Contingent-Claim-Based Expected Stock Returns

Transcription

1 Contingent-Claim-Based Expected Stock Returns Zhiyao Chen University of Reading Ilya Strebulaev Stanford University and NBER February 6, 2013 Abstract We develop and test a parsimonious contingent claim model for cross-sectional stock returns. Because stocks are residual claims on firms assets that generate operating cash flows (Merton, 1974), stock returns are cash flow rates scaled by the sensitivities of stocks to cash flows. Our model outperforms the capital asset pricing model (CAPM) and the Fama French model in explaining the returns of stock portfolios formed on market leverage, book-to-market ratio, asset growth rate and equity size. The success of our model is attributed to our innovative structural estimation of the stock-cash flow sensitivity that carries information about a firm s financial leverage and probability of default. Specifically, value stocks, high-leverage stocks and low-asset-growth stocks are more sensitive to cash flows than growth stocks, low-leverage stocks and high-assetgrowth stocks, particularly in recessions when default probabilities are high. Keywords: continuous time contingent claim model, structural estimation, GMM, financial leverage, default probability, stock returns JEL Classification: G12, G13 We thank Harjoat Bhamra, Chris Brooks, Wayne Ferson, Vito Gala, Patrick Gagliardini, Chris Hennessy, Chris Juilliard, Stavros Panageas, Marcel Prokopczuk, Stephan Siegel, Kathy Yuan for their helpful comments. Zhiyao Chen: ICMA Centre, University of Reading, UK, n.chen@icmacentre.ac.uk; Ilya Strebulaev: Graduate School of Business, Stanford University, and NBER, istrebulaev@stanford.edu. 1

2 1 Introduction Equity is a residual claim contingent on a firm s assets that generate operating cash flows (Merton, 1974). We build a parsimonious contingent claim model for cross-sectional stock returns. In the model stock returns are cash flow rates scaled by the sensitivities of stocks to cash flows. Our model outperforms the capital asset pricing model (CAPM) and the Fama French three-factor model in explaining stock returns of portfolios formed on market leverage, book-to-market equity, asset growth rate and market capitalization. The better performance of our model is attributed to our innovative structural estimation of the stockcash flow sensitivity that captures variation in default probabilities across firms and over business cycles. Our model is parsimonious because it has only one state variable and two policy parameters related to dividend payout and default policies that determine the stock-cash flow sensitivities. Instead of looking for an unobservable market return (Roll, 1977), we take the observable cash flows as our state variable. Our choice of an observable state variable follows Cochrane (1996) who essentially uses observable investment returns as his main state variable. In our one-factor contingent-claim model we use the operating cash flows as the only state variable to keep the model tractable. 1 The basic idea of our paper is to infer the underlying market movement by watching firms operating cash flows. We explicitly model the dividend payout and default policies that affect the amount of cash flows accruing to stock holders and therefore the stock-cash flow sensitivities. While the dividend payout policy determines how much stock holders can claim on cash flows when the firms are solvent, the strategic default policy affects how much they receive at bankruptcy. Therefore, both policies have an impact on the sensitivities of stock holders to the changing cash flows. The sensitivity is not only determined by corporate policies but also associated with firm characteristics, such as financial leverage, book-to-market equity ratio, size and asset growth rate. In other words, firms with different characteristics 1 Our model is slightly different from Campbell and Shiller (1988) and Campbell and Vuolteenaho (2004) who use dividend cash flows as their state variable. 2

3 can strategically choose different payout and strategic default policies. While the cash flow rates can be estimated from the data, the stock-cash flow sensitivity is difficult to measure. For instance, Garlappi and Yan (2011) and Favara, Schroth, and Valta (2011) use linear regressions to estimate the sensitivity of stocks to market returns after deriving theoretical implications of the strategic default policy for stock returns. In contrast, we directly evaluate both policies in a structural estimation and then examine their impacts on time-varying stock-cash flow sensitivities and cross-sectional stock returns. Inspired by the success of Schaefer and Strebulaev (2008) who use the bond-stock sensitivities to explain bond returns, we employ generalized method of moments (GMM) to estimate the stock-cash flow sensitivities and explain the cross-sectional stock returns. Following Cochrane (1996), we match average predicted returns with realized returns for four sets ofequal-weighted quintileportfolios inourstructuralestimation. 2 Thefirstset included in this study is market leverage portfolios. They are natural choices for testing portfolios because equity is a contingent claim on operating cash flows after contractual debt payments. As expected, our model performs well in explaining the cross-sectional returns of the market leverage portfolios. The pricing error of the high-minus-low (H L) portfolio is 0.34% per year, substantially lower than 11.74% from the CAPM and 3.13% from the Fama French model. The mean absolute error (m.a.e.) of this set of portfolios is 1.27% per year, compared with 6.83% from the CAPM and 3.66% from the Fama French Model. We take book-to-market and size portfolios as our next two sets of testing portfolios for two reasons. First, both value and size premiums are related to a firm s debt financing policy. Gomes and Schmid (2010) demonstrate that firms with high book-to-market equity are mature firms that have accumulated their debt during their expansions and hence have high financial leverage. Second, both value and size premiums have been found associated with default risk by Fama and French (1996). Griffin and Lemmon (2002) and Vassalou and Xing (2004) document that both value and size premiums are more significant in firms with high default risk. Garlappi and Yan (2011) and Avramov, Chordia, Jostova, and Philipov 2 Recent other papers that use GMM to study investment models for stock returns include Liu, Whited, and Zhang (2009), Liu and Lu (2011) and Li and Liu (2011). 3

4 (2011) further associate the value premium with default risk. Because the contingent claim model is a standard valuation framework for default risk, it is interesting to examine its performance for these two sets of portfolios. Our model successfully captures the value and size premiums. For the book-to-market portfolios, the pricing error of the H L portfolio from our model is 1.80% per year, lower than 14.81% from the CAPM and 7.56% from the Fama French model. For the size portfolios, the pricing error of the small-minus-big (S B) portfolio from our model is 0.58% per year, which is much lower than 8.28% from the CAPM and 3.14% from the Fama French model. The last set of portfolios of interest is asset growth portfolios. 3 The reason for including the asset growth portfolios is that the low-asset-growth firms are more likely to be mature firms with high book equity in place and high default risk. Additionally, Avramov, Chordia, Jostova, and Philipov(2011) show that the asset growth premium is associated with financial distress risk. The pricing error of the H L portfolio from our model is 5.06% per year. Although it is the greatest among the four sets of testing portfolios, this error is still considerably lower than 11.74% from the CAPM and 10.65% from the Fama French model. Our work contributes to an emerging literature on contingent claim models that investigate cross-sectional stock returns. The first theoretical paper that applies a contingent claim model to studying asset prices can be dated back to Galai and Masulis (1976). Since Berk, Green, and Naik (1999) who take a dynamic model to study cross-sectional returns, recent research papers either employ linear regressions to test their predications or take simulated method of moments (SMM) to estimate the model. A non-exclusive list includes Gomes, Kogan, and Zhang (2003), Carlson, Fisher, and Giammarino (2004), Gomes and Schmid (2010), Bhamra, Kuehn, and Strebulaev (2010), Eisfeldt and Papanikolaou (2010) and Ai and Kiku (2011). 4 Different from them, we perform a structural estimation via 3 Cooper, Gulen, and Schill (2008) show that firms with low asset growth rates outperform their counterparts with high growth rates by 8% per year for value-weighted portfolios and 20% per year for equal-weighted portfolios. 4 Our paper is also related to the recent literature on dynamic models of capital structure in the contingent claim framework. A partial list of recent papers includes Goldstein, Ju, and Leland(2001), Strebulaev(2007), 4

5 GMM. Compared with SMM that simulates a firm s dynamic paths, GMM uses all the relevant data points. To the best of our knowledge, our work is the first study that uses GMM to evaluate a contingent claim model for cross-sectional stock returns in the structural estimation. Our main empirical contributions and results can be summarized as follows. First, we propose an empirical procedure that embeds the KMV method (Crosbie and Bohn, 2003) into the GMM framework in our structural estimation. We estimate the latent risk-neutral rate and cash flow volatility, and then use them to estimate the time-varying stock-cash flow sensitivity. Compared with the conventional linear regression approach, our structural estimation of the stock-cash flow sensitivity is innovative. Second, we find that stocks are more sensitive to the changing cash flows in bad times when default probabilities are high than they are in good times when default probabilities are low. More important, the spread in stock-cash flow sensitivity is a manifestation of the cross-sectional difference in default probability over business cycles. Third, the spreads in the stock-cash flow sensitivities are able to explain a large portion of cross-sectional stock returns for the market leverage, book-to-market and asset growth portfolios, while the spread in the cash flow rates plays an important role for the size portfolios. The remainder of this paper proceeds as follows. Section 2 presents the parsimonious contingent claim model. Section 3 explains the empirical specifications and procedures. Section 4 describes the data and the empirical measures. Section 5 uses GMM to estimate the model, examines the cross-section and the time series of stock-cash flow sensitivity and default probability, and examines the impacts of the sensitivities on the average predicted stock returns. Section 6 concludes the paper. 2 A Parsimonious Contingent Claim Model We consider an economy with a large number of firms, indexed by subscript i. Assets that generate cash flows are traded continuously in arbitrage-free markets. Under a risk-neutral Chen (2009), Morellec, Nikolov, and Schurhoff (2008) and Glover (2011). 5

6 probability measure, the operating cash flow X it is governed by dx it X it = µ i dt+σ i dw it, (1) where µ i is an expected risk-neutral rate, σ i is a volatility parameter, and W it is a standard Brownian motion. The counterpart of µ i under the physical probability measure is ˆµ i = µ i +λ i, where λ i is an individual risk premium. It is worth noting that we do not explicitly specify the risk premium throughout this study. 5 The cash flow is independent of capital structure choices and investment policies. At the beginningof time t, the firmifinances its asset investments with equity and debt. Itissues aconsol bondof B i with a fixed couponpayment of C i. After the firmpays coupons and taxes, it distributes a fraction θ of its net income back to its equity holders, where θ 1 is the dividend net income ratio, and the remainder of the net income is used for capital investments, cash retention, etc. We assume that the payout policy θ and tax structure are equal for all firms. The cash flow is taxed at a rate of τ c, dividend is taxed at τ d and interest income is taxed at τ i. Hence, the effective tax rate is τ eff = 1 (1 τ c )(1 τ d ) and the final payoff that accrues to equity holders is the dividend, D it = θ(x it C i )(1 τ eff ). When the firm s condition deteriorates, it has an option to go bankrupt, which leads to immediate liquidation or debt renegotiation. Upon liquidation debt holders take over the remaining assets and liquidate them at a fractional cost of α. Renegotiation costs a constant fraction κ < α of the assets. Because liquidation is more costly than renegotiation, debt holders are willing to renegotiate with equity holders. The renegotiation surplus α κ > 0 is shared between equity and debt holders. 6 Given their bargaining power η 1, equity holders are able to extract a fraction η of the renegotiation surplus (α κ). Anticipating the outcome of renegotiation, equity holders determine an optimal bankruptcy 5 In a CAPM setup, a risk premium could be specified as λ i = β iλ M, where β i is the market beta for firm i and λ M is the market premium. We do not measure the market premium in this study because it is empirically unobservable (Roll, 1977). 6 Recent studies that make the same assumption include Fan and Sundaresan (2000), Garlappi and Yan (2011), Morellec, Nikolov, and Schurhoff (2008) and Favara, Schroth, and Valta (2011). 6

7 threshold X ib to maximize the equity value S it (X it ) according to the following conditions: S it (X ib ) = η(α κ) X ib (2) r µ i S it 1 = η(α κ), (3) X it Xit =X ib r µ i Equation (2) is the value-matching condition, which states that equity holders extract η(α κ)x ib from the renegotiation surplusat bankruptcy X ib. Equation (3) is the smoothpasting condition that enables equity holders to choose the optimal X ib to exercise their bankruptcy option. When the costs of paying interests to keep the firm alive exceed the benefits from debt renegotiation and tax reduction, equity holders decide to declare bankruptcy at X ib (Leland, 1994). Proposition 1 For X it > X ib, at time t the contingent-claim (CC) stock return r cc it of a firm, i, is r cc it = rdt+ǫ it(r X it µ idt), (4) where r is the risk-free rate, r X it = dx it/x it is the physical operating cash flow rate and ǫ it is the sensitivity of stocks to cash flows ǫ it = X it S it S it X it =1+ C i/r θ(1 τ eff ) (1 ω [ i) Ci S } it S {{} it r θ+ X ] ib (η(α κ) θ) (1 τ eff )π it. r µ i }{{} Financial leverage (+) Option to go bankrupt (+) (5) π it ( X it X ib ) ω i is the risk-neutral default probability and S it is the equity value S it = [ ( X it C i r µ i r )θ +(C i r θ + X ] ib (η(α κ) θ))π it (1 τ eff ). (6) r µ i The optimal bankruptcy threshold X ib and ω i < 0 are provided in Appendix A. Proof: See Appendix A. Equation (4) states that the instantaneous stock return r cc it in this contingent claim 7

8 framework is the risk-free rate rdt plus an excess cash flow rate scaled by the stock-cash flow sensitivity ǫ it. The excess cash flow rate is the physical cash flow rate rit X, defined in equation (A1), in excess of the expected risk-neutral rate µ i dt. Our model is not against the standard capital asset pricing model. In this partial equilibrium model, the risk premium λ i dt of the cash flow rate is the difference between the expected physical rate r X it and the risk neutral rate µ i dt of cash flows. In a structural equilibrium framework, Bhamra, Kuehn, and Strebulaev (2010) embed the contingent-claim-based capital structure model into the consumption-based asset pricing model and derive the following result for the risk premium: λ i dt = E(r X it µ i dt) = γρ i σ i σ c dt (7) where γ is the relative risk aversion of an Epstein-Zin-Weil agent, ρ i is the correlation coefficient between cash flows X it and aggregate consumption, and σ c is the volatility of aggregate consumption growth rate. 7 Following Cochrane (1996) who infers real macroeconomic shocks from firms investment returns, we directly use the observable operating cash flow X it as our state variable because consumption data is hard to measure. Additionally, we do not need to estimate ρ i which can be different from different estimation windows and frequencies. The stock-cash flow sensitivity ǫ it in equation (5) plays an important role in characterizing the stock return. It consists of three components: The first one is the cash flow sensitivity normalized to one. The second component is the well-known financial leverage effect because C i /r is equivalent to the value of a risk-free perpetual bond. Intuitively, the more coupon payments C i equity holders distribute back to debt holders, the less residual claim D it they can receive. Hence, the stock-cash flow sensitivity increases with the coupon 7 Similarly, in the CAPM framework, the risk premium λ idt of the cash flow rate can be exogenously specified as: λ idt = β X i E(r m t rdt), (8) where β X i is the beta of cash flows X it and r m t is the market return. It follows that the market beta of stock i is β s i = ǫ itβ X i. Gomes and Schmid (2010), Garlappi and Yan (2011) and Favara, Schroth, and Valta (2011) assume the same β X i across all the stocks and call ǫ it the market beta. Different from them, we do not make intermediate assumptions about β X i and do not attempt to estimate β X i both because the market return r m t is unobservable (Roll, 1977) and because different estimation windows could result in different estimates. 8

9 payments. More importantly, the dividend net income ratio, θ, amplifies this financial leverage effect. The more dividends equity holders are able to claim on the residual income after their debt service, the more cautious they become about their contractual coupon payment and the more sensitive they are to the net incomes. To illustrate the impact of the dividend net income ratio on the stock-cash flow sensitivity, we calibrate this simple model with standard parameter values from the literature. Panel A of Figure 1 shows that the stock-cash flow sensitivity significantly increases with the dividend net income ratio, thereby confirming our intuition. The last component of equation (5) is the option to go bankrupt. The equity holders strategic default policy, X ib, is affected by their bargainingpower η at bankruptcy. Because of the costly liquidation following bankruptcy, debt holders are willing to share the renegotiation surplus with equity holders. The more bargaining power equity holders have, the more asset value they can extract through debt renegotiation at bankruptcy. Hence, equity holders with greater bargaining power are willing to file for bankruptcy earlier than their counterparts with relatively lower power. Because such equity holders with greater power extract more rents from debt holders at bankruptcy, they have less exposure to downside risk and, consequently, become less sensitive to the changing cash flows. Garlappi and Yan (2011) show that the bargaining power helps us understand the hump-shaped relationship between default probability and cross-sectional stock returns. Favara, Schroth, and Valta (2011) provide international evidence regarding the negative impact of bargaining power on equity risk. Consistent with the reasoning and the literature, the stock-cash flow sensitivity declines monotonically with the bargaining power as shown in Panel B of Figure 1. More important, the stock-cash flow sensitivity is partially determined by default probability, π it (X it /X ib ) ω i, which increases with default threshold X ib. In bad times when X it is low and is close to the threshold X ib, the firm has a higher default probability π it. On the one hand, high default probability implies high financial leverage because equation (A12) shows that default threshold increases with the risk-free equivalent debt C i /r. We have seen that the stock-cash flow sensitivity increases with its second component financial 9

10 leverage. Hence, π it increases ǫ it because it implies a high financial leverage. On the other hand, the option to go bankrupt becomes more valuable to the distressed firm when π it arises. With such a protective option, stock holders become less sensitive to the changing cash flows as shown in the third component of equation (5). Because default probability is usually low for most of the firms in reality, the direct negative effect of default probability on the sensitivity from the option to go bankrupt is small. Therefore, the default probability and the stock-cash flow sensitivity are positively associated and both of them are counter-cyclical. We provide empirical evidence in Section 5.5 to support this argument. 3 Empirical Specification and Design We test the equality between the observed stock returns, rit+1 S, and the contingent-claimbased returns, rit+1 cc, at time t+1 at the portfolio level as follows: E[r S it+1 r cc it+1] = 0, (9) where E[.] is the unconditional mean operator for a time series. We incorporate the KMV method (Crosbie and Bohn, 2003) into GMM to test the model. To calculate the predicted return r cc it+1 in equation (4), X it, S it and C i, are directly observed fromthedata andr,α,κ aredrawnfromtheexistingliterature. Because themodel prediction for r cc it+1 holds for each period t and for every state, we assume that the firms restructure their debt every time at time t and use the time-varying C it in our empirical implementation. It is worth noting that the time-varying C it is not an additional state variable because, given X it, it is endogenously chosen by equity holders. The derivation of the optimal coupon can be found in Leland (1994) and Goldstein, Ju, and Leland (2001). The two important parameters, θ and η, are to be estimated because they are the determinants of the residual cash flows stock holders can receive when the firm is solvent and at bankruptcy, respectively. After estimating θ and η, the risk-neutral rate µ i and the cash flow volatility σ i are implied from the he KMV method (Crosbie and Bohn, 2003). 10

11 Finally, we used the implied µ i and σ i to calculate ǫ it+1 and r cc it Latent Risk-Neutral Cash Flow Rate and Volatility Thelatent parameters, µ i and σ i, are not observable. Following theliterature (Vassalou and Xing (2004), Bharath and Shumway (2008) and Davydenko and Strebulaev (2007)), we use the widely accepted KMV method (Crosbie and Bohn, 2003) to calculate these two latent variables. The way we back out the latent µ i and σ i is similar to the implied state GMM procedure proposed by Pan (2002). The difference is that σ i is constant in our American option framework, while it is time-varying and the second state variable in her European stock option model. To estimate the expected cash flow volatility σ it+1, we need the historical stock volatility σ S it. We use the past one-year daily returns of stock portfolios to estimate σs it as in the literature. Observing the set of information Θ it = (X it,s it,c it,σ S it,r,α,κ,τ eff) up to time t and assuming the trial values of θ and η are true in each GMM estimation loop, we solve the following system of equations to obtain µ it+1 (θ,η,θ it ) and σ it+1 (θ,η,θ it ): σit S = E t[σ it+1 ǫ it+1 ] σ it+1 ǫ it+1 (10) [ X it S it = ( C it r µ it+1 r )θ +(C it r θ+ X ib (η(α κ) θ))( X ] it ) ω it+1 (1 τ eff ), (11) r µ it+1 X ib where the expected stock-cash flow sensitivity is ǫ it+1 = 1+ C it/r θ(1 τ eff )+ (ω [ it+1 1) Cit S it S it r θ+ X ] ib (η(α κ) θ) (1 τ eff )( X it ) ω it+1. r µ it+1 X ib (12) and ω it+1 is the negative root of the following equation 1 2 (σx it+1) 2 ω it+1 (ω it+1 1)+µ it+1 ω it+1 r = 0. (13) Equation (10) is implied by Ito s lemma. Equation (11) is to calculate the equity value S it, defined in equation (6), given the information set Θ it. The estimates of µ it+1 and σ it+1 11

12 from this procedure are expected values because the observed equity value is forward-looking and, in theory, is the present value of future cash flows discounted by the expected discount rate. The time-varying sensitivity ǫ it+1 is an expected value as well. In short, this procedure obtains the estimates of µ it+1 E t [µ it+1 ] and σ it+1 E t [σ it+1 ] at time t. Their unconditional expectations are assumed to converge to their true values, µ i and σ i, respectively. Our model potentially suffers from the specification problem if µ i and σ i arestochastic. However, theobjective of this studyisto investigate potential explanatory power of a simplest contingent-claim model for the cross-sectional stock returns, in the same spirit as Schaefer and Strebulaev (2008) who deliberately use a simplest model of Merton (1974) to explain the bond returns. Another related study is by Morellec, Nikolov, and Schurhoff (2008). Although assuming constant µ i and σ i and deriving the closed-form solutions for their dynamic capital structure model, Morellec et al. use the time varying values in their structural estimation as well GMM Testing Framework Given the estimates of µ it+1 and σ it+1 for each period t, the discrete-time version of the contingent-claim-based return from equation (4) is r cc it+1 = r t+ǫ it+1 ( X it+1 X it µ it+1 t). (14) The model is tested at the annual frequency. Hence, t = 1. From equation (14), the 8 Alternative choices are maximum likelihood method (MLM) proposed by Duan (1994) and GMM by Huang and Zhou (2008). For each firm, both the MLM and GMM procedures take the entire time series of the equity and debt values to obtain the point estimates of µ i and σ i. However, we test the model at the portfolio level. Those two alternatives are not consistent with the standard portfolio formation procedure in Fama and French (1992). Because they use the entire data sample to estimate these parameters, the MLM and GMM procedures obtain the estimates of µ i and σ i that contain information beyond time t when we form the portfolio and use them to calculate the predicted stock returns. Hence, although those two alternatives involve less computation, they are not appropriate for this study when the portfolio formation is based on the past observed accounting information. 12

13 conditional expectation of the instantaneous contingent-claim-based return is E t [r cc it+1] = r +E t [ǫ it+1 ( X it+1 X it µ it+1 )]. (15) In addition to potential specification errors mentioned previously, this discretization might suffer from measurement errors (Lo, 1986). However, we can still test the weak condition of equations (9) as in Cochrane (1991) and Liu, Whited, and Zhang (2009). Denote b [θ,η]. The pricing error for each portfolio i at time t is e cc it(b,θ it,µ it+1 (b,θ it ),σ it+1 (b,θ it )) = r S it+1 E t [r cc it+1] (16) and the expected pricing error for each portfolio, i, is e cc i = E[e cc it(b,θ it,µ it+1 (b,θ it ),σ it+1 (b,θ it ))] = E[r S it+1 E t [r cc it+1]] (17) = E[r S it+1 (r +ǫ it+1(r X it+1 µ it+1))]. according to the law of iterated expectation. The sample moments of pricing errors are g T = [e cc 1...ecc n ], where n is the number of testing portfolios. If the model is correctly specified and if empirical measures are accurate, g T converges to zero, theoretically, given an infinite sample size. Both measurement and specification errors contribute to the expected pricing errors. Under the weak condition of equation (9), the objective of the GMM procedure is to choose the optimal parameter vector, b, to minimize a weighted sum of squared errors (Hansen, 1982): J T = g TWg T, (18) s.t. 0 < θ 1, (19) 0 < η 1. (20) 13

14 where W is a positive-definite symmetric weighting matrix. Until the optimal parameter vector b [θ, η] is found, both µ it+1 and σ it+1 are recalculated for each trial set of b in the GMM optimization loops. Following Cochrane (1991) and Liu, Whited, and Zhang (2009), we choose an identity matrix W = I in one-stage GMM. By weighting the pricing errors from individual portfolios equally, the identity weighting matrix preserves the economic structure of the testing assets (Cochrane, 1996). A robustness check using two-stage GMM is provided in the Internet Appendix. In summary, the values of the firm- and time- specific variables, such as X it,s it,c it and σit S, are obtained from the data. The constant values of the market-wide variables, including r,α,κ and τ eff, are drawn from the extant literature. We imply µ it+1 and σ it+1 from the KMV method and estimate the optimal values of θ and η from the following GMM procedure: 1. A trial set of b 0 [θ,η] is initialized. 2. Given the initial values of b 0 and information set of Θ it, µ it+1 and σ it+1 are solved from the system of equations (10) and (11) for each portfolio-year observation. 3. Given b 0 and Θ it as well as the implied µ it+1 (b 0,Θ it ) and σ it+1 (b 0,Θ it ), ǫ it+1 and rit+1 cc are calculated based on equations (12) and (14) respectively. 4. The pricing error e cc i for each portfolio is obtained from (17) and the objective value J T in equation (18) across all the portfolios is calculated. 5. Repeat from Step 1 until the optimal vector b [θ,η] is found that minimizes J T. 4 Data The data universe for this study includes daily and monthly stock returns from the Center for Research in Security Prices (CRSP) as well as the Compustat annual industrial files from 1963 to We exclude firms from the financial (SIC ) and utility (SIC ) sectors and include all the common stocks listed on the NYSE, AMEX, and 14

15 NASDAQ with CRSP codes 10 or 11. For the Compustat data, we restrict our sample of annual data to firm-year observations with non-missing values for operating income, debt and total assets and with positive total assets and debt. 4.1 Variable Measurement and Parameter Values We follow Fama and French (1995) and Liu, Whited, and Zhang (2009) and aggregate firmspecific characteristics to portfolio-level characteristics. The most important state variable in this study is the operating cash flows X it. Following Glover (2011), we use operating income after depreciation (Compustat item OIADP) to proxy for the operating cash flows without considering capital depreciation. Because of their importance, the operating income observations are trimmed at the upper and lower one-percentiles to eliminate outliers and eradicate errors. S it is the equity value (price per share times the number of shares outstanding) and Coupon C it is the total interest expenses (item XINT). X it, S it and C it in year t are aggregated for all the firms in portfolio i formed in June of year t. r X t+1 is the percentage change of the aggregate operating cash flows from year t to year t+1. The corporate tax rate is set to τ c = 35% and the tax rates on dividend and interest income are set to τ d = 11.5% and τ i = 29.3%, respectively. The renegotiation cost is small and is set to κ = 0. The after-tax annual risk-free rate, r, is 4.10% and the Fama French factors are obtained from Kenneth French s website. 9 The expected liquidation cost is set to α = 0.25 according to the estimate by Korteweg (2010). A different value of the liquidation cost is used for robustness check as reported in the Internet Appendix. 4.2 Testing Portfolios We employ four sets of testing portfolios: five market leverage portfolios, five book-to-market portfolios, five asset growth portfolios and five size portfolios. We choose five portfolios for each asset pricing anomaly to ensure that the simultaneous equations (10) and (11) are 9 We use the constant risk-free rate to avoid the ultra low interest rates in recessions, which cause the model unsolvable for certain portfolio-year observations. 15

16 solvable for all portfolio-year observations. All the portfolios are equal-weighted as in Liu, Whited, and Zhang (2009). We take standard procedures to calculate ranking variables and form stock portfolios (Fama and French, 1992, 1993). The first ranking variable is market leverage measured as a ratio of total debt over the sum of total debt and the market value of equity. It is calculated as book debt for the fiscal year ending in calendar year t 1 divided by the sum of book debt and market equity (ME) at the end of December of year t 1. Book debt is the sum of short term debt (Computstat item DLC) and long term debt (item DLTT). ME is price per share (CRSP item PRC) times the number of shares outstanding (item SHROUT). Book-to-market equity ratio is the second variable of interest for the BE/ME portfolios. It is the ratio of book equity (BE) of the fiscal year ending in calendar year t 1over the ME at the end of December of year t 1. The BE is the book value of equity (Computstat item CEQ), plus balance sheet deferred taxes (item TXDB) and investment tax credit (ITCB, if available), minus the book value of preferred stock. Depending on availability, we use redemption (item PSTKRV), liquidation (item RSTKL), or par value (item PSTK) in that order to estimate the book value of preferred stock. Observations with negative BE/ME are excluded. The third variable considered is the asset growth rate for the asset growth portfolios. Following Cooper, Gulen, and Schill (2008), the asset growth rate is calculated as the percentage change in total assets (Compustat item AT). The growth rate for year t 1 is the percentage change from fiscal year ending in calendar year t 2 to fiscal year ending in calendar year t 1. The last ranking variable is market equity (ME) for the size portfolios. The ME is obtained at the end of each December of calendar year t 1. The firms with a stock price lower than $5 are excluded at the portfolio formation. This condition is to ensure that the model is solvable because those firms are possibly insolvent already. Other studies use the same requirement to avoid market microstructure noise. We follow Fama and French (1992) and construct stock portfolios with NYSE break- 16

17 points for every set of portfolios. Based on the ranking variables calculated at the end of year t 1, we first sort firms into quintiles to form equal-weighted portfolios at the end of each June of year t. Then, we rebalance them each June. Raw returns of equal-weighted portfolios are computed from the beginning of July of year t to the end of June of year t+1. To match the observed stock returns r S it+1 with the predicted returns rcc it+1 from our model, we follow Liu, Whited, and Zhang (2009) and align the state variable and firm characteristics with the observed stock returns. The state variable X it in the model is a flow variable. The operating cash flow rate r X it+1 is calculated as the percentage change in operating cash flows from the end of year t to t+1. Hence, the cash flow rates largely match with the stock returns. Appendix B contains further details for the timing alignment. 5 Empirical Results In this section we take the model to the data. We incorporate the KMV method into the GMM framework to perform a structural estimation for the model and then use comparative statics analysis to identify the key driving force of our model. After fitting the model into cross-sectional stock returns, we use the estimated policy parameters to calculate the expected risk-neutral rate and cash flow volatility, and then analyze the cross-sectional and time series properties of stock-cash flow sensitivity. Finally, in our comparative statics analysis, we demonstrate that the spread in stock-cash flow sensitivities is crucial for us to understand the cross-sectional stock returns. 5.1 Pricing Errors from Traditional Models We first confirm the well-known pricing errors in our data sample. Table 1 reports the average returns in annual percent for equal-weighted quintile portfolios, sorted on the increasing rank of the anomaly variables, and for the high-minus-low (H L) and small-minus-big (S B) hedge portfolios. The pricing errors, such as e C from the CAPM and e FF from the Fama French model, are estimated by regressing the time series of portfolio returns on the 17

18 market factor and on the Fama French three factors. Market leverage portfolio Panel A shows that stocks with high market leverage earn 12.21% per year more than do stocks with low leverage. The pricing error of the H L portfolio from the CAPM is 11.74% (t = 4.05). This error decreases to 3.13% (t = 1.54) and becomes statistically insignificant for the Fama French model. This significant drop is consistent with the conclusion of Fama and French (1992) that the book-to-market factor is capable of explaining the cross-sectional returns of the market leverage portfolios. Additionally, the mean absolute errors (m.a.e.) is 6.83% per year for the CAPM and decreases to 3.66% for the Fama French model. BE/ME portfolios The average returns in Panel B monotonically increase with the book-to-market ratio from 12.61% to 26.71% per year. After controlling for the market factor, the H L portfolio earns 14.81% (t = 5.97) per year and the m.a.e. is 6.76%. The performance of the Fama French model improves as its error of the H L portfolio decreases to 7.56% (t = 4.13) and its m.a.e. declines to 3.90%. Asset growth portfolios As shown in Panel C, high-growth firms earn 12.13% lower stock returns per year than low-growth firms. 10 This finding can not be explained by the standard CAPM and the Fama French model. The errors of the H L portfolio from the CAPM and the Fama French model are 11.74% (t = 6.16) and 10.65% (t = 4.90), respectively. The m.a.e. s for asset growth portfolios are the greatest among all the four sets of testing portfolios. The m.a.e. is 7.08% from the CAPM and 4.22% from the Fama French model. Size portfolios Panel D confirms the size effect. Small firms earn 8.28% greater returns per year than big firms, even if we exclude small firms with a price lower than $5 at the portfolio formation. The decrease in average returns with the equity size remains the same after controlling the market factor and Fama French three factors. The errors of the smallminus-big (S B) portfolio from the CAPM and the Fama French model are 8.28% (t = 2.67) and 3.14% (t = 2.36), respectively. The m.a.e. is 4.22% for the CAPM and is 2.09% 10 The difference is smaller than the difference of 20% per year documented by Cooper, Gulen, and Schill (2008) because our sample requires positive debt and has other restrictions. 18

19 for the Fama French model. Overall, we demonstrate that the well-documented pricing errors from the traditional models are largely the same in our data sample as in the literature. In the next two sections, we summarize the model inputs and compare the traditional models with our model. 5.2 Summary Statistics of Model Inputs and Portfolio Characteristics Table 2 reports portfolio characteristics for the four sets of quintile portfolios. Instead of reporting the dollar amount for X it, C it and S it, we report the earnings price ratio X it /S it and the interest coverage ratio X it /C it. X it /S it is considered because the equity value S it is contingent on the underlying earning X it in the model. X it /C it measures the financial health of the firms and provides preliminary information about the financial leverage effect on stock-cash flow sensitivity, as shown in the second component of equation (5). Market leverage portfolios Unlike the monotonically increasing stock returns across the market leverage portfolios, the average cash flow rates r X t+1 and their correlations with the stock returns rt+1 S are slightly U-shaped. More interestingly, the correlations between them are relatively weak. While X it /S it increases from 0.09 to 0.23, X it /C it dramatically declines from to 2.12 with market leverage. It is interesting to note that the stock volatility σ S it of the low-leverage portfolio is 26.33% per year is almost identical to that of the highleverage portfolio. This result suggests that debt financing behavior is more complicated than our conventional wisdom that stock volatility increases with financial leverage (Gomes and Schmid, 2010). BE/ME portfolios Similar to the market leverage portfolios, both rt+1 X and σs it are slightly U-shaped. The magnitude of the increase in the earnings price ratio across the book-to-market portfolios is comparable to that across the market leverage portfolios as well. The interest coverage ratio declines from 9.63 to Hence, the decrease in X it /C it in the BE/ME portfolios is considerably smaller than the decline in the market leverage portfolios. Asset growth portfolios The patterns of portfolio characteristics in the asset growth 19

20 portfolios are generally contrary to those observed in the book-to-market portfolios. The difference occurs because the firms with a low asset growth rate are more likely to be the firms with more book equity-in-place. The increment in the interest coverage ratio from the low-growth firms to the high-growth firms is only 1.58, the smallest change among the four sets of portfolios. Size portfolios Different from all other three sets of portfolios, r X it+1 declines significantly from 15.09% to 7.53% per year across the size portfolios. The magnitude of the decrease in cash flow rates is comparable to that in stock returns. Its correlation with the stock returns decreases as well. The monotonic decline in σ S it is the most evident among the four sets of portfolios. While the earnings price ratio slightly decreases, the interest coverage increases significantly with the market capitalization. This contrast implies that small firms face greater interest payment pressures and are more likely to become distressed. This observation is consistent with Vassalou and Xing (2004). In short, the average cash flow rates increase or decrease in the same direction as the average stock returns do with the ranking variables for all the four sets of portfolios. Except for the size portfolios, the magnitude of the changes in the average cash flow rates is considerably smaller than that in the average stock returns, and the stock volatilities are all slightly U-shaped for the other three sets of portfolios. Moreover, the decline in the interest coverage ratio is the most evident for the market leverage portfolios. 5.3 Model Estimation and Pricing Errors from Structural Model We estimate two parameters, dividend net income ratio θ and shareholder bargaining power η, for this parsimonious contingent claim model within the GMM framework. Table 3 reports the parameter estimates and χ 2 statistics for model fitness when matching the predicted returns with the observed returns as in equation (9). The estimates of θ are 0.88, 0.83 and 1.00 for the market leverage, BE/ME and asset growth portfolios. Those estimates suggest that % of the net incomes are distributed back to equity holders. The associated t-statistics indicate that these point estimates are statistically significant at 20

21 a 95% confidence level. In contrast, the estimate of θ for the size portfolios is only 0.38 and is statistically insignificant. The estimates of η are very close across the four sets of portfolios. They are around 0.60, greater than the Nash equilibrium value of 0.5 chosen by Morellec, Nikolov, and Schurhoff (2008) and close to the value of 0.6 assumed in Favara, Schroth, and Valta (2011). 11 Although they are not statistically significant, the estimates of η consistently indicate that equity holders have greater bargaining power than debt holders do. The χ 2 statistic, which tests whether all the model errors are jointly zero, gives an overall evaluation of the model performance. For the four sets of portfolios, the degrees of freedom (d.f.) are three because the number of the moments (or portfolios) is five and the number of parameters is two. The p-values of the χ 2 tests indicate that the model can not be rejected for all the four sets of testing portfolios, with the asset growth portfolios having the lowest p-value. Overall, the model performs well for all the sets of testing portfolios with a modest performance for the asset growth portfolios. However, the t-statistics of the two estimates and the p-values are relatively low. The relatively weak statistical significance could be attributed to the small data sample because each testing set has only five portfolios in annual frequency. Additionally, it is well-known that the consistent one-stage GMM estimation gives relatively weaker statistical performance, compared with the efficient two-stage GMM estimation as shown in the Appendix. Given the estimates of θ and η, we construct the contingent-claim-based returns r cc it+1 as in equation (14) and calculate the expected pricing error e cc i as in equation (16) for each individual portfolio. Table 4 reports the pricing errors from the model estimation. Market leverage portfolios The first row shows that the pricing errors vary from 1.57% to 1.77% per year. Additionally, the pricing error of the H L portfolios is 0.34% (t = 0.29) and is not statistically significant. This error is smaller than 11.74% from the CAPM and 11 For the size portfolio, the estimate is only 0.25 if we do not exclude the firms with a stock price lower than $5 at the portfolio formation. The possible reason for this lower estimate for a sample with more small firms is that smaller equity size is an indicator of weaker bargaining power (Garlappi and Yan, 2011). 21

22 3.13% from the Fama French model in Table 1. Figure 3 visually illustrates the model fitness and pricing errors. We plot the average predicted returns against their realized returns for the contingent claim model, the CAPM and the Fama French model. If a model fits the data perfectly, all the predicted returns should lie on the 45-degree line. As shown in the scatter plot in Panel A, the predicted average returns from the contingent claim model reside on the 45-degree line. In contrast, the predicted returns from the CAPM in Panel B are almost flat. Although the predicted returns from the Fama French model in Panel C show some improvement, none of the predicted returns lie on the 45-degree line. BE/ME portfolios From the third row, the H L portfolio has a pricing error of 1.80% per year, which is smaller than 14.81% in the CAPM and 7.56% per year in the Fama French model. This error is mostly due to the large deviation of 2.11% from the growth portfolio. The small error of 0.31% in value portfolio implies that our model is able to capture the default risk associated with value firms. The mean absolute error (m.a.e) is 0.99% per year, much lower than 6.76% from the CAPM and 3.90% from the Fama French model. Figure 4 provides a visual confirmation. As shown in Panel A, the largest deviation from the 45-degree line is from the growth portfolio. The predicted returns from the CAPM are almost horizontal in Panel B and those from the Fama French model in Panel C are quite similar. Asset growth portfolios The difference in the pricing errors between the high- and lowgrowth portfolios is 5.06% per year, which however is much less than 11.74% from the CAPM and 10.65% from the Fama French model in Table 1. Panel A of Figure 5 shows that the average predicted returns generally align with the realized returns. The predicted returns for the low- and high- growth portfolios are slightly out of line. In a sharp contrast, the predicted returns from the CAPM and the Fama French model are almost flat. Size portfolios The pricing errors range from 1.49% to 1.47%. The error of the S B portfolio is 0.58% per year. It is evident that, in Panel A of Figure 6, the predicted returns are aligned very well with the realized stock returns, even for the portfolio of small stocks. The performance of the CAPM remains poor, as shown by the horizonital line of 22

23 its predicted returns. Although the Fama French model performs much better than the CAPM, it still fails to capture a big outlier from the small portfolio. In summary, our model does a good job for all the sets of portfolios and outperforms the CAPM and the Fama French model. The model performs best for the market leverage portfolios and predicts the expected returns of value firms and small firms well. Although the model does not fit the asset growth portfolios very well, it gives a much better fit than the CAPM and the Fama French model. 5.4 Cross-Sectional Properties of Default Probabilities and Stock-Cash Flow Sensitivities Given the optimal estimates of θ and η, we calculate the implied risk-neutral rate µ it+1 and cash flow volatility σ it+1 according to equations (10) and (11) for each portfolio-year observation. Then, wecalculate therisk-neutral default probability π it+1 andthestock-cash flow sensitivity ǫ it+1 according to equation (12). It is worth noting again that ǫ it+1 from our method is a structural estimate instead of a reduced-form estimate from rolling regressions in other studies. Moreover, µ it+1 does not contain information on the riskiness of the underlying operating cash flows and that it is negatively correlated with the stocks returns according to equation (4). Table 5 reports the distribution of the estimates. Market leverage portfolios Three observations from Panel A are worth noting. First, the means and medians of µ it+1 are all small and negative. The median decreases from 0.51% to 0.53% per year along with the increasing rank of the debt ratios. The small and negative average risk-neutral rates are generally consistent with the results obtained by Glover (2011). Second, the fact that σ it+1 monotonically declines with leverage confirmsour conventional wisdom that firms with low operating risk have better access to debt markets and therefore have greater financial leverage. However, the decreasing cash flow volatility differs from the U-shaped stock volatility in Table 1. This difference implies that the stock volatility is not necessarily a good proxy for the underlying cash flow volatility. Third, firms 23