mauer (1996), Johnson (2003), Custódio et al. (2013), and Gopalan et al. (2014). The conceptual framework employed in our paper is motivated by trade-off models of optimal capital structure in the spirit of Fischer et al. (1989), Leland (1994b, 1998) or Leland and Toft (1996). These models endogenize a firms optimal leverage and default decisions. Bhamra et al. (2010a,b) and Chen (2010) are among the first to discuss the asset pricing implications of dynamic leverage models and relate leverage and default decisions to the timeseries patterns of equity returns and credit spreads. More recently, these frameworks are applied in the structural debt pricing literature that elaborates on the relation between rollover risk and credit risk. He and Xiong (2012) show that short-term debt exacerbates default risk via the rollover channel due to its higher sensitivity to shocks to debt funding costs. Other models that feature a mechanism where debt refinancing costs are borne by equity holders include, among others, Acharya et al. (2011), Cheng and Milbradt (2012), Chen et al. (2013, 2017), and He and Milbradt (2014). Alternatively, Diamond and He (2014) examine the role of debt maturity in determining future debt overhang. Interestingly, most studies that rely on these structural frameworks treat leverage, debt 5

8 maturity, or both as exogenous. Notable exceptions are Dangl and Zechner (2016) or He and Milbradt (2016), however, their focus is very different compared to the objective of this paper. Dangl and Zechner (2016) study the role of bankruptcy costs for leverage and debt maturity dynamics. He and Milbradt (2016) study a firm s optimal choice of debt maturity structure and default timing, both without commitment. Our paper is the first to explore how refinancing risk associated with the rollover of debt affects equity returns, specifically through its interaction with leverage. Finally, we revisit the relation between leverage, book-to-market and size from a new perspective, without relying on arguments related to a firm s investment policy, operating leverage, and/or profitability (e.g. Fama and French, 1993; Carlson et al., 2004; Zhang, 2005; Novy-Marx, 2011, 2013; Fama and French, 2015). In our model, the firm s book-to-market ratio and size are both directly related to its leverage and its debt maturity structure and can be interpreted as measures of how far a firm s capital structure deviates from its (initial) optimum, i.e. the capital structure arising from jointly choosing leverage and the mix of short- and long-term debt. Consistent with the model implications, our empirical results show that size and bookto-market capture leverage effects on equity returns but that a firm s refinancing intensity conveys information for returns beyond these characteristics. The remainder of the paper is organized as follows. Section II describes the structural model and Section III discusses the model s implications for expected equity returns as well as for book-to-market and size in a setting without heterogeneity in debt benefits across firms. In Section IV we relax this assumption and discuss the implications if firms differ in their relative debt benefits. Section V describes the data. In Section VI, we present and discuss the results of our empirical analysis. Section VII concludes, and the Appendix contains technical details. II. Structural model In this section, we present a simple model of the firm s capital structure that follows the spirit of Leland (1998) but endogenizes the firm s optimal choice of leverage and debt maturity. In the next section, we then use this framework to derive the implications of the interaction between leverage and debt refinancing for firm s expected equity returns. 6

9 The standard trade-off theory of capital structure postulates that a firm maximizes its value by levering up to the extent that the benefits of debt equal its costs. Understanding a firm s debt maturity profile is important because the optimal choice of debt maturity is also subject to a tradeoff. Previous research provides various arguments as to why short-term debt offers benefits compared to long-term debt. Such benefits may arise from short-term debt typically being issued at lower fixed costs compared to long-term debt (see e.g., He and Milbradt, 2014; Chen et al., 2017), reducing information asymmetries (e.g., Flannery, 1986; Diamond, 1991; Custódio et al., 2013) and/or mitigating agency conflicts (Datta et al., 2005; Brockman et al., 2010). These benefits of short-term debt, however, come at the cost of frequently rolling over the firm s debt, which exposes the firm to refinancing risk. In what follows, we illustrate how this tradeoff matters for the optimal capital structure of a firm that raises debt capital by issuing short-term bonds and long-term bonds. The general implication of our model is that firms with comparably low (high) cash flow volatility optimally choose higher (lower) levels of leverage with longer (shorter) debt maturities. These model-implied patterns are consistent with empirical evidence on the link between firm risk and debt financing policies as, e.g., in Barclay and Smith (1995), Stohs and Mauer (1996), Johnson (2003), and Custódio et al. (2013). Moreover, the higher a firm s benefits of shortterm compared to long-term debt, the more short-term debt the firm issues, which in turn reduces its overall capacity for leverage due to increased debt rollover risk. A. Short-term and long-term debt We assume that the firm s cumulative cash flow (X t ) follows a Geometric Brownian Motion (GBM) under the risk-neutral probability measure (Q) with drift µ Q and volatility σ. The instantaneous risk-free rate is denoted by r. As in He and Xiong (2010), the firm has access to two types of debt instruments: a shortterm zero-coupon bond (S) and a long-term zero-coupon bond (L). At time t = 0, the firm raises a principal amount P i from issuing bond i {S, L}, thus, the aggregate principal amount of debt is given by P = P S + P L. We model the maturity of bond i by a Poisson process with intensity φ i, and φ S > φ L reflects the earlier redemption of S relative to L. Assuming a stationary debt structure, refinancing short-term and long-term debt can be equivalently 7

10 thought of as continuously refinancing the amounts φ S P S and φ L P L, respectively. 1 The key question for a value-maximizing firm is to decide on the amounts of short-term and long-term debt to raise. To determine the market value of type-i debt (D i ), we start from the required return on debt (rd i ), which is given by rd i = µ Q XDX i σ2 X 2 DXX i + φ i (P i D i ). (1) }{{}}{{} sensitivity of D i debt refinancing to cash flow The above equation illustrates the two driving forces behind changes in debt value. The first is the sensitivity of D i to the firm s cash flows. type-i debt due to the firm refinancing the fraction φ i characteristics. The second captures the value change in by issuing new debt with identical To solve Equation (1) for the value of debt D i, we impose two standard boundary conditions by evaluating the limits of the cash flow at infinity and at the default boundary. We discuss the solution below but delegate technical details to Appendix A. In the first case (X t = ), the firm never defaults and the associated default-free value of debt (p i ) is given by p i = P i 1 + r/φ i. (2) In the second case, shareholders choose to optimally default (when X t = X B, where X B is the endogenous default boundary) and bondholders take over the firm with a debt value given by where λ i = P i /P. 2 D i (X B ) = X B r µ Q λi, (3) The difference in debt values in the two boundary scenarios in Equations (3) and (2), D i (X B ) p i < 0, reflects the bondholders loss given default. The market value of bond i is given by D i (X t ) = p i + [ D i (X B ) p i] π i,q t, (4) 1 The assumption that the firm commits to a stationary debt structure follows Leland and Toft (1996), Leland (1998), and He and Xiong (2012) who argue that tight covenants prohibit the firm from changing its debt structure; Fama and French (2002), Baker and Wurgler (2002), Welch (2004), Strebulaev (2007), and Lemmon et al. (2008) provide empirical evidence that firms debt structures are indeed stationary over time. 2 Note that the boundary condition implies that short-term and long-term bondholders share the remaining value of the firm proportionally to P in the event of default, i.e. there is no maturity-related debt seniority. 8

11 where π i,q t is the risk-neutral default probability of type i debt, approaching zero when X t = or one when X t = X B. Thus, the bond value is given by its default-free value adjusted by the expected loss due to default risk. B. Equity valuation With shareholders being the residual claimants of the firm, the value of equity (E) is given by the differential of the levered firm value (F ) minus the value of debt (D), i.e., E(X t ) = F (X t ) i Di (X t ). Changes in the equity value (re), thus, depend on the firm s current cash flow, the sensitivity of the equity value to the underlying cash flow process, and debt-related flows (debt benefits and refinancing of, both, short-term and long-term debt). In particular, the equity value satisfies the equation re = }{{} X + µ Q XE X σ2 X 2 E XX cash flow }{{} sensitivity of E to cash flow + k i φ i P i i }{{} debt benefits φ i (P i D i ), (5) } {{ } debt refinancing where k > 0 is a scaling factor to model debt benefits in reduced form. With this specification we ensure that debt benefits are inversely proportional to debt maturity with k i = kφ i P i, in line with previous theoretical and empirical evidence for benefits of short-term debt. Equation (5) shows how the tradeoff between short- and long-term debt affects the value of equity: It increases with debt benefits and decreases with costs related to debt refinancing. The cost of refinancing depends on the fraction of debt that has to be rolled over, i.e. the refinancing intensity (φ i ), and on the discount at which debt is refinanced, i.e. on the difference between the principal (P i ) and debt value (D i t). Since φ i remains constant over time, any time-variation in debt-related flows that matters for the equity value arises from the bond s discount P i D i t, which depends on the firm s current cash flow X t. In periods with high (low) cash flows, the firm moves further away from (closer to) the default boundary and hence the discount P i D i t is small (large). This trade-off between the benefits of a high level of short-term debt and increased refinancing risk determines the firms optimal leverage and debt maturity choice. 9

12 C. Optimal capital structure Based on the valuation of the firm s debt and equity, we now explore the implications for the firm s optimal capital structure. At time t = 0, the firm chooses the principal amounts for short-term and long-term debt to maximize the initial value of the firm. By simultaneously choosing P S and P L, the firm decides on the overall amount of debt to issue as well as on the maturity structure of its debt. In other words, the firm jointly optimizes its leverage and refinancing intensity. With P S and P L being the only decision variables, we fix all other parameters in accordance with the structural equity and bond pricing literature. 3 In particular, we set the initial cash flow level X 0 = 1, the riskless rate r = 5%, and the risk-neutral drift of the cash flow process µ Q = 1%. Furthermore, we assume short- and long-term debt maturities of one and ten years, respectively, implying refinancing intensities of φ S = 1 and φ L = 0.1. The idea is to choose a maturity range consistent with empirical maturities. For example, Barclay and Smith (1995) and Custódio et al. (2013) document that typically around two-thirds of a firm s debt matures within five years. 4 For now, to illustrate the model s basic tradeoff, we set the scaling factor for debt benefits in Equation (5) to k = 0.01, which implies for the maturities chosen with one and ten years that the benefit of short-term debt exceeds the one of long-term debt by a factor of ten. Figure I illustrates the tradeoff between issuing short-term and long-term debt for a given cash flow volatility σ = 10%, showing that the benefits of short-term debt outweigh the costs arising from rollover risk when issuing too much short-term debt. Figure I about here In the next step, we study the tradeoff between short-term and long-term debt for different levels of cash flow risk σ. Figure II summarizes firms optimal choices of leverage and debt 3 See, e.g., Leland (1994b), Leland and Toft (1996), Goldstein et al. (2001), Dangl and Zechner (2016), Garlappi and Yan (2011), He and Xiong (2012), Chen et al. (2017), Chen et al. (2013), He and Milbradt (2014) or Diamond and He (2014). 4 In our model, we do not allow for early repayment/refinancing of debt; in reality, issuers may have a call option on their debt. Empirical evidence suggests that such early refinancing is rather limited and the evidence is mixed with respect to motives for early refinancing. For example, Mian and Santos (2017) and Xu (2017) both find procyclical patterns in refinancing activities, but the former argue that these are driven by firms with high credit quality who want to secure favorable borrowing rates for longer maturities, whereas the latter argues that these are driven by low credit quality firms who want to mitigate refinancing risk. 10

13 maturity for σ-values in the range from 5% to 25%. We measure the firm s leverage as L(X t ) = P P + E(X t ), (6) which corresponds to the leverage measure applied in empirical research such as, e.g., Strebulaev and Yang (2013) or Danis et al. (2014). The firm s aggregate refinancing intensity (Φ) is determined by its optimal debt maturity mix and given by Φ = λ S φ S + λ L φ L. (7) Panels A and B in Figure II show that a firm s optimal level of short-term debt as well as its fraction of short-term to total debt increases with cash flow risk. Panel C shows that the optimal leverage (L) chosen by a firm decreases in cash flow risk, which reflects that firms with lower cash flow risk have a higher debt bearing capacity. This result matches the implications of standard structural models (e.g, Leland, 1994b) and lines up well with empirical research that finds an inverse relation between firms cash flow volatility and leverage (e.g., Lemmon et al., 2008). 5 Panel D shows that the firm s optimal refinancing intensity (Φ) increases with cash flow volatility, reflecting the insights from Panels A and B. The intuition is that it is too costly for a firm with volatile cash flows to issue large amounts of long-term debt, because the discount (P i D i t) of the long-term bond is more sensitive to cash flow volatility than the discount of the short-term bond. Recent empirical work provides evidence that supports this notion by documenting that riskier firms issue relatively more short-term debt (e.g., Gopalan et al., 2014). Overall, our model implies that firms with comparably low (high) cash flow volatility optimally choose higher (lower) levels of leverage with longer (shorter) debt maturities, consistent with the empirical evidence in, e.g., Barclay and Smith (1995), Stohs and Mauer (1996), Johnson (2003), and Custódio et al. (2013). 5 In our analysis, we are interested in isolating the overall structural relations between leverage and debt refinancing and their implications for equity returns. Therefore, we keep our model tractable by restricting the number of parameters to the key primitives. This comes at the well-documented cost (for a discussion we refer to Strebulaev and Whited, 2012) that it is difficult to numerically match real-world quantities, most importantly leverage ratios. The quantitative properties of structural models can be improved by adding additional frictions, such as e.g., direct bankruptcy costs or different layers of taxes, etc. For the sake of the model s tractability, we abstain from adding such frictions, because they do not affect the overall structural relations between leverage, debt maturity, and equity returns. 11

14 Figure II about here Finally, we also allow for heterogeneity in debt benefits and summarize the firms optimal leverage and maturity choices for k (0.005, 0.01, 0.02) in Figure III. Recall from above that a higher value of k drives a larger wedge between the benefits of short-term debt (k S ) and long-term debt (k L ). Accordingly, we find that, for a given level of cash flow risk, firms with higher debt benefits k choose to issue relatively more short-term debt which in turn reduces their overall leverage due to the associated increase in refinancing risk: Panel A shows that optimal leverage is inversely related to k whereas Panel B shows that the optimal refinancing intensity increases with k. Panel C shows that firms with different debt benefits optimally choose different debt maturity structures. A firm with higher k, and therefore higher benefits of short-term relative to long-term debt, will choose more short-term debt and hence have a higher refinancing intensity. Finally, Panel D plots the inverse relation between the firm s leverage and its equity value. The figure illustrates that, for a given leverage, equity valuations increase with debt benefits. Figure III about here D. Expected equity returns at t = 0 Following previous research that derives equity return implications from structural models, such as Gomes and Schmid (2010) and Garlappi and Yan (2011), we define expected returns on equity via the sensitivity of equity to cash flows. The intuition is that, since equity can be viewed as a contingent claim on the firm s assets and cash flows represent the only source of risk, any change in the equity value must be driven by innovations in cash flows. Defining the asset risk premium as the difference in the real-world and risk-neutral drift of the cash flow process, i.e. ξ t = µ P t µ Q t, one can express the time-t expected stock return as E P [R t ] = r + b X t ξ t (8) where b X t = dloge(x t) dx t (9) 12

15 measures the sensitivity of the equity value to cash flows. 6 In our analysis below, we assume a time-invariant risk premium ξ = 5%; for all other parameters, we keep the values that we have used for the optimization of the firm s capital structure in the previous section. To understand the link between a firm s time-t expected equity returns and its debt structure, we need to distinguish model-implications at the point in time when the firm chooses its optimal capital structure (i.e. at t = 0) and implications after the capital structure has been determined (i.e. at t > 0). This distinction is important because changes in the equity value affect the firm s leverage ratio (L) over time. Now, we discuss the firm s expected returns at t = 0 and present the implications in Figure IV. Panel A shows that expected equity returns generally decrease with cash flow risk and that for a given σ expected returns are lower for firms that have higher benefits of short-term relative to long-term debt. The reason is that firms with the same cash flow risk but differences in debt benefits choose different levels of leverage; as discussed above, firms with higher k issue more short-term debt which in turn reduces their overall debt capacity. Panel B shows that the expected equity return is directly related to the level of leverage, independent of the underlying combination of σ and k: the higher a firm s leverage the higher its expected equity return. Figure IV about here Our empirical analysis naturally focuses on the more general model predictions for equity returns at any time t > 0, on which we elaborate in the subsequent Sections III and IV. In Section III, we discuss the model predictions when all firms face the same benefits of short-term relative to long-term debt, i.e. when there is no cross-sectional variation in k. We relax this assumption in Section IV, where we allow for heterogeneity in debt benefits across firms. III. Equity returns when all firms have the same debt benefits In this section we study the model implications for expected equity returns at times t > 0 under the assumption that the benefits of short-term relative to long-term debt are equal for 6 We use the structural model to derive expected equity return implications based on the firm s total risk, similar to e.g. Garlappi and Yan (2011) or Friewald et al. (2014). If we were to specify the systematic risk structure of the firm s cash flow process, the return-implications of our model would be based on the systematic risk component of the firm s total risk. Since such a distinction would not affect our subsequent empirical analysis, we prefer not to introduce an additional parameter/degree of freedom. 13

16 all firms, i.e. there is no cross-sectional variation in k. First, we show that expected equity returns increase with the firm s time-t leverage and its refinancing intensity. We then discuss the model-implied relations of book-to-market and size to expected equity returns. We show that in a world without heterogeneity in debt benefits these characteristics are sufficient to explain the cross-section of stock returns. Specifically, we show that the joint distribution of book-to-market and size contains the same information for stock returns as the combination of leverage and refinancing intensity. In other words, book-to-market and size subsume leverage effects in stock returns, as suggested by Fama and French (1992). By contrast, once we allow for variation in debt benefits across firms (in Section IV below), firms debt refinancing intensities contain return-relevant information that cannot be subsumed by other characteristics. A. Leverage, debt refinancing and expected equity returns After the firm chooses its optimal capital structure at t = 0, any changes in the equity value directly affect the firm s leverage ratio (L), whereas the firm s refinancing intensity does not change over time. 7 As a consequence and contrary to t = 0, there is no one-to-one mapping between leverage, refinancing intensity, and expected equity returns at times t > 0. intuition is that time-t leverage is not informative about the debt refinancing policies that a firm has committed to at t = 0. Conversely, the refinancing intensity does not take into account that refinancing costs, that is, bond discounts, change in response to cash flow realizations, whereas these changes are reflected in the firm s time-t leverage. To see how stock returns relate to leverage and debt maturity structure, Figure V plots expected equity excess returns for different combinations of leverage and refinancing intensity. Stock returns increase with leverage (for a given refinancing intensity) and increase with refinancing intensity (for a given leverage). More specifically, the figure shows that neither the leverage nor the refinancing intensity alone is sufficient to understand how a firm s debt-related risks affect expected equity returns. For instance, a firm with high leverage (solid line) but low refinancing intensity may have the same expected return as a firm with medium lever- 7 Given that the firm commits to a stationary debt structure, changes in the leverage ratio are only driven by changes in the equity value. This feature of the model is consistent with empirical evidence provided by Welch (2004) who concludes that variation in equity value is the primary determinant of changes in a firm s leverage ratio. It is also consistent with the notion that capital structure adjustments occur infrequently; see e.g., Leary and Roberts (2005) and Strebulaev (2007). The 14

17 age (dashed line) and medium refinancing intensity or a firm with low leverage (dotted line) and high refinancing intensity. Hence, an (empirical) analysis on how stock returns relate to leverage should account for differences in firms refinancing risk due to differences in their debt maturity profiles. Figure V about here The big symbols in Figure V mark firms whose time-t capital structure is the same as their initial capital structure, optimally chosen commensurate with the firm s cash flow risk. All other combinations of leverage and refinancing intensity represent firms whose leverage ratios have changed since t = 0, that is, L(X t ) L(X 0 ). Hence, an equivalent interpretation of our results is that the refinancing intensity conveys complementary information to L(X t ), because it allows to assess the change in the firm s leverage. We define the change in leverage as L(X t ) = L(X t) L(X 0 ), (10) L(X 0 ) and illustrate the relation between refinancing intensity, change in leverage, L, and expected returns in Figure VI. Panel A shows how the combination of the firm s refinancing intensity and its time-t leverage relate to changes in leverage, and Panel B plots expected equity returns as a function of time-t leverage and changes in leverage. Firms with time-t leverage lower than their initial leverage (jointly chosen with their refinancing intensity), i.e. L(X t ) < 0, have expected returns that are lower than expected returns at t = 0, and vice versa. The expected equity return of a firm depends on its leverage and on the change in leverage. In other words, two firms that have the same leverage today have different expected returns when they differ in their deviation from the initial optimal capital structure. Figure VI about here Below, we rely on both of these conceptually equivalent interpretations. Using the notion of changes in leverage allows us to connect expected returns to the firm s book-to-market ratio and its size. The advantage of arguments building on the firm s refinancing intensity is that these allow for direct empirical tests, because choices of firms debt maturity structures are 15

18 observable, whereas deviations from firms optimal capital structures are not. 8 B. Book-to-market and expected equity returns Assuming that equity is priced at its book value when the firm decides on its capital structure at time t = 0, the model-implied book-to-market ratio is given by BM(X t ) = E(X 0 )/E(X t ). Hence, all firms start from an initial book-to-market ratio of one, but BM changes over time as the value of equity evolves in response to cash flow realizations, similar to the firm s leverage L. More specifically, at t > 0, a book-to-market ratio of one corresponds to the firm s timet leverage being at the level initially chosen in the joint optimization of leverage and debt maturity. Conversely, BM > 1 (BM < 1) corresponds to leverage having increased (decreased) over time, and hence BM reflects the firm s leverage evolution relative to its initial level chosen in accordance with its debt maturity profile. To see that BM captures essentially the same information as L(X t ), note that we can write that BM(X t ) = 1 + δ t L(X t ) (11) with δ t = 1 + P/E(X t ). In other words BM is centered around one, its initial value at t = 0, adjusted for the relative change in leverage since t = 0. The adjustment is proportional to δ t which is one plus the debt-to-equity ratio, P/E(X t ). Panels A and B in Figure VII illustrate that the relation of BM to leverage and refinancing intensity is the same as for changes in leverage shown in Figure VI (Panel A) and so is the link to expected returns (Panel B). BM increases with leverage (for a given refinancing intensity) and increases with refinancing intensity (for a given leverage). The highest BM-values are associated with high leverage and high refinancing intensity firms, while the lowest BM-values come from firms with low leverage and low refinancing intensity. Other things equal, stock returns increase with BM, consistent with empirical evidence that firms with high BM earn higher returns than low-bm firms. However, to achieve a strictly monotonic, positive relation of expected returns to BM in our model, we either have to hold leverage or refinancing intensity 8 Given that a firms optimal/target capital structure is unknown, leverage deviations have to be estimated, e.g. following the approach of Flannery and Rangan (2006). Using their and other regression-based approaches, Ippolito et al. (2017) find that over levered (under levered) firms earn higher (lower) returns. 16

19 fixed across firms. For a given refinancing intensity, two firms may have different BM due to different levels of leverage, and the firm with higher BM (due to higher leverage) has higher expected returns than the firm with lower BM (which has lower leverage). This result is consistent with empirical evidence that book-to-market captures the leverage effect on stock returns to some extent (e.g., Choi, 2013; Doshi et al., 2014). On the other hand, two firms with the same leverage may differ in BM values and expected returns due to differences in the refinancing intensities, which suggests that the link between BM and stock returns cannot be understood in terms of leverage alone. Figure VII about here C. Size and expected equity returns We follow the convention in empirical (asset pricing) studies and define size as the firm s market value of equity E(X t ) at time t. In our model the evolution of leverage is driven by changes in the equity value, hence, there is a direct link between size and leverage. At time t = 0, there is a one-to-one mapping between leverage, size, and expected returns: high leverage firms are small and earn high returns whereas low leverage firms are big and earn low returns (see Panel D in Figure III and Panel B in Figure IV). At times t > 0, the relation between size, leverage, and expected returns depends on the firm s refinancing intensity. Panel A of Figure VIII illustrates that for a given refinancing intensity firm size decreases with leverage, and, for a given leverage firm size decreases with refinancing intensity. In other words, the model implies that firms are small (big) when they have high (low) leverage and a high (low) refinancing intensity. Panel B shows that the relation of size to leverage and refinancing intensity gives rise to a size effect in stock returns, consistent with empirical evidence that small firms have higher returns than big firms. The smallest firms with the highest expected returns are firms with high leverage and high refinancing intensity whereas the biggest firms earning lowest returns are those with low leverage and low refinancing intensity. For two firms with the same leverage, their refinancing intensities determine their size and their expected equity return, and vice versa when we fix the refinancing intensity. Figure VIII about here 17

20 D. Explaining the cross-section of returns with book-to-market and size Under the assumption of this section that there is no variation in k across firms, BM and size are sufficient to explain the cross-section of expected equity returns. As shown above, a firm s expected return depends on its leverage and its refinancing intensity, or equivalently, the change in leverage relative to the firm s initial, optimal capital structure. Hence, all that we need to determine expected returns is information about the firm s current and its initial capital structure. This allows us to recover the underlying riskiness of the firm, i.e. it s cash flow volatility, which is in this setup the only parameter that determines the firm s optimal capital structure. The firm s refinancing intensity conveys the relevant information about the firm s initial capital structure, because the firm is committed to a stationary debt maturity structure. This information is also embedded in the combination of book-to-market and size. The intuition is as follows. As discussed above, BM contains essentially the same information as changes in leverage because it reflects the firm s change in equity value, which is the only driver of changes in leverage. Further, size measured as the time-t equity value, combined with BM allows us to infer the level of the firm s initial equity value, E(X 0 ) = BM t E(X t ), and therefore the firm s initial capital structure, since there is a one-to-one mapping between equity value, leverage, and refinancing intensity at t = 0. To illustrate that book-to-market and size are informative about this mapping, Figure IX shows that there is a direct relation of BM size to the firm s initial leverage and its refinancing intensity. Figure IX about here Put differently, knowing book-to-market and size is equivalent to knowing the firm s initial capital structure because it allows to infer the firm s cash flow risk (σ). This also implies that the combination of BM and size contains all information necessary to compute the firm s time-t leverage via Equation (11). Hence, leverage represents redundant information when book-to-market and size are known, consistent with the arguments of Fama and French (1992) that leverage effects on stock returns are subsumed by these characteristics. In the setup of the current section, without cross-sectional variation in debt benefits, the refinancing intensity does not convey any additional information either. In other words, the joint distribution of BM and size is sufficient to explain the cross-section of expected equity returns, because it 18

21 spans the same information as leverage and refinancing intensity. By contrast, book-to-market and size are not sufficient to understand firms expected returns when there are differences in debt benefits across firms. IV. Equity returns with variation in debt benefits across firms When firms differ in debt benefits, the mapping of firm characteristics into stock returns becomes more complex. The reason is that the initial optimal capital structure depends on the particular combination of cash flow risk (σ) and debt benefits (k), which cannot be recovered from book-to-market and size alone. While the information embedded in leverage remains redundant, the refinancing intensity is informative about the firms benefits of short-term relative to long-term debt, which are otherwise unobservable to the empiricist. With k being an additional determinant of the optimal capital structure, Figure X shows that expected returns generally increase with leverage and with refinancing intensity. But even for a given leverage and a given refinancing intensity there can be cross-sectional differences in returns due to differences in k. Hence the combination of leverage and refinancing intensity is not sufficient to assess expected returns. Figure X about here Similarly, cross-sectional variation in k induces cross-sectional variation in BM and size, even when we fix leverage and refinancing intensity. We illustrate this in Figure XI for a fixed level of time-t leverage (medium leverage from Panel C, Figure X). 9 For the same leverage and refinancing intensity, firms with lower debt benefits tend to have higher book-to-market ratios and are smaller in size. Moreover, the k-related link between book-to-market and size to expected returns appears to be mostly opposite to the standard BM and size effects. For BM we find that higher k is associated with lower BM but mostly with higher returns, unless BM is very low. For size, we find opposite to the standard size effect that higher (lower) k is associated with bigger (smaller) size and higher (lower) returns. Figure XI about here 9 The patterns for low and high leverage firms are very similar and presented in the Internet Appendix in Figures IA.I and IA.II, respectively. 19